Suppose $H_1$ and $K_1$ are subgroups of $G_1$ and $H_2$ and $K_2$ are subgroups of $G_2$ such that $H_1 \cong H_2$ and $K_1 \cong K_2$.
When can we say $H_1 \cap K_1 \cong H_2 \cap K_2$?
If this holds and if those subgroups are normal, can we conclude $H_1K_1 \cong H_2K_2$? I also would like to know when this isomorphism holds.
Finally, if $H_1 \lhd K_1$ and $H_2 \lhd K_2$, do we have conditions to gurantee $K_1/H_1 \cong K_2/H_2?$
Edit
My motivation to ask this question comes from the fact that if $I$ and $J$ are ideals of a ring then $$(I+J)/(I\cap J) \cong (I+J)/I\ \oplus (I+J)/J$$ Also, $I/(I\cap J) \cong (I+J)/J$ and $J/(I\cap J) \cong (I+J)/I$