Examples of Second Isomorphism Theorem for Groups I was wondering if there were any standard insightful applications of the second isomorphism for groups:
THM: Let $ G $ be a group with $ H, N $ as subgroups and $ N $ normal in $G$. Then $ H \cap N $ is normal in $ H $ and $ \frac{HN}{N} \cong \frac{H}{H\cap N} $. 
I have read through some helpful posts on the intuition of the theorem and feel that I grasp it now, but am still having trouble concocting a clear example. Any solid examples would be appreciated.

actually there are a lot of examples, sometimes working with $\frac{H}{H\cap N}$ doesn't give the answer so you'll be obliged to use the 2nd thm of isomorphism.
i'll give you an exmple 
whene you want to prove that a sub group of a solvable group is also solvable you'll take your quotients as $\frac{H}{H\cap G_{i}}$  where $G_{i}$  are the termes of the abelian serie of G
