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).

The uncountable subgroup of $\Bbb Z^{[0,1]}$ of maps $[0,1]\to\Bbb Z$ vanishing at $0$ has index $|\Bbb Z|=\aleph_0$.
