Number of elements in a group and its subgroups (GS 2013) 
Every countable group has only countably many distinct subgroups.

The above statement is false. How to show it? One counterexample may be sufficient, but I am blind to find it out. I have considered some counterexample only like $(\mathbb{Z}, +)$, $(\mathbb{Q}, +)$ and $(\mathbb{R}, +)$. 
Is there any relationship between number of elements in a group and number of its subgroup? I do not know. Please discuss a little. What will be if the group be uncountable?
Thank you for your help.

Take $\bigoplus_{i \in \Bbb N}\Bbb Z_i $ direct sum of countable many copies of $(\Bbb Z,+)$. It's free group with countable base. $Card(\bigoplus_{i \in \Bbb N}\Bbb Z_i)=Card(\Bbb N^*)$. It's easy to see that set of subgroups of $\bigoplus_{i \in \Bbb N}\Bbb Z_i $, $\{\bigoplus_{i \in A}\Bbb Z_i \mid A\in \mathcal P(\Bbb N)\}$ is uncountable.
$\bigoplus_{i \in \Bbb N}\Bbb Z_i $ is a subgroup of $\prod_{i \in \Bbb N}\Bbb Z_i$ consisting of those sequences which are $0$ everywhere but finite number of places. Thats why its countable. Now for different $A,B\subset \Bbb N$ we get different subgroups $\bigoplus_{i \in A}\Bbb Z_i $, $\bigoplus_{i \in B}\Bbb Z_i $. There is continuum different subset of $\Bbb N$ and we are done.
