Let $\Lambda \subset \mathbb{Z}^n$ be a proper sublattice (so that $\Lambda \ne \mathbb{Z}^n$). We say that $\Lambda$ is cyclically generated if there exists a matrix $M \in \text{GL}_n(\mathbb{Z})$ and an element $\mathbf{u} \in \mathbb{Z}^n$ such that $\Lambda$ is equal the $\mathbb{Z}$-span of $\{\mathbf{u}, M \mathbf{u}, M^2 \mathbf{u}, \cdots \}$. Is it true that all proper sublattices are cyclically generated? If so, how would one prove this, and if not, what's a counterexample?