How do I determine the zero elements of a Hamiltonian in a 4 ket space?

The Hamiltonian matrix of particle subject to a central potential is described by

$$
H=begin{pmatrix}
H_{11} & H_{12} & H_{13} & H_{14}
H_{21} & H_{22} & H_{23} & H_{24}
H_{31} & H_{32} & H_{33} & H_{34}
H_{41} & H_{42} & H_{43} & H_{44}
end{pmatrix}
$$

where $$ H_{ij}= langlephi_i| H |phi_j rangle $$

and each component has a wave function on the coordinate space given by

$$
begin{align}
langletextbf{x}|phi_1rangle & = e^{-alpha r}
langletextbf{x}|phi_2rangle & = xe^{-beta r}
langletextbf{x}|phi_3rangle & = ye^{-beta r}
langletextbf{x}|phi_3rangle & = ze^{-beta r}
end{align}
$$

How can I identify which elements are zero?

One hint I received was trying to use the parity ($ pi $) and the plane reflection ($ mathcal{O}_{xy}, mathcal{O}_{xz}, mathcal{O}_{yz} $) operators, but that didn’t really help.