These factions make up the bulk of the Hashed al-Shaabi military network, founded in 2014 to fight the Islamic State group (IS) and now largely incorporated into Iraq's government forces.

But after protests outside the US embassy organised by the Hashed, US air strikes on them and a string of attacks on US forces in Iraq, the groups face fresh scrutiny.

- Kataeb Hezbollah -

This faction was targeted by the US on Sunday night and has been blamed for a series of rocket attacks on Iraqi bases where US forces are deployed.

Kataeb Hezbollah first began fighting US troops in 2003 during the American-led invasion to topple Saddam Hussein, according to its spokesman Jaafar al-Husseini.

It joined up with other paramilitary units in 2014 to fight IS as part of the Hashed al-Shaabi, now officially part of the government's security forces.

But some Kataeb Hezbollah units still operate independently of Baghdad, including those fighting in neighbouring Syria on behalf of President Bashar al-Assad.

The group does not publicise its precise force size but claims they are armed with "drones, guided missiles and other rockets of various sizes," according to Husseini.

The group's chief is known by his nom de guerre Abu Ahmad al-Basri, but his real name and identity remain unknown.

It is the top armed Iraqi ally of Iran's Revolutionary Guard Corps, according to specialist Michael Knights, who described Kataeb Hezbollah as Tehran's "third major militant force" after the Guards' Quds Force and Lebanon's Hezbollah.

- Nujaba, Shuhada, and more -

Thousands more Iraqi fighters are members of Harakat al-Nujaba, the Imam Ali Brigades, Sayyed al-Shuhada Brigades and the Kharasani Brigades.

Together with Kataeb Hezbollah, these forces make up the hardline wing of the Hashed and are known to have especially close ties to Iran's Revolutionary Guards.

The factions largely steer clear of politics and keep most of their operations secret -- but their links to Iran have worried the United States.

Washington has blacklisted Nujaba, which is led by Akram al-Kaabi, as well as the head of the Imam Ali Brigades Shibil al-Zaydi.

- Asaib Ahl al-Haq -

Another leading faction in the Hashed, Asaib was founded after splintering from the Mahdi Army, the powerful militia led by Shiite cleric Moqtada Sadr which fiercely fought US troops after 2003.

Asaib then carried out its own deadly operations against US forces and its units have also received training and weapons from Iran.

It has since ballooned into one of the Hashed's most powerful units, with around 10,000 fighters.

Asaib Ahl al-Haq was accused of firing some rockets at US assets and its chief Qais al-Khazaali was recently blacklisted by the US for "widespread forced disappearances, abductions, killings, and torture".

Asaib is more politically active, with its political arm represented in parliament by the Sadiqun bloc and a share of government posts in Iraq's ministries.

- Badr Organisation -

Founded in 1982 in Iran, the Badr Organisation is the oldest of the Hashed's factions and was established in opposition to Iraq's then-dictator Saddam Hussein.

Badr even fought on Iran's side in its war against Iraq between 1980 and 1988.

It is led by Hadi al-Ameri, who is known to have close personal ties to Qasem Soleimani, the head of the Quds Force and Iran's pointman on Iraqi affairs.

It is the largest of the Hashed factions and is the most integrated into the state's regular security forces, and always holds at least one post in Iraq's cabinet.