Motivated by Gromov’s subgroup conjecture (GSC), a fundamental open conjecture in the area of geometric group theory, we tackle the problem of the existence of particular types of subgroups---arising from so-called periodic apartments---for a specific set of hyperbolic groups with respect to which GSC is currently open. This problems is equivalent to determining whether specific types of graphs with a non-trivial combination of properties exist. The existence of periodic apartments allows for ruling the groups out as some of the remaining potential counterexamples to GSC. Our approach combines both automated reasoning techniques---in particular, Boolean satisfiability (SAT) solving---with problem-specific orderly generation. Compared to earlier attempts to tackle the problem through computational means, our approach scales several magnitudes better, and allows for both confirming results from a previous computational treatment for smaller parameter values as well as ruling out further groups out as potential counterexamples to GSC.