Let $G$ be a compact Hausdorff group, $H \leq G$ a closed subgroup of infinite index in $G$. 


Is it possible that the conjugates of $H$ cover some open neighbourhood of $1$ in $G$ (or the whole of $G$)?


If this is possible, I would like to know whether there are conditions on $G$ rendering this impossible.



Union of conjugates of a closed subgroup of a compact group
The usual example is $G = {\rm SU}\_2({\bf C})$ and $H$ the diagonal subgroup.
Every unitary matrix is diagonalizable, and thus contained in a
conjugate of $H$.


