Assume that $G$ is a group with a subgroup $H$ s.t. $|H|=6, [G:H]> 4$, and $|G| < 50$. What are the possibilities for $|G|$?

Question

Assume that $G$ is a group with a subgroup $H$ s.t. $|H|=6, [G:H]> 4$, and $|G| < 50$. What are the possibilities for $|G|$?

$|G|$ has to be a multiple of $6$ so either $12, 18, 24, 30, 36, 42$ or $48.$ But since $[G:H]$ has to be greater than $4$, that only leaves $|G|$ to be $30,36,42$, and $48$. Does this seem right?

Follow up question? What if $[G:H]>3, |G|<45$, and $|H|>10$ Would $|G|$ only be able to be $44$?

Answer 1

Yes, you're right.

According to this comment of @Bungo's, if $\lvert H\rvert>10$ and $[G:H]>3$, then $\lvert G\rvert=44$ by $|G|=[G:H]|H|$ and $|G|<45$.