Captain reported an in flight shut down of the #1 Engine as a result of an oil filter bypass light illuminating. The flight Crew elected to divert and make a precautionary landing.

Narrative

Aircraft X; ZZZ-ZZZ1. I was the Pilot Flying. Upon arrival to the aircraft in ZZZ; we noticed that the #1 Engine Hydraulic Pump had been replaced the day before and was signed off in the logbook. Normal taxi and departure procedures to cruise altitude FL360. Shortly after level off; we got a #1 Engine Hydraulic pump low pressure light and associated Caution. We followed the QRH procedures and turned off the #1 Engine driven Hydraulic pump. We sent an ACARS message to ZZZ1 Maintenance and also our Dispatcher. We continued to ZZZ1.Approximately 20-30 minutes later; we noticed the #1 Engine alerts flashing followed by a steady #1 Engine oil filter bypass light. We ran the QRH for this condition and determined that the #1 engine would need to be shut down. We [requested priority handling] with ATC and requested a lower altitude since we could not maintain airspeed at that altitude on one engine. We also informed them that we would like to divert to ZZZ2. This was the nearest most suitable airport.Flight Attendants noticed a momentary yaw as we shut down the engine and called the flight deck. We told them we were running checklists; that we had to shut down the #1 engine and were diverting to ZZZ2 for a 'normal' landing (no bracing or evacuation). They proceeded to secure the cabin and make an announcement to the Passengers that we were diverting to ZZZ2. ATC was asked to notify our Dispatcher to reduce our task loading as we were busy finishing checklists and setting up for an approach to Runway XX in ZZZ2. The FO (First Officer) reprogrammed the FMC for ZZZ2 and ran non-normal performance data for a single-engine approach to Runway XX. It was not an overweight landing and no brake cooling was required. The weather conditions were VMC and we landed without any further incidents. Passengers were deplaned and rebooked by the ZZZ2 Operations Agent and Supervisor.Although our precautionary engine shutdown was due to the #1 Engine oil filter bypass light; when Contract Maintenance arrived at the aircraft; our #1 Hydraulic quantity was at 11%. We were informed by Contract Maintenance that hydraulic fluid was all over the interior of the engine cowling. When they tried to refill the hydraulic fluid; it essentially just ran out onto the ramp. Not sure how these two issues may or may not be related; but other than the initial #1 engine driven hydraulic pump pressure being low and subsequent QRH direction to turn off that pump; we had no apparent hydraulic issues.I called Dispatch and he set me up with a five-way conference call with Maintenance; ZZZ1 Chief Pilot; Chief Dispatcher; and the Supervisor. I talked/debriefed with the Flight Attendants as this was a relatively normal divert event for them. They handled everything with great professionalism and customer service. The First Officer and myself were pulled from our trip and were placed on a deadhead back to our base in ZZZ.