There are Jesuits as well, but they focus on political intrigue and questionable magical research. The original Jesuits were founded to counter Protestantism; in Ytarria, Protestantism is almost entirely suppressed apart from a handful of frightened Huguenots here and there.
The Hospitallers are anti-magic human-supremacist anti-Islamic crusaders based in New Jerusalem; the Templars are magic-loving political schemers with a military force that can match the Hospitallers. In Caithness, the Order of the Dragon serve the schismatic Church of Caithness, while the Order of the Stone serve the King.
The Franciscans look after the peasants, while the Michaelites enforce the law. The Thomasites study magic, natural philosophy and healing. The Constantinians specialise in administration and archive-keeping for the Imperial Court. The Olybrians are the main evangelists, the Alycites the main healers for the common people.
There’s also the Friedrichites, but they’re just weird. Hermits and ascetics; the regular Church tries to ignore them as much as possible.