  In a sample of 58 million $J/\psi$ events collected with the BES II detector,
the process J/$\psi\to\gamma\eta_c$ is observed in five decay channels: $\eta_c
\to K^+K^-\pi^+\pi^-$, $\pi^+\pi^-\pi^+\pi^-$, $K^\pm K^0_S \pi^\mp$ (with
$K^0_S\to\pi^+\pi^-$), $\phi\phi$ (with $\phi\to K^+K^-$) and $p\bar{p}$. From
these signals, we determine $Br(J/\psi\to\gamma\eta_c)\times Br(\eta_c\to
K^+K^-\pi^+\pi^-)$ $=(1.5\pm0.2\pm0.2)\times10^{-4}$,
$Br(J/\psi\to\gamma\eta_c)\times Br(\eta_c\to \pi^+\pi^-\pi^+\pi^-)$
$=(1.3\pm0.2\pm0.4)\times10^{-4}$, $Br(J/\psi\to\gamma\eta_c)\times
Br(\eta_c\to K^\pm K_{S}^{0}\pi^\mp)$ $=(2.2\pm0.3\pm0.5)\times10^{-4}$,
$Br(J/\psi\to\gamma\eta_c)\times Br(\eta_c\to \phi\phi)$
$=(3.3\pm0.6\pm0.6)\times10^{-5}$ and $Br(J/\psi\to\gamma\eta_c)\times
Br(\eta_c\to p\bar{p})$ $=(1.9\pm0.3\pm0.3)\times10^{-5}$.
