Also, in the sentence "They wake up early to arrive on time," early to arrive is not a subject. As a matter of fact, the sentence does not contain any object. Let's look at it together:
Subject: They
Verb: Wake up
Adverb: early
inifinitive phrase: to arrive on time
Subject: They
Verb: Wake up
Adverb: early
inifinitive phrase: to arrive on time