  High-resolution Zeeman spectroscopy of electronic-nuclear hyperfine levels of
${^5\mathrm{I}_8}\rightarrow{^5\mathrm{I}_7}$ transitions in
Ho$^{3+}$:Y$_2$SiO$_5$ is reported. Crystal-field parameters determined for the
two $C_1$ symmetry sites in Er$^{3+}$:Y$_2$SiO$_5$ are successfully used to
model the Zeeman-hyperfine data, including the prediction of avoided crossings
between hyperfine levels under the influence of an external magnetic field. The
two six- and seven-coordinate substitutional sites may be distinguished by
comparing the spectra with crystal-field calculations.
