  With the aim of understanding how the magnetic properties of active regions
(ARs) control the eruptive character of solar flares, we analyze 719 flares of
Geostationary Operational Environmental Satellite (GOES) class $\geq$C5.0
during 2010$-$2019. We carry out the first statistical study that investigates
the flare-coronal mass ejections (CMEs) association rate as function of the
flare intensity and the AR characteristics that produces the flare, in terms of
its total unsigned magnetic flux ($\Phi$$_{AR}$). Our results show that the
slope of the flare-CME association rate with flare intensity reveals a steep
monotonic decrease with $\Phi$$_{AR}$. This means that flares of the same GOES
class but originating from an AR of larger $\Phi$$_{AR}$, are much more likely
confined. Based on an AR flux as high as 1.0$\times$$10^{24}$ Mx for solar-type
stars, we estimate that the CME association rate in X100-class ``superflares"
is no more than 50\%. For a sample of 132 flares $\geq$M2.0 class, we measure
three non-potential parameters including the length of steep gradient polarity
inversion line (L$_{SGPIL}$), the total photospheric free magnetic energy
(E$_{free}$) and the area with large shear angle (A$_{\Psi}$). We find that
confined flares tend to have larger values of L$_{SGPIL}$, E$_{free}$ and
A$_{\Psi}$ compared to eruptive flares. Each non-potential parameter shows a
moderate positive correlation with $\Phi$$_{AR}$. Our results imply that
$\Phi$$_{AR}$ is a decisive quantity describing the eruptive character of a
flare, as it provides a global parameter relating to the strength of the
background field confinement.
