If G is a group with subgroup H and m, k are two fixed nonnegative integers, H is called an $(m,k)$ -subnormal subgroup of G if it has index at most m in a subnormal subgroup of G of defect less than or equal to k. We study the behaviour of uncountable groups of cardinality $\aleph $ where all subgroups of cardinality $\aleph $ are $(m,k)$ -subnormal.

c-Sections of maximal subgroups in a finite group and their relation to solvability have been extensively researched in recent years. A fundamental result due to Wang [‘C-normality of groups and its properties’, J. Algebra 180 (1998), 954–965] is that a finite group is solvable if and only if the c-sections of all its maximal subgroups are trivial. In this paper we prove that if for each maximal subgroup of a finite group G, the corresponding c-section order is smaller than the index of the maximal subgroup, then each composition factor of G is either cyclic or isomorphic to the O’Nan sporadic group (the converse does not hold). Furthermore, by a certain ‘refining’ of the latter theorem we obtain an equivalent condition for solvability. Finally, we provide an existence result for large subgroups in the sense of Lev [‘On large subgroups of finite groups’ J. Algebra 152 (1992), 434–438].