So I know that subgroups of $(\mathbb{R},+)$ are either isomorphic to $x \mathbb{Z}$ for $x\in \mathbb{R}$, or dense in $\mathbb{R}$.
I don't see how (if?) this descends to the quotient, though. Denote $G = \mathbb{R}/\mathbb{Z}$. If $H \leq G$, then by the third iso theorem there is a group $\mathbb{Z} \leq \tilde{H} \leq \mathbb{R}$ such that $H \cong \tilde{H}/\mathbb{Z}$. How to proceed? I'm very lost.
(Also, from this it supposedly follows that for $x\in\mathbb{R}$ the image of $$ \phi_x:\mathbb{Z} \to G, \quad n \mapsto nx + \mathbb{Z} $$ is dense in $G$ if $x$ is irrational. I also don't see this: For any $x$, the image of $\phi_x$ is $x\mathbb{Z} + \mathbb{Z}$, is this not cyclic? The above result would then imply that the image is not dense. Somewhere is an error in my thought process.)
