Let $K$ be a knot in
${{S}^{3}}$
. This paper is devoted to Dehn surgeries which create 3-manifolds containing a closed non-orientable surface
$\hat{S}$
. We look at the slope $p/q$ of the surgery, the Euler characteristic
$\mathcal{X}(\hat{S})$
of the surface and the intersection number $s$ between
$\hat{S}$
and the core of the Dehn surgery. We prove that if
$\mathcal{X}(\hat{S})\,\ge \,15\,-3q$, then $s\,=\,1$. Furthermore, if $s\,=\,1$ then
$q\,\le \,4\,-\,3\,\mathcal{X}(\hat{S})$
or $K$ is cabled and
$q\,\le \,8\,-5\mathcal{X}(\hat{S})$
. As consequence, if $K$ is hyperbolic and
$\mathcal{X}(\hat{S})\,=\,-1$
, then $q\,\le \,7$.