  Let $G$ be a finite, connected, simple graph. The critical group $K(G)$, also
known as the sandpile group, is the torsion subgroup of the cokernel of the
graph Laplacian $\operatorname{cok}(L)$. We investigate a family of graphs with
relatively simple non-cyclic critical group with an end goal of understanding
whether multiple divisors, i.e., formal linear combinations of vertices of $G$,
generate $K(G)$. These graphs, referred to as hinge graphs, can be intuitively
understood by taking multiple base shapes and ``gluing" them together by a
single shared edge and two corresponding shared vertices. In the case where all
base shapes are identical, we compute the explicit structure of the critical
group. Additionally, we compute the order of three special divisors. We prove
the structure of the critical group of hinge graphs when variance in the number
of vertices of each base shape is allowed, generalizing many of the
aforementioned results.
