C-symplectic poset structure on a simply connected space

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For a field of characteristic zero, we introduce a cohomologically symplectic poset structure P(X) on a simply connected space X from the viewpoint of -homotopy theory. It is given by the poset of inclusions of subgroups preserving c-symplectic structures in the group E(X) of -homotopy classes of -homotopy self-equivalences of X, which is defined by the -Sullivan model of X. We observe that the height of the Hasse diagram of P(X) added by 1, denoted by c-s- depth(X), is finite and often depends on the field. In this paper, we will give some examples of P(X).