The sentence is: "Each of the students have submitted their assignments on time."
Subject-verb agreement requires the verb to match the subject in number. The subject, "Each of the students," is singular because "each" refers to individual students. Consequently, the verb must be singular.
The phrase "have submitted" should be changed to "has submitted" for correct subject-verb agreement. The revised sentence is:
Each of the students has submitted their assignments on time.