I want to find the condition of a group $G$ such that for all $a\big||G|$, there exists $H\leq G, |H|=a$. For instance, If $G=C_n$, then for all $a|n$, $C_a\leq C_n$, so $C_n$ satisfies the condition.
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A sufficient condition is if the group is a product of its Sylow subgroups, but I doubt this is necessary. – Michael Burr Jul 04 '20 at 04:40
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@MichaelBurr Thanks! – LoonyMelodica Jul 04 '20 at 05:06