The structure of infinite groups in which any two (proper) subgroups of the same cardinality are isomorphic is described within the universe of locally graded groups. The corresponding problem for finite groups was considered by R. Armstrong (1958).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm6779-1-2016, author = {Francesco de Giovanni and Alessio Russo}, title = {A note on groups with few isomorphism classes of subgroups}, journal = {Colloquium Mathematicae}, volume = {144}, year = {2016}, pages = {265-271}, zbl = {06575004}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6779-1-2016} }
Francesco de Giovanni; Alessio Russo. A note on groups with few isomorphism classes of subgroups. Colloquium Mathematicae, Tome 144 (2016) pp. 265-271. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6779-1-2016/