Theorem
A group of order $pqr$ where $p$,$q$ and $r$ are primes and $p<q<r$ always has a normal subgroup of order $qr$.
I found this theorem in a book. But a normal subgroup should have an index of $2$ w.r.t . the group. Now in case of a group of order $3×5×7$, it must have a normal subgroup of order $7×5$ and for this subgroup its index w.r.t. the group is not $2$. Is it not always necessary for a normal subgroup to have an index of 2 w.r.t. the group ?
