What About The Converse of Lagrange's Theorem?

See these two links to have additional information:

The inverse of Lagrange's Theorem is true for finite supersolvable group.

and

A kind of converse of Lagrange's Theorem. In the second one, there is a great classification for the problem done by missed @Arturo Magidin.

Sylow's theorems deal with the case when $d$ is a prime power. One can show that if $d$ is not a prime power, then there's a multiple $n$ of $d$ and a group $G$ of order $n$ with no subgroups of order $d$.

Here's a reference (please let me know if it is not accessible): Donald McCarthy, Sylow's theorem is a sharp partial converse to Lagrange's theorem. Math. Z. 113 (1970) 383-384.

Two general theorem are:

Cauchy's Theorem: If $G$ is a finite group and $p$ is a prime number dividing $|G|$ then $G$ has an element of order $p$ (and thus a subgroup of order $p$).

Every $p$ group $G$ has a subgroup of any order dividing $|G|$.