Finite groups admitting maximal subgroup series with certain normality

Let $G$ be a finite group and $H$ be a subgroup of $G$. Then $H$ is said to be $S$-quasinormally embedded in $G$ if for each prime $p$ dividing the order of $H$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $S$-quasinormal subgroup of $G$. $H$ is said to be $c$-$c$-permutable in $G$ if for each subgroup $A$ of $G$, there exists an element $ g \in \langle A, H \rangle $ such that $ AH^g = H^gA $. $H$ is said to be an $SS$-quasinormal subgroup of $G$ if there is a supplement $B$ of $H$ to $G$ such that $H$ permutes with every Sylow subgroup of $B$. A subgroup series $\Omega:G=G_{0}>G_{1}>\cdots>G_i>\cdots > G_{n-1}>G_{n} = 1$ is said to be a maximal subgroup series of $G$ if $G_i$ is a maximal subgroup of $G_{i-1}$ for each $i\in\{1,2,\ldots,n\}$. In this paper, we first prove that $G$ is supersolvable if and only if $G$ possesses subnormal maximal series $\Omega$ such that either $G_i$ is $S$-quasinormally embedded in $G$, or $G_i$ is $SS$-quasinormal in $G$ for each $i\in\{1,2,\ldots,n\}$. Second, we prove that if $G$ possesses a maximal subgroup series $\Omega$ such that either $G_i$ is $c$-$c$-permutable in $G$, or $G_i$ is $SS$-quasinormal in $G$, then $G$ is solvable.