Does Putin need the support of the Russian oligarchs in order to stay in power? What power does the Russian oligarchs have over Putin? I heard a lot of the oligarchs have ties to Putin and a lot of them are ex-KGB members, so I am wondering if they have a lot of political power and if they can somehow use their political power against Putin. If so, in what ways?
Short answer: No. To think the oligarchs are important political actors is to misunderstand Russia's internal politics.
Since he became Russia's president, Vladimir Putin's stated aim was to create a 'power vertical': an authoritarian system to ensure security and stability for Russian people.
Russia's oligarchs became infamously wealthy during the privatisation of state assets in the Yeltsin era, but they hold no political power. Putin in comparison has gradually merged all important institutions into his power vertical. For example, the Orthodox Church has become a servile state organ, and patriarch Kirill often heaps sycophantic praise upon Putin. In 2012 Kirill said Putin's rule was a "gift from God".
Another important example of how the political structure of Russia has changed is the National Guard, who were created in 2016. The Guard are a gendarmerie force with the power to arrest anyone anywhere in Russia. The Guard replaced the Internal Troops of Russia, who had been a paramilitary force under the Ministry of the Interior. The Guard are uniquely subservient to the executive branch, not the Ministry of Defence or the Ministry of Internal Affairs, and are thus loyal to the presidency and not regular military or police. Their commander is also Viktor Zolotov, who has been Putin's bodyguard for years.
Putin is very sensitive to perceived insults, and any oligarchs who have crossed such a line quickly lose their rights. The most obvious example is Mikhail Khodorkovsky. Khodorkovsky was a famous oil baron, and one of the richest men in Russia. On the 19th of February 2003 he attended a televised meeting with Putin and Russian business leaders. During this meeting Khodorkovsky was critical of Putin's record on state corruption, arguing that it was now the biggest obstacle to Russian economic growth, and that senior public officials must be taking millions of dollars worth of bribes. Later that year his company was investigated for tax irregularities. He was then charged with fraud, and imprisoned for over a decade. After his release he left for Germany.
Political rivals who pose a real threat to Putin end up in prison (like Alexei Navalny) or worse. For example, opposition politician Boris Nemtsov was shot dead in Moscow in 2015. This happened as he was preparing to publish a report exposing Russian military activity in Ukraine, thus contradicting Moscow's position that the annexation of Crimea, Donetsk, and Luhansk were due to local volunteers, and had nothing to do with the Russian armed forces.
In conclusion, Russian oligarchs may be super rich, but they lack the political power to meaningfully challenge Putin's authority. Those who have done less, and simply asked Putin to do more to tackle corruption, have been prosecuted and forced out.
Edit: It is important to stress that Russia's oligarchs are dependant upon Putin, and have zero influence. Oligarch is not a synonym for Boyar, they are not feudal lords with the power to raise armies. Suppose all of them decide to team up against Putin. Now what? They have no popular support or command and control capability. The Russian people regard them as thieves and hate them. Even if this impossible oligarch revolt occurs, and they somehow manage a double impossibility by inspiring units of the army or police to join them, Putin will then order his aforementioned National Guard to arrest the plotters, and that will be the end of that.
Putin's command of Russian society could be challenged by a few things, but not the oligarchs. The most likely possibilities are a popular revolution or coup, for example: Czechoslovakia in 1989 or Egypt in 2011.
To cite one opinion piece from The Guardian by Angus Roxburgh:
"The only real value in punishing oligarchs is that it makes the west feel and look good because it is doing something. Economic sanctions – cancelling the Nordstream II gas pipeline, or restricting Russia’s access to the Swift system – may have a greater effect, but they may also rebound on us in the form of higher energy prices.
However, there is a much bolder and more imaginative approach. Russia’s ruling class – the members of the Duma, the Senate, the presidential council, the top echelons of the security and defence services, top state television employees – is several thousands strong. These men (and some women) draft, rubber-stamp, promote and carry out Putin’s decisions. Some of them also – unlike the oligarchs – actually advise him.
Being a member of the Duma or Senate is a pretty cushy number – you are well-paid, you can make an occasional speech if you wish, but you are basically there to vote for the Kremlin’s decisions, and, above all, you can extort as many bribes as you can cope with. (For this reason they are detested by a majority of Russians.) Members of the presidential council are civil servants, essential for the preparation of legislation. The security services play crucial roles in executing Putin’s vision. And TV propagandists spread disinformation.
If any kind of sanctions are going to have an effect, it is these. Forget the oligarchs; go for the political elite."
Angus Roxburgh is a former BBC Moscow correspondent and consultant to the Kremlin. He is the author of The Strongman: Vladimir Putin and the Struggle for Russia and Moscow Calling: Memoirs of a Foreign Correspondent.