
COMMENTS

The converse to Lagrange's theorem does not hold. A340511 lists the numbers n such that there exists a group of order n which has no subgroup of order d, for some divisor d of n; they are called "nonconverse Lagrange theorem" (NCLT) orders.
A finite group is supersolvable (SS) if it has a normal series of subgroups with cyclic factors; A066085 lists the numbers for which there exists a group of order n that is not supersolvable; they are called a "nonsupersolvable" (NSS) order.
Theorem: Every NCLT order is an NSS order (see MacHaleManning, 2016, Theorem 3, page 2); hence A340511 is a subsequence of A066085.
However, there exist infinitely many NSS orders that are not NCLT orders (see MacHaleManning, 2016, Corollary 19, page 6) and these NSSCLT orders are proposed in this sequence.
Theorem: The number 224*p is an NSSCLT order for all primes p <> 2, 3, 5, 7, 31 (see MacHaleManning, 2016, Theorem 18, page 6). So, 10528, 11872, 13216, 13664, 15008, 15904, ... are other terms.


EXAMPLE

There exist 197 groups of order 224, and one of these groups is NSSCLT (see MacHaleManning, 2016, Theorem 8, page 5); this group is NSS (A066085) but satisfies the converse of Lagrange theorem (CLT): for all divisors d of 224, this group has at least one subgroup of order d; hence, 224 is a term.
