We establish the largest eruptive/confined flare database to date and analyze 322 flares of Geostationary Operational Environmental Satellite class M1.0 and larger that occurred during 2010-2019, i.e., almost spanning all of solar cycle 24. We find that the total unsigned magnetic flux ( ${{\rm{\Phi }}}_{\mathrm{AR}}$ ) of active regions (ARs) is a key parameter governing the eruptive character of large flares, with the proportion of eruptive flares exhibiting a strong anticorrelation with ${{\rm{\Phi }}}_{\mathrm{AR}}$ . This means that an AR containing a large magnetic flux has a lower probability that the large flares it produces will be associated with a coronal mass ejection (CME). This finding is supported by the high positive correlation we obtained between the critical decay index height and ${{\rm{\Phi }}}_{\mathrm{AR}}$ , implying that ARs with a larger ${{\rm{\Phi }}}_{\mathrm{AR}}$ have a stronger magnetic confinement. Moreover, the confined flares originating from ARs larger than $1.0\times {10}^{23}$ Mx have several characteristics in common: stable filament, slipping magnetic reconnection, and strongly sheared post-flare loops. Our findings reveal new relations between the magnetic flux of ARs and the occurrence of CMEs in association with large flares. The relations obtained here provide quantitative criteria for forecasting CMEs and adverse space weather, and have important implications for "superflares" on solar-type stars and stellar CMEs.