Finite groups with submodular Sylow subgroups

A subgroup $H$ of a finite group $G$ is submodular in $G$ if $H$ can be joined with $G$ by a chain of subgroups each of which is modular in the subsequent subgroup. We reveal some properties of groups with submodular Sylow subgroups. A group $G$ is called strongly supersoluble if $G$ is supersoluble and every Sylow subgroup of $G$ is submodular. We show that $G$ is strongly supersoluble if and only if $G$ is metanilpotent and every Sylow subgroup of $G$ is submodular. The following are proved to be equivalent: (1) every Sylow subgroup of a group is submodular; (2) a group is Ore dispersive and its every biprimary subgroup is strongly supersoluble; and (3) every metanilpotent subgroup of a group is supersoluble.