  We explore a large class of F-theory compactifications to four dimensions. We
find evidence that gauge groups that cannot be Higgsed without breaking
supersymmetry, often accompanied by associated matter fields, are a ubiquitous
feature in the landscape of ${\cal N} = 1$ 4D F-theory constructions. In
particular, we study 4D F-theory models that arise from compactification on
threefold bases that are $\mathbb{P}^1$ bundles over certain toric surfaces.
These bases are one natural analogue to the minimal models for base surfaces
for 6D F-theory compactifications. Of the roughly 100,000 bases that we study,
only 80 are weak Fano bases in which there are no automatic singularities on
the associated elliptic Calabi-Yau fourfolds, and 98.3% of the bases have
geometrically non-Higgsable gauge factors. The $\mathbb{P}^1$-bundle threefold
bases we analyze contain a wide range of distinct surface topologies that
support geometrically non-Higgsable clusters. Many of the bases that we
consider contain $SU(3)\times SU(2)$ seven-brane clusters for generic values of
deformation moduli; we analyze the relative frequency of this combination
relative to the other four possible two-factor non-Higgsable product groups, as
well as various other features such as geometrically non-Higgsable candidates
for dark matter structure and phenomenological ($SU(2)$-charged) Higgs fields.
