When I have a finite group $G$ and some subgroup $H$, Lagrange's Theorem tells me that the number of cosets of $H$ in $G$ is $\frac{|G|}{|H|}$. Is there any analog of this for infinite groups?
For example $3\mathbb{Z}$ has three distinct cosets in $\mathbb{Z}$, which I can tell by just taking a moment to think about it, but I can't see a general approach to deduce anything similar for a infinite group that is not so convenient as $\mathbb{Z}$.
(I suspect the answer to this is no, but references to any relevant theorems would be appreciated)