  We consider composition operators $\mathscr{C}_\varphi$ on the Hardy space of
Dirichlet series $\mathscr{H}^2$, generated by Dirichlet series symbols
$\varphi$. We prove two different subordination principles for such operators.
One concerns affine symbols only, and is based on an arithmetical condition on
the coefficients of $\varphi$. The other concerns general symbols, and is based
on a geometrical condition on the boundary values of $\varphi$. Both principles
are strict, in the sense that they characterize the composition operators of
maximal norm generated by symbols having given mapping properties. In
particular, we generalize a result of J. H. Shapiro on the norm of composition
operators on the classical Hardy space of the unit disc. Based on our
techniques, we also improve the recently established upper and lower norm
bounds in the special case that $\varphi(s) = c + r2^{-s}$. A number of other
examples are given.
