Laughing loudly breaks the prayer according to four popular sects but it also breaks the wudu according to the Hanafi Sect. The narrations, which were given as evidence by Hanafi sect were not considered as precise information by hadith scholars. Hence laughing loudly does not break wudu during or out of prayer.
Allah the Almighty commands:
“(…) But if you are ill or on a journey or one of you comes from toilet or you have contacted women and do not find water, then seek clean earth and wipe over your faces and hands with it. (…)” (5:6)
According to this verse, things that break wudu are what we do in the toilet. Laughing is not one of those things.