Suppose that $G_1$ has $m_1$ number of subgroups and $G_2$ has $m_2$ number of subgroups. Can we find any closed formula on the number of subgroups of $G_1\times G_2$ ? Here $G_1\times G_2$ is the external direct product of the groups $G_1, G_2$.
I tried to start with finite abelian groups $G_1, G_2$. But in vain. I mean manually step by step it is possible for me to derive the answer but I am unable to derive a closed formula. I know fundamental theory of finite abelian groups and I know how to use it to find step by step the ans. But in case, if I want to find a closed formula how shall I finish the task ?
Any help will be appreciated.