44 Magnums with a 4-screw frame first appeared in early 1957 at approximately serial number S167900. However, in mid-1957, S&W shipped some 44 Magnums with a 5-screw frame that have a serial number in the S171000 to S175000 range. Apparently, S&W found a number of 5-screw frames and used them to make 44 Magnums. Your 44 Magnum is one of these.
I had one in my collection for a number of years. It has serial number S174042 and was shipped on June 3, 1957.
Bill