Finite groups whose $n$-maximal subgroups are modular

Let $G$ be a finite group. If $M_n<M_{n-1}<\dots<M_1<M_0=G$ with $M_i$ a maximal subgroup of $M_{i-1}$ for all $i=1,\dots,n$, then $M_n$ ($n>0$) is an $n$-maximal subgroup of $G$. A subgroup $M$ of $G$ is called modular provided that (i) $\langle X,M\cap Z\rangle=\langle X,M\rangle\cap Z$ for all $X\leq G$ and $Z\leq G$ such that $X\leq Z$, and (ii) $\langle M,Y\cap Z\rangle=\langle M,Y\rangle\cap Z$ for all $Y\leq G$ and $Z\leq G$ such that $M\leq Z$. In this paper, we study finite groups whose $n$-maximal subgroups are modular.