  Coulomb branches of a set of $3d\ \mathcal{N}=4$ supersymmetric gauge
theories are closures of nilpotent orbits of the algebra $\mathfrak{so}(n)$.
From the point of view of string theory, these quantum field theories can be
understood as effective gauge theories describing the low energy dynamics of a
brane configuration with the presence of orientifold planes. The presence of
the orientifold planes raises the question to whether the orthogonal factors of
a the gauge group are indeed orthogonal $O(N)$ or special orthogonal $SO(N)$.
In order to investigate this problem, we compute the Hilbert series for the
Coulomb branch of $T^\sigma(SO(n)^\vee)$ theories, utilizing the monopole
formula. The results for all nilpotent orbits from $\mathfrak {so} (3)$ to
$\mathfrak{so}(10)$ which are special and normal are presented. A new
relationship between the choice of $SO/O(N)$ factors in the gauge group and the
Lusztig's Canonical Quotient of the corresponding nilpotent orbit is observed.
We also provide a new way of projecting several magnetic lattices of different
$SO(N)$ gauge group factors by the simultaneous action of a $\mathbb Z_2$
group.
