Subgroups — C₂×C₇²×D₁₂ #72245608

Order 2352.

540 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃×C₇²	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₇²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇²×D₁₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇²×D₄	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₇²	C₂²	1	1	yes	characteristic
6	C₂×D₃×C₇²	—	2	4	no	—
7	C₂×D₃×C₇²	C₂²	1	4	yes	complemented
8	C₃×C₄×C₇²	C₂²	1	2	yes	complemented
9	C₂³×C₇²	—	3	2	no	—
10	C₇²×D₄	—	3	4	no	—
11	C₂×C₄×C₇²	D₃	1	1	yes	characteristic, complemented
12	C₂×C₇×D₁₂	C₇	1	8	yes	maximal, direct product factor
13	C₂×C₃×C₇²	D₄	1	2	yes	complemented
14	C₂×C₃×C₇²	C₂³	1	1	yes	characteristic
15	D₃×C₇²	—	2	4	no	—
16	C₂²×C₇²	—	3	4	no	—
17	C₂²×C₇²	—	6	4	no	—
18	C₂²×C₇²	C₂×D₃	1	1	yes	characteristic
19	C₄×C₇²	C₂×D₃	1	2	yes	complemented
20	C₂²×D₃×C₇	C₂×C₇	1	16	yes	complemented
21	C₂×C₃×C₄×C₇	C₂×C₇	1	8	yes	complemented
22	C₇×D₁₂	C₂×C₇	1	32	yes	direct product factor
23	C₃×C₇²	C₂×D₄	1	1	yes	characteristic, complemented, Hall
24	C₂×C₇×D₄	—	3	8	no	—
25	C₂×C₇²	—	6	4	no	—
26	C₂×C₇²	D₁₂	1	2	yes	direct product factor
27	C₂×C₇²	C₂²×D₃	1	1	yes	characteristic
28	C₂²×C₃×C₇	C₂²×C₇	1	8	yes	—
29	C₂×D₃×C₇	—	2	32	no	—
30	C₂×D₃×C₇	C₂²×C₇	1	32	yes	complemented
31	C₃×C₄×C₇	C₂²×C₇	1	16	yes	complemented
32	C₂³×C₇	—	3	16	no	—
33	C₇×D₄	—	3	32	no	—
34	C₂×C₄×C₇	D₃×C₇	1	8	yes	complemented
35	C₇²	C₂×D₁₂	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×D₁₂	C₇²	1	1	yes	characteristic, direct product factor, Hall
37	C₂×C₃×C₇	C₇×D₄	1	16	yes	complemented
38	C₂×C₃×C₇	C₂³×C₇	1	8	yes	—
39	D₃×C₇	—	2	32	no	—
40	C₂²×C₇	—	3	32	no	—
41	C₂²×C₇	—	6	32	no	—
42	C₂²×C₇	C₂×D₃×C₇	1	8	yes	—
43	C₄×C₇	C₂×D₃×C₇	1	16	yes	complemented
44	C₂²×D₃	C₂×C₇²	1	2	yes	complemented
45	C₂×C₃×C₄	C₂×C₇²	1	1	yes	characteristic, complemented
46	D₁₂	C₂×C₇²	1	4	yes	direct product factor
47	C₃×C₇	C₂×C₇×D₄	1	8	yes	complemented
48	C₂×D₄	—	3	1	no	Sylow
49	C₂×C₇	—	6	32	no	—
50	C₂×C₇	C₇×D₁₂	1	16	yes	direct product factor
51	C₂×C₇	C₂²×D₃×C₇	1	8	yes	—
52	C₂²×C₃	C₂²×C₇²	1	1	yes	characteristic
53	C₂×D₃	—	2	4	no	—
54	C₂×D₃	C₂²×C₇²	1	4	yes	complemented
55	C₃×C₄	C₂²×C₇²	1	2	yes	complemented
56	C₂³	—	3	2	no	—
57	D₄	—	3	4	no	—
58	C₂×C₄	D₃×C₇²	1	1	yes	characteristic, complemented
59	C₇	C₂×C₇×D₁₂	1	8	yes	direct product factor
60	C₂×C₃	C₇²×D₄	1	2	yes	complemented
61	C₂×C₃	C₂³×C₇²	1	1	yes	characteristic
62	D₃	—	2	4	no	—
63	C₂²	—	3	4	no	—
64	C₂²	—	6	4	no	—
65	C₂²	C₂×D₃×C₇²	1	1	yes	characteristic
66	C₄	C₂×D₃×C₇²	1	2	yes	complemented
67	C₃	C₂×C₇²×D₄	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	6	4	no	—
69	C₂	C₇²×D₁₂	1	2	yes	direct product factor
70	C₂	C₂²×D₃×C₇²	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇²	C₂×D₃
commutator, cosocle	C₂×C₃	C₂³×C₇²
Frattini	C₂	C₂²×D₃×C₇²
Fitting	C₂×C₃×C₄×C₇²	C₂
socle	C₂²×C₃×C₇²	C₂²
hypercenter	C₂×C₄×C₇²	D₃
hypocenter	C₃	C₂×C₇²×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇² ◁ C₂×C₄×C₇²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₃×C₇²	C₂	semidirect
2. C₂×C₃×C₄×C₇²	C₂	semidirect
3. C₇²×D₁₂	C₂	direct (N × H)
7. C₂×D₃×C₇²	C₂²	semidirect
8. C₃×C₄×C₇²	C₂²	semidirect
11. C₂×C₄×C₇²	D₃	semidirect
12. C₂×C₇×D₁₂	C₇	direct (N × H)
13. C₂×C₃×C₇²	D₄	semidirect
19. C₄×C₇²	C₂×D₃	semidirect
20. C₂²×D₃×C₇	C₂×C₇	semidirect
21. C₂×C₃×C₄×C₇	C₂×C₇	semidirect
22. C₇×D₁₂	C₂×C₇	direct (N × H)
23. C₃×C₇²	C₂×D₄	semidirect
26. C₂×C₇²	D₁₂	direct (N × H)
30. C₂×D₃×C₇	C₂²×C₇	semidirect
31. C₃×C₄×C₇	C₂²×C₇	semidirect
34. C₂×C₄×C₇	D₃×C₇	semidirect
35. C₇²	C₂×D₁₂	direct (N × H)
36. C₂×D₁₂	C₇²	direct (N × H)
37. C₂×C₃×C₇	C₇×D₄	semidirect
43. C₄×C₇	C₂×D₃×C₇	semidirect
44. C₂²×D₃	C₂×C₇²	semidirect
45. C₂×C₃×C₄	C₂×C₇²	semidirect
46. D₁₂	C₂×C₇²	direct (N × H)
47. C₃×C₇	C₂×C₇×D₄	semidirect
50. C₂×C₇	C₇×D₁₂	direct (N × H)
54. C₂×D₃	C₂²×C₇²	semidirect
55. C₃×C₄	C₂²×C₇²	semidirect
58. C₂×C₄	D₃×C₇²	semidirect
59. C₇	C₂×C₇×D₁₂	direct (N × H)
60. C₂×C₃	C₇²×D₄	semidirect
66. C₄	C₂×D₃×C₇²	semidirect
67. C₃	C₂×C₇²×D₄	semidirect
69. C₂	C₇²×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×D₃×C₇²	2
C₂×C₃×C₄×C₇²	2
C₇²×D₁₂	2
C₂×C₇²×D₄	3
C₂×C₇×D₁₂	7

Subgroups — C2×C72×D12 #72245608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇²×D₁₂ #72245608