Subgroups — Q₈×D₁₈₈ #15689790

Order 3008.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₄₇	C₂	1	2	yes	maximal, complemented
2	#13094024	C₂	1	3	yes	maximal, complemented
3	#13094004	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₈₈	C₂	1	3	yes	maximal
5	C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
7	Q₈×D₄₇	—	2	8	no	—
8	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄₇	C₂²	1	6	yes	—
10	C₂×Dic₉₄	C₂²	1	6	yes	—
11	C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
12	C₄²×C₄₇	C₂²	1	3	yes	—
13	(C₂×D₄₇)⋊C₄	C₂²	1	6	yes	—
14	C₄⋊Dic₄₇	C₂²	1	3	yes	—
15	C₄⋊Dic₄₇	C₂²	1	6	yes	—
16	C₂²×D₄₇	C₂³	1	2	yes	—
17	Q₈×C₄₇	D₄	1	4	yes	complemented
18	C₂×C₄×C₄₇	C₂³	1	3	yes	—
19	C₂×C₄×C₄₇	C₂³	1	3	yes	—
20	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₄₇	C₂³	1	6	yes	—
22	D₁₈₈	Q₈	1	4	yes	direct product factor
23	C₄×D₄₇	—	2	12	no	—
24	Dic₉₄	—	2	12	no	—
25	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
26	C₂×D₄₇	—	2	2	no	—
27	C₂×D₄₇	—	2	2	no	—
28	C₂×D₄₇	C₂×Q₈	1	4	yes	complemented
29	C₄×C₄₇	—	2	3	no	—
30	C₄×C₄₇	C₂×D₄	1	6	yes	—
31	C₄×C₄₇	C₂×Q₈	1	2	yes	complemented
32	Dic₄₇	—	2	6	no	—
33	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₄₇	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
36	D₄₇	—	2	4	no	—
37	D₄×Q₈	—	47	1	no	maximal, Sylow
38	C₄₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
39	C₂²×Q₈	—	47	2	no	—
40	C₄⋊Q₈	—	47	3	no	—
41	C₂²⋊Q₈	—	47	6	no	—
42	C₄×Q₈	D₄₇	1	1	yes	characteristic, complemented
43	C₄×D₄	—	47	3	no	—
44	C₂×Q₈	—	47	6	no	—
45	C₂×Q₈	—	94	8	no	—
46	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
47	C₂×D₄	—	47	1	no	—
48	C₂²×C₄	—	47	6	no	—
49	C₄⋊C₄	—	47	3	no	—
50	C₄⋊C₄	—	47	6	no	—
51	C₄⋊C₄	C₂×D₄₇	1	3	yes	—
52	C₂²⋊C₄	—	47	6	no	—
53	C₄²	C₂×D₄₇	1	3	yes	—
54	C₂³	—	47	2	no	—
55	Q₈	—	94	12	no	—
56	Q₈	D₁₈₈	1	4	yes	direct product factor
57	D₄	—	47	4	no	—
58	C₂×C₄	—	47	6	no	—
59	C₂×C₄	—	94	12	no	—
60	C₂×C₄	C₂²×D₄₇	1	3	yes	—
61	C₂×C₄	C₂²×D₄₇	1	3	yes	—
62	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
63	C₂²	—	47	4	no	—
64	C₂²	—	94	2	no	—
65	C₂²	—	94	2	no	—
66	C₂²	C₂³×D₄₇	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	94	6	no	—
69	C₄	C₂×D₁₈₈	1	6	yes	—
70	C₄	Q₈×D₄₇	1	2	yes	complemented
71	C₂	—	94	4	no	—
72	C₂	C₂²×D₁₈₈	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₄₇	1	1	yes	characteristic
74	C₂	#13094071	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₄×Q₈×C₄₇	C₂
prime radical	C₂×D₁₈₈	C₂²
hypercenter	C₄×Q₈	D₄₇
hypocenter	C₄₇	D₄×Q₈
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₄₇	C₂	semidirect
2. #13094024	C₂	semidirect
3. #13094004	C₂	semidirect
5. C₄×Q₈×C₄₇	C₂	semidirect
17. Q₈×C₄₇	D₄	semidirect
22. D₁₈₈	Q₈	direct (N × H)
28. C₂×D₄₇	C₂×Q₈	semidirect
31. C₄×C₄₇	C₂×Q₈	semidirect
38. C₄₇	D₄×Q₈	semidirect
42. C₄×Q₈	D₄₇	semidirect
56. Q₈	D₁₈₈	direct (N × H)
70. C₄	Q₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₄₇	2
#13094024	2
#13094004	2
C₄×D₁₈₈	2
C₄×Q₈×C₄₇	2
D₄×Q₈	47

Subgroups — Q8×D188 #15689790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₁₈₈ #15689790