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Claim. Let $a_{n,j}$ be the number of subgroups of $S_n$ of order $2^j$. Uniformly for $\lfloor n/4\rfloor\le j\le\lfloor n/2\rfloor$,

$$\log_2 a_{n,j}=\frac{n^2}{16}+O(n\log n),$$

so every order in that range is attained by subgroups whose count has the full quadratic exponent. The lower bound comes from constructions that combine permutation groups on two, four and eight points with counts of subspaces of vector spaces over $\mathbb{F}_2$; the upper bound of the same leading order follows from Roney-Dougal and Tracey's bound on the number of 2-subgroups of $S_n$ (their Theorem 2), and the note also cites Kovács and Praeger. The write-up is a web page titled "Subgroups of prescribed power-of-two order — Erdős 1163 partial result", dated 2026-09-04, with no manuscript record or license statement; no check of its argument is recorded.

Submission note. Posted to erdosproblems.com as a proof claim by tienxion (account tienxion) on 4 September 2026, giving "OpenAI GPT-6 (proof development and audits; summary drafting assistance)" as the AI used:

We count subgroups of $S_n$ having a prescribed order $2^j$. Uniformly for $\lfloor n/4\rfloor\le j\le\lfloor n/2\rfloor$, we obtain (\log_2 a_{n,j}=n^2/16+O(n\log n)). The construction combines permutation groups on two, four, and eight points with binary subspace counting; the matching leading upper bound follows from Roney-Dougal–Tracey. This gives a partial counting result, while the order distribution of a uniformly chosen unrestricted subgroup remains unresolved as of now. Notes: Here a_{n,j} counts actual subgroups of S_n of order 2^j, not conjugacy classes. The linked note includes an explicit uniform lower bound with an n log n term and a complete construction. Novelty has not been established. AI assistance is disclosed above; the summary was drafted with AI assistance and edited by the submitter. No claim of independent human expert verification or of a full solution is made.

Covers. A count of the subgroups of each order $2^j$ with $j$ between $n/4$ and $n/2$, to within a factor $2^{O(n\log n)}$. It does not describe the distribution of the orders of a uniformly random subgroup of $S_n$, which the note says remains unresolved, and the note observes that an error of order $n\log n$ in the exponent can still change relative probabilities substantially and does not imply equidistribution among those orders. The note also says that whether this refinement already appears in the literature has not been established. What a complete answer to Problem 1163 would consist of is itself unclear, as the problem page records.

Standing. The claimant is the forum user tienxion, who filed the result on the site's proof-claims page on 2026-09-04; the claim's entry names the system OpenAI GPT-6 for proof development and audits and for drafting the summary, and the note says it was developed with that system's assistance, including independent agent proof audits. The site shows OPEN with no verdict and no comments; the write-up is not refereed and no reviewer is recorded, so the claim stays claimed.

Depends on. Nothing in this wiki; the upper bound rests on Theorem 2 of Roney-Dougal and Tracey's unrefereed preprint (arXiv:2503.05416, 2025), which has a library card but no result page.