First of all, I think that the two need not come into conflict in most cases.
However, I would say that historically it would have been disastrous if loyalty to family had trumped loyalty to country. This can be seen most clearly in the case of wars.
When men (and now women) are called to the armed services in time of war, it is surely bad for their families. They lose touch with their families for long periods of time and may, of course, even die or be maimed in the war.
Therefore, it would be better for their families if they would not go to war. But if all men chose to do that, wars would be lost.
So that is, at least, one opinion.