Nonmaximal symmetry breaking patterns in the supersymmetric $\widehat{\mathfrak{s}\mathfrak{u}}(8{)}_{k_U=1}$ theory

We study the nonmaximal symmetry breaking patterns of $\mathfrak{s}\mathfrak{u}(8)\to {\mathfrak{g}}_{531}\mathrm{/}{\mathfrak{g}}_{351}$ allowed in a supersymmetric theory with the affine Lie algebra of $\widehat{\mathfrak{s}\mathfrak{u}}(8{)}_{k_U=1}$, which was recently proposed to explain the three-generational standard model quark/lepton mass hierarchies as well as the Cabbibo-Kobayashi-Maskawa mixing pattern. Three gauge couplings are found to achieve the unification below the Planck scales, according to the unification relations determined by the affine Lie algebra at level of $1$. We also prove that a separate nonmaximal symmetry breaking pattern of $\mathfrak{s}\mathfrak{u}(8)\to {\mathfrak{g}}_{621}$ is unrealistic due to the additional massless vectorlike quarks in the spectrum.