Structure of сonjugacy subgroup intersection graph

Let $$G$$ be a finite group, $$H$$ is proper subgroup and $${\cal H}(H)$$ the set of all subgroups of conjugate H. Construct the graph $$\Gamma$$, with vertices $${\cal H}(H)$$ and two subgroups adjacent if they have trival intersection.

Its true (if graph not trivial), that this graph regular and have diamert 2?

For example in case $$G = A_5$$, $$H = S_3$$ (generated by $$(12)(34)$$ and $$(125)$$), $$\Gamma$$ is complement of Petersen graph. For other $$H$$, $$\Gamma$$ is complete multipartite graph (complenet of $$mK_n$$ for some $$n$$ and $$m$$) (for $$H \in \{C_2, C_3, C_2\times C_2, C_5\}$$) and empty graph (for $$H \in \{D_5, A_4\}$$)

Next interest example gave $$G = A_6$$, and $${\cal H}(H) = Syl_2(A_6)$$. Then a $$\Gamma$$ has order $$45$$, degree $$32$$, spectrum $$\{32^1, 2^{19}, -1^{16}, -6^9\}$$, $$\Gamma$$ is vertex transitive and has automorphism group isomorphic to $$(A_{6} \rtimes C_{2}) \rtimes C_{2}$$.

• It's certainly always regular, because $G$ acts transitively by conjugation on its vertices, but why do you think that it should have diameter $2$? – Derek Holt Jun 9 at 16:30
• Can't you just pick a group where all the conjugates intersect nontrvially? For example, they could be Sylow p-subgroups of a group $G$ with $O_p(G)\neq 1$. – verret Jun 9 at 21:07
• Take for example $G=Q_8\rtimes C_3\cong SL(2,3)$, with the $C_3$ acting by cyclically permuting $\{i,j,k\}$. Then the group $\langle i\rangle$ has three conjugates in $G$, and they all intersect in $\langle -1\rangle$ so the graph is disconnected. – verret Jun 9 at 21:10