I wouldn't really say that's a universally held notion, especially in the modern world. Historically, yes, but these days it stands more on its own. In similar vein, you could say biology is a subfield of physics, but most people don't think of them that way.
To a librarian, Computer Science and Library Science are just different aspects of information systems, which are (a librarian need hardly tell HN) humanity's highest calling, thus justifying their shared Dewey classification of 0.
It's not really philosophically, though. As a pure mathematician, I would not consider CS a subfield of math. The spirit of math is really studying abstract axiomatic systems, and while CS in theory is sort of like that, in actual practice, CS is too tied and focused on the actual instantiation of computers to really have the spirit of math.
Many would disagree with your characterisation of pure mathematics, computer science, or both. And I am certain that computer science yields nothing to pure mathematics in her pursuit of abstraction.
I think that statement held true before widespread availability of computing; when most computer science was really theoretical. I once skimmed a few chapters of a graduate-level textbook on Category Theory and realized that it was the foundation of object-oriented programming.
The biggest issue is that a lot of "Computer Science" is really applied software engineering; much like confusing physics and mechanical engineering.
Or, a different way to say it: Most students studying "Computer Science" really should be studying "Software Engineering."
More practically, I have a degree in Computer Science. When I was in school, it was clear that most of my professors couldn't program their way out of a paper bag, nor did they understand how to structure a large software program. It was the blind leading the blind.
Meanwhile at my ABET acredited, 2nd rate state school, my Computer Science college professors were talented people who had been doing this shit for decades and clearly understood what they were talking about.
I had ONE class that was about Software engineering.
Meanwhile I had an entire years worth of curriculum that was just "Go take various unrelated science and math classes so you have a strong understanding of the fundamentals in both science and math"
People so often generalize their very specific college experience to the entire world. Meanwhile you'd be lucky to find a consistent college experience just from crossing state lines.
Of course these terms aren’t well-defined, but some (me) would say both math and CS are subfields of “formal systems” (but “formal systems” could be named CS, which would put it at the top of the hierarchy)
The Joy computer language and Scheme with SICP are prime examples.
Reading SICP it's the 'easy way' and the Joy docs plus trying to code even simple
as a quadratic eqn solver it's a deep, hard task which requires lots of knowledge on cathegory theory.
Software Engineering and Computer Science don't necessarily mean the same thing (even though at this point, most schools with CS programs just teach software engineering anyway).
…and I thought math branched off from computing science sometime in the 19th century. Before that, what was called „math“ was mostly algorithms to compute stuff.