How do I cut into something with a shape?

[rare salmon]

To better word it, I want to make a circular hole (like the smaller cylinder) in the big cylinder. How would i go about doing this?

[willow aurora]

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[rare salmon]

update to what im doing, i want to cut this shape on the top into the cylinder below, basically making a claw shape

i have found a somewhat obtuse way to make the circle hole, but i would still like to cut out that rectangle part to make it a sort of claw shape.

continued to find an obtuse way to get something thats close enough to what i need, so i i guess i dont need this anymore