Subgroups — Q₈×D₁₆₈₄ #41003507

Order 26944.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₄₂₁	C₂	1	2	yes	maximal, complemented
2	#27212366	C₂	1	3	yes	maximal, complemented
3	#27212346	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₆₈₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₄₂	C₂²	1	6	yes	—
7	Q₈×D₄₂₁	—	2	8	no	—
8	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄₂₁	C₂²	1	6	yes	—
10	C₄⋊Dic₄₂₁	C₂²	1	3	yes	—
11	C₄⋊Dic₄₂₁	C₂²	1	6	yes	—
12	#20269748	C₂²	1	6	yes	—
13	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₄₂₁	C₂²	1	3	yes	—
15	C₄²×C₄₂₁	C₂²	1	3	yes	—
16	C₂²×D₄₂₁	C₂³	1	2	yes	—
17	Dic₈₄₂	—	2	12	no	—
18	D₁₆₈₄	Q₈	1	4	yes	direct product factor
19	C₂×Dic₄₂₁	C₂³	1	6	yes	—
20	C₄×D₄₂₁	—	2	12	no	—
21	Q₈×C₄₂₁	D₄	1	4	yes	complemented
22	C₂×C₄×C₄₂₁	C₂³	1	3	yes	—
23	C₂×C₄×C₄₂₁	C₂³	1	3	yes	—
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
26	C₂×D₄₂₁	—	2	2	no	—
27	C₂×D₄₂₁	—	2	2	no	—
28	C₂×D₄₂₁	C₂×Q₈	1	4	yes	complemented
29	C₄×C₄₂₁	—	2	3	no	—
30	C₄×C₄₂₁	C₂×D₄	1	6	yes	—
31	C₄×C₄₂₁	C₂×Q₈	1	2	yes	complemented
32	Dic₄₂₁	—	2	6	no	—
33	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₄₂₁	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₄₂₁	2₋¹⁺⁴	1	1	yes	characteristic
36	D₄₂₁	—	2	4	no	—
37	C₄₂₁	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
38	D₄×Q₈	—	421	1	no	maximal, Sylow
39	C₂²×Q₈	—	421	2	no	—
40	C₄⋊Q₈	—	421	3	no	—
41	C₂²⋊Q₈	—	421	6	no	—
42	C₄×Q₈	D₄₂₁	1	1	yes	characteristic, complemented
43	C₄×D₄	—	421	3	no	—
44	C₂×Q₈	—	842	8	no	—
45	C₂×Q₈	—	421	6	no	—
46	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
47	C₂×D₄	—	421	1	no	—
48	C₂²×C₄	—	421	6	no	—
49	C₄⋊C₄	—	421	3	no	—
50	C₄⋊C₄	—	421	6	no	—
51	C₄⋊C₄	C₂×D₄₂₁	1	3	yes	—
52	C₂²⋊C₄	—	421	6	no	—
53	C₄²	C₂×D₄₂₁	1	3	yes	—
54	C₂³	—	421	2	no	—
55	Q₈	—	842	12	no	—
56	Q₈	D₁₆₈₄	1	4	yes	direct product factor
57	D₄	—	421	4	no	—
58	C₂×C₄	—	842	12	no	—
59	C₂×C₄	—	421	6	no	—
60	C₂×C₄	C₂²×D₄₂₁	1	3	yes	—
61	C₂×C₄	C₂²×D₄₂₁	1	3	yes	—
62	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
63	C₂²	—	842	2	no	—
64	C₂²	—	842	2	no	—
65	C₂²	—	421	4	no	—
66	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	842	6	no	—
69	C₄	C₂×D₁₆₈₄	1	6	yes	—
70	C₄	Q₈×D₄₂₁	1	2	yes	complemented
71	C₂	—	842	4	no	—
72	C₂	C₂²×D₁₆₈₄	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₄₂₁	1	1	yes	characteristic
74	C₂	#27212413	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	C₄×Q₈×C₄₂₁	C₂
prime radical	C₂×D₁₆₈₄	C₂²
hypercenter	C₄×Q₈	D₄₂₁
hypocenter	C₄₂₁	D₄×Q₈
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₄₂₁	C₂	semidirect
2. #27212366	C₂	semidirect
3. #27212346	C₂	semidirect
5. C₄×Q₈×C₄₂₁	C₂	semidirect
18. D₁₆₈₄	Q₈	direct (N × H)
21. Q₈×C₄₂₁	D₄	semidirect
28. C₂×D₄₂₁	C₂×Q₈	semidirect
31. C₄×C₄₂₁	C₂×Q₈	semidirect
37. C₄₂₁	D₄×Q₈	semidirect
42. C₄×Q₈	D₄₂₁	semidirect
56. Q₈	D₁₆₈₄	direct (N × H)
70. C₄	Q₈×D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₄₂₁	2
#27212366	2
#27212346	2
C₄×D₁₆₈₄	2
C₄×Q₈×C₄₂₁	2
D₄×Q₈	421

Subgroups — Q8×D1684 #41003507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₁₆₈₄ #41003507