I am interested in finding a method for determining all the matrix subgroups of a matrix group that have a specific connected component. This is what I thought would work from what I have read so far is:
Suppose we have a matrix group $G$ (for example $SO(n)$, $USp(2n)$, $U(n)$ etc.). Further we have a connected subgroup of $G$, $G_1$. I want to find a way to determine all the subgroups $H \subset G$ that have connected component $G_1$.
From what I have read it seems to me that $H \subset N_G(G_1)$, the normaliser of $G_1$ in $G$. Further, there would exists a subgroup $H_1 \subset N_G(G_1)/G_1$ such that all the components of $H$ would be $hG_1$ for $h \in H_1$. Thus to determine all possible $H$ it is sufficient to determine all possible subgroups of $N_G(G_1)/G_1$.
Does this work? Is it true that $H \subset N_G(G_1)$? If not, is there a method known for doing something like this? Any solution or reference would be greatly appreciated.