  In this paper we will be classifying some regular upper-triangular
subalgebras of $\mathfrak{sl}(n, \mathbb C)$ up to conjugacy by matrices in
$SL(n, \mathbb C)$. We do so for dimension 2, codimension 1, and codimension 2
subalgebras. We prove some general results for codimension $k$. The approach we
use reduces an abstract classification problem to a combinatorial one, which we
solve through a mixture of inductive and computational approaches.
