Non-normal subgroups of M16

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) M16 (see subgroup structure of M16).
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Definition

We consider the group:

$G=M_{16}=⟨a,x\mid a^8=x^2=e,xax=a^5⟩$

with $e$ denoting the identity element.

This is a group of order 16, with elements:

$\{e,a,a^2,a^3,a^4,a^5,a^6,a^7,x,ax,a^{2}x,a^{3}x,a^{4}x,a^{5}x,a^{6}x,a^{7}x\}$

We are interested in the following two conjugate subgroups:

$H_1=\{e,x\},H_2=\{e,a^{4}x\}$

The two subgroups are conjugate by any element not centralizing either of them. Specifically, we can choose any of the elements $a,a^3,a^5,a^7,ax,a^{3}x,a^{5}x,a^{7}x$ to conjugate either subgroup into the other.

Subgroup properties

Normality

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	invariant under inner automorphisms	No	The two subgroups are conjugate to each other via $a$
2-subnormal subgroup	normal subgroup of normal subgroup	Yes	normal inside $⟨a^4,x⟩$ which is normal
subnormal subgroup	normal subgroup of normal subgroup of ... normal subgroup	Yes	follows from being 2-subnormal, also from being a subgroup of nilpotent group