Subgroup structure of generalized quaternion group:Q16

This article gives specific information, namely, subgroup structure, about a particular group, namely: generalized quaternion group:Q16.
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The generalized quaternion group:Q16, denoted $Q_{16}$is a group of order $16$, is a generalized quaternion group. It can be described by the following presentation:

$G:=\langle a,b,c\mid a^{4}=b^{2}=c^{2}=abc\rangle$.

Note that $c=ab=ba^{-1}$ from these relations, and $bab^{-1}=a^{-1}$. This in turn forces that $b^{2}=b(b^{2})b^{-1}=ba^{4}b^{-1}=a^{-4}=b^{-2}$, forcing $b^{2}=a^{4}=c^{2}=abc$ to have order two. We shall denote this element of order two, which is clearly central, as $z$.

Here is a list of subgroups:

1. The trivial subgroup. Isomorphic to trivial group. (1)
2. The center, which is a subgroup of order two, generated by $z=a^{4}=b^{2}=c^{2}$. Isomorphic to cyclic group:Z2. (1)
3. The cyclic subgroup of order four generated by $a^{2}$. Isomorphic to cyclic group:Z4. (1)
4. The four cyclic subgroups of order four, namely: $\langle b\rangle$, $\langle ab\rangle$, $\langle a^{2}b\rangle$ and $\langle a^{3}b\rangle$. These come in two conjugacy classes of 2-subnormal subgroups, one conjugacy class comprising $\langle ab\rangle$ and $\langle a^{3}b\rangle$ and the other comprising $\langle b\rangle$ and $\langle a^{2}b\rangle$. Isomorphic to cyclic group:Z4. (4)
5. The cyclic subgroup of order eight, generated by $a$. This is characteristic; in fact, it equals the centralizer of commutator subgroup. Isomorphic to cyclic group:Z8. (1)
6. Two quaternion groups of order eight, namely $\langle a^{2},b\rangle$ and $\langle a^{2},ab\rangle$. Isomorphic to quaternion group. (2)
7. The whole group. (1)

Table classifying isomorphism types of subgroups

Group name	GAP ID	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
Trivial group	$(1,1)$	1	1	1	1
Cyclic group:Z2	$(2,1)$	1	1	1	1
Cyclic group:Z4	$(4,1)$	5	3	1	1
Cyclic group:Z8	$(8,1)$	1	1	1	1
Quaternion group	$(8,4)$	2	2	2	0
Generalized quaternion group:Q16	$(16,9)$	1	1	1	1
Total	--	11	9	7	5

Table listing number of subgroups by order

Group order	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
$2$	1	1	1	1
$2$	1	1	1	1
$4$	5	3	1	1
$8$	3	3	3	1
$16$	1	1	1	1
Total	11	9	7	5