Subgroups — D₈×Dic₄₂₁ #41002923

Order 26944.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₄₂₁	C₂	1	2	yes	maximal, complemented
2	#27212300	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212287	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₁	—	2	2	no	—
7	C₂×#16809890	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
9	C₂²⋊Dic₄₂₁	—	2	2	no	—
10	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₄₂₁	C₂²	1	2	yes	—
12	D₈×C₄₂₁	C₄	1	4	yes	complemented
13	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₂₁	—	2	2	no	—
15	C₂×Dic₄₂₁	—	4	4	no	—
16	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
17	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
18	#16809890	—	2	1	no	—
19	C₂³×C₄₂₁	—	2	2	no	—
20	D₄×C₄₂₁	—	2	2	no	—
21	D₄×C₄₂₁	C₂×C₄	1	4	yes	complemented
22	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
23	C₈×C₄₂₁	C₂×C₄	1	2	yes	complemented
24	C₂²×C₄₂₁	—	2	4	no	—
25	C₂²×C₄₂₁	—	4	2	no	—
26	C₂²×C₄₂₁	—	4	2	no	—
27	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₄₂₁	#41	1	1	yes	characteristic
30	Dic₄₂₁	—	2	1	no	—
31	Dic₄₂₁	—	4	2	no	—
32	Dic₄₂₁	D₈	1	2	yes	direct product factor
33	C₂×C₄₂₁	—	4	4	no	—
34	C₂×C₄₂₁	C₄×D₄	1	1	yes	characteristic
35	C₂×C₄₂₁	C₂×D₈	1	1	yes	characteristic
36	C₂×C₄₂₁	#135	1	1	yes	characteristic
37	C₄₂₁	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
38	C₄×D₈	—	421	1	no	maximal, Sylow
39	C₂×D₈	D₄₂₁	1	1	yes	characteristic
40	C₄×D₄	—	421	2	no	—
41	#107	—	421	1	no	—
42	#102	—	421	2	no	—
43	C₄×C₈	—	421	1	no	—
44	C₂×D₄	C₂×D₄₂₁	1	2	yes	—
45	C₂²×C₄	—	842	2	no	—
46	D₈	Dic₄₂₁	1	4	yes	direct product factor
47	C₂×C₈	—	421	1	no	—
48	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
49	C₄⋊C₄	—	421	2	no	—
50	C₂²⋊C₄	—	842	2	no	—
51	C₄²	—	421	1	no	—
52	C₂³	—	2	2	no	—
53	D₄	—	2	2	no	—
54	D₄	C₂×Dic₄₂₁	1	4	yes	complemented
55	C₂×C₄	—	842	2	no	—
56	C₂×C₄	—	1684	4	no	—
57	C₂×C₄	—	421	1	no	—
58	C₂×C₄	—	421	1	no	—
59	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
60	C₈	—	842	1	no	—
61	C₈	C₂×Dic₄₂₁	1	2	yes	complemented
62	C₂²	—	2	4	no	—
63	C₂²	—	4	2	no	—
64	C₂²	—	4	2	no	—
65	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
66	C₄	—	842	1	no	—
67	C₄	—	1684	2	no	—
68	C₄	—	421	2	no	—
69	C₄	C₂²×Dic₄₂₁	1	1	yes	characteristic
70	C₄	#20269770	1	1	yes	characteristic
71	C₂	—	4	4	no	—
72	C₂	D₄×Dic₄₂₁	1	1	yes	characteristic
73	C₂	D₈×D₄₂₁	1	1	yes	characteristic
74	C₂	#27212385	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator	C₄×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting, prime radical	C₂×D₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	C₂×D₈	D₄₂₁
hypocenter	C₄₂₁	C₄×D₈
LCS	G ▷ C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. D₄×Dic₄₂₁	C₂	semidirect
2. #27212300	C₂	semidirect
3. #27212287	C₂	semidirect
4. C₈×Dic₄₂₁	C₂	semidirect
12. D₈×C₄₂₁	C₄	semidirect
21. D₄×C₄₂₁	C₂×C₄	semidirect
23. C₈×C₄₂₁	C₂×C₄	semidirect
32. Dic₄₂₁	D₈	direct (N × H)
37. C₄₂₁	C₄×D₈	semidirect
46. D₈	Dic₄₂₁	direct (N × H)
54. D₄	C₂×Dic₄₂₁	semidirect
61. C₈	C₂×Dic₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₄₂₁	2
#27212300	2
#27212287	2
C₈×Dic₄₂₁	2
C₂×D₈×C₄₂₁	2
C₄×D₈	421

Subgroups — D8×Dic421 #41002923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×Dic₄₂₁ #41002923