Transitive subgroup of $S_n$

Let \(n\geq 1\). A subgroup \(H\leq S_n\) is transitive if its action on \(\theset{1, \ldots, n}\) is transitive, that is, if for all \(i, j\in\theset{1,\ldots,n}\) there exists \(\sigma\in H\) with \(\sigma(i) = j\).