On $NH$-embedded and $SS$-quasinormal subgroups of finite groups

Let $G$ be a finite group. A subgroup $H$ is called $S$-semipermutable in $G$ if $HG_p$ = $G_pH$ for any $G_p\in Syl_p(G)$ with $(|H|, p) = 1$, where $p$ is a prime number divisible $|G|$. Furthermore, $H$ 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 $H \cap T \leq H_{\overline{s}G}$, where $H_{\overline{s}G}$ is the largest $S$-semipermutable subgroup of $G$ contained in $H$, and $H$ is said to be $SS$-quasinormal in $G$ provided there is a supplement $B$ of $H$ to $G$ such that $H$ permutes with every Sylow subgroup of $B$. 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 $SS$-quasinormal subgroups.