Let us be given the three normal subgroups $N_1,N_2,N_3$ of a group $G$ together with an isomorphism of $G$ onto $G/K_1\times G/K_2\times G/K_3$ defined by $$\pi(g)=(gK_1,gK_2,gK_3),$$ where $K_1=\langle N_2\cup N_3\rangle$, $K_2=\langle N_1\cup N_3\rangle$ and $K_3=\langle N_1\cup N_2\rangle$.
What is the relationship between each pair of subgroups $N_i$ and $K_i$? e.g. how do they interact with each other in this example (as quotient groups or otherwise)? Is the $K_i$ some sort of "complement" of the $N_i$?
(Is this a useless isomorphism?)
edit:
well, I can use this isomorphism to construct another isomorphism: $$N_1\times N_2\times N_3\rightarrow\pi^{-1}(\pi_1(N_1)\times\pi_2(N_2)\times\pi_3(N_3))$$ which sends $(x_1,x_2,x_3)$ to $x_1x_2x_3$. But I would like for the codomain to be G. Is that possible? Note that $\pi_i:G\rightarrow G/K_i$ is the canonical homomorphism.