In general, I feel like the group $\mathbb{Z} \times \mathbb{Z}$ of integers under addition contain subgroups of the form $a\mathbb{Z} \times b\mathbb{Z}$.
In particular, I am trying to find how many such subgroups contain the product group of the even integers. I tried defining the surjective homomorphism $\varphi\colon \mathbb{Z}\times\mathbb{Z} \to \{(-1, -1), (-1, 1), (1, -1), (1, 1)\}$ under multiplication. Intuitively, I would think that this would lead to $4$ such subgroups. Is that correct?