  Using the newly measured masses of $B_c(1S)$ and $B_c(2S)$ from the CMS
Collaboration and the $1S$ hyperfine splitting determined from the lattice QCD
as constrains, we calculate the $B_c$ mass spectrum up to the $6S$ multiplet
with a nonrelativistic linear potential model. Furthermore, using the wave
functions from this model we calculate the radiative transitions between the
$B_c$ states within a constituent quark model. For the higher mass $B_c$ states
lying above $DB$ threshold, we also evaluate the Okubo-Zweig-Iizuka (OZI)
allowed two-body strong decays with the $^{3}P_{0}$ model. Our study indicates
that besides there are large potentials for the observations of the low-lying
$B_c$ states below the $DB$ threshold via their radiative transitions, some
higher mass $B_c$ states, such as $B_c(2^3P_2)$, $B_c(2^3D_1)$, $B_c(3^3D_1)$,
$B_c(4^3P_0)$, and the $1F$-wave $B_c$ states, might be first observed in their
dominant strong decay channels $DB$, $DB^*$ or $D^*B$ at the LHC for their
relatively narrow widths.
