On NH-embedded and S-quasinormally embedded subgroups of finite groups

Let G be a finite group and H a subgroup of G . H is said to be NH -embedded in G if there exists a normal subgroup T of G such that HT is a Hall subgroup of G and H ∩ T ≤ H s ¯ G , where H s ¯ G is the largest s -semipermutable subgroup of G contained in H , and H is said to be S -quasinormally embedded in G provided every Sylow subgroup of H is a Sylow subgroup of some S -quasinormal subgroup of G . In this paper, we obtain some criteria for p -nilpotency and Supersolvability of a finite group and extend some known results concerning NH -embedded and S -quasinormally embedded subgroups.