Subgroups — C₂×C₅×D₈₄ #13132872

Order 1680.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₂₁	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₈₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×D₂₈	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₅×D₂₁	—	2	4	no	—
7	C₂×C₅×D₂₁	C₂²	1	4	yes	complemented
8	C₃×C₄×C₅×C₇	C₂²	1	2	yes	complemented
9	C₂×D₈₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₅×D₇	—	3	2	no	—
11	C₂×C₄×C₅×C₇	D₃	1	1	yes	characteristic, complemented
12	C₅×D₂₈	—	3	4	no	—
13	C₂×C₅×D₁₂	—	7	1	no	maximal, Hall
14	C₂×C₃×C₅×C₇	D₄	1	2	yes	complemented
15	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
16	C₅×D₂₁	—	2	4	no	—
17	C₂²×D₂₁	C₂×C₅	1	2	yes	complemented
18	C₂×C₃×C₄×C₇	C₂×C₅	1	1	yes	characteristic, complemented
19	D₈₄	C₂×C₅	1	4	yes	direct product factor
20	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
21	C₂×C₅×D₇	—	3	4	no	—
22	C₂×C₅×D₇	—	6	4	no	—
23	C₄×C₅×C₇	C₂×D₃	1	2	yes	complemented
24	C₂²×C₅×D₃	—	7	2	no	—
25	C₂×C₃×C₄×C₅	D₇	1	1	yes	characteristic, complemented
26	C₅×D₁₂	—	7	4	no	—
27	C₂×D₂₈	—	3	1	no	Hall
28	C₃×C₅×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×C₃×C₇	C₂²×C₅	1	1	yes	characteristic
30	C₂×D₂₁	—	2	4	no	—
31	C₂×D₂₁	C₂²×C₅	1	4	yes	complemented
32	C₃×C₄×C₇	C₂²×C₅	1	2	yes	complemented
33	C₂×C₅×D₄	—	21	1	no	Hall
34	C₂×C₅×C₇	D₁₂	1	2	yes	complemented
35	C₂×C₅×C₇	C₂²×D₃	1	1	yes	characteristic
36	C₅×D₇	—	6	4	no	—
37	C₂²×C₃×C₅	C₂×D₇	1	1	yes	characteristic
38	C₂×C₅×D₃	—	7	4	no	—
39	C₂×C₅×D₃	—	14	4	no	—
40	C₃×C₄×C₅	C₂×D₇	1	2	yes	complemented
41	C₂²×D₇	—	3	2	no	—
42	C₂×C₄×C₇	C₅×D₃	1	1	yes	characteristic, complemented
43	D₂₈	—	3	4	no	—
44	C₂×D₁₂	—	7	1	no	Hall
45	C₂×C₃×C₇	C₅×D₄	1	2	yes	complemented
46	C₂×C₃×C₇	C₂³×C₅	1	1	yes	characteristic
47	D₂₁	—	2	4	no	—
48	C₂³×C₅	—	21	2	no	—
49	C₅×D₄	—	21	4	no	—
50	C₂×C₄×C₅	D₂₁	1	1	yes	characteristic, complemented
51	C₅×C₇	C₂×D₁₂	1	1	yes	characteristic, complemented, Hall
52	C₂×C₃×C₅	D₂₈	1	2	yes	complemented
53	C₂×C₃×C₅	C₂²×D₇	1	1	yes	characteristic
54	C₅×D₃	—	14	4	no	—
55	C₂²×C₇	C₂×C₅×D₃	1	1	yes	characteristic
56	C₂×D₇	—	3	4	no	—
57	C₂×D₇	—	6	4	no	—
58	C₄×C₇	C₂×C₅×D₃	1	2	yes	complemented
59	C₂²×D₃	—	7	2	no	—
60	C₂×C₃×C₄	C₅×D₇	1	1	yes	characteristic, complemented
61	D₁₂	—	7	4	no	—
62	C₃×C₇	C₂×C₅×D₄	1	1	yes	characteristic, complemented, Hall
63	C₂²×C₅	—	21	4	no	—
64	C₂²×C₅	—	42	4	no	—
65	C₂²×C₅	C₂×D₂₁	1	1	yes	characteristic
66	C₄×C₅	C₂×D₂₁	1	2	yes	complemented
67	C₂×D₄	—	21	1	no	Sylow
68	C₃×C₅	C₂×D₂₈	1	1	yes	characteristic, complemented, Hall
