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In this paper, we investigate the suborbital graphs for the normalizer of Γ_0 (N) in PSL(2,R), where N will be of the form 2^5 p^2, p is a prime and p > 3. It is known that the action of the normalizer Nor(N) on the extended rational numbers Q ̂ is non transitive. The edge conditions of the graphs arising from this non transitive action and then using these edge conditions, which kind of circuits the suborbital graphs have are investigated. Finally, we show that these circuits are only quadrilaterals. [ABSTRACT FROM AUTHOR]