We prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of Lie type of bounded rank is bounded. For exceptional groups this solves a long-standing open problem. The proof uses, among other tools, some methods from geometric invariant theory. Using this result, we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of Lie-type groups of fixed rank, drawing conclusions regarding the behaviour of the corresponding ``zeta function'' , which appears in many probabilistic applications. More specifically, we are able to show that for simple groups and for any fixed real number , as . This confirms a conjecture made in [27, page 84]. We also apply these results to prove the conjecture made in [28, Conjecture 1, page 343, that the symmetric group has conjugacy classes of primitive maximal subgroups.
"On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function." Duke Math. J. 128 (3) 541 - 557, 15 June 2005. https://doi.org/10.1215/S0012-7094-04-12834-9