  We explore a variety of composite topological structures that arise from the
spontaneous breaking of $SO(10)$ to $SU(3)_c \times U(1)_{em}$ via one of its
maximal subgroups $SU(5) \times U(1)_\chi$, $SU(4)_c \times SU(2)_L \times
SU(2)_R$, and $SU(5) \times U(1)_X$ (also known as flipped $SU(5)$). They
include i) a network of $\mathbb{Z}$ strings which develop monopoles and turn
into necklaces with the structure of $\mathbb{Z}_2$ strings, ii) dumbbells
connecting two different types of monopoles, or monopoles and antimonpoles,
iii) starfish-like configurations, iv) polypole configurations, and v) walls
bounded by a necklace. We display these structures both before and after the
electroweak breaking. The appearance of these composite structures in the early
universe and their astrophysical implications including gravitational wave
emission would depend on the symmetry breaking patterns and scales, and the
nature of the associated phase transitions.
