Cyclic maximal subgroups of quaternion group

The quaternion group is defined by the following presentation:

$$\langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle$$

We use the symbol $$1$$ for the identity element and the symbol $$-1$$ for $$i^2$$. $$i^3,j^3,k^3$$ are denoted by $$-i,-j,-k$$ respectively.

It has the following multiplication table:

The subgroups that we are interested in studying are the three four-element subgroups of this group, namely the subgroups:

$$\{ 1,-1,i,-i \}, \qquad \{ 1,-1,j,-j \}, \qquad \{1,-1,k,-k \}$$.

Normal subgroup
All three of these subgroups are normal. The quotient group in each case is a cyclic group of order two.

Retraction-invariant subgroup
All three of these subgroups are invariant under all retractions of the whole group. This is because of the trivial reason that the group has no retractions under that the identity map and the trivial homomorphism.

Characteristic subgroup
None of these three subgroups is characteristic. In fact, there is an automorphism cyclically permuting $$i,j,k$$ that permutes these subgroups cyclically.

Coprime automorphism-invariant and cofactorial automorphism-invariant
None of these three subgroups is invariant either under the automorphisms whose order is a power of two or under the automorphisms whose order is relatively prime to two. Specifically:


 * There is an automorphism of order three cyclically permuting $$i,j,k$$.
 * There is an automorphism of order two that sends $$i$$ to $$-j$$, $$j$$ to $$-i$$, and $$k$$ to $$-k$$. This has order two and it interchanges two of the three subgroups while sending the third subgroup to itself.

Resemblance notions
All three subgroups are satisfies property::isomorph-automorphic subgroups. In fact, they are satisfies property::order-automorphic subgroups.

Generic maximality notions
All three subgroups are maximal subgroups of the group of prime power order. Thus, they satisfy all these properties: satisfies property::maximal normal subgroup, satisfies property::maximal subgroup, satisfies property::subgroup of index two, satisfies property::order-normal subgroup, satisfies property::isomorph-normal subgroup, satisfies property::maximal subgroup of finite nilpotent group.

Abelian subgroups of maximum order
All three subgroups are abelian subgroups of maximum order, and they are the only ones. Also, they are satisfies property::maximal among abelian subgroups and satisfies property::maximal among abelian normal subgroups.

Abelian subgroups of maximum rank
All three subgroups are abelian subgroups of maximum rank. However, they are not the only ones -- so is the center, which has order two.

Finding these subgroups in a black-box quaternion group
Suppose $$G$$ is a group we know to be abstractly isomorphic to the quaternion group. Then, we can define a three-element list of the subgroups of $$G$$ using the MaximalSubgroups function as:

L := MaximalSubgroups(G);

Alternatively, we can use the NormalSubgroups function:

L := Filtered(NormalSubgroups(G),H -> Order(H) = 4);

The individual members are accessed as $$L[1], L[2], L[3]$$ respectively.