  Using a sample of $(10087\pm44)\times10^{6}$ $J/\psi$ events collected with
the BESIII detector at the BEPCII collider, a partial wave analysis on the
decay $\gamma\gamma\phi$ is performed to investigate the intermediate
resonances in $J/\psi\rightarrow\gamma X, X\rightarrow\gamma\phi$. The
resonances $f_{1}(1285)$, $\eta(1405)$, $f_{1}(1420)$, $f_{1}(1510)$,
$f_{2}(1525)$, $X(1835)$, $f_{2}(1950)$, $f_{2}(2010)$, $f_{0}(2200)$ and
$\eta_{c}$ are observed with statistical significance greater than 5$\sigma$.
The product branching fractions $\mathcal{B}(J/\psi\rightarrow\gamma X,
X\rightarrow \gamma \phi)$ are reported. The resonance parameters of
$\eta(1405)$ and $X(1835)$ are also measured.
