  This paper provides a bridge between two active areas of research, the
spectrum (set of element orders) and the power graph of a finite group.
  The order sequence of a finite group $G$ is the list of orders of elements of
the group, arranged in non-decreasing order. Order sequences of groups of order
$n$ are ordered by elementwise domination, forming a partially ordered set. We
prove a number of results about this poset, among them the following.
  M.~Amiri recently proved that the poset has a unique maximal element,
corresponding to the cyclic group. We show that the product of orders in a
cyclic group of order $n$ is at least $q^{\phi(n)}$ times as large as the
product in any non-cyclic group,where $q$ is the smallest prime divisor of $n$
and $\phi$ is Euler's function, with a similar result for the sum.
  The poset of order sequences of abelian groups of order $p^n$ is naturally
isomorphic to the (well-studied) poset of partitions of $n$ with its natural
partial order.
  If there exists a non-nilpotent group of order $n$, then there exists such a
group whose order sequence is dominated by the order sequence of any nilpotent
group of order $n$.
  There is a product operation on finite ordered sequences, defined by forming
all products and sorting them into non-decreasing order. The product of order
sequences of groups $G$ and $H$ is the order sequence of a group if and only if
$|G|$ and $|H|$ are coprime.
  The paper concludes with a number of open problems.
