# How to calculate all the subgroups of the fundamental group of the Klein bottle?

A problem asks me to find all the covering spaces of a Klein bottle. This needs to calculate all the subgroups of the fundamental group of the Klein bottle. But I don't have any idea how to do it.

I googled it and an article says

The subgroups of the fundamental group of the Klein bottle are either trivial, free of rank one, free Abelian of rank two, or non-Abelian of rank two.

I don't know how to get the result and what is the concrete form of the subgroups (which is needed to calculate the covering spaces.)

The Klein bottle group is a semi-direct product $\mathbb Z \rtimes \mathbb Z$ where $\mathbb Z$ acts on $\mathbb Z$ by its sole non-trivial involution. So any subgroup of the Klein bottle group is a semi-direct product $A \rtimes B$ where $A, B \subset \mathbb Z$, and there's basically just three possibilities for each of $A$ and $B$, up to isomorphism and the action of $B$ on $A$, etc.. – Ryan Budney Apr 21 '11 at 17:57
@Ryan: Well, I'm not quite familiar with semi-direct product. But your statement is not true when it is direct product. Not all subgroups of $\mathbb{Z}\times\mathbb{Z}$ are in the form $A\times B$ where $A,B\subset\mathbb{Z}$. – Roun Apr 22 '11 at 1:06