  We introduce Sylow subgroups and $0$-groups to the theory of complex
algebraic supergroups, which mimic Sylow subgroups and $p$-groups in the theory
of finite groups. We prove that Sylow subgroups are always $0$-groups, and show
that they are unique up to conjugacy. Further, we give an explicit
classification of $0$-groups which will be very useful for future applications.
Finally, we prove an analogue of Sylow's third theorem on the number of Sylow
subgroups of a supergroup.
