A mutineer is a soldier or sailor who openly rebels against or refuses to obey the orders of a superior officer or person in authority. This word matches the definition given in the question exactly.
Option A, radical, describes a person who holds extreme political or social views. It does not specifically mean disobeying military orders.
Option B, anarchist, is a person who believes there should be no government or ruling authority at all. This is about a political belief, not about a soldier disobeying orders.
Option D, revolutionary, refers to a person who takes part in or supports a revolution, aiming to overthrow a government or system. This is broader than the specific act of a soldier or sailor rebelling against orders.
Since mutineer is the precise term for a soldier or sailor who rebels against authority, the correct answer is option C, Mutineer.