Date: 201911
Local time of day: 1201-1800
Location: ZZZ.Airport, US
Flight conditions: Mixed
Weather elements / visibility: 8
Light: Night
Aircraft: B737-700
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Climb
Airspace: Class B ZZZ
ATC / advisory: TRACON ZZZ
Aircraft component: Autoflight System
Component problem: Malfunctioning
Reporter organization: Air Carrier
Reporter function: Pilot Flying; Captain
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Human factors: Troubleshooting; Workload; Other / Unknown
Anomaly: Aircraft Equipment Problem Less Severe; Deviation - Speed All Types; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance
Detector: Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Provided Assistance; Flight Crew Diverted; Flight Crew Landed in Emergency Condition
Contributing factors: Aircraft
Primary problem: Aircraft

Narrative: After a normal takeoff and complying with the 'noise abatement procedures;' the aircraft would not allow for the autopilot or autothrottles to be engaged. In addition; after a normal flap and gear retraction; all three green lights illuminated. The flaps in the up position; we could not reduce the thrust lever setting lower than approximately 60% without the 'Configuration Warning Horn' sounding. So; initially we were obligated to keep the power setting higher than required for maintaining the assigned altitude of 5;000 feet to silence the horn; but in doing so; we exceeded the speed on the SID and the 250 kts restriction below 10;000 feet.We also deviated from the SID and had difficulties complying with ATC multiple heading and altitude changes as we were dealing with the associated problems and figuring out our best course of action to stabilize the aircraft. We did not have any master caution or warning lights illuminate to help assist in identifying the problem until we would try to engage the autopilot and then we would receive an 'Autopilot and Autothrottle' Disconnect Warning horn and lights. We recycled the landing gear but it had no effect on the Warning horn. We were only able to control the aircraft speed; without the horn sounding; when the landing gear was in the 'down and locked' position.Even then; we were not able to engage the autopilot; autothrottles; LNAV/VNAV and we were resigned to fly the aircraft by hand with the landing gear in the down position. We initially elected to return to ZZZ but after discussing the deteriorating weather decided to go to ZZZ1; where the weather was better; longer runways and more ground support. Unfortunately; with the highly congested airspace; multiple ATC course; altitude and speed changes; TA; lights; and horns in the cockpit; we had difficulty in complying with ATC instructions and ATC provided priority handling on our behalf.If ATC would engage the aircraft 'less' during a difficult situation; allow pilots time to correct; trouble-shoot or deal with an aircraft exhibiting unusually characteristics; I think the crew would be in a better position to not only handle the problem; but also be able to comply more accurately with heading; altitude and speed changes and only if there were 'fewer' of them.




Synopsis: B737 Captain reported Autoflight system malfunctioned erroneously; activating the warning horn and along with the high workload departure procedure; prevented the use of LNAV/VNAV; autopilot; and autothrottles.