On T-groups, supersolvable groups, and maximal subgroups

A group is called a T -group if all its subnormal subgroups are normal. Finite T -groups have been widely studied since the seminal paper of Gaschütz (J. Reine Angew. Math. 198 ( 1957 ), 87–92), in which he described the structure of finite solvable T -groups. We call a finite group G an NNM -group if each non-normal subgroup of G is contained in a non-normal maximal subgroup of G . Let G be a finite group. Using the concept of NNM -groups, we give a necessary and sufficient condition for G to be a solvable T -group (Theorem 1), and sufficient conditions for G to be supersolvable (Theorems 5, 7 and Corollary 6).