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.

This wont answer to your question, but still it is interesting. consider $S_{\mathbb{N}}$ , the set of all bijections from $\mathbb{N}$ to $\mathbb{N}$ which is uncountable in cardinality. 
Now for any non empty subset $A$ of $\mathbb{N}$  we can find a subgroup of $S_{\mathbb{N}}$ which is isomorphic to $S_A$ namely the collection of all bijections in $S_{\mathbb{N}}$ which fixes all the elements of $\mathbb{N} - A$. 
Hence $S_{\mathbb{N}}$ has uncountably many subgroups. 
