Suppose $G$ is a compact connected Lie group and $H$ is a subgroup of $G$. Then is $G/H$ isomorphic to some subgroup of $G$?
Browsing around the internet, I came across the following link: https://groupprops.subwiki.org/wiki/Quotient_group_need_not_be_isomorphic_to_any_subgroup
Here it is stated that
1) If $G$ is a finite abelian group, then any quotient group is isomorphic to some subgroup.
2) If we quotient the Quarternion group $G$ by its center $H$, this is not isomorphic to any subgroup of $G$.
Can anything be said in my setting above? Thanks!
The group ${\rm Spin}(n)$ is connected and is the universal cover of $SO(n)$ if $n>2$ it is simply connected. ${\rm Spin}(n), n>2$ does not have a subgroup isomorphic to $SO(n)$ since such a subgroup would have been open (dimension) closed (as a Lie subgroup) and equal to ${\rm Spin}(n)$ since ${\rm Spin}(n)$ is connected contradiction since $SO(n)$ is not ${\rm Spin}(n), n>2$ since $SO(n)$ is not simply connected.
https://en.wikipedia.org/wiki/Spin_group