Group Order a product of primes

Hello!

If there is a group G of order pqr

(for distict primes p, q and r)

with subgroups H and K of G,

where the order of H is pq

and the order of K is qr

I am trying to show that the order of $\displaystyle H \cap K$ must be q.

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Not sure where to run with this one?

I thought maybe that an intersection of two subgroups may be a subgroup of either of those subgroups,

ie, $\displaystyle H \cap K$ is a subgroup of H, so the order of $\displaystyle H \cap K$ must divide the order of H,

and similarily,

and also $\displaystyle H \cap K$ is a subgroup of K,

so the order of $\displaystyle H \cap K$ must divide the order of K

but I am not sure if that is valid or not (is the intersection of two subgroups (H and K) a subgroup itself of either of the two orginal subgroups (H or K)?

Any help appreciated!

Thank you!!