On one generalization of modular subgroups

Abstract: Let $G$ be a finite group. If $M_n < M_{n-1} < \ldots < M_1 < M_{0}=G $ where $M_i$ is a maximal subgroup of $M_{i-1}$ for all $i=1, \ldots ,n$, then $M_n $ ($n > 0$) is an \emph{$n$-maximal subgroup} of $G$. A subgroup $M$ of $G$ is called \emph{modular} if the following conditions are held: (i) $\langle X, M \cap Z \rangle=\langle X, M \rangle \cap Z$ for all $X \leq G, Z \leq G$ such that $X \leq Z$, and (ii) $\langle M, Y \cap Z \rangle=\langle M, Y \rangle \cap Z$ for all $Y \leq G, Z \leq G$ such that $M \leq Z$. In this paper, we study finite groups whose $n$-maximal subgroups are modular.