How does the inclusion of a subprime q value affect the Diffie Hellman equation for key exchange?

This is a follow on from this question:

“Diffie-Hellman Key Exchange” in plain English

In the answer to that question, the standard Diffie Hellman key exchange equation is derived:

(g^a mod p)^b mod p = g^ab mod p
(g^b mod p)^a mod p = g^ba mod p

In this answer, only the generator g and a prime number p are used. What I would like to know is how this equation is affected/modified when we include the optional subprime q value to the initial parameters?

I have found this:

The order of G should have a large prime factor to prevent use of the Pohlig–Hellman algorithm to obtain a or b. For this reason, a Sophie Germain prime q is sometimes used to calculate p = 2q + 1, called a safe prime, since the order of G is then only divisible by 2 and q. g is then sometimes chosen to generate the order q subgroup of G, rather than G, so that the Legendre symbol of ga never reveals the low order bit of a. A protocol using such a choice is for example IKEv2.(11)


But it goes a bit over my head. The above makes it sound as though p is derived directly from q, but to use a practical example of some parameters I am given:

        "p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
        "q": "8000000092C37D5F1106277105CB36B6E775199D9075B6D6934444D1EE78D78D",
        "g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

The p value does not equal 2q + 1

Side note, I have also seen this q value referred to as

“q is an (odd) prime divisor of p – 1”