Let $k$ be an algebraic-closed field and let $G$ be a soluble subgroup of $GL_n(k)$. By a result of Mal'cev (Bertram A.F. Wehrfritz - Infinite Linear Groups, Theorem 3.6), $G$ contains a normal triangularizable subgroup of finite index, i.e. a subgroup which is conjugated inside $GL_n(k)$ with a (lower) triangular subgroup.
Assume that $G$ itself is triangularizable and that contains an element $g$ such that its first row has $1$ in the first position and $0$ elsewhere. My question is the following: is it possible to find a triangular subgroup $T$ of $GL_n(k)$ and an element $x$ of $GL_n(k)$ such that $G^x=H$ and $g^x$ has its first row equal to that of $g$?