As the title, I would like to ask the dimension of the group of all motions in $\mathbb{R}^n$ which leaves a fixed r-plane $L^0_r$ invariant.

Here is my observation, but I don't know if it is useful to help solving the problem.

(1) Since the group of motions $M_n$ in $\mathbb{R}^n$ consists of the group of rotations, $SO(n)$, and the group of translations which is isomorphic to $\mathbb{R}^n$, we have $$M_n= SO(n)\times \mathbb{R}^n, $$ and $$dim(M_n)=\frac{n(n-1)}{2}+n=\frac{n^2+n}{2}.$$

(2) Given a fixed r-plane $L^0_r$ in $\mathbb{R}^n$ ($L^0_r$ may as a subspace, passes through the origin $O$ or not (as an affine subspace of $\mathbb{R}^n$). Let $h_r$ be the subgroup of $M_n$ such that any element in $h_r$ leaves $L^0_r$ invariant. By the fundamental knowledge of homogeneous spaces, we have the fact that there is an one-to-one correspondence between the space of r-planes $L_r$ in $\mathbb{R}^n$ and the homogeneous space $M_n/h_r$ by $$L_r=g \cdot L^0_r\leftrightarrow g\in M_n/h_r,$$ where $g\cdot L^0_r$ is the left translation in $\mathbb{R}^n$ regarded as a Lie group $(\mathbb{R}^n, \cdot)$.