Non-normal subgroups of dihedral group:D8

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) dihedral group:D8 (see subgroup structure of dihedral group:D8).
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Definition

Suppose $G$ is the dihedral group of order eight (degree four) given by the presentation below, where $e$ denotes the identity element of $G$:

$G := \langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle$.

Then, we are interested in the following four subgroups:

$A_0 := \langle x \rangle = \{ x, e \}, A_1 := \langle ax \rangle = \{ ax, e \}, A_2 := \langle a^2x \rangle = \{ a^2x, e \}, A_3 = \langle a^3x \rangle = \{ a^3x, e \}$.

$A_0$ and $A_2$ are conjugate subgroups (via $a$, for instance). $A_1$ and $A_3$ are conjugate subgroups (via $a$, for instance). $A_0$ and $A_1$ are not conjugate but are related by an outer automorphism that fixes $a$ and sends $x$ to $ax$. Thus, all four subgroups are automorphic subgroups. These are the only non-normal subgroups of $G$ and they are all 2-subnormal subgroups.

Arithmetic functions

Function	Value	Explanation
order of whole group	8
order of subgroup	2
index	2
size of conjugacy class	2
number of conjugacy classes in automorphism class	2
size of automorphism class	2
subnormal depth	2
hypernormalized depth	2

Effect of subgroup operators

Specific values (in the second column) are for $A_0 = \langle x \rangle$.

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	$\langle a^2, x \rangle$	Klein four-subgroups of dihedral group:D8	Klein four-group
centralizer	$\langle a^2,x \rangle$	Klein four-subgroups of dihedral group:D8	Klein four-group
normal core	$\{ e \}$	--	trivial group
normal closure	$\langle a^2,x \rangle$	Klein four-subgroups of dihedral group:D8	Klein four-group
characteristic core	$\{ e \}$	--	trivial group
characteristic closure	$G$, i.e., $\langle a,x \rangle$	--	dihedral group:D8

Related subgroups

Intermediate subgroups

We use $A_0 = \langle x\rangle$ here.

Value of intermediate subgroup (descriptive)	Isomorphism class of intermediate subgroup	Small subgroup in intermediate subgroup	Intermediate subgroup in big group
$\langle a^2, x \rangle$	Klein four-group	Z2 in V4	Klein four-subgroups of dihedral group:D8

Subgroup properties

Invariance under automorphisms and endomorphisms

Suppose $c_a$ and $c_x$ denote conjugation by $a$ and $x$ respectively. Let $\sigma$ denote the automorphism that sends $a$ to $a^3$ and $x$ to $ax$. Then, $\langle c_a, c_x\rangle$ is the inner automorphism group and $\langle c_a, c_x, \sigma \rangle$ is the automorphism group.

The automorphism $c_x$ fixes $A_0$ and $A_2$ while interchanging $A_1$ and $A_3$. The automorphism $c_a$ interchanges $A_0$ and $A_2$ while also interchanging $A_1$ and $A_3$. The automorphism $c_{ax} = c_a \circ c_x$ fixes $A_1$ and $A_3$ while interchanging $A_0$ and $A_2$. The automorphism $\sigma$ interchanges $A_0$ and $A_1$ and also interchanges $A_2$ and $A_3$.

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	invariant under inner automorphisms	No	See above description of conjugation automorphisms that permute the subgroups
coprime automorphism-invariant subgroup	invariant under automorphisms of coprime order to group	Yes	there are no nontrivial automorphisms of coprime order
cofactorial automorphism-invariant subgroup	invariant under all automorphisms whose order has prime factors only among those of the group	No	follows from not being normal
2-subnormal subgroup	normal subgroup of normal subgroup	Yes	normal inside Klein four-subgroups of dihedral group:D8 (of the form $\langle a^2,x \rangle$ and $\langle a^2,ax\rangle$) that are normal in the whole group.
subnormal subgroup		Yes	follows from being 2-subnormal, also from being subgroup of nilpotent group.