Examples of uncountable subgroups, but with countably many cosets. My question is pretty much as stated in the title: what examples are there (or does there exist) uncountable subgroups of a group but which have countably-infinite many cosets.  I can only think of examples in the other direction (countable subgroups giving uncountably-many cosets).

How about $\mathbb{R}$ in $\mathbb{Q} \times \mathbb{R}$? Or take any countably infinite group $H$, uncountable group $K$ and consider $K$ in  $H \times K$.
