Groups with many modular or self-normalizing subgroups

In this paper, locally graded group satisfying the minimal condition on subgroups which are neither modular nor self-normalizing are described; locally (soluble-by-finite) groups of infinite rank in which every subgroup is either modular or self-normalizing are also characterized in terms of their subgroups of infinite rank. Moreover, the results obtained are used to study groups satisfying similar restrictions on subgroups which are neither permutable nor self-normalizing.