  Describing the conjugacy classes of the unipotent upper triangular groups
$\mathrm{UT}_{n}(\mathbb{F}_{q})$ uniformly (for all or many values of $n$ and
$q$) is a nearly impossible task. This paper takes on the related problem of
describing the normal subgroups of $\mathrm{UT}_{n}(\mathbb{F}_{q})$. For $q$ a
prime, a bijection will be established between these subgroups and pairs of
combinatorial objects with labels from $\mathbb{F}_{q}^{\times}$. Each pair
comprises a loopless binary matroid and a tight splice, an apparently new kind
of combinatorial object which interpolates between nonnesting partitions and
shortened polyominoes. For arbitrary $q$, the same approach describes a natural
subset of normal subgroups: those which correspond to the ideals of the Lie
algebra $\mathfrak{ut}_{n}(\mathbb{F}_{q})$ under an approximation of the
exponential map.
