Subgroups — C₅×C₇×#67 #11719813

Order 840.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×D₃×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₇×D₄	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅×C₇	—	2	1	no	—
6	C₂×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₅×D₃×C₇	—	2	1	no	—
8	C₇×#67	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₅×C₇	—	3	1	no	—
10	C₂²×C₅×C₇	D₃	1	1	yes	characteristic, complemented
11	C₄×C₅×C₇	—	3	1	no	—
12	C₅×#67	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₃×C₅×C₇	D₄	1	1	yes	characteristic, complemented, Hall
14	C₂²×C₃×C₇	C₂×C₅	1	1	yes	characteristic, complemented
15	C₂×D₃×C₇	C₂×C₅	1	1	yes	characteristic, complemented
16	C₇×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
17	C₂×C₅×C₇	—	2	1	no	—
18	C₂×C₅×C₇	—	6	1	no	—
19	C₂×C₅×C₇	C₂×D₃	1	1	yes	characteristic
20	C₂²×C₃×C₅	C₂×C₇	1	1	yes	characteristic, complemented
21	C₂×C₅×D₃	C₂×C₇	1	1	yes	characteristic, complemented
22	C₅×Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
23	C₇×D₄	—	3	1	no	Hall
24	C₂×C₃×C₇	—	2	1	no	—
25	C₂×C₃×C₇	C₂²×C₅	1	1	yes	characteristic
26	D₃×C₇	—	2	1	no	—
27	C₅×D₄	—	3	1	no	Hall
28	C₅×C₇	#67	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₃×C₅	—	2	1	no	—
30	C₂×C₃×C₅	C₂²×C₇	1	1	yes	characteristic
31	C₅×D₃	—	2	1	no	—
32	C₂²×C₇	—	3	1	no	—
33	C₂²×C₇	C₅×D₃	1	1	yes	characteristic, complemented
34	C₄×C₇	—	3	1	no	—
35	#67	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
36	C₃×C₇	C₅×D₄	1	1	yes	characteristic, complemented, Hall
37	C₂²×C₅	—	3	1	no	—
38	C₂²×C₅	D₃×C₇	1	1	yes	characteristic, complemented
39	C₄×C₅	—	3	1	no	—
40	C₃×C₅	C₇×D₄	1	1	yes	characteristic, complemented, Hall
41	C₂×C₇	—	2	1	no	—
42	C₂×C₇	—	6	1	no	—
43	C₂×C₇	C₂×C₅×D₃	1	1	yes	characteristic
44	C₂²×C₃	C₂×C₅×C₇	1	1	yes	characteristic, complemented
45	C₂×D₃	C₂×C₅×C₇	1	1	yes	characteristic, complemented
46	Dic₃	C₂×C₅×C₇	1	1	yes	characteristic, complemented
47	C₂×C₅	—	2	1	no	—
48	C₂×C₅	—	6	1	no	—
49	C₂×C₅	C₂×D₃×C₇	1	1	yes	characteristic
50	D₄	—	3	1	no	Sylow
51	C₇	C₅×#67	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	C₂²×C₅×C₇	1	1	yes	characteristic
54	D₃	—	2	1	no	—
55	C₅	C₇×#67	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²	—	3	1	no	—
57	C₂²	C₅×D₃×C₇	1	1	yes	characteristic, complemented
58	C₄	—	3	1	no	—
59	C₃	C₅×C₇×D₄	1	1	yes	characteristic, complemented, Sylow
60	C₂	—	2	1	no	—
61	C₂	—	6	1	no	—
62	C₂	C₂×C₅×D₃×C₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×C₇	C₂	semidirect
2. C₂×C₅×D₃×C₇	C₂	semidirect
3. C₅×C₇×Dic₃	C₂	semidirect
8. C₇×#67	C₅	direct (N × H)
10. C₂²×C₅×C₇	D₃	semidirect
12. C₅×#67	C₇	direct (N × H)
13. C₃×C₅×C₇	D₄	semidirect
14. C₂²×C₃×C₇	C₂×C₅	semidirect
15. C₂×D₃×C₇	C₂×C₅	semidirect
16. C₇×Dic₃	C₂×C₅	semidirect
20. C₂²×C₃×C₅	C₂×C₇	semidirect
21. C₂×C₅×D₃	C₂×C₇	semidirect
22. C₅×Dic₃	C₂×C₇	semidirect
28. C₅×C₇	#67	direct (N × H)
33. C₂²×C₇	C₅×D₃	semidirect
35. #67	C₅×C₇	direct (N × H)
36. C₃×C₇	C₅×D₄	semidirect
38. C₂²×C₅	D₃×C₇	semidirect
40. C₃×C₅	C₇×D₄	semidirect
44. C₂²×C₃	C₂×C₅×C₇	semidirect
45. C₂×D₃	C₂×C₅×C₇	semidirect
46. Dic₃	C₂×C₅×C₇	semidirect
51. C₇	C₅×#67	direct (N × H)
55. C₅	C₇×#67	direct (N × H)
57. C₂²	C₅×D₃×C₇	semidirect
59. C₃	C₅×C₇×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×C₇	2
C₂×C₅×D₃×C₇	2
C₅×C₇×Dic₃	2
C₅×C₇×D₄	3
C₇×#67	5
C₅×#67	7

Subgroups — C5×C7×#67 #11719813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇×#67 #11719813