As Lie group, the isometry group of Riemannian manifold and its subgroup.

Question

Myers-Steenrod theorem says that the isometry group $\text{Iso}(M)$ of a Riemannian manifold $M$ is a Lie group. I am interested in the subgroup $$\text{Iso}_+(M)=\lbrace g\in \text{Iso}(M):\text{$g$ preserves the orientation of $M$} \rbrace$$ for an oriented Riemannian manifold $M$. Is it a (closed) Lie subgroup of $\text{Iso}(M)$? And I ask you in what conditions for $M$ the subgroup is compact.