As is well known, every finite (natural) number can be associated with a finite set of that cardinality. So in particular, the cardinality of a famous or special finite set must be a famous or special number. Here's a list of all the special numbers less than or equal to 9999 and contains quite a few items between $2^{10}$ and $2^{20}$.
Oh, what, you actually want the sets, and not just their cardinality, because there is more than one way of realising a set of a given cardinality? (Me grumbles something about bijective maps and isomorphisms of sets.) Fine:
- 1132 is the number of 3-valent trees with 15 vertices
- 1144 is the number of non-invertible knots with 12 crossings.
- 1165 is the number of conjugacy classes in the automorphism group of the 12 dimensional hypercube.
- 1205 is the number of fullerenes with 58 carbon atoms
- 1294 is the number of 4 dimensional polytopes with 8 vertices.
- 1378 is the number of symmetric idempotent 6×6 matrices over GF(2).
- 1411 is the number of quasi-groups of order 5.
- ...
- 3240 is the number of 3×3×3 Rubik's cube positions that require exactly 3 moves to solve.
- 3286 is the number of stable patterns with 16 cells in Conway's game of Life.
- ...
- 4535 is the number of unlabeled topologies with 7 elements.
- ...
As beauty is in the eye of the beholder, I'm sure there are mathematicians out there who think each of the above numbers ought to be better known.