The subgroup lattice of a group G is the poset consisting of the subgroups of G, ordered by inclusion.  Can you recognize whether a group is in a class from its subgroup lattice?  For some nice families of groups such as abelian, Hamiltonian, or nilpotent, the answer is “no”; while for other classes such as supersolvable or solvable, the answer is “yes”.  I’ll give several conditions on a subgroup lattice that are equivalent to solvability of the originating group: combinatorial, topological, and commutative algebraic.