Subgroups of order $21$ of symmetric group $S_{10}$

Question

I am seeking the subgroups of order $21$ of symmetric group $S_{10}$ .

I find one which the subgroup $H$ generated by two elements $(1,2,3,4,5,6,7)$ and $(8,9,10)$. Since their order $7$ and $3$ respectively, so the order of the subgroup $H$ generated by these two is $\operatorname{lcm}(7,3)=21$.

Are there any other subgroups of order $21$ which are not of this form ?

Answer 1

I'll concretize the comments.

If $a=(1234567)$ and $b=(235)(467)$, then $$ bab^{-1}=a^2. $$ It follows that if $H=\langle a,b\rangle$, then $|H|=21$. And obviously $H<S_7<S_{10}$.

Note that the permutations here are multiplied from right to left, i.e. $(12)(23)=(123)$.