Subgroups — C₂³×C₇×D₁₂ #13062350

Order 1344.

2724 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₃×C₇	C₂	1	2	yes	maximal, complemented
2	C₂³×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×D₁₂	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₇×D₄	—	3	1	no	maximal, Hall
5	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₂³×D₃×C₇	—	2	16	no	—
7	C₂³×D₃×C₇	C₂²	1	28	yes	complemented
8	C₂²×C₃×C₄×C₇	C₂²	1	14	yes	complemented
9	C₂×C₇×D₁₂	C₂²	1	112	yes	direct product factor
10	C₂⁵×C₇	—	3	2	no	—
11	C₂²×C₇×D₄	—	3	28	no	—
12	C₂³×C₄×C₇	D₃	1	1	yes	characteristic, complemented
13	C₂³×D₁₂	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂³×C₃×C₇	D₄	1	8	yes	complemented
15	C₂³×C₃×C₇	C₂³	1	7	yes	—
16	C₂²×D₃×C₇	—	2	112	no	—
17	C₂²×D₃×C₇	C₂³	1	56	yes	complemented
18	C₂×C₃×C₄×C₇	C₂³	1	28	yes	complemented
19	C₇×D₁₂	C₂³	1	64	yes	direct product factor
20	C₂⁴×C₇	—	3	28	no	—
21	C₂⁴×C₇	—	6	16	no	—
22	C₂⁴×C₇	C₂×D₃	1	1	yes	characteristic
23	C₂×C₇×D₄	—	3	112	no	—
24	C₂²×C₄×C₇	C₂×D₃	1	14	yes	complemented
25	C₂⁴×D₃	C₂×C₇	1	2	yes	complemented
26	C₂³×C₃×C₄	C₂×C₇	1	1	yes	characteristic, complemented
27	C₂²×D₁₂	C₂×C₇	1	28	yes	direct product factor
28	C₂²×C₃×C₇	C₂×D₄	1	28	yes	complemented
29	C₂²×C₃×C₇	C₂⁴	1	7	yes	—
30	C₂×D₃×C₇	—	2	112	no	—
31	C₂×D₃×C₇	C₂⁴	1	16	yes	complemented
32	C₃×C₄×C₇	C₂⁴	1	8	yes	complemented
33	C₂³×D₄	—	3	1	no	Sylow
34	C₂³×C₇	—	3	56	no	—
35	C₂³×C₇	—	6	112	no	—
36	C₂³×C₇	D₁₂	1	8	yes	direct product factor
37	C₂³×C₇	C₂²×D₃	1	7	yes	—
38	C₇×D₄	—	3	64	no	—
39	C₂×C₄×C₇	C₂²×D₃	1	28	yes	complemented
40	C₂⁴×C₃	C₂²×C₇	1	1	yes	characteristic
41	C₂³×D₃	—	2	16	no	—
42	C₂³×D₃	C₂²×C₇	1	28	yes	complemented
43	C₂²×C₃×C₄	C₂²×C₇	1	14	yes	complemented
44	C₂×D₁₂	C₂²×C₇	1	112	yes	direct product factor
45	C₂×C₃×C₇	C₂²×D₄	1	14	yes	complemented
46	C₂×C₃×C₇	C₂⁵	1	1	yes	characteristic
47	D₃×C₇	—	2	16	no	—
48	C₂⁵	—	3	2	no	—
49	C₂²×D₄	—	3	28	no	—
50	C₂³×C₄	D₃×C₇	1	1	yes	characteristic, complemented
51	C₂²×C₇	—	3	16	no	—
52	C₂²×C₇	—	6	112	no	—
53	C₂²×C₇	C₂×D₁₂	1	28	yes	direct product factor
54	C₂²×C₇	C₂³×D₃	1	7	yes	—
55	C₄×C₇	C₂³×D₃	1	8	yes	complemented
56	C₂³×C₃	C₇×D₄	1	8	yes	complemented
57	C₂³×C₃	C₂³×C₇	1	7	yes	—
58	C₂²×D₃	—	2	112	no	—
59	C₂²×D₃	C₂³×C₇	1	56	yes	complemented
60	C₂×C₃×C₄	C₂³×C₇	1	28	yes	complemented
61	D₁₂	C₂³×C₇	1	64	yes	direct product factor
62	C₃×C₇	C₂³×D₄	1	1	yes	characteristic, complemented, Hall
63	C₂⁴	—	3	28	no	—
64	C₂⁴	—	6	16	no	—
65	C₂⁴	C₂×D₃×C₇	1	1	yes	characteristic
66	C₂×D₄	—	3	112	no	—
67	C₂²×C₄	C₂×D₃×C₇	1	14	yes	complemented
68	C₂×C₇	—	6	16	no	—
69	C₂×C₇	C₂²×D₁₂	1	14	yes	direct product factor
70	C₂×C₇	C₂⁴×D₃	1	1	yes	characteristic
71	C₂²×C₃	C₂×C₇×D₄	1	28	yes	complemented
72	C₂²×C₃	C₂⁴×C₇	1	7	yes	—
73	C₂×D₃	—	2	112	no	—
74	C₂×D₃	C₂⁴×C₇	1	16	yes	complemented
75	C₃×C₄	C₂⁴×C₇	1	8	yes	complemented
76	C₂³	—	3	56	no	—
77	C₂³	—	6	112	no	—
78	C₂³	C₇×D₁₂	1	8	yes	direct product factor
79	C₂³	C₂²×D₃×C₇	1	7	yes	—
80	D₄	—	3	64	no	—
81	C₂×C₄	C₂²×D₃×C₇	1	28	yes	complemented
82	C₇	C₂³×D₁₂	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₃	C₂²×C₇×D₄	1	14	yes	complemented
84	C₂×C₃	C₂⁵×C₇	1	1	yes	characteristic
85	D₃	—	2	16	no	—
86	C₂²	—	3	16	no	—
87	C₂²	—	6	112	no	—
88	C₂²	C₂×C₇×D₁₂	1	28	yes	direct product factor
89	C₂²	C₂³×D₃×C₇	1	7	yes	—
90	C₄	C₂³×D₃×C₇	1	8	yes	complemented
91	C₃	C₂³×C₇×D₄	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	6	16	no	—
93	C₂	C₂²×C₇×D₁₂	1	14	yes	direct product factor
94	C₂	C₂⁴×D₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₇	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁵×C₇
Frattini	C₂	C₂⁴×D₃×C₇
Fitting	C₂³×C₃×C₄×C₇	C₂
socle	C₂⁴×C₃×C₇	C₂²
hypercenter	C₂³×C₄×C₇	D₃
hypocenter	C₃	C₂³×C₇×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂³×C₄×C₇

