Finite groups whose $n$-maximal subgroups are generalized $S$-quasinormal

Let $G$ be a finite group and $M$ a subgroup of $G$. Then $M$ is called: modular in $G$ if the following conditions are held: (i) $\langle X, M\cap Z\rangle=\langle X, M\rangle\cap Z$ for all $X\leqslant G$, $Z\leqslant G$ such that $X\leqslant Z$, and (ii) $\langle M, Y\cap Z\rangle=\langle M, Y\rangle\cap Z$ for all $Y\leqslant G$, $Z\leqslant G$ such that $M\leqslant Z$; quasinormal (respectively $S$-quasinormal) in $G$ if $MP=PM$ for all subgroups (respectively for all Sylow subgroups) $P$ of $G$. We say that $M$ is a generalized subnormal (respectively generalized $S$-quasinormal) subgroup of $G$ if $H=\langle A, B\rangle$ for some modular subgroup $A$ and subnormal (respectively $S$-quasinormal) subgroup $B$ of $G$. If $M_n< M_{n-1}<\dots<M_1<M_0=G$, where $M_i$ is a maximal subgroup of $M_{i-1}$ for all $i=1,\dots,n$, then $M_n$ ($n>0$) is an $n$-maximal subgroup of $G$. In this paper, we study finite groups whose $n$-maximal subgroups are generalized subnormal or generalized $S$-quasinormal.