Subgroups — C₂×C₃×C₇×D₂₀ #13132842

Order 1680.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×D₅	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×D₂₀	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×D₂₀	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₇×D₅	—	2	4	no	—
7	C₂×C₃×C₇×D₅	C₂²	1	4	yes	complemented
8	C₃×C₄×C₅×C₇	C₂²	1	2	yes	complemented
9	C₂×C₃×C₇×D₄	—	5	1	no	maximal, Hall
10	C₂²×C₇×D₅	C₂×C₃	1	2	yes	complemented
11	C₂×C₄×C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented
12	C₇×D₂₀	C₂×C₃	1	4	yes	direct product factor
13	C₂×C₃×D₂₀	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₃×C₅×C₇	D₄	1	2	yes	complemented
15	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
16	C₃×C₇×D₅	—	2	4	no	—
17	C₂³×C₃×C₇	—	5	2	no	—
18	C₃×C₇×D₄	—	5	4	no	—
19	C₂×C₃×C₄×C₇	D₅	1	1	yes	characteristic, complemented
20	C₂²×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
21	C₂×C₇×D₅	—	2	4	no	—
22	C₂×C₇×D₅	C₂²×C₃	1	4	yes	complemented
23	C₄×C₅×C₇	C₂²×C₃	1	2	yes	complemented
24	C₂²×C₃×D₅	C₂×C₇	1	2	yes	complemented
25	C₂×C₃×C₄×C₅	C₂×C₇	1	1	yes	characteristic, complemented
26	C₃×D₂₀	C₂×C₇	1	4	yes	direct product factor
27	C₂×C₇×D₄	—	5	1	no	Hall
28	C₃×C₅×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×C₃×C₇	—	5	4	no	—
30	C₂²×C₃×C₇	—	10	4	no	—
31	C₂²×C₃×C₇	C₂×D₅	1	1	yes	characteristic
32	C₃×C₄×C₇	C₂×D₅	1	2	yes	complemented
33	C₂×D₂₀	C₃×C₇	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₅×C₇	C₃×D₄	1	2	yes	complemented
35	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
36	C₇×D₅	—	2	4	no	—
37	C₂²×C₃×C₅	C₂²×C₇	1	1	yes	characteristic
38	C₂×C₃×D₅	—	2	4	no	—
39	C₂×C₃×D₅	C₂²×C₇	1	4	yes	complemented
40	C₃×C₄×C₅	C₂²×C₇	1	2	yes	complemented
41	C₂³×C₇	—	5	2	no	—
42	C₇×D₄	—	5	4	no	—
43	C₂×C₄×C₇	C₃×D₅	1	1	yes	characteristic, complemented
44	C₂×C₃×D₄	—	5	1	no	Hall
45	C₂×C₃×C₇	—	10	4	no	—
46	C₂×C₃×C₇	D₂₀	1	2	yes	direct product factor
47	C₂×C₃×C₇	C₂²×D₅	1	1	yes	characteristic
48	C₂²×D₅	C₂×C₃×C₇	1	2	yes	complemented
49	C₂×C₄×C₅	C₂×C₃×C₇	1	1	yes	characteristic, complemented
50	D₂₀	C₂×C₃×C₇	1	4	yes	direct product factor
51	C₅×C₇	C₂×C₃×D₄	1	1	yes	characteristic, complemented, Hall
52	C₂×C₃×C₅	C₇×D₄	1	2	yes	complemented
53	C₂×C₃×C₅	C₂³×C₇	1	1	yes	characteristic
54	C₃×D₅	—	2	4	no	—
55	C₂²×C₇	—	5	4	no	—
56	C₂²×C₇	—	10	4	no	—
57	C₂²×C₇	C₂×C₃×D₅	1	1	yes	characteristic
58	C₄×C₇	C₂×C₃×D₅	1	2	yes	complemented
59	C₂³×C₃	—	5	2	no	—
60	C₃×D₄	—	5	4	no	—
61	C₂×C₃×C₄	C₇×D₅	1	1	yes	characteristic, complemented
62	C₃×C₇	C₂×D₂₀	1	1	yes	characteristic, direct product factor, Hall
63	C₂²×C₅	C₂²×C₃×C₇	1	1	yes	characteristic
64	C₂×D₅	—	2	4	no	—
65	C₂×D₅	C₂²×C₃×C₇	1	4	yes	complemented
66	C₄×C₅	C₂²×C₃×C₇	1	2	yes	complemented
67	C₂×D₄	—	5	1	no	Sylow
68	C₃×C₅	C₂×C₇×D₄	1	1	yes	characteristic, complemented, Hall
69	C₂×C₇	—	10	4	no	—
70	C₂×C₇	C₃×D₂₀	1	2	yes	direct product factor
71	C₂×C₇	C₂²×C₃×D₅	1	1	yes	characteristic
