Overall, this class is surprisingly balanced.
It's pretty obviously that someone decided to port the Pathfinder Alchemist over to 5e, and they did a remarkably good job. Only a few things stand out to me:
The entire Mutator subclass is fairly weak without investing basically all of your discoveries into it, cutting down on your utility pretty heavily. Unfortunate, but not too bad.
The other thing I've noticed is that the Discoveries can vary widely in power. For example, one gives proficiency in Brewers Tools (thematic, yes, but not actually good) while another gives the ability to permanently detect the presence of magic within 30', which is directly comparable to the nearly identical Warlock Invocation that lets you cast Detect Magic at will, but notably still requires your concentration.
The only other thing that I noticed, and this is more of a clerical nitpick rather than an actual mechanics issue, is that some of the discoveries say "This discovery is permanent, and makes X discovery permanent". Which is weird wording to use, because you would assume that all of them are permanent, rather than temporary. What I assume the author meant was "Once you select this discovery, you cannot change your choice later, and you cannot change out one of the discoveries this one has as a prerequisite".
So the base class and all of the sub-classes are pretty OK power-level wise, but the discoveries are hit or miss. I would suggest comparing them against similar Warlock Invocations since that's what they are obviously modeled after, and adjust accordingly, especially since the Warlock has to worry about only having 2 spells available at any given time, while the Alchemist has a much larger pool of spells available past level 3.