What is the role of genealogy in The Children of Men?

The Children of Men by P.D. James is a dystopian novel that depicts a world where all men are infertile and mankind is on the brink of extinction. Since the survival of the human race depends on a child being born, there is a desire for any possible children to be bred from the healthiest and strongest men and women possible. Because of this, there is a need for identifying citizens whose genealogies reveal strong and healthy families. Qualifying men and women are given compulsory fertility tests, while unhealthy or disabled people are excluded. 
Besides this very pragmatic need, genealogy seems to be surprisingly unimportant in The Children of Men. Since there is no chance of continuing, the family unit has almost disintegrated. Sex is no longer used for procreation. The elderly are murdered in state-sanctioned rituals called the "Quietus." People live for the pleasure of the moment in order to defend themselves against the grief of childlessness and their coming extinction. 
Even British aristocrats, who are typically notorious for being proud of their rich heritage, care little now for the land and manors they have inherited from their ancestors. Theo explains this phenomenon in the following passage:

I can understand how the aristocrats and great landowners with no hope of posterity leave their estates untended. We can experience nothing but the present moment, live in no other second of time, and to understand this is as close as we can get to eternal life.

It seems that the need to appreciate the past is dependent on humanity's ability to appreciate the future. Without a coming generation to carry on the family history, the genealogy of these citizens is forgotten. Only the present is valued.