Subgroups — C₇×#1220 #11719823

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×C₇×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×C₂²⋊D₅	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅×C₇	—	2	1	no	—
6	C₂×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₇×D₁₅	—	2	1	no	—
8	C₇×#67	—	5	1	no	maximal, Hall
9	C₂²×C₅×C₇	D₃	1	1	yes	characteristic, complemented
10	C₂×C₇×D₅	—	3	1	no	—
11	C₇×Dic₅	—	3	1	no	—
12	#1220	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	D₅	1	1	yes	characteristic, complemented
15	C₂×D₃×C₇	—	5	1	no	—
16	C₇×Dic₃	—	5	1	no	—
17	C₂×C₅×C₇	—	2	1	no	—
18	C₂×C₅×C₇	C₂×D₃	1	1	yes	characteristic
19	C₇×D₅	—	6	1	no	—
20	C₂²×C₃×C₅	C₂×C₇	1	1	yes	characteristic, complemented
21	C₂×D₁₅	C₂×C₇	1	1	yes	characteristic, complemented
22	Dic₁₅	C₂×C₇	1	1	yes	characteristic, complemented
23	C₇×D₄	—	15	1	no	Hall
24	C₂×C₃×C₇	—	2	1	no	—
25	C₂×C₃×C₇	C₂×D₅	1	1	yes	characteristic
26	D₃×C₇	—	10	1	no	—
27	C₂²⋊D₅	—	3	1	no	Hall
28	C₅×C₇	#67	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₅	—	2	1	no	—
30	C₂×C₃×C₅	C₂²×C₇	1	1	yes	characteristic
31	D₁₅	—	2	1	no	—
32	C₂²×C₇	—	15	1	no	—
33	C₂²×C₇	D₁₅	1	1	yes	characteristic, complemented
34	C₄×C₇	—	15	1	no	—
35	#67	—	5	1	no	Hall
36	C₃×C₇	C₂²⋊D₅	1	1	yes	characteristic, complemented, Hall
37	C₂²×C₅	D₃×C₇	1	1	yes	characteristic, complemented
38	C₂×D₅	—	3	1	no	—
39	Dic₅	—	3	1	no	—
40	C₃×C₅	C₇×D₄	1	1	yes	characteristic, complemented, Hall
41	C₂×C₇	—	2	1	no	—
42	C₂×C₇	—	30	1	no	—
43	C₂×C₇	C₂×D₁₅	1	1	yes	characteristic
44	C₂²×C₃	C₇×D₅	1	1	yes	characteristic, complemented
45	C₂×D₃	—	5	1	no	—
46	Dic₃	—	5	1	no	—
47	C₂×C₅	—	2	1	no	—
48	C₂×C₅	C₂×D₃×C₇	1	1	yes	characteristic
49	D₅	—	6	1	no	—
50	D₄	—	15	1	no	Sylow
51	C₇	#1220	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	C₂×C₇×D₅	1	1	yes	characteristic
54	D₃	—	10	1	no	—
55	C₅	C₇×#67	1	1	yes	characteristic, complemented, Sylow
56	C₂²	—	15	1	no	—
57	C₂²	C₇×D₁₅	1	1	yes	characteristic, complemented
58	C₄	—	15	1	no	—
59	C₃	C₇×C₂²⋊D₅	1	1	yes	characteristic, complemented, Sylow
60	C₂	—	2	1	no	—
61	C₂	—	30	1	no	—
62	C₂	C₂×C₇×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	C₂×D₁₅
commutator, cosocle	C₂×C₃×C₅	C₂²×C₇
Frattini	C₂	C₂×C₇×D₁₅
Fitting	C₂²×C₃×C₅×C₇	C₂
socle	C₂×C₃×C₅×C₇	C₂²
hypercenter	C₂²×C₇	D₁₅
hypocenter	C₃×C₅	C₇×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×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×C₇×D₁₅	C₂	semidirect
3. C₇×Dic₁₅	C₂	semidirect
9. C₂²×C₅×C₇	D₃	semidirect
12. #1220	C₇	direct (N × H)
13. C₃×C₅×C₇	D₄	semidirect
14. C₂²×C₃×C₇	D₅	semidirect
20. C₂²×C₃×C₅	C₂×C₇	semidirect
21. C₂×D₁₅	C₂×C₇	semidirect
22. Dic₁₅	C₂×C₇	semidirect
28. C₅×C₇	#67	semidirect
33. C₂²×C₇	D₁₅	semidirect
36. C₃×C₇	C₂²⋊D₅	semidirect
37. C₂²×C₅	D₃×C₇	semidirect
40. C₃×C₅	C₇×D₄	semidirect
44. C₂²×C₃	C₇×D₅	semidirect
51. C₇	#1220	direct (N × H)
55. C₅	C₇×#67	semidirect
57. C₂²	C₇×D₁₅	semidirect
59. C₃	C₇×C₂²⋊D₅	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×C₇×D₁₅	2
C₇×Dic₁₅	2
C₇×C₂²⋊D₅	3
C₇×#67	5
#1220	7

Subgroups — C7×#1220 #11719823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#1220 #11719823