Subgroups — C₂×C₇×#67 #66724

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#67	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×D₄	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₇	—	2	2	no	—
7	C₂²×C₃×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₇	C₂²	1	2	yes	complemented
9	C₂×D₃×C₇	—	2	2	no	—
10	C₂×D₃×C₇	C₂²	1	2	yes	complemented
11	C₇×Dic₃	C₂²	1	2	yes	complemented
12	C₂³×C₇	—	3	1	no	—
13	C₂³×C₇	D₃	1	1	yes	characteristic, complemented
14	C₇×D₄	—	3	4	no	—
15	C₂×C₄×C₇	—	3	1	no	—
16	C₂×#67	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₃×C₇	—	2	2	no	—
18	C₂×C₃×C₇	D₄	1	2	yes	complemented
19	C₂×C₃×C₇	C₂³	1	1	yes	characteristic
20	D₃×C₇	—	2	2	no	—
21	C₂²×C₇	—	2	2	no	—
22	C₂²×C₇	—	3	2	no	—
23	C₂²×C₇	—	6	2	no	—
24	C₂²×C₇	C₂×D₃	1	1	yes	characteristic
25	C₂²×C₇	C₂×D₃	1	2	yes	complemented
26	C₄×C₇	—	3	2	no	—
27	C₂³×C₃	C₂×C₇	1	1	yes	characteristic, complemented
28	C₂²×D₃	C₂×C₇	1	1	yes	characteristic, complemented
29	#67	C₂×C₇	1	4	yes	direct product factor
30	C₂×Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
31	C₃×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
32	C₂×D₄	—	3	1	no	Sylow
33	C₂×C₇	—	2	2	no	—
34	C₂×C₇	—	6	2	no	—
35	C₂×C₇	#67	1	2	yes	direct product factor
36	C₂×C₇	C₂²×D₃	1	1	yes	characteristic
37	C₂²×C₃	—	2	2	no	—
38	C₂²×C₃	C₂²×C₇	1	1	yes	characteristic
39	C₂²×C₃	C₂²×C₇	1	2	yes	complemented
40	C₂×D₃	—	2	2	no	—
41	C₂×D₃	C₂²×C₇	1	2	yes	complemented
42	Dic₃	C₂²×C₇	1	2	yes	complemented
43	C₂³	—	3	1	no	—
44	C₂³	D₃×C₇	1	1	yes	characteristic, complemented
45	D₄	—	3	4	no	—
46	C₂×C₄	—	3	1	no	—
47	C₇	C₂×#67	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×C₃	—	2	2	no	—
49	C₂×C₃	C₇×D₄	1	2	yes	complemented
50	C₂×C₃	C₂³×C₇	1	1	yes	characteristic
51	D₃	—	2	2	no	—
52	C₂²	—	2	2	no	—
53	C₂²	—	3	2	no	—
54	C₂²	—	6	2	no	—
55	C₂²	C₂×D₃×C₇	1	1	yes	characteristic
56	C₂²	C₂×D₃×C₇	1	2	yes	complemented
57	C₄	—	3	2	no	—
58	C₃	C₂×C₇×D₄	1	1	yes	characteristic, complemented, Sylow
59	C₂	—	2	2	no	—
60	C₂	—	6	2	no	—
61	C₂	C₇×#67	1	2	yes	direct product factor
62	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	C₂³×C₃×C₇	C₂
socle	C₂²×C₃×C₇	C₂²
hypercenter	C₂³×C₇	D₃
hypocenter	C₃	C₂×C₇×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₇

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₇	C₂	semidirect
2. C₂²×D₃×C₇	C₂	semidirect
3. C₇×#67	C₂	direct (N × H)
4. C₂×C₇×Dic₃	C₂	semidirect
8. C₂²×C₃×C₇	C₂²	semidirect
10. C₂×D₃×C₇	C₂²	semidirect
11. C₇×Dic₃	C₂²	semidirect
13. C₂³×C₇	D₃	semidirect
16. C₂×#67	C₇	direct (N × H)
18. C₂×C₃×C₇	D₄	semidirect
25. C₂²×C₇	C₂×D₃	semidirect
27. C₂³×C₃	C₂×C₇	semidirect
28. C₂²×D₃	C₂×C₇	semidirect
29. #67	C₂×C₇	direct (N × H)
30. C₂×Dic₃	C₂×C₇	semidirect
31. C₃×C₇	C₂×D₄	semidirect
35. C₂×C₇	#67	direct (N × H)
39. C₂²×C₃	C₂²×C₇	semidirect
41. C₂×D₃	C₂²×C₇	semidirect
42. Dic₃	C₂²×C₇	semidirect
44. C₂³	D₃×C₇	semidirect
47. C₇	C₂×#67	direct (N × H)
49. C₂×C₃	C₇×D₄	semidirect
56. C₂²	C₂×D₃×C₇	semidirect
58. C₃	C₂×C₇×D₄	semidirect
61. C₂	C₇×#67	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×C₇	2
C₂²×D₃×C₇	2
C₇×#67	2
C₂×C₇×Dic₃	2
C₂×C₇×D₄	3
C₂×#67	7

Subgroups — C2×C7×#67 #66724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#67 #66724