(with (that + indicative) or intransitive, with on or upon) To hold up a claim emphatically.
The defendant insisted on his innocence.
I insist that my secretary dresses nicely.
(sometimes with (that + subjunctive) or intransitive, with on or upon) To demand continually that something happen or be done; to reiterate a demand despite requests to abandon it.
The Prime Minister insisted on his Chancellor's resignation.
The Prime Minister insisted that his Chancellor resign.
I insist that my secretary dress nicely.
I know I promised to pay you back tomorrow, but it's not very convenient for me. Can we put it off to Friday? —I'm afraid I have to insist on what we agreed.
(obsolete, chiefly geometry) To stand (on); to rest (upon); to lean (upon).