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.
Martin W. Liebeck. Benjamin M. S. Martin. Aner Shalev. "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