Normal subgroups of direct product

Question

Does anybody have an example of a normal subgroup N of a group $G\times H$ such that N is not equal to some product $N_1\times N_2$ while $N_1$ is normal in G and $N_2$ is normal in H.

I've already proofed the analog statement for rings-ideals, but I've understood that this is not necessary so using groups-normal subgroups and I'm having a hard time to find an example.

Answer 1

Even in the simplest of cases: take $\;N_1=N_2=\Bbb Z_2=\{0,1\}\pmod 2\;$, and now take the Klein group $\;\Bbb Z_2\times \Bbb Z_2$ and in it the diagonal subgroup $\;N:=\langle\,(1,1)\;\rangle\;$ ...

Answer 2

Hint: What are the subgroups of the Klein 4-group?

There are three proper, non-trivial subgroups. List them.