conjugation

Suppose $E = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$ . Suppose we have:

$\psi_{\sqrt{2}, -\sqrt{2}}: \mathbb{Q}(\sqrt{3}, \sqrt{5}))(\sqrt{2}) \to (\mathbb{Q}(\sqrt{3}, \sqrt{5}))(-\sqrt{2})$

$\psi_{\sqrt{3}, -\sqrt{3}}: \mathbb{Q}(\sqrt{2}, \sqrt{5}))(\sqrt{3}) \to (\mathbb{Q}(\sqrt{2}, \sqrt{5}))(-\sqrt{3})$

$\psi_{\sqrt{5}, -\sqrt{5}}: \mathbb{Q}(\sqrt{2}, \sqrt{3}))(\sqrt{5}) \to (\mathbb{Q}(\sqrt{2}, \sqrt{3}))(-\sqrt{5})$

Let these equal $\tau_2$ , $\tau_3$ and $\tau_5$ respectively. Show that these elements have order 2 in the group $G(E/\mathbb{Q})$ . So just square the elements and verify that you get the identity?

What is the subgroup $H$ of $G(E/ \mathbb{Q})$ generated by the elements?