Complex sentence: For example, Hunter College High School admits students to its middle and high schools via a rigorous competitive examination, however it is run by Hunter College, not the NYC Department of Education, and therefore cannot be one of the Specialized High Schools.
Split into simpler sentences: For example, Hunter College High School admits students to its middle and high schools via a rigorous competitive examination. However, Hunter College High School is run by Hunter College, not the NYC Department of Education, and therefore cannot be one of the Specialized High Schools.