My post is motivated at least in part by this MO question.

Has there been any work done on realizable order sequences for finite groups? By an "order sequence" I mean a non-decreasing list of the orders of the elements of the group. By "realizable" I mean there is some finite group that has that particular order sequence.

For example, $(1, 2, 4, 4)$ is a realizable order sequence; it is the order sequence of $\mathbb{Z}/4\mathbb{Z}$.

Is this something that has been studied in any depth? (Perhaps under a different name?) Are there any non-trivial theorems about realizable order sequences? (Examples of trivial theorems: ones that fall immediately out of Lagrange's Theorem; the degree sequence of $\mathbb{Z}/n\mathbb{Z}$)

I know that there are some nice theorems about degree sequences in graph theory (e.g., Erdős-Gallai theorem; Havel-Hakimi theorem), and though I have seen "order sequence" defined in a few Abstract Algebra texts, I have yet to come across any results of much interest.

I also wonder whether results such as this problem solution (Problem 6636, F. Schmidt, Amer. Math. Monthly, Vol. 98, No. 10 (Dec., 1991), pp. 970-972) or this paper (Isaacs et al (2009). Sums of element orders in finite groups. Commun. Alg. 37(9):2978-2980) contain ideas that would be applicable to such a topic.

Finally, is there an easily accessible (and organized) database that lists order sequences for all finite groups up to a certain not-too-small size?

**Edit 1:**Is anyone up to computing such a list and posting it somewhere accessible?

**Edit 2:**Now that Alexander Gruber has kindly posted computations for a fair number of order sequences, I wonder:

**(a)**when does the same order sequence correspond to more than one group?

**(b)**given an order sequence that corresponds to precisely one group, how difficult is it to recover the corresponding group's structure?

**Edit 3:**Mr. Gruber has most recently pointed me toward a related area of research on "OD-characterizability." One mathematician who has done a fair bit of work in this area is AR Moghaddamfar. See, for example, Recognizing Finite Groups Through Order and Degree Pattern.