$k\leq m$,

-   $p_1$ is a divisor of $|GO^-_{2m}(p)|$ if and only if $k\leq m-1$.

If $p$ is odd and $k$ is even, then

-   $p_1$ is a divisor of $|GO_{2m+1}(p)|$ if and only if $k\leq 2m$,

-   $p_1$ is a divisor of $|GO^+_{2m}(p)|$ if and only if $k\leq 2m-2$,

-   $p_1$ is a divisor of $|GO^-_{2m}(p)|$ if and only if $k\leq 2m$.

If $p=2$ and $k$ is odd, then

-   $p_1$ is a divisor of $|\mathrm{Sp}_{2m}(2)|$ if and only if $k\leq m$,

-   $p_1$ is a divisor of $|\mathrm{O}(2t+1,2)|$ if and only if $k\leq t$,

-   $p_1$ is a divisor of $|\mathrm{O}(2t,2)|$ if and only if $k\leq t-1$.

If $p=2$ and $k$ is even, then

-   $p_1$ is a divisor of $|\mathrm{Sp}_{2m}(2)|$ if and only if $k\leq 2m$,

-   $p_1$ is a divisor of $|\mathrm{O}(2t+1,2)|$ if and only if $k\leq 2t$,

-   $p_1$ is a divisor of $|\mathrm{O}(2t,2)|$ if and only if $k\leq 2(t-1)$.

\[decomposition\][Let $p_1,p_2$ be two distinct prime numbers. Let $k_1$ be the order of $p_1$ in $({\mathbb{Z}}/(p_2))^*$. Note that the field ${\mathbb{F}}_{p_1^{k_1}}$ of $p_1^{k_1}$ elements is the smallest field of characteristic $p_1$ that contains an element of multiplicative order $p_2$. Consider $x^{p_2}-1\in {\mathbb{Z}}/(p_1)[x]$ and its decomposition 