I have stumbled across a related question asking which large cardinal properties can hold for $\aleph_1$. This question is probably also related, asking in what ways $\aleph_0$ is a "large" cardinal.

To state my question:

For which large cardinal properties is it consistent with ZFC that $\frak{c}$, the cardinality of continuum, has this property? How does the answer change if we abandon choice?

Here are results I'm aware of:

$\frak{c}$ can be real-valued measurable (relative to existence of a measurable cardinal), which apparently implies it's weakly Mahlo. I believe all of these hold in ZFC.

Possibility of $\frak{c}$ being weakly inaccessible and weakly Mahlo (or, I believe, weakly 1-inaccessible or whatever intermediate condition we put on it) is also consistent relative to existence of respective cardinals, which can be established by forcing which doesn't disrupt in any way structure of ordinals, only the size of continuum.

Clearly $\frak{c}$ can't be strongly inaccessible or strongly Mahlo, because these directly require a cardinal to be strongly limit.

In ZFC, every measurable is strongly limit, so $\frak{c}$ can't be measurable. But it is also true in ZF, because we have $\aleph_0<\frak{c}$ and $2^{\aleph_0}\geq\frak{c}$, and standard proof that this can't happen for measurable goes through.

By same means, $\frak{c}$ can't be the critical point of any elementary embedding, so it can't land in any of the higher entries of large cardinals list.

Continuum can't be weakly compact since weak compactness implies strong limitness. I'm sure this is true in ZFC, but not sure about ZF.

What other properties can or can't continuum have? I believe axiom of choice disallows it to be in anywhere above weakly compact (correct me if I'm wrong), but I have hopes for ZF itself giving $\frak{c}$ more possibilites to be large.

Thank you in advance.

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