TITLE: Can an uncountable group $G$ have countably many distinct non isomorphic subgroups?
QUESTION [6 upvotes]: Can an uncountable group $G$ have only countably many distinct non isomorphic subgroups?I'm unable to produce a example.Any ideas?

REPLY [6 votes]: Given any set $X$, the free Abelian group $\Bbb Z[X]$ is of size $|X|+\aleph_0$, and it has exactly $|X|$ generators. Moreover, if $G$ is a free Abelian group with $|X|$ generators, then they are isomorphic.
So if $X$ is any set such that $\{|Y|\mid Y\subseteq X\}$ is a countable set, $\Bbb Z[X]$ will have exactly that many non-isomorphic subgroups. As bof notes in his answer, taking any $|X|<\aleph_{\omega_1}$ will do, and if $\aleph_1\leq|X|$ then the resulting group is indeed uncountable.