Just like the secular government is based on the hierarchy of the feudal system, so is the religious leadership. They collect tithes and expect service from their priestly vassals. They rule over all matters of religious law, called Canon.
At the top is the Solar, a priest who is elected by the Lanterns.
Each Lantern is the head of an organization devoted to one of the Fire Gods. There are eight of them. They hold the sole power to elect a new Solar to replace one who has died.
A Lantern oversees a number of Brands, each of whom serves a Fire Temple in a territory called a Brandright. Usually a Fire Temple is built in a city large enough to handle the task, and the brandright includes other cities and the surrounding lands.
Each Brand oversees priests called Torches, who have a small temple and a territory to minister.
Each Torch’s temple has dozens of priests, monks, nuns, and laypeople serving them. There are ranks and titles for most of those, too (Father/Mother for priests, Brother/Sister for monks and nuns, and so on).