Finite groups and $k$-submodular subgroups

For natural numbers $n$ and $k$, the concepts of $n$-modularly embedded subgroup, $k$-submodular subgroup and $k$-$\mathrm{L}\mathrm{M}$-group are given, which generalize, respectively, the concepts of modular subgroup, submodular subgroup and $\mathrm{L}\mathrm{M}$-group. Classes of groups with given systems of $k$-submodular subgroups are investigated.