In this paper, we report an experimental observation of the large anomalous Hall effect (AHE) in a hexagonal ferromagnetic $\mathrm{F}{\mathrm{e}}_{5}\mathrm{S}{\mathrm{n}}_{3}$ single crystal with current along the $b$ axis and a magnetic field normal to the $bc$ plane. The intrinsic contribution of the anomalous Hall conductance ${\ensuremath{\sigma}}_{AH}^{\mathrm{int}}$ was approximately $613\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Omega}}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}\mathrm{c}{\mathrm{m}}^{\ensuremath{-}1}$, which was more than 3 times the maximum value in the frustrated kagome magnet $\mathrm{F}{\mathrm{e}}_{3}\mathrm{S}{\mathrm{n}}_{2}$ and nearly independent of the temperature over a wide range between 5 and 350 K. The analysis results revealed that the large AHE was dominated by a common, intrinsic term, while the extrinsic contribution, i.e., the skew scattering and side jump, turned out to be small. In addition to the large AHE, it was found that the types of majority carriers changed at approximately 275 and 30 K, consistent with the critical temperatures of the spin reorientation. These findings suggest that the hexagonal ferromagnetic $\mathrm{F}{\mathrm{e}}_{5}\mathrm{S}{\mathrm{n}}_{3}$ single crystal is an excellent candidate to use for the study of the topological features in ferromagnets.