Is the space of conjugacy classes of algebraic subgroups of a fixed group a standard Borel space?

Suppose $H$ is an algebraic group (let's say over $\mathbb{R}$ or $\mathbb{C}$). I'm interested in the space $\mathrm{Sub}_{\text{alg}}(H)$ whose elements are conjugacy classes of algebraic subgroups of $H$. Is it true that $\mathrm{Sub}_{\text{alg}}(H)$ can be realized as a standard Borel space?

As a bonus, is there also more structure on here? I would be a little surprised if it was a variety.

----EDIT----

After some thought, I believe the best way to go may be to use the following theorem of Chevalley (see, for example, 3.1.4 in Zimmer's book Ergodic Theory & Semisimple Groups) which says: If $L<H$ is an algebraic subgroup then there is a rational representation $\pi:H\to GL_n\mathbb{R}$ and a point $p\in \mathbb{P}^{n-1}$ such that $L$ is the stabilizer of $p$.

Switching to descriptive set theory mode, one should maybe be able to say that the set of all finite dimensional rational representations is countable because rationality means you are cut out by polynomials. Then each $L$ is in correspondence with a point in one of countably many projective spaces (which are of course very nice) or something? I'm not sure whether anything I have described gives me a Borel description of $\mathrm{Sub}_\mathrm{alg}(H)$. If someone can make this formal, that's great, and if you have another way, also great.