On critical conditions of overgroups of weak second maximal subgroups

A subgroup H of a finite group G is called a weak second maximal subgroup of G if there exists a maximal subgroup M of G such that H is a maximal subgroup of M. Let m(G, H) denote the number of maximal subgroups of G containing H. In this paper the classification of finite groups G is given if G/CoreG(H) is solvable and H is a weak second maximal subgroup of G such that m(G,H)−1 is equal to the index of some maximal subgroup of G, which make us understand the difference between the weak second maximal subgroups and the second maximal subgroups.