Is there a good example of a subgroup of an infinitely generated abelian group that is not isomorphic to a quotient of that group?

Whilst I understand the classification of the finitely generated abelian groups, this had me wondering whether there is a subgroup $H$ of a general (necessarily infinitely generated) abelian group $G$ such that $H$ is not isomorphic to any quotient $G/N$ of $G$.

Solution 1:

For a very simple example, let $G=\mathbb{Q}$ and $H=\mathbb{Z}$. Since $\mathbb{Q}$ is divisible (for any $x\in\mathbb{Q}$ and any nonzero integer $n$, there exists $y\in\mathbb{Q}$ such that $x=ny$), any quotient of $\mathbb{Q}$ is also divisible, so $\mathbb{Z}$ is not isomorphic to any quotient of $\mathbb{Q}$.