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₃×C₇	C₂	semidirect
2. C₂³×C₃×C₄×C₇	C₂	semidirect
3. C₂²×C₇×D₁₂	C₂	direct (N × H)
7. C₂³×D₃×C₇	C₂²	semidirect
8. C₂²×C₃×C₄×C₇	C₂²	semidirect
9. C₂×C₇×D₁₂	C₂²	direct (N × H)
12. C₂³×C₄×C₇	D₃	semidirect
13. C₂³×D₁₂	C₇	direct (N × H)
14. C₂³×C₃×C₇	D₄	semidirect
17. C₂²×D₃×C₇	C₂³	semidirect
18. C₂×C₃×C₄×C₇	C₂³	semidirect
19. C₇×D₁₂	C₂³	direct (N × H)
24. C₂²×C₄×C₇	C₂×D₃	semidirect
25. C₂⁴×D₃	C₂×C₇	semidirect
26. C₂³×C₃×C₄	C₂×C₇	semidirect
27. C₂²×D₁₂	C₂×C₇	direct (N × H)
28. C₂²×C₃×C₇	C₂×D₄	semidirect
31. C₂×D₃×C₇	C₂⁴	semidirect
32. C₃×C₄×C₇	C₂⁴	semidirect
36. C₂³×C₇	D₁₂	direct (N × H)
39. C₂×C₄×C₇	C₂²×D₃	semidirect
42. C₂³×D₃	C₂²×C₇	semidirect
43. C₂²×C₃×C₄	C₂²×C₇	semidirect
44. C₂×D₁₂	C₂²×C₇	direct (N × H)
45. C₂×C₃×C₇	C₂²×D₄	semidirect
50. C₂³×C₄	D₃×C₇	semidirect
53. C₂²×C₇	C₂×D₁₂	direct (N × H)
55. C₄×C₇	C₂³×D₃	semidirect
56. C₂³×C₃	C₇×D₄	semidirect
59. C₂²×D₃	C₂³×C₇	semidirect
60. C₂×C₃×C₄	C₂³×C₇	semidirect
61. D₁₂	C₂³×C₇	direct (N × H)
62. C₃×C₇	C₂³×D₄	semidirect
67. C₂²×C₄	C₂×D₃×C₇	semidirect
69. C₂×C₇	C₂²×D₁₂	direct (N × H)
71. C₂²×C₃	C₂×C₇×D₄	semidirect
74. C₂×D₃	C₂⁴×C₇	semidirect
75. C₃×C₄	C₂⁴×C₇	semidirect
78. C₂³	C₇×D₁₂	direct (N × H)
81. C₂×C₄	C₂²×D₃×C₇	semidirect
82. C₇	C₂³×D₁₂	direct (N × H)
83. C₂×C₃	C₂²×C₇×D₄	semidirect
88. C₂²	C₂×C₇×D₁₂	direct (N × H)
90. C₄	C₂³×D₃×C₇	semidirect
91. C₃	C₂³×C₇×D₄	semidirect
93. C₂	C₂²×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
C₂⁴×D₃×C₇	2
C₂³×C₃×C₄×C₇	2
C₂²×C₇×D₁₂	2
C₂³×C₇×D₄	3
C₂³×D₁₂	7

Subgroups — C23×C7×D12 #13062350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×D₁₂ #13062350