The natural filtrations of the infinite-dimensional modular Lie superalgebra SHO are proved to be invariant under automorphisms of SHO. The proof involves the investigation of the ad-nilpotent elements of the even part, and the determination of the subalgebras generated by certain ad-nilpotent elements. A property of automorphisms of these Lie superalgebras can be established, and an intrinsic characterization of SHO can be obtained.