  We calculate ordinal $L_p$ index defined in "An ordinal L_p index for Banach
spaces with an application to complemented subspaces of L_p" authored by J.
Bourgain, H. P. Rosenthal and G. Schechtman, for Rosenthal's space $X_p$,
$\ell_p$ and $\ell_2$. We show a subspace of $L_p$ $(2 < p < \infty)$ non
isomorphic to $\ell_2$ embeds in $\ell_p$ if and only if its ordinal index is
minimum possible. We also give a sufficient condition for a $\mathcal{L}_p$
subspace of $\ell_p\oplus\ell_2$ to be isomorphic to $X_p$.
