The influence of $s$-$c$-permutably embedded subgroups on the structure of finite groups

A subgroup $H$ of a group $G$ is said to be $s$-$c$-permutably embedded in $G$ if every Sylow subgroup of $H$ is a Sylow subgroup of some $s$-conditionally permutable subgroup of $G$. In this paper, some new characterizations for a finite group to be $p$-supersoluble or $p$-nilpotent are obtained under the assumption that some of its maximal subgroups or 2-maximal subgroups of Sylow subgroups are $s$-$c$-permutably embedded. A series of known results are generalized.