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We explore a large class of F-theory compactifications to four dimensions. Wefind evidence that gauge groups that cannot be Higgsed without breakingsupersymmetry, often accompanied by associated matter fields, are a ubiquitousfeature in the landscape of ${\cal N} = 1$ 4D F-theory constructions. Inparticular, we study 4D F-theory models that arise from compactification onthreefold bases that are $\mathbb{P}^1$ bundles over certain toric surfaces.These bases are one natural analogue to the minimal models for base surfacesfor 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 onthe associated elliptic Calabi-Yau fourfolds, and 98.3% of the bases havegeometrically non-Higgsable gauge factors. The $\mathbb{P}^1$-bundle threefoldbases we analyze contain a wide range of distinct surface topologies thatsupport geometrically non-Higgsable clusters. Many of the bases that weconsider contain $SU(3)\times SU(2)$ seven-brane clusters for generic values ofdeformation moduli; we analyze the relative frequency of this combinationrelative to the other four possible two-factor non-Higgsable product groups, aswell as various other features such as geometrically non-Higgsable candidatesfor dark matter structure and phenomenological ($SU(2)$-charged) Higgs fields.

$\mathbb{P}^1$-bundle bases and the prevalence of non-Higgsable structure in 4D F-theory models