69	C₂×C₇	C₅×D₁₂	1	2	yes	complemented
70	C₂×C₇	C₂²×C₅×D₃	1	1	yes	characteristic
71	D₇	—	6	4	no	—
72	C₂²×C₃	C₂×C₅×D₇	1	1	yes	characteristic
73	C₂×D₃	—	7	4	no	—
74	C₂×D₃	—	14	4	no	—
75	C₃×C₄	C₂×C₅×D₇	1	2	yes	complemented
76	C₂×C₅	—	42	4	no	—
77	C₂×C₅	D₈₄	1	2	yes	direct product factor
78	C₂×C₅	C₂²×D₂₁	1	1	yes	characteristic
79	C₂³	—	21	2	no	—
80	D₄	—	21	4	no	—
81	C₂×C₄	C₅×D₂₁	1	1	yes	characteristic, complemented
82	C₇	C₂×C₅×D₁₂	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	C₅×D₂₈	1	2	yes	complemented
84	C₂×C₃	C₂²×C₅×D₇	1	1	yes	characteristic
85	D₃	—	14	4	no	—
86	C₅	C₂×D₈₄	1	1	yes	characteristic, direct product factor, Sylow
87	C₂²	—	21	4	no	—
88	C₂²	—	42	4	no	—
89	C₂²	C₂×C₅×D₂₁	1	1	yes	characteristic
90	C₄	C₂×C₅×D₂₁	1	2	yes	complemented
91	C₃	C₂×C₅×D₂₈	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	42	4	no	—
93	C₂	C₅×D₈₄	1	2	yes	direct product factor
94	C₂	C₂²×C₅×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₁
commutator, cosocle	C₂×C₃×C₇	C₂³×C₅
Frattini	C₂	C₂²×C₅×D₂₁
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle	C₂²×C₃×C₅×C₇	C₂²
hypercenter	C₂×C₄×C₅	D₂₁
hypocenter	C₃×C₇	C₂×C₅×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×D₂₁	C₂	semidirect
2. C₂×C₃×C₄×C₅×C₇	C₂	semidirect
3. C₅×D₈₄	C₂	direct (N × H)
7. C₂×C₅×D₂₁	C₂²	semidirect
8. C₃×C₄×C₅×C₇	C₂²	semidirect
9. C₂×D₈₄	C₅	direct (N × H)
11. C₂×C₄×C₅×C₇	D₃	semidirect
14. C₂×C₃×C₅×C₇	D₄	semidirect
17. C₂²×D₂₁	C₂×C₅	semidirect
18. C₂×C₃×C₄×C₇	C₂×C₅	semidirect
19. D₈₄	C₂×C₅	direct (N × H)
23. C₄×C₅×C₇	C₂×D₃	semidirect
25. C₂×C₃×C₄×C₅	D₇	semidirect
28. C₃×C₅×C₇	C₂×D₄	semidirect
31. C₂×D₂₁	C₂²×C₅	semidirect
32. C₃×C₄×C₇	C₂²×C₅	semidirect
34. C₂×C₅×C₇	D₁₂	semidirect
40. C₃×C₄×C₅	C₂×D₇	semidirect
42. C₂×C₄×C₇	C₅×D₃	semidirect
45. C₂×C₃×C₇	C₅×D₄	semidirect
50. C₂×C₄×C₅	D₂₁	semidirect
51. C₅×C₇	C₂×D₁₂	semidirect
52. C₂×C₃×C₅	D₂₈	semidirect
58. C₄×C₇	C₂×C₅×D₃	semidirect
60. C₂×C₃×C₄	C₅×D₇	semidirect
62. C₃×C₇	C₂×C₅×D₄	semidirect
66. C₄×C₅	C₂×D₂₁	semidirect
68. C₃×C₅	C₂×D₂₈	semidirect
69. C₂×C₇	C₅×D₁₂	semidirect
75. C₃×C₄	C₂×C₅×D₇	semidirect
77. C₂×C₅	D₈₄	direct (N × H)
81. C₂×C₄	C₅×D₂₁	semidirect
82. C₇	C₂×C₅×D₁₂	semidirect
83. C₂×C₃	C₅×D₂₈	semidirect
86. C₅	C₂×D₈₄	direct (N × H)
90. C₄	C₂×C₅×D₂₁	semidirect
91. C₃	C₂×C₅×D₂₈	semidirect
93. C₂	C₅×D₈₄	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×C₅×D₂₁	2
C₂×C₃×C₄×C₅×C₇	2
C₅×D₈₄	2
C₂×C₅×D₂₈	3
C₂×D₈₄	5
C₂×C₅×D₁₂	7

Subgroups — C2×C5×D84 #13132872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₈₄ #13132872