Suppose $L=[1,\tau]$ is an integer lattice with $\tau$ in the upper half-plane, $\gamma=\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ is an integer $2\times 2$ matrix with determinant $n$, and $L' = [a\tau+b, c\tau+d]$ is a sublattice. Cox provides a proof (in an exercise) that $L/L'$ is cyclic if and only if $\gcd(a,b,c,d)=1$. He also gives a reference to Lang's Elliptic Functions that supposedly provides another proof.
The relevant theorem in Lang is: Let $\Delta_n^*$ be the set of integral matrices with determinant $n$ whose entries are relatively prime (in the sense above). Then $SL(2,\mathbb{Z})$ operates left transitively on the right $SL(2,\mathbb{Z})$-cosets of $\Delta_n^*$, and operates right transitively on the left $SL(2,\mathbb{Z})$-cosets. The proof is pretty straightforward, using the elementary divisor theorem on the lattice $L=[1,\tau]$ and sublattice $L'=[a\tau+b,c\tau+d]$ to show that $\Delta_n^* = SL(2,\mathbb{Z})\,\gamma\, SL(2,\mathbb{Z})$ for any $\gamma\in \Delta_n^*$.
I'm sure I'm missing something obvious, but I just don't see how this implies that $L/L'$ is cyclic.