Suppose you have sets $S_1, \dots S_m$ such that $\sum_i |S_i| = n$. The goal is to arrange all the sets into a (possible unconnected) DAG such that $S_i$ is a parent (or ancestor) of $S_j$ iff $S_j \subseteq S_i$. The domain of the elements over which $S_i$ are constructed can have size as large as $O(n)$ (so no word packing tricks or orthogonal vector style lower bounds).
Note that any set $S_i$ that is not a subset of some other set will be an isolated vertex in the graph. I wish to know the time complexity of solving this problem and possible lower bounds.