What is the extent of the IRGC's control over Iran?

On January 7, Iran fired 15 missiles at two US military bases in Iraq, the Ain al-Asad base and the Erbil base. The attacks were promised as 'revenge' for the US drone attack that killed top military commander General Qassem Soleimani. The missile attack was carried out by IRGC—the Islamic Revolutionary Guard Corps. After the attack, the IRGC issued a statement saying they would retaliate against US allies if they allowed the US to launch more attacks on Iran.

So, who exactly are the Revolutionary Guard Corps? They are a branch of Iran's military, separate from the mainstream military and established in 1979 to protect Iran's Islamic republic political system. 1979 was the year that Iran's first supreme leader Sayyid Ruhollah Khomeini came to power, overthrowing the last Shah of Iran, Mohammad Reza Pahlavi. After the revolution, the IRGC was formed alongside the regular military, and has its own air, land and sea divisions. It reports directly to Iran's supreme leader, Ayatollah Ali Khameini.

Amidst the standoff over belligerence on Iran's part about the nuclear deal and the US imposing sanctions on the nation, nearly excluding it from the world economy, the US in April labelled the Revolutionary Guard Corps a terrorist organisation.

Going by a broad brief, the group is endowed with a lethal mix of political, legal and religious power. The Revolutionary Guards are one of the branches of the Iranian Armed Forces and while it has certain control over the ground forces and the navy, it also has control of the Quds forces (which Soleimani had led) and Basij.

In comparison, Quds forces are a rough amalgamation of the CIA (Central Intelligence Agency) and the Joint Special Operations Command (JSOC) in the United States. The Quds also support proxy groups in the region like Hezbollah in Lebanon, Shia militias in Iraq and Houthis in Yemen, to name a few. Basij is an auxiliary paramilitary force made up of part-time civilian volunteers. Basij was often deployed to quell 2009 Iranian election protests and the 2017–18 Iranian protests.

And it is due to this, that western analysts often say that the IRGC's powers may have exceeded that of the country's Shia clerical system. 

The IRGC is also said to have economic interests in form of some control over certain construction and oil companies like Oriental Oil Kish, Imensazen Consultant Engineers Institute and Hara Company, an excavation and tunnel construction company. 

The IRGC is also known to have a certain influence over the conventional armed forces, which has approximately 470,000 personnel. The Corps is also said to have a hand in rebuilding the nation after its gruelling 8-year-war with Iraq. Another area that the IRGC has a say over is the ballistic missile and nuclear programs. And this causes the US to say that the Corps is responsible for much tension and volatility in the Mideast region. 

An Al Jazeera article on the IRGC and its influence said that “Such threats [When in 2017, IRGC commander Mohammad Ali Jafari said that his troops 'will consider the American army to be like Islamic State all around the world'] are particularly ominous for US forces in places such as Iraq, where Iran-aligned Shia militia are located in close proximity to US troops.”