Let $G/\mathbb{C}$ be a connected linear algebraic group. Let $\rho:G\to \mathrm{GL}(V)$ be a rational representation. If $G$ is unipotent, then does $\rho$ has at most countably many subrepresentations?
The question is motivated by the following. If $G$ is reductive, then it has at most countably many representations. If $G=\mathbb{G}_a$, then a representation of $G$ on $V$ is a nilpotent linear operator on $V$. From the Jordan normal form, $G$ also has at most countably many representations. But I don't know what happens for noncommutative unipotent groups.