Boosting an analogue of Jordan's theorem for finite groups

Let CC be a set of finite groups which is closed under taking subgroups and let d and M   be positive integers. Suppose that for every G∈CG∈C whose order is divisible by at most two distinct primes there exists an abelian subgroup A⊆GA⊆G such that A is generated by d   or fewer elements and [G:A]≤M[G:A]≤M. We prove that there exists a positive constant C0C0 such that every G∈CG∈C has an abelian subgroup A   satisfying [G:A]≤C0[G:A]≤C0, and A can be generated by d or fewer elements. We also prove some related results. Our proofs use the Classification of Finite Simple Groups.