  Motivated by recent experiments and density functional theory calculations on
choloalite PbCuTe$_2$O$_6$, which possesses a Cu-based three-dimensional
hyperkagome lattice, we propose and study a $J_1$-$J_2$-$J_3$ antiferromagnetic
Heisenberg model on a hyperkagome lattice. In the classical limit, possible
ground states are analyzed by two triangle rules, i.e., the "hyperkagome
triangle rule" and the "isolated triangle rule," and classical Monte Carlo
simulations are exploited to identify possible classical magnetic ordering and
explore the phase diagram. In the quantum regime, Schwinger boson theory is
applied to study possible quantum spin liquid states and long-range
magnetically ordered states on an equal footing. These quantum states with
bosonic partons are classified and analyzed by using projective symmetry groups
(PSGs). It is found that there are only four types of algebraic PSGs allowed by
the space group $P4_{1}32$ on a hyperkagome lattice. Moreover, there are only
two types of PSGs that are compatible with the $J_1$-$J_2$-$J_3$ Heisenberg
model. These two types of $Z_2$ bosonic states are distinguished by the
gauge-invariant flux on the elementary ten-site loops on the hyperkagome
network, called zero-flux state and $\pi$-flux state respectively. Both the
zero-flux state and the $\pi$-flux state are able to give rise to quantum spin
liquid states as well as magnetically ordered states, and the zero-flux states
and the $\pi$-flux states can be distinguished by the lower and upper edges of
the spectral function $S(\bm{q},\omega)$, which can be measured by inelastic
neutron scattering experiments.
