Subgroups — C₂²×D₂₂₀ #13183442

Order 1760.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅₅	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×C₅×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₂₂₀	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅×C₁₁	C₂²	1	1	yes	characteristic
5	C₂²×D₅₅	—	2	8	no	—
6	C₂²×D₅₅	C₂²	1	12	yes	complemented
7	C₂×C₄×C₅×C₁₁	C₂²	1	6	yes	complemented
8	D₂₂₀	C₂²	1	16	yes	direct product factor
9	C₂²×D₄₄	—	5	1	no	maximal, Hall
10	C₂²×C₅×C₁₁	D₄	1	4	yes	complemented
11	C₂²×C₅×C₁₁	C₂³	1	3	yes	—
12	C₂×D₅₅	—	2	24	no	—
13	C₂×D₅₅	C₂³	1	8	yes	complemented
14	C₄×C₅×C₁₁	C₂³	1	4	yes	complemented
15	C₂³×D₁₁	—	5	2	no	—
16	C₂²×C₄×C₁₁	D₅	1	1	yes	characteristic, complemented
17	C₂×D₄₄	—	5	12	no	—
18	C₂²×D₂₀	—	11	1	no	maximal, Hall
19	C₂×C₅×C₁₁	C₂×D₄	1	6	yes	complemented
20	C₂×C₅×C₁₁	C₂⁴	1	1	yes	characteristic
21	D₅₅	—	2	8	no	—
22	C₂³×C₁₁	C₂×D₅	1	1	yes	characteristic
23	C₂²×D₁₁	—	5	12	no	—
24	C₂²×D₁₁	—	10	8	no	—
25	C₂×C₄×C₁₁	C₂×D₅	1	6	yes	complemented
26	D₄₄	—	5	16	no	—
27	C₂³×D₅	—	11	2	no	—
28	C₂²×C₄×C₅	D₁₁	1	1	yes	characteristic, complemented
29	C₂×D₂₀	—	11	12	no	—
30	C₅×C₁₁	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
31	C₂²×C₁₁	D₂₀	1	4	yes	complemented
32	C₂²×C₁₁	C₂²×D₅	1	3	yes	—
33	C₂×D₁₁	—	5	8	no	—
34	C₂×D₁₁	—	10	24	no	—
35	C₄×C₁₁	C₂²×D₅	1	4	yes	complemented
36	C₂³×C₅	C₂×D₁₁	1	1	yes	characteristic
37	C₂²×D₅	—	11	12	no	—
38	C₂²×D₅	—	22	8	no	—
39	C₂×C₄×C₅	C₂×D₁₁	1	6	yes	complemented
40	D₂₀	—	11	16	no	—
41	C₂²×D₄	—	55	1	no	Sylow
42	C₂×C₁₁	C₂×D₂₀	1	6	yes	complemented
43	C₂×C₁₁	C₂³×D₅	1	1	yes	characteristic
44	D₁₁	—	10	8	no	—
45	C₂²×C₅	D₄₄	1	4	yes	complemented
46	C₂²×C₅	C₂²×D₁₁	1	3	yes	—
47	C₂×D₅	—	11	8	no	—
48	C₂×D₅	—	22	24	no	—
49	C₄×C₅	C₂²×D₁₁	1	4	yes	complemented
50	C₂⁴	—	55	2	no	—
51	C₂×D₄	—	55	12	no	—
52	C₂²×C₄	D₅₅	1	1	yes	characteristic, complemented
53	C₁₁	C₂²×D₂₀	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₅	C₂×D₄₄	1	6	yes	complemented
55	C₂×C₅	C₂³×D₁₁	1	1	yes	characteristic
56	D₅	—	22	8	no	—
57	C₂³	—	55	12	no	—
58	C₂³	—	110	8	no	—
59	C₂³	C₂×D₅₅	1	1	yes	characteristic
60	D₄	—	55	16	no	—
61	C₂×C₄	C₂×D₅₅	1	6	yes	complemented
62	C₅	C₂²×D₄₄	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	55	8	no	—
64	C₂²	—	110	24	no	—
65	C₂²	D₂₂₀	1	4	yes	direct product factor
66	C₂²	C₂²×D₅₅	1	3	yes	—
67	C₄	C₂²×D₅₅	1	4	yes	complemented
68	C₂	—	110	8	no	—
69	C₂	C₂×D₂₂₀	1	6	yes	direct product factor
70	C₂	C₂³×D₅₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₅
commutator, cosocle	C₂×C₅×C₁₁	C₂⁴
Frattini	C₂	C₂³×D₅₅
Fitting	C₂²×C₄×C₅×C₁₁	C₂
socle	C₂³×C₅×C₁₁	C₂²
hypercenter	C₂²×C₄	D₅₅
hypocenter	C₅×C₁₁	C₂²×D₄
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₅₅	C₂	semidirect
2. C₂²×C₄×C₅×C₁₁	C₂	semidirect
3. C₂×D₂₂₀	C₂	direct (N × H)
6. C₂²×D₅₅	C₂²	semidirect
7. C₂×C₄×C₅×C₁₁	C₂²	semidirect
8. D₂₂₀	C₂²	direct (N × H)
10. C₂²×C₅×C₁₁	D₄	semidirect
13. C₂×D₅₅	C₂³	semidirect
14. C₄×C₅×C₁₁	C₂³	semidirect
16. C₂²×C₄×C₁₁	D₅	semidirect
19. C₂×C₅×C₁₁	C₂×D₄	semidirect
25. C₂×C₄×C₁₁	C₂×D₅	semidirect
28. C₂²×C₄×C₅	D₁₁	semidirect
30. C₅×C₁₁	C₂²×D₄	semidirect
31. C₂²×C₁₁	D₂₀	semidirect
35. C₄×C₁₁	C₂²×D₅	semidirect
39. C₂×C₄×C₅	C₂×D₁₁	semidirect
42. C₂×C₁₁	C₂×D₂₀	semidirect
45. C₂²×C₅	D₄₄	semidirect
49. C₄×C₅	C₂²×D₁₁	semidirect
52. C₂²×C₄	D₅₅	semidirect
53. C₁₁	C₂²×D₂₀	semidirect
54. C₂×C₅	C₂×D₄₄	semidirect
61. C₂×C₄	C₂×D₅₅	semidirect
62. C₅	C₂²×D₄₄	semidirect
65. C₂²	D₂₂₀	direct (N × H)
67. C₄	C₂²×D₅₅	semidirect
69. C₂	C₂×D₂₂₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₅₅	2
C₂²×C₄×C₅×C₁₁	2
C₂×D₂₂₀	2
C₂²×D₄₄	5
C₂²×D₂₀	11

Subgroups — C22×D220 #13183442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₂₂₀ #13183442