TITLE: countable group, uncountably many distinct subgroup?
QUESTION [33 upvotes]: I need to know whether the following statement is true or false?
Every countable group $G$ has only countably many distinct subgroups.
I have not gotten any counter example to disprove the statement but an vague idea to disprove like: if it has uncountably many distinct subgroup then It must have uncountable number of element?

REPLY [6 votes]: There is another one: $F_\infty$, the free group on countably many letters $x_1,x_2, \ldots$.
It is countable because each of its elements is a finite string of symbols from a countable alphabet. At the same time, for every subset $A\subset \mathbb{N}$, there is a subgroup $H_A$ generated by the set $\{x_i | i \in A\}$. These subgroups are distinct, so we get uncountably many of them.
Also, $F_\infty$ embeds into $F_2$, so we can get these uncountably many subgroups within a finitely presented group!