Subgroups — Q₈×D₇₆₄ #27133330

Order 12224.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₁₉₁	C₂	1	2	yes	maximal, complemented
2	#20259322	C₂	1	3	yes	maximal, complemented
3	#20259302	C₂	1	6	yes	maximal, complemented
4	C₄×D₇₆₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₈₂	C₂²	1	6	yes	—
7	Q₈×D₁₉₁	—	2	8	no	—
8	C₂×D₇₆₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₉₁	C₂²	1	6	yes	—
10	C₄⋊Dic₁₉₁	C₂²	1	3	yes	—
11	C₄⋊Dic₁₉₁	C₂²	1	6	yes	—
12	(C₂×D₁₉₁)⋊C₄	C₂²	1	6	yes	—
13	C₂×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₁	C₂²	1	3	yes	—
15	C₄²×C₁₉₁	C₂²	1	3	yes	—
16	C₂²×D₁₉₁	C₂³	1	2	yes	—
17	Dic₃₈₂	—	2	12	no	—
18	D₇₆₄	Q₈	1	4	yes	direct product factor
19	C₂×Dic₁₉₁	C₂³	1	6	yes	—
20	C₄×D₁₉₁	—	2	12	no	—
21	Q₈×C₁₉₁	D₄	1	4	yes	complemented
22	C₂×C₄×C₁₉₁	C₂³	1	3	yes	—
23	C₂×C₄×C₁₉₁	C₂³	1	3	yes	—
24	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₁	C₂⁴	1	1	yes	characteristic
26	C₂×D₁₉₁	—	2	2	no	—
27	C₂×D₁₉₁	—	2	2	no	—
28	C₂×D₁₉₁	C₂×Q₈	1	4	yes	complemented
29	C₄×C₁₉₁	—	2	3	no	—
30	C₄×C₁₉₁	C₂×D₄	1	6	yes	—
31	C₄×C₁₉₁	C₂×Q₈	1	2	yes	complemented
32	Dic₁₉₁	—	2	6	no	—
33	C₂×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
36	D₁₉₁	—	2	4	no	—
37	C₁₉₁	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
38	D₄×Q₈	—	191	1	no	maximal, Sylow
39	C₂²×Q₈	—	191	2	no	—
40	C₄⋊Q₈	—	191	3	no	—
41	C₂²⋊Q₈	—	191	6	no	—
42	C₄×Q₈	D₁₉₁	1	1	yes	characteristic, complemented
43	C₄×D₄	—	191	3	no	—
44	C₂×Q₈	—	382	8	no	—
45	C₂×Q₈	—	191	6	no	—
46	C₂×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
47	C₂×D₄	—	191	1	no	—
48	C₂²×C₄	—	191	6	no	—
49	C₄⋊C₄	—	191	3	no	—
50	C₄⋊C₄	—	191	6	no	—
51	C₄⋊C₄	C₂×D₁₉₁	1	3	yes	—
52	C₂²⋊C₄	—	191	6	no	—
53	C₄²	C₂×D₁₉₁	1	3	yes	—
54	C₂³	—	191	2	no	—
55	Q₈	—	382	12	no	—
56	Q₈	D₇₆₄	1	4	yes	direct product factor
57	D₄	—	191	4	no	—
58	C₂×C₄	—	382	12	no	—
59	C₂×C₄	—	191	6	no	—
60	C₂×C₄	C₂²×D₁₉₁	1	3	yes	—
61	C₂×C₄	C₂²×D₁₉₁	1	3	yes	—
62	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
63	C₂²	—	382	2	no	—
64	C₂²	—	382	2	no	—
65	C₂²	—	191	4	no	—
66	C₂²	C₂³×D₁₉₁	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	382	6	no	—
69	C₄	C₂×D₇₆₄	1	6	yes	—
70	C₄	Q₈×D₁₉₁	1	2	yes	complemented
71	C₂	—	382	4	no	—
72	C₂	C₂²×D₇₆₄	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
74	C₂	#20259369	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₁
commutator, socle, cosocle	C₂²×C₁₉₁	C₂⁴
Fitting	C₄×Q₈×C₁₉₁	C₂
prime radical	C₂×D₇₆₄	C₂²
hypercenter	C₄×Q₈	D₁₉₁
hypocenter	C₁₉₁	D₄×Q₈
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₁₉₁	C₂	semidirect
2. #20259322	C₂	semidirect
3. #20259302	C₂	semidirect
5. C₄×Q₈×C₁₉₁	C₂	semidirect
18. D₇₆₄	Q₈	direct (N × H)
21. Q₈×C₁₉₁	D₄	semidirect
28. C₂×D₁₉₁	C₂×Q₈	semidirect
31. C₄×C₁₉₁	C₂×Q₈	semidirect
37. C₁₉₁	D₄×Q₈	semidirect
42. C₄×Q₈	D₁₉₁	semidirect
56. Q₈	D₇₆₄	direct (N × H)
70. C₄	Q₈×D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₁₉₁	2
#20259322	2
#20259302	2
C₄×D₇₆₄	2
C₄×Q₈×C₁₉₁	2
D₄×Q₈	191

Subgroups — Q8×D764 #27133330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₇₆₄ #27133330