  For a connected reductive group $G$ over a finite field, we define partial
Hasse invariants on the stack of $G$-zip flags. We obtain similar sections on
the flag space of Shimura varieties of Hodge-type. They are mod $p$ automorphic
forms which cut out a single codimension one stratum. We study their properties
and show that such invariants admit a natural factorization through higher rank
automorphic vector bundles. We define the socle of an automorphic vector
bundle, and show that partial Hasse invariants lie in this socle.
