It is proved that if a locally soluble group of infinite rank has only finitely many non-trivial conjugacy classes of subgroups of infinite rank, then all its subgroups are normal.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-2-6, author = {Maria De Falco and Francesco de Giovanni and Carmela Musella}, title = {Groups with finitely many conjugacy classes of non-normal subgroups of infinite rank}, journal = {Colloquium Mathematicae}, volume = {131}, year = {2013}, pages = {233-239}, zbl = {1286.20038}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-2-6} }
Maria De Falco; Francesco de Giovanni; Carmela Musella. Groups with finitely many conjugacy classes of non-normal subgroups of infinite rank. Colloquium Mathematicae, Tome 131 (2013) pp. 233-239. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-2-6/