On m-S-complemented subgroups of finite groups

Let G be a finite group and H a subgroup of G. We say that H: is generalized S-quasinormal in G if H = < A, B > for some modular subgroup A and S-quasinormal subgroup B of G; m-S-complemented in G if there are a generalized S-quasinormal subgroup S and a subgroup T of G such that G = HT and H boolean AND T <= S <= H. In this paper, we study finite groups with given systems of m-S-complemented subgroups. In particular, we prove the following result: Let F be a saturated formation containing all supersoluble groups, and let E be a normal subgroup of a finite group G such that G/E is an element of F. If for any Sylow subgroup P of E every maximal subgroup of P not having a supersoluble supplement in G is m-S-complemented in G, then G is an element of F.