A note on groups with many locally supersoluble subgroups

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It is proved here that if $G$ is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble, then $G$ is either locally supersoluble or a vCernikov group. The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups. As a consequence, it is shown that any infinite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble.