MathOverflow

Let $G$ be a finite group and $Sub(G)$ denote the number of subgroups of $G$ including the trivial subgroup and $G$ itself. I believe the following is true:
$Sub(H \times K)\leq Sub(H \rtimes K)$ for all finite groups $H$ and $K$ with coprime orders.
However, I was not able to prove it, may be due to lack of Goursat-like results in semidirect products. I tried to use the results of a paper by V.M.Usenko. But I could not manage to do it. Any counterexample or a way leading to the proof will be helpful...

I encountered this mathematical structure but need helpdecoding the notation:
[T = \prod_{i=1}^{\infty} \mathbb{N}]
with hierarchy levels:

Let f be a function in the Hardy space H^2(\mathbb{C}^+). Consider a region \Omega \subset \mathbb{C} that contains infinitely many zeros of f, where neither the density nor the distribution of these zeros is known a priori.
The question is:
Does there exist an analytic continuation of f into \Omega under such minimal assumptions on the zeros?...

I am studying a variational framework that I call a generator–filter transform (GFT), and I would like to understand whether the mathematical structure falls inside an existing branch of geometry (e.g., symplectic, Kähler, information geometry, topos...