Subgroups — C₂²×C₇×D₂₀ #11765438

Order 1120.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×D₅	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×D₂₀	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅×C₇	C₂²	1	1	yes	characteristic
5	C₂²×C₇×D₅	—	2	8	no	—
6	C₂²×C₇×D₅	C₂²	1	12	yes	complemented
7	C₂×C₄×C₅×C₇	C₂²	1	6	yes	complemented
8	C₇×D₂₀	C₂²	1	16	yes	direct product factor
9	C₂²×C₇×D₄	—	5	1	no	maximal, Hall
10	C₂²×D₂₀	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₅×C₇	D₄	1	4	yes	complemented
12	C₂²×C₅×C₇	C₂³	1	3	yes	—
13	C₂×C₇×D₅	—	2	24	no	—
14	C₂×C₇×D₅	C₂³	1	8	yes	complemented
15	C₄×C₅×C₇	C₂³	1	4	yes	complemented
16	C₂⁴×C₇	—	5	2	no	—
17	C₂×C₇×D₄	—	5	12	no	—
18	C₂²×C₄×C₇	D₅	1	1	yes	characteristic, complemented
19	C₂³×D₅	C₂×C₇	1	2	yes	complemented
20	C₂²×C₄×C₅	C₂×C₇	1	1	yes	characteristic, complemented
21	C₂×D₂₀	C₂×C₇	1	12	yes	direct product factor
22	C₂×C₅×C₇	C₂×D₄	1	6	yes	complemented
23	C₂×C₅×C₇	C₂⁴	1	1	yes	characteristic
24	C₇×D₅	—	2	8	no	—
25	C₂³×C₇	—	5	12	no	—
26	C₂³×C₇	—	10	8	no	—
27	C₂³×C₇	C₂×D₅	1	1	yes	characteristic
28	C₇×D₄	—	5	16	no	—
29	C₂×C₄×C₇	C₂×D₅	1	6	yes	complemented
30	C₂³×C₅	C₂²×C₇	1	1	yes	characteristic
31	C₂²×D₅	—	2	8	no	—
32	C₂²×D₅	C₂²×C₇	1	12	yes	complemented
33	C₂×C₄×C₅	C₂²×C₇	1	6	yes	complemented
34	D₂₀	C₂²×C₇	1	16	yes	direct product factor
35	C₅×C₇	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
36	C₂²×D₄	—	5	1	no	Sylow
37	C₂²×C₇	—	5	8	no	—
38	C₂²×C₇	—	10	24	no	—
39	C₂²×C₇	D₂₀	1	4	yes	direct product factor
40	C₂²×C₇	C₂²×D₅	1	3	yes	—
41	C₄×C₇	C₂²×D₅	1	4	yes	complemented
42	C₂²×C₅	C₇×D₄	1	4	yes	complemented
43	C₂²×C₅	C₂³×C₇	1	3	yes	—
44	C₂×D₅	—	2	24	no	—
45	C₂×D₅	C₂³×C₇	1	8	yes	complemented
46	C₄×C₅	C₂³×C₇	1	4	yes	complemented
47	C₂⁴	—	5	2	no	—
48	C₂×D₄	—	5	12	no	—
49	C₂²×C₄	C₇×D₅	1	1	yes	characteristic, complemented
50	C₂×C₇	—	10	8	no	—
51	C₂×C₇	C₂×D₂₀	1	6	yes	direct product factor
52	C₂×C₇	C₂³×D₅	1	1	yes	characteristic
53	C₂×C₅	C₂×C₇×D₄	1	6	yes	complemented
54	C₂×C₅	C₂⁴×C₇	1	1	yes	characteristic
55	D₅	—	2	8	no	—
56	C₂³	—	5	12	no	—
57	C₂³	—	10	8	no	—
58	C₂³	C₂×C₇×D₅	1	1	yes	characteristic
59	D₄	—	5	16	no	—
60	C₂×C₄	C₂×C₇×D₅	1	6	yes	complemented
61	C₇	C₂²×D₂₀	1	1	yes	characteristic, direct product factor, Sylow
62	C₅	C₂²×C₇×D₄	1	1	yes	characteristic, complemented, Sylow
63	C₂²	—	5	8	no	—
64	C₂²	—	10	24	no	—
65	C₂²	C₇×D₂₀	1	4	yes	direct product factor
66	C₂²	C₂²×C₇×D₅	1	3	yes	—
67	C₄	C₂²×C₇×D₅	1	4	yes	complemented
68	C₂	—	10	8	no	—
69	C₂	C₂×C₇×D₂₀	1	6	yes	direct product factor
70	C₂	C₂³×C₇×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₇	C₂×D₅
commutator, cosocle	C₂×C₅	C₂⁴×C₇
Frattini	C₂	C₂³×C₇×D₅
Fitting	C₂²×C₄×C₅×C₇	C₂
socle	C₂³×C₅×C₇	C₂²
hypercenter	C₂²×C₄×C₇	D₅
hypocenter	C₅	C₂²×C₇×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇×D₅	C₂	semidirect
2. C₂²×C₄×C₅×C₇	C₂	semidirect
3. C₂×C₇×D₂₀	C₂	direct (N × H)
6. C₂²×C₇×D₅	C₂²	semidirect
7. C₂×C₄×C₅×C₇	C₂²	semidirect
8. C₇×D₂₀	C₂²	direct (N × H)
10. C₂²×D₂₀	C₇	direct (N × H)
11. C₂²×C₅×C₇	D₄	semidirect
14. C₂×C₇×D₅	C₂³	semidirect
15. C₄×C₅×C₇	C₂³	semidirect
18. C₂²×C₄×C₇	D₅	semidirect
19. C₂³×D₅	C₂×C₇	semidirect
20. C₂²×C₄×C₅	C₂×C₇	semidirect
21. C₂×D₂₀	C₂×C₇	direct (N × H)
22. C₂×C₅×C₇	C₂×D₄	semidirect
29. C₂×C₄×C₇	C₂×D₅	semidirect
32. C₂²×D₅	C₂²×C₇	semidirect
33. C₂×C₄×C₅	C₂²×C₇	semidirect
34. D₂₀	C₂²×C₇	direct (N × H)
35. C₅×C₇	C₂²×D₄	semidirect
39. C₂²×C₇	D₂₀	direct (N × H)
41. C₄×C₇	C₂²×D₅	semidirect
42. C₂²×C₅	C₇×D₄	semidirect
45. C₂×D₅	C₂³×C₇	semidirect
46. C₄×C₅	C₂³×C₇	semidirect
49. C₂²×C₄	C₇×D₅	semidirect
51. C₂×C₇	C₂×D₂₀	direct (N × H)
53. C₂×C₅	C₂×C₇×D₄	semidirect
60. C₂×C₄	C₂×C₇×D₅	semidirect
61. C₇	C₂²×D₂₀	direct (N × H)
62. C₅	C₂²×C₇×D₄	semidirect
65. C₂²	C₇×D₂₀	direct (N × H)
67. C₄	C₂²×C₇×D₅	semidirect
69. C₂	C₂×C₇×D₂₀	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×C₇×D₅	2
C₂²×C₄×C₅×C₇	2
C₂×C₇×D₂₀	2
C₂²×C₇×D₄	5
C₂²×D₂₀	7

Subgroups — C22×C7×D20 #11765438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×D₂₀ #11765438