I already know that if m and n are coprime, then $\mathbb{Z}_{mn}^*$ is isomorphic to $\mathbb{Z}_m^* \times \mathbb{Z}_n^*$ using the Chinese remainder theorem.
Now, Is it possible to describe specifically all the subgroups of $\mathbb{Z}_{mn}^*$ isomorphic to $\mathbb{Z}_m^*$ and $\mathbb{Z}_n^*$, respectively?
I have tried to find such a subgroup using the isomorphism between two groups. (the image of a group is isomorphic to the group if the map is injective) But, I think the image may be impossible to describe specifically.
Do you have any ideas? I need your help. ;(