A group is called metahamiltonian if all its non-abelian subgroups are normal. It is known that any infinite locally graded group whose proper subgroups are metahamiltonian is likewise metahamiltonian, and the aim of this paper is to describe the structure of locally graded groups whose proper subgroups contain a metahamiltonian subgroup of finite index.
"Groups whose proper subgroups are metahamiltonian-by-finite." Rocky Mountain J. Math. 50 (1) 153 - 162, Febuary 2020. https://doi.org/10.1216/rmj.2020.50.153