Let $G = Gl_{2}(\mathbb{C})$ and let $H = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} \ |\ a,b,c \in \mathbb{C}, ac \neq 0 \right\}$. Prove that every element of G is conjugate to some element of the subgroup H.
That is we need to prove that following $G = gHg^{-1}$, so we need to show that for every matrix in G it is similar to matrices of H or that is at least what I think we should do ? However I am not sure if my reasoning is correct as its leading me to big calculations and I got lost while calculating general eigenvector of element $A \in Gl_2(\mathbb{C})$.
This results, say, from Jordan normal form over $\mathbf C$. Or from the theorem that a matrix can be put in triangular form by a change of basis if and only if its characteristic/minimal polynomials split over the base field into linear factors, which is the case when the base field is the algebraically closed field $\mathbf C$.