Subgroups — Q₈×#13182004 #28364936

Order 13952.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13182004	C₂	1	3	yes	maximal
2	C₄×#13182004	C₂	1	3	yes	maximal
3	C₈×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182004	C₂²	1	3	yes	—
5	C₂×#13182004	C₂²	1	1	yes	characteristic
6	C₄×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₁₀₉	C₄	1	3	yes	—
8	C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
9	#13182004	—	2	3	no	—
10	#13182004	Q₈	1	2	yes	direct product factor
11	C₂×Q₈×C₁₀₉	C₈	1	1	yes	characteristic
12	C₂×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
14	C₄⋊C₄×C₁₀₉	C₈	1	3	yes	—
15	C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
16	Q₈×C₁₀₉	C₁₆	1	4	yes	complemented
17	C₂×C₄×C₁₀₉	C₂×C₈	1	3	yes	—
18	C₂×C₄×C₁₀₉	C₂×C₈	1	3	yes	—
19	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
20	C₈×C₁₀₉	—	2	3	no	—
21	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
22	C₈×C₁₀₉	#41	1	1	yes	characteristic
23	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
24	C₄×C₁₀₉	—	2	3	no	—
25	C₄×C₁₀₉	C₂×C₁₆	1	6	yes	—
26	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
27	C₄×C₁₀₉	#131	1	1	yes	characteristic
28	C₂×C₁₀₉	C₈×Q₈	1	1	yes	characteristic
29	C₂×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
30	C₂×C₁₀₉	#504	1	1	yes	characteristic
31	Q₈×C₁₆	—	109	1	no	maximal, Sylow
32	C₁₀₉	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
33	C₈×Q₈	D₁₀₉	1	1	yes	characteristic
34	C₄⋊C₁₆	—	109	3	no	—
35	C₄×C₁₆	—	109	3	no	—
36	C₄×Q₈	Dic₁₀₉	1	1	yes	characteristic
37	C₂×C₁₆	—	109	1	no	—
38	C₂×C₁₆	—	109	3	no	—
39	C₄⋊C₈	Dic₁₀₉	1	3	yes	—
40	C₄×C₈	C₂×D₁₀₉	1	3	yes	—
41	C₂×Q₈	#11724784	1	1	yes	characteristic
42	C₂×C₈	C₂×Dic₁₀₉	1	3	yes	—
43	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
44	C₄⋊C₄	#11724784	1	3	yes	—
45	C₄²	C₂×Dic₁₀₉	1	3	yes	—
46	C₁₆	—	109	2	no	—
47	C₁₆	—	218	3	no	—
48	Q₈	#13182004	1	4	yes	direct product factor
49	C₂×C₄	C₂×#11724784	1	3	yes	—
50	C₂×C₄	C₂×#11724784	1	3	yes	—
51	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
52	C₈	—	2	3	no	—
53	C₈	Q₈×D₁₀₉	1	1	yes	characteristic
54	C₈	#13182030	1	1	yes	characteristic
55	C₂²	C₂²×#11724784	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×#13182004	1	6	yes	—
58	C₄	Q₈×Dic₁₀₉	1	1	yes	characteristic
59	C₄	#16812706	1	1	yes	characteristic
60	C₂	Q₈×#11724784	1	1	yes	characteristic
61	C₂	C₂²×#13182004	1	1	yes	characteristic
62	C₂	#20295443	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₁₆
Fitting	C₈×Q₈×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂²×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₈×Q₈	D₁₀₉
hypocenter	C₁₀₉	Q₈×C₁₆
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
10. #13182004	Q₈	direct (N × H)
16. Q₈×C₁₀₉	C₁₆	semidirect
32. C₁₀₉	Q₈×C₁₆	semidirect
48. Q₈	#13182004	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄⋊#13182004	2
C₄×#13182004	2
C₈×Q₈×C₁₀₉	2
Q₈×C₁₆	109

Subgroups — Q8×#13182004 #28364936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13182004 #28364936