Toral subgroup acting regularly on the homogeneous space

Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $H_x\subset G$ be the stability subgroup at $x$.

Question: Does there exist a toral subgroup $\mathbb{T}\subset G$ such that the subset $\{x\in G/H\,|\quad\mathbb{T}\cap H_x\neq\{1\}\}$ is at most countable?

Here $\mathbb{T}\subset G$ is a closed subgroup isomorphic to the usual torus $U(1)$.

This may turn out to be a very easy question, I just can't figure out an efficient approach. Thank you.

• Some questions : what do you mean by toral subgroup ? abelian connected subgroup ? compact ? The intersection $T\cap H_x$ contains the neutral element, you may not want to require the intersection to be empty ? – user120527 Feb 16 '18 at 12:13
• Yes, what is "a toral subgroup"? Note: The Pontryagin dual of the rationals doesn't have any subgroup isomorphic to $S^1$. – Uri Bader Feb 16 '18 at 13:45
• Rationals with which topology? Discrete? Is the dual connected in that case? Note that I explicitly required $G$ to be connected. Every pro-torus in a connected compact group is a projective limit of finite dimensional tori, and in particular contain a 1-torus. – Bedovlat Feb 16 '18 at 13:53
• Let $G$ be the solenoid and $H=\{1\}$. Then $G$ contains no total subgroups. – Misha Feb 16 '18 at 14:53
• I just realized Uri Bader mentioned the same example since the solenoid is isomorphic to the Pontryagin dual of the additive group of rational numbers (with the standard topology). – Misha Feb 16 '18 at 15:13