In a group, does $[G:H] \leq |N|$ always imply $[G:N] \leq|H|$?

Question

Let $G$ be a group and let $H$ and $N$ be subgroups of $G$. Suppose that $[G:H] \leq |N|$. Does this always imply that $[G:N] \leq |H|\ $? Lagrange's theorem tells us that this is true in the finite case. What about in general?

Answer 1

It doesn't hold in general. For example Consider $G =Z \times C_2=\langle a,b\rangle$. We take $H = \langle a^4\rangle$ and $N = \langle b\rangle$, so that $|G| = |H| = |G:N| = \aleph_0$, while $|G:H| = 4 \not\le |N| = 2$.