72	C₂²×C₃	—	5	4	no	—
73	C₂²×C₃	—	10	4	no	—
74	C₂²×C₃	C₂×C₇×D₅	1	1	yes	characteristic
75	C₃×C₄	C₂×C₇×D₅	1	2	yes	complemented
76	C₂×C₅	C₃×C₇×D₄	1	2	yes	complemented
77	C₂×C₅	C₂³×C₃×C₇	1	1	yes	characteristic
78	D₅	—	2	4	no	—
79	C₂³	—	5	2	no	—
80	D₄	—	5	4	no	—
81	C₂×C₄	C₃×C₇×D₅	1	1	yes	characteristic, complemented
82	C₇	C₂×C₃×D₂₀	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₃	—	10	4	no	—
84	C₂×C₃	C₇×D₂₀	1	2	yes	direct product factor
85	C₂×C₃	C₂²×C₇×D₅	1	1	yes	characteristic
86	C₅	C₂×C₃×C₇×D₄	1	1	yes	characteristic, complemented, Sylow
87	C₂²	—	5	4	no	—
88	C₂²	—	10	4	no	—
89	C₂²	C₂×C₃×C₇×D₅	1	1	yes	characteristic
90	C₄	C₂×C₃×C₇×D₅	1	2	yes	complemented
91	C₃	C₂×C₇×D₂₀	1	1	yes	characteristic, direct product factor, Sylow
92	C₂	—	10	4	no	—
93	C₂	C₃×C₇×D₂₀	1	2	yes	direct product factor
94	C₂	C₂²×C₃×C₇×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₇	C₂×D₅
commutator, cosocle	C₂×C₅	C₂³×C₃×C₇
Frattini	C₂	C₂²×C₃×C₇×D₅
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle	C₂²×C₃×C₅×C₇	C₂²
hypercenter	C₂×C₃×C₄×C₇	D₅
hypocenter	C₅	C₂×C₃×C₇×D₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ C₂×C₃×C₄×C₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×D₅	C₂	semidirect
2. C₂×C₃×C₄×C₅×C₇	C₂	semidirect
3. C₃×C₇×D₂₀	C₂	direct (N × H)
4. C₂×C₇×D₂₀	C₃	direct (N × H)
7. C₂×C₃×C₇×D₅	C₂²	semidirect
8. C₃×C₄×C₅×C₇	C₂²	semidirect
10. C₂²×C₇×D₅	C₂×C₃	semidirect
11. C₂×C₄×C₅×C₇	C₂×C₃	semidirect
12. C₇×D₂₀	C₂×C₃	direct (N × H)
13. C₂×C₃×D₂₀	C₇	direct (N × H)
14. C₂×C₃×C₅×C₇	D₄	semidirect
19. C₂×C₃×C₄×C₇	D₅	semidirect
22. C₂×C₇×D₅	C₂²×C₃	semidirect
23. C₄×C₅×C₇	C₂²×C₃	semidirect
24. C₂²×C₃×D₅	C₂×C₇	semidirect
25. C₂×C₃×C₄×C₅	C₂×C₇	semidirect
26. C₃×D₂₀	C₂×C₇	direct (N × H)
28. C₃×C₅×C₇	C₂×D₄	semidirect
32. C₃×C₄×C₇	C₂×D₅	semidirect
33. C₂×D₂₀	C₃×C₇	direct (N × H)
34. C₂×C₅×C₇	C₃×D₄	semidirect
39. C₂×C₃×D₅	C₂²×C₇	semidirect
40. C₃×C₄×C₅	C₂²×C₇	semidirect
43. C₂×C₄×C₇	C₃×D₅	semidirect
46. C₂×C₃×C₇	D₂₀	direct (N × H)
48. C₂²×D₅	C₂×C₃×C₇	semidirect
49. C₂×C₄×C₅	C₂×C₃×C₇	semidirect
50. D₂₀	C₂×C₃×C₇	direct (N × H)
51. C₅×C₇	C₂×C₃×D₄	semidirect
52. C₂×C₃×C₅	C₇×D₄	semidirect
58. C₄×C₇	C₂×C₃×D₅	semidirect
61. C₂×C₃×C₄	C₇×D₅	semidirect
62. C₃×C₇	C₂×D₂₀	direct (N × H)
65. C₂×D₅	C₂²×C₃×C₇	semidirect
66. C₄×C₅	C₂²×C₃×C₇	semidirect
68. C₃×C₅	C₂×C₇×D₄	semidirect
70. C₂×C₇	C₃×D₂₀	direct (N × H)
75. C₃×C₄	C₂×C₇×D₅	semidirect
76. C₂×C₅	C₃×C₇×D₄	semidirect
81. C₂×C₄	C₃×C₇×D₅	semidirect
82. C₇	C₂×C₃×D₂₀	direct (N × H)
84. C₂×C₃	C₇×D₂₀	direct (N × H)
86. C₅	C₂×C₃×C₇×D₄	semidirect
90. C₄	C₂×C₃×C₇×D₅	semidirect
91. C₃	C₂×C₇×D₂₀	direct (N × H)
93. C₂	C₃×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
C₂²×C₃×C₇×D₅	2
C₂×C₃×C₄×C₅×C₇	2
C₃×C₇×D₂₀	2
C₂×C₇×D₂₀	3
C₂×C₃×C₇×D₄	5
C₂×C₃×D₂₀	7

Subgroups — C2×C3×C7×D20 #13132842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×D₂₀ #13132842