Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that there exists a subgroup of $G$ with order $d$ or $n/d$?
Of course this does not hold in full generality. -- In particular one quickly finds that e.g. ${\rm PSL}(2,8)$, ${\rm PSL}(2,11)$, ${\rm PSL}(2,13)$, ${\rm SL}(2,11)$, ${\rm SL}(2,13)$, ${\rm PSL}(2,17)$, ${\rm A}_7$, ${\rm PSL}(2,19)$, ${\rm A}_5 \times {\rm A}_5$, etc. are counterexamples.
Though does the assertion hold for
solvable groups?
solvable groups with derived length less than a certain bound $> 2$?
groups whose order has at most 2 distinct prime divisors?
A quick computation with GAP shows that any counterexample must have order $\geq 192$.
