On the type of conjugacy classes and the set of indices of maximal subgroups

Let G be a finite groupý. ýBy MT(G)=(m1,⋯,mk) we denote the type ofý ýconjugacy classes of maximal subgroups of Gý, ýwhich implies that G has exactly k conjugacy classes ofý ýmaximal subgroups and m1,…,mk are the numbers of conjugatesý ýof maximal subgroups of Gý, ýwhere m1≤⋯≤mký. ýIn this paperý, ýweý ýgive some new characterizations of finite groups by the type ofý ýconjugacy classes of maximal subgroupsý. By πt(G) we denoteý ýthe set of indices of all maximal subgroups of G ý. ýWe also investigateý ýthe influence of the set of indices of all maximal subgroups on the structure ofý ýfinite groupsý.