  The lepton flavor mixing matrix having one row or one column in common with
the bimaximal mixing up to permutations is still compatible with the present
neutrino oscillation data. We provide a thorough exploration of generating such
a mixing matrix from $S_4$ family symmetry and generalized CP symmetry
$H_{CP}$. Supposing that $S_4\rtimes H_{CP}$ is broken down to
$Z^{ST^2SU}_2\times H^{\nu}_{CP}$ in the neutrino sector and
$Z^{TST^{2}U}_4\rtimes H^{l}_{CP}$ in the charged lepton sector, one column of
the PMNS matrix would be of the form $\left(1/2, 1/\sqrt{2}, 1/2\right)^{T}$ up
to permutations, both Dirac CP phase and Majorana CP phases are trivial in
order to accommodate the observed lepton mixing angles. The phenomenological
implications of the remnant symmetry $K^{(TST^2, T^2U)}_4\rtimes H^{\nu}_{CP}$
in the neutrino sector and $Z^{SU}_{2}\times H^{l}_{CP}$ in the charged lepton
sector are studied. One row of PMNS matrix is determined to be $\left(1/2, 1/2,
-i/\sqrt{2}\right)$, and all the three leptonic CP phases can only be trivial
to fit the measured values of the mixing angles. Two models based on $S_4$
family symmetry and generalized CP are constructed to implement these model
independent predictions enforced by remnant symmetry. The correct mass
hierarchy among the charged leptons is achieved. The vacuum alignment and
higher order corrections are discussed.
