For $n\geq 5$, let $\mathcal {P}_n$ be the set of all isomorphism classes of graphs with n vertices. Give this set the poset structure given by $G \le H$ if and only if $G$ is a subgraph of $H$.

Is it true that $\mathcal {P}_n$ has no nontrivial automorphisms?

**Remarks:**

This follows if one can recognize a graph from the set of isomorphism classes of its edge-deleted subgraphs. However, since recognizing a graph from the set of isomorphism classes of its edge-deleted subgraphs is stronger than edge reconstruction, I'm wondering if there is an alternative way of proving this.