  Let $M(N_{h,n})$ denote the mapping class group of a compact nonorientable
surface of genus $h\ge 7$ and $n\le 1$ boundary components, and let
$T(N_{h,n})$ be the subgroup of $M(N_{h,n})$ generated by all Dehn twists. It
is known that $T(N_{h,n})$ is the unique subgroup of $M(N_{h,n})$ of index $2$.
We prove that $T(N_{h,n})$ (and also $M(N_{h,n})$) contains a unique subgroup
of index $2^{g-1}(2^g-1)$ up to conjugation, and a unique subgroup of index
$2^{g-1}(2^g+1)$ up to conjugation, where $g=\lfloor(h-1)/2\rfloor$. The other
proper subgroups of $T(N_{h,n})$ and $M(N_{h,n})$ have index greater than
$2^{g-1}(2^g+1)$. In particular, the minimum index of a proper subgroup of
$T(N_{h,n})$ is $2^{g-1}(2^g-1)$.
