Set of invertible matrices with real entries

[LagrangeEuler]

$GL(n,\mathbb{R})$ is set of invertible matrices with real entries. We know that
SO(n,\mathbb{R}) \subset O(n,\mathbb{R}) \subset GL(n,\mathbb{R})
is there any specific subgroups of $GL(n,\mathbb{R})$ that is highly important.

 

[jgens]

You named most of the big ones. I would definitely add SL(n,R) to that list. Some of the subgroups consisting of diagonal matrices and say upper triangular matrices are important too.

 

[George Jones]

Other important subgroups are the pseudo-orthogonal groups O(p,g) and SO(p,q) (e.g., the Lorentz group) and the symplectic groups.

 

[jgens]

There is an embedding GL(n,C) < GL(2n,R) so all the important subgroups of GL(n,C) like the unitary groups can be regarded as subgroups of GL(2n,R) as well.