If you visit this link, you'll see at the top of the PDF view. Basic properties of finite abelian groups:

Every quotient group of a finite abelian group is isomorphic to a subgroup.

If the above statement true, it would make some proofs in Serge Lang's Algebra easier, particularly in the p-Sylow groups section.

I know that there is a correspondence between subgroups of G/N and subgroups of G containing N, but the corresponding groups are not necessarily isomorphic or are they?