Understanding $p$-rank, $p$-Sylow subgroup, $2$-part, odd and even part of a group.

Question

As the title says, I would like to be clear about what each of those terms mean in an ideal class group setting. I would like someone to correct me where I am wrong. The reason for this clarification is at the end. Maybe some ideas on how to approach the problem would also be great.

Let $G = C(\mathcal{O}_{\Delta_K})$ be the class group of a maximal order where $\Delta_K\equiv 1\pmod{4}$, a fundamental discriminant.

1. The odd part of the class group is the subgroup $H$ of $C(\mathcal{O}_{\Delta_K})$ whose elements $h_i$ have odd orders in $G$
2. The even part of the class group is the subgroup $H'$ of $C(\mathcal{O}_{\Delta_K})$ whose elements $h'_i$ have even orders in $G$

3. The two part of the class group is the subgroup $F$ of $C(\mathcal{O}_{\Delta_K})$ whose elements $f_i$ have powers of two orders in $G$ (shouldn't this be called something like 'power of two-part'?)

4. The $p$- rank of $C(\mathcal{O}_{\Delta_K})$ is the largest integer $k$ such that $C(\mathcal{O}_{\Delta_K})$ has an Abelian subgroup of order $p^k$

5. Let $p$ be a prime divisor of $\lvert C(\mathcal{O}_{\Delta_K})\rvert$ , i.e. $p$ divides $h_{\Delta_K}$. If $p^k$ divides $h_{\Delta_K}$ and $p^{k+1}$ does not divide $h_{\Delta_K}$ then any subgroup of order $p^k$ is a $p$-Sylow subgroup of $C(\mathcal{O}_{\Delta_K})$.

Consider the following two examples taken from here, page 29

1. $\Delta_K = -93920213643973848047$ $h_K = 5602080000$ $2$-part $=(2,2,2,2,2,8)$

2. $\Delta_K = -106994885997905465007$ $h_K = 6485160000$ $2$-part $= (2,2,2,2,2,2)$

In example 1, the $2$-rank is $6$ and the size of the $2$-Sylow subgroup is $8$.

In example 2, the $2$-rank is $6$ again and the size of the $2$-Sylow subgroup is $2$.

Questions

1. Is everything that I wrote correct, esp. the sizes of the $2$-Sylow subgroups in the examples?
2. Essentially, given the factorization of $\Delta_K$, I would like to restrict the size of the $2$-Sylow subgroup to $2$. It is well known that if $\Delta_K=-pq$ and $\left(\frac{p}{q}\right)=-1$ then it can be restricted to a size of $2$.

I want to figure out the conditions on the $\Delta_K$ prime divisors when $\Delta_K=-p_1\cdots p_n$ and $\Delta_K=-p_1^{e_1}\cdots p_n^{e_n}$ so that the $2$-Sylow subgroup has size $2$ like in example 2.