The distaste（主语） that she felt （定语从句,修饰distaste）for what at Verrieres goes by the name of gaiety （distaste的后置定语,其中what引导的宾语从句做for的宾语）had won（谓语） her（间接宾语） the reputation of being extremely proudof her birth（直接宾语）.
句子主干是distaste had won her the reputation,然后有个定语从句修饰主语,有个宾语从句做介宾词组的宾语.句子里没有主语从句.
the distaste for sth. ：对...的厌恶,for sth.是介宾词组做distaste的后置定语.
宾语从句中：what（主语）at Verrieres(地点状语） goes（谓语） by the name of gaiety（方式状语).