  Recently, a new pairing state with the mixing between s-wave singlet channel
and isotropic d-wave quintet channel induced by centrosymmetric spin-orbit
coupling has been theoretically proposed in the superconducting materials with
$j=\frac{3}{2}$ electrons. In this work, we derive the expressions of the
zero-temperature spin susceptibility, the upper critical field close to the
zero-field critical temperature $T_c$ and the critical temperature with weak
random non-magnetic disorders for the singlet-quintet mixed state based on the
Luttinger model. Our study revealed the following features of the
singlet-quintet mixing. (1) The zero-temperature spin susceptibility remains
zero for the singlet-quintet mixed state if only the centrosymmetric spin-orbit
coupling is taken into account, and will deviate from zero when the
non-centrosymmetric spin-orbit coupling is introduced. (2) The singlet-quintet
mixing can help enhance the upper critical field roughly because it can
increase $T_c$. (3) Although the quintet channel is generally suppressed by the
non-magnetic disorder scattering, we find the strong mixing between singlet and
quintet channels can help to stabilize the quintet channel. As a result, we
still find a sizable quintet component mixed into the singlet channel in the
presence of weak random non-magnetic disorders. Our work provides the guidance
for future experiments on spin susceptibility and upper critical field of the
singlet-quintet mixed superconducting states, and illustrates the stability of
the singlet-quintet mixing against the weak random non-magnetic disorder.
