  Using data samples collected with the BESIII detector at center-of-mass
energies $\sqrt{s} = 4.23, 4.26, 4.36,$ and $4.42$~\rm{GeV}, we measure the
branching fractions of $\eta_c\to K^+K^-\pi^0$, $K^0_S K^{\pm}\pi^{\mp}$,
$2(\pi^+\pi^-\pi^0)$, and $p \bar{p}$, via the process
$e^+e^-\to\pi^+\pi^-h_c$, $h_c\to\gamma\eta_c$. The corresponding results are
$(1.15\pm0.12\pm0.10)\%$, $(2.60\pm0.21\pm0.20)\%$, $(15.2\pm1.8\pm1.7)\%$, and
$(0.120\pm0.026\pm0.015)\%$, respectively. Here the first uncertainties are
statistical, and the second ones systematic. Additionally, the charged track
multiplicity of $\eta_c$ decays is measured for the first time.
