There has been substantial interest in the indices $0 \leq \beta''
\leq \beta' \leq \beta \leq 2$, defined by Blumenthal and Getoor, determined by
a general Lévy process in $\mathbf{R}^d$. Pruitt defined an index
$\gamma$ which determines the covering dimension and Taylor showed that an
index $\gamma'$, first considered by Hendricks, determines the packing
dimension for the trajectory. In the present paper we prove that
$$\frac{\beta}{2} \le \gamma' \le \min(\beta, d),
and give examples to show that the whole range is attainable.
However, we cannot completely determine the set of values of $(\gamma, \gamma',
\beta)$ which can be attained as indices of some Lévy process.