From my experience most company 401k plans give you options for some general index funds. Usually there's something like S&P500 or something close. Some international fund, some Small/Mid/Large cap funds, and a basket of various target date funds. I also usually find the fees on them to be higher than if I choose my own fund at the brokerage of my choice. Everyone always mentions Vanguard VTSAX here, and Fidelity has FZROX which are either no fees or so low of fees that they are truly negligible.
For me that eliminates keeping the 401k with the old employer or moving it to my new employer since self directed I can find better indexing choices.
I try not to have too many different accounts all over the place to I would move it to a rollover IRA at the brokerage you prefer.
In the rare case your new/old company has the extremely low/no fee indexing options then perhaps keeping it with them isn't the worst idea. Just keep in mind that companies will change up their 401k plans from time to time and you will have no option but to roll with whatever changes they decide to implement. My old company changed their plan multiple times over the 8 years I was there, new brokerage, new options, etc, etc. Each time I had to re-visit all the generic indexing options they felt like offering for what I felt was my best option of what was available.