  We study Dark Matter (DM) abundance in the framework of the extension of the
Standard Model (SM) with an additional $U(1)_X$ gauge symmetry. One complex
singlet is included to break the $U(1)_X$ gauge symmetry, meanwhile one of the
doublets is considered inert to introduce a DM candidate. The stability of the
DM candidate is analyzed with a continuous $U(1)_X$ gauge symmetry as well as
discrete $Z_2$ symmetry. We find allowed regions for the free model parameters
which are in agreement with the most up-to-date experimental results reported
by CMS and ATLAS collaborations, the upper limit on WIMP-nucleon cross section
imposed by XENON1T collaboration and the upper limit on the production
cross-section of a $Z^{\prime}$ gauge boson times the branching ratio of the
$Z^{\prime}$ boson decaying into $\ell^-\ell^+$. We also obtain allowed regions
for the DM candidate mass from the relic density reported by the PLANCK
collaboration including light, intermediate and heavy masses; depending mainly
on two parameters of the scalar potential, $\lambda_{2x}$ and
$\lambda_{345}=\lambda_3+\lambda_4+2\lambda_5$. We find that trough
$pp\rightarrow \chi\chi \gamma$ production, it may only be possible for a
future hadron-hadron Circular Collider (FCC-hh) to be able to detect a DM
candidate within the range of masses 10-60 GeV.
