Subgroups — C₂×C₅×C₇×D₁₂ #13132862

Order 1680.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₅×C₇	C₂×D₃
commutator, cosocle	C₂×C₃	C₂³×C₅×C₇
Frattini	C₂	C₂²×C₅×D₃×C₇
Fitting	C₂×C₃×C₄×C₅×C₇	C₂
socle	C₂²×C₃×C₅×C₇	C₂²
hypercenter	C₂×C₄×C₅×C₇	D₃
hypocenter	C₃	C₂×C₅×C₇×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅×C₇ ◁ C₂×C₄×C₅×C₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×D₃×C₇	C₂	semidirect
2. C₂×C₃×C₄×C₅×C₇	C₂	semidirect
3. C₅×C₇×D₁₂	C₂	direct (N × H)
7. C₂×C₅×D₃×C₇	C₂²	semidirect
8. C₃×C₄×C₅×C₇	C₂²	semidirect
9. C₂×C₇×D₁₂	C₅	direct (N × H)
12. C₂×C₄×C₅×C₇	D₃	semidirect
13. C₂×C₅×D₁₂	C₇	direct (N × H)
14. C₂×C₃×C₅×C₇	D₄	semidirect
17. C₂²×D₃×C₇	C₂×C₅	semidirect
18. C₂×C₃×C₄×C₇	C₂×C₅	semidirect
19. C₇×D₁₂	C₂×C₅	direct (N × H)
23. C₄×C₅×C₇	C₂×D₃	semidirect
24. C₂²×C₅×D₃	C₂×C₇	semidirect
25. C₂×C₃×C₄×C₅	C₂×C₇	semidirect
26. C₅×D₁₂	C₂×C₇	direct (N × H)
28. C₃×C₅×C₇	C₂×D₄	semidirect
31. C₂×D₃×C₇	C₂²×C₅	semidirect
32. C₃×C₄×C₇	C₂²×C₅	semidirect
35. C₂×C₅×C₇	D₁₂	direct (N × H)
39. C₂×C₅×D₃	C₂²×C₇	semidirect
40. C₃×C₄×C₅	C₂²×C₇	semidirect
43. C₂×C₄×C₇	C₅×D₃	semidirect
44. C₂×D₁₂	C₅×C₇	direct (N × H)
45. C₂×C₃×C₇	C₅×D₄	semidirect
50. C₂×C₄×C₅	D₃×C₇	semidirect
51. C₅×C₇	C₂×D₁₂	direct (N × H)
52. C₂×C₃×C₅	C₇×D₄	semidirect
58. C₄×C₇	C₂×C₅×D₃	semidirect
59. C₂²×D₃	C₂×C₅×C₇	semidirect
60. C₂×C₃×C₄	C₂×C₅×C₇	semidirect
61. D₁₂	C₂×C₅×C₇	direct (N × H)
62. C₃×C₇	C₂×C₅×D₄	semidirect
66. C₄×C₅	C₂×D₃×C₇	semidirect
68. C₃×C₅	C₂×C₇×D₄	semidirect
70. C₂×C₇	C₅×D₁₂	direct (N × H)
74. C₂×D₃	C₂²×C₅×C₇	semidirect
75. C₃×C₄	C₂²×C₅×C₇	semidirect
77. C₂×C₅	C₇×D₁₂	direct (N × H)
81. C₂×C₄	C₅×D₃×C₇	semidirect
82. C₇	C₂×C₅×D₁₂	direct (N × H)
83. C₂×C₃	C₅×C₇×D₄	semidirect
86. C₅	C₂×C₇×D₁₂	direct (N × H)
90. C₄	C₂×C₅×D₃×C₇	semidirect
91. C₃	C₂×C₅×C₇×D₄	semidirect
93. C₂	C₅×C₇×D₁₂	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×D₃×C₇	2
C₂×C₃×C₄×C₅×C₇	2
C₅×C₇×D₁₂	2
C₂×C₅×C₇×D₄	3
C₂×C₇×D₁₂	5
C₂×C₅×D₁₂	7

Subgroups — C2×C5×C7×D12 #13132862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×C₇×D₁₂ #13132862