Let $G = \prod_{i=1}^\infty \mathbb{Z}_2$ with addition mod 2. I am trying to find subgroups of index 2. I see that taking the entire space and removing all sequences which have a 1 in a certain position gives a subgroup of index 2. For example the set of all sequences $\{(0,\cdot,\cdot,\ldots)\}$ which have a 0 in the first position forms a subgroup. Taking the coset $\{(0,\cdot,\cdot,\ldots)\} + (1,0,0,\ldots)$ gives all other sequences, so there are two cosets.
This gives me infinitely many subgroups of index 2. However, I have read there are actually uncountably many such subgroups. How can I find the others? Thank you!