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:

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

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 $\langle a^{4},x\rangle$ 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