S-embedded subgroups of finite groups

A subgroup H of G is said to be S-embedded in G if G has a normal subgroup N such that HN is s-permutable in G and H ∩ N ⩽ H _sG , where H _sG is the largest s-permutable subgroup of G contained in H. S-embedded subgroups are used to give novel characterizations for some classes of groups. New results are obtained and a number of previously known ones are generalized.