In Call of Duty Advanced Warfare a lot of the maps are large and have wide, open areas. Therefore any weapon good weapon must powerful at medium and the edge of long range. Very few Submachine guns are good even at medium range. Hence there are only very few contenders for the best Submachine gun in Call of Duty Advanced Warfare. I believe the best Submachine Gun is the MP11 for its versatility and range.
The MP11 is the Best Submachine Gun in Call of Duty Advanced Warfare
The stock MP11 is already very good with decent damage and reasonable fire rate. It has very little recoil and is therefore suited for even the edge of long range, however Assault Rifles do have an advantage over the MP11 there.
There are many class setups that work well with the MP11. I personally like getting every last bit of damage out of it I can by putting on Rapid Fire. This does mean that you probably need Extended Mags and the Foregrip to keep this version of the MP11 under control. Above is a link to the recommended class setup for the MP11.
The other contender was the ASM1 which also works well at medium range, probably slightly better than the MP11, but at close range the MP11 is much better. Also the ASM1 is hard to keep under control and you have to use a red dot sight with it since the iron sights are very poor.
So there you go, the best Submachine Gun in Call of Duty Advanced Warfare is, in my opinion, the MP11.