Since it began last year, Russia’s war in Ukraine has hinged not just on battlefield results, but also a question in Moscow: Could President Vladimir V. Putin’s grip on power withstand the strain of fighting a long and costly war, with no end in sight?

The events of the last few days, in which Yevgeny V. Prigozhin, the head of a notorious private army called Wagner, mounted a brief rebellion against Russia’s military leadership, are not enough to answer that question. But they do suggest that Mr. Putin’s hold over the elite coalition that keeps him in power is under stress, with unpredictable consequences.

Even though authoritarian leaders may appear to rule by fiat, they all rely on coalitions of powerful elites to stay in power, analysts say. The specifics vary by country and situation: Some count on the military, others on a single ruling party, the religious authorities, or wealthy business leaders.

In Syria, for instance, the military is dominated by members of Bashar al-Assad’s Alawite religious minority, and officers have long relied on the government for housing and other benefits, entangling their lives with the survival of the regime. Even when a 2011 popular uprising turned into a bloody, protracted civil war, Mr. Assad’s supporters within the military kept him in power: The benefits of loyalty, to them, far outweighed the costs.

Mr. Putin’s alliance had until recently seemed very robust, centered around the “siloviki,” a group of officials who came to politics after serving in the K.G.B. or other security services, and who now occupy key roles in Russia’s intelligence services, oil and gas industry and ministries.

His high public support has long been another major source of strength, and Mr. Putin had structural advantages as well. He does not answer to a political party whose leadership could band together and replace him, as was the case in the Soviet Union. And by dividing power between different agencies, ministers and wealthy businessmen, he ensured that no person or institution was strong enough to overthrow him.

But when Russia first launched its invasion of Ukraine last year, experts said that the war had the potential to undermine his hold on power.

“The relationship between authoritarian rulers and their core of elite supporters can be strained when dictators wage war abroad — particularly where elites view the conflict as misguided,” said Erica de Bruin, a political scientist at Hamilton College and the author of a recent book on coups.

For a while, Mr. Prigozhin looked like a solution to many of the president’s problems. The Wagner group joined the fighting last summer, as Russia’s military sought to recover from heavy losses. Wagner led an offensive in eastern Ukraine, and for a time was allowed to recruit thousands from Russian prisons.

The growing power of the mercenary force was a counterbalance to that of the regular armed forces, too — an additional tool with which Mr. Putin protected his own power.

But it soon became clear that Wagner was creating problems. Mr. Prighozhin began publicly criticizing the conduct of the war, excoriating a close ally of Mr. Putin, Defense Minister Sergei Shoigu. In profane social media posts, he accused Mr. Shoigu and the military’s chief of the general staff of cowardice and corruption, and of sending Russians into slaughter.

The ministry’s leaders, he said last year, “should go with machine guns barefoot to the front.”

As his online following grew, so did his populist appeal, giving him a level of political celebrity that was essentially unheard-of in Mr. Putin’s Russia. Some analysts wondered if he might challenge the president himself.

But Mr. Shoigu moved to curtail Wagner, cutting off its access to prisons and, this month, ordering its fighters to sign a contract with the military by July — a move that would have effectively dismantled the private group’s autonomy. Mr. Prigozhin refused, while maintaining his loyalty to Mr. Putin.

With Mr. Prigozhin’s group threatened by the military, things escalated rapidly. In a series of social media posts on Friday, he accused Mr. Shoigu of ordering deadly strikes on Wagner fighters, saying “The evil borne by the country’s military leadership must be stopped.”

That night, he and his forces took the city of Rostov-on-Don. The next morning, they began marching on Moscow.

The uprising was a mutiny, not a coup: Mr. Prigozhin’s stated goal was to oust the senior military leadership, not to take over the country himself, and on Monday he called it a “protest” over the order to make Wagner fighters sign contracts.

It also ended quickly. By late Saturday night, the Kremlin announced that Mr. Prigozhin would leave Russia for Belarus, and his troops would not face repercussions.

Now, the question is what the mutiny tells the elites who keep Mr. Putin in power, and whether it has changed their incentives.

“Mutinies can signal dissatisfaction within the ranks that future coup plotters can capitalize on,” Dr. de Bruin said. One large-scale study of military mutinies in Africa, for instance, found that they rarely escalate directly into coups, but they are associated with an increased likelihood of coups in the near future.

Sometimes the opposite is true: In the aftermath of a failed coup, leaders often take the opportunity to purge those whom they suspect of disloyalty, strengthening their hold on power. President Recep Tayyip Erdogan of Turkey, for instance, cracked down on tens of thousands after a failed coup attempt in 2016, purging the military as well as institutions like the police, schools and the courts.

But that may not be possible in this case, Dr. de Bruin said. Because Mr. Prigozhin withdrew, rather than being defeated by Russia’s army, “Putin doesn’t come out of this looking like he won the confrontation,” she said. The public saw that Wagner troops could race toward Moscow, and that they now seem to face little punishment.

Even if there was more going on behind the scenes, appearances matter. After making a brief statement on Saturday, Mr. Putin vanished from sight, making no further appearances during the dramatic uprising and its aftermath. Then his government announced a deal with Mr. Prigozhin, even though the president had publicly called Mr. Prigozhin’s actions “traitorous.”

Mr. Putin’s response, analysts said, may signal that disloyalty is not as costly as many might have imagined.

Mr. Prigozhin is an “exceptional phenomenon” and isolated among Russia’s elites, according to Tatiana Stanovaya, a senior fellow at the Carnegie Russia Eurasia Center, but she wrote over the weekend that he still dealt Mr. Putin a blow. “I won’t discount the possibility of future imitators, but there will never be another one like him.”

None of that means that Mr. Putin’s days as president are numbered. But his hold on power looks less certain than ever before. Mr. Putin “is now marked as weak enough to challenge,” said Naunihal Singh, a professor at the Naval War College and the author of a book on the strategic logic of military coups. “I think there may be other challengers now.”

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