Finite Simple Groups of Bounded Subgroup Chain Length

Given a finite group G, we denote by l(G) the length of the longest chain of subgroups of G. We study whether certain sets of non-isomorphic finite simple groups S with bounded l(S) are finite or infinite. We prove, in particular, that there exists an infinite number of non-isomorphic non-abelian finite simple groups S with l(S) bounded.