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Unfortunately, no. It is known that the maximum number of mutually disjoint $S(4,5,11)$s on the same point set is $2$. Any such pair are always isomorphic. So, you can't find $7$ copies of an $S(4,5,11)$ in the complete $5$-uniform hypergraph on $11$ vertices; you can find only two of them at most.

E. S. Kramer, D. M. Mesner, Intersections among Steiner systems, J. Combin. Theory, Ser. A, 16 (1974), 273–285