Investing $400k at once can be an emotional task, not just a financial one. I'd recommend you invest that money over a number of months so each purchase doesn't seem like that big of a deal. But if you have an iron stomach and lots of investment experience, you could invest it more quickly. Key thing is to start...
Fidelity has some zero fee index funds for both U.S. and international. You should take a look at those, first, since that might meet your needs.
I think iShares are $0/trade at Fidelity, so you can buy a number of iShares ETFs...
ITOT is U.S. Total Stock Market
IXUS is international - everywhere except U.S.
SUB is a short-term tax-exempt bond ETF (you need to pay CA/NY/etc state tax, but not Federal tax)
Those ETFs are also $0/trade at Vanguard, since almost all ETFs are now $0/trade at Vanguard. But if your money is already at Fidelity, it might be easier to start there.
Does your work have a 401(k) plan? Another use for cash is to pay living expenses while you ramp up your retirement accounts. And then with that done, if your income doesn't exceed IRS limits, you can contribute to an IRA (Roth or Traditional). Once you've used up tax advantaged space, it makes sense to invest in taxable.