For #1, let be normal subgroups of orders respectively. Using the identity and the fact that we find that is a subgroup of order . Let then with this means . Therefore this group is cyclic of order .
The general result says that the subgroups of (the cyclic subgroup of order ) are precisely where is a positive divisor of . And furthermore, is a subgroup of if and only if .
For example, the complete list of subgroups of are given below.