I would agree that fanlights are fixed. They're particularly popular in Georgian architecture and you get some people who make whole studies about them. They come in a wide variety of patterns - mine has a number of interlocking oval shapes filling in the semi-circle, others look a bit like the sun with rays coming out of them, some are much fancier. If it's just got plain glass in it, then odds on the original was removed at some point and you've just got a poor imitation.
The fanlight is also usually where the house number is put.