Overgroups of second maximal subgroups (Q1805510)

A subgroup \(H\) in a finite group \(G\) is called second maximal if for every subgroup \(M\) with \(H<M<G\) both \(H\) is maximal in \(M\) and \(M\) is maximal in \(G\). Fix a second maximal subgroup \(H\) and denote by \(r\) the number of maximal subgroups containing \(H\) and by \(m\) the largest index \(|G:M|\) for a maximal subgroup \(M\) containing \(H\). The author proves that \(r\leq m+1\) and determines all situations where equality holds.