Finding all subgroups of $\mathbb{Z}^2$ using algebraic topology.
I want to find all the subgroups of $\mathbb{Z}^2$ using algebraic topology, I know that $\mathbb{Z}^2$ acts in a properly discontinuous way in $\mathbb{R}^2$ by means of translation and so $p:\mathbb{R}^2\to \mathbb{R}^2/\mathbb{Z}^2\cong \mathbb{S}^1\times\mathbb{S}^1$ is a universal covering space, so that any $H\subset \mathbb{Z}^2$ subgroup corresponds to a $p: \mathbb{R}^2/H\to \mathbb{R}^2/\mathbb{Z}^2\cong \mathbb{S}^1\times\mathbb{S}^1$ covering space. .