The Dark Knight wrote:
Exactly right, the football club is usually the bigger club and therefore has more power.
In an instance that you'd be referring to though the football club would struggle to survive without the cricket club as there would be no money coming in over summer, that's why so many are called "Sports & Social Clubs".
I had similar issues when I was running a "cricket" club, trying to get incentives etc. from the sporting body was like extracting teeth, it took a couple of years but we ended up with 4 senior sides, 2 womens teams and 4 junior sides not only becoming self-sufficient, we were making a tidy profit, it only shot me in the foot though as I ended up becoming the chairman of the whole sporting body.
Football brings in more money via bar takings & canteen, moreso if they are winning, they also cost a shit load more to run via player payments, insurance, affiliation fees, medical consumable along with umpires etc.