Subgroups — C₂×C₇×D₁₂ #66717

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃×C₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₁₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×D₄	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₂×D₃×C₇	—	2	4	no	—
7	C₂×D₃×C₇	C₂²	1	4	yes	complemented
8	C₃×C₄×C₇	C₂²	1	2	yes	complemented
9	C₂³×C₇	—	3	2	no	—
10	C₇×D₄	—	3	4	no	—
11	C₂×C₄×C₇	D₃	1	1	yes	characteristic, complemented
12	C₂×D₁₂	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇	D₄	1	2	yes	complemented
14	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
15	D₃×C₇	—	2	4	no	—
16	C₂²×C₇	—	3	4	no	—
17	C₂²×C₇	—	6	4	no	—
18	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
19	C₄×C₇	C₂×D₃	1	2	yes	complemented
20	C₂²×D₃	C₂×C₇	1	2	yes	complemented
21	C₂×C₃×C₄	C₂×C₇	1	1	yes	characteristic, complemented
22	D₁₂	C₂×C₇	1	4	yes	direct product factor
23	C₃×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
24	C₂×D₄	—	3	1	no	Sylow
25	C₂×C₇	—	6	4	no	—
26	C₂×C₇	D₁₂	1	2	yes	direct product factor
27	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
28	C₂²×C₃	C₂²×C₇	1	1	yes	characteristic
29	C₂×D₃	—	2	4	no	—
30	C₂×D₃	C₂²×C₇	1	4	yes	complemented
31	C₃×C₄	C₂²×C₇	1	2	yes	complemented
32	C₂³	—	3	2	no	—
33	D₄	—	3	4	no	—
34	C₂×C₄	D₃×C₇	1	1	yes	characteristic, complemented
35	C₇	C₂×D₁₂	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₃	C₇×D₄	1	2	yes	complemented
37	C₂×C₃	C₂³×C₇	1	1	yes	characteristic
38	D₃	—	2	4	no	—
39	C₂²	—	3	4	no	—
40	C₂²	—	6	4	no	—
41	C₂²	C₂×D₃×C₇	1	1	yes	characteristic
42	C₄	C₂×D₃×C₇	1	2	yes	complemented
43	C₃	C₂×C₇×D₄	1	1	yes	characteristic, complemented, Sylow
44	C₂	—	6	4	no	—
45	C₂	C₇×D₁₂	1	2	yes	direct product factor
46	C₂	C₂²×D₃×C₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×D₃×C₇	C₂	semidirect
2. C₂×C₃×C₄×C₇	C₂	semidirect
3. C₇×D₁₂	C₂	direct (N × H)
7. C₂×D₃×C₇	C₂²	semidirect
8. C₃×C₄×C₇	C₂²	semidirect
11. C₂×C₄×C₇	D₃	semidirect
12. C₂×D₁₂	C₇	direct (N × H)
13. C₂×C₃×C₇	D₄	semidirect
19. C₄×C₇	C₂×D₃	semidirect
20. C₂²×D₃	C₂×C₇	semidirect
21. C₂×C₃×C₄	C₂×C₇	semidirect
22. D₁₂	C₂×C₇	direct (N × H)
23. C₃×C₇	C₂×D₄	semidirect
26. C₂×C₇	D₁₂	direct (N × H)
30. C₂×D₃	C₂²×C₇	semidirect
31. C₃×C₄	C₂²×C₇	semidirect
34. C₂×C₄	D₃×C₇	semidirect
35. C₇	C₂×D₁₂	direct (N × H)
36. C₂×C₃	C₇×D₄	semidirect
42. C₄	C₂×D₃×C₇	semidirect
43. C₃	C₂×C₇×D₄	semidirect
45. C₂	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×D₃×C₇	2
C₂×C₃×C₄×C₇	2
C₇×D₁₂	2
C₂×C₇×D₄	3
C₂×D₁₂	7

Subgroups — C2×C7×D12 #66717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×D₁₂ #66717