Subgroups — C₅×#4268 #11719818

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×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₂²⋊D₇	—	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×D₂₁	—	2	1	no	—
8	#4268	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₅×C₇	D₃	1	1	yes	characteristic, complemented
10	C₂×C₅×D₇	—	3	1	no	—
11	C₅×Dic₇	—	3	1	no	—
12	C₅×#67	—	7	1	no	maximal, 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×D₂₁	C₂×C₅	1	1	yes	characteristic, complemented
16	Dic₂₁	C₂×C₅	1	1	yes	characteristic, complemented
17	C₂×C₅×C₇	—	2	1	no	—
18	C₂×C₅×C₇	C₂×D₃	1	1	yes	characteristic
19	C₅×D₇	—	6	1	no	—
20	C₂²×C₃×C₅	D₇	1	1	yes	characteristic, complemented
21	C₂×C₅×D₃	—	7	1	no	—
22	C₅×Dic₃	—	7	1	no	—
23	C₂²⋊D₇	—	3	1	no	Hall
24	C₂×C₃×C₇	—	2	1	no	—
25	C₂×C₃×C₇	C₂²×C₅	1	1	yes	characteristic
26	D₂₁	—	2	1	no	—
27	C₅×D₄	—	21	1	no	Hall
28	C₅×C₇	#67	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₅	—	2	1	no	—
30	C₂×C₃×C₅	C₂×D₇	1	1	yes	characteristic
31	C₅×D₃	—	14	1	no	—
32	C₂²×C₇	C₅×D₃	1	1	yes	characteristic, complemented
33	C₂×D₇	—	3	1	no	—
34	Dic₇	—	3	1	no	—
35	#67	—	7	1	no	Hall
36	C₃×C₇	C₅×D₄	1	1	yes	characteristic, complemented, Hall
37	C₂²×C₅	—	21	1	no	—
38	C₂²×C₅	D₂₁	1	1	yes	characteristic, complemented
39	C₄×C₅	—	21	1	no	—
40	C₃×C₅	C₂²⋊D₇	1	1	yes	characteristic, complemented, Hall
41	C₂×C₇	—	2	1	no	—
42	C₂×C₇	C₂×C₅×D₃	1	1	yes	characteristic
43	D₇	—	6	1	no	—
44	C₂²×C₃	C₅×D₇	1	1	yes	characteristic, complemented
45	C₂×D₃	—	7	1	no	—
46	Dic₃	—	7	1	no	—
47	C₂×C₅	—	2	1	no	—
48	C₂×C₅	—	42	1	no	—
49	C₂×C₅	C₂×D₂₁	1	1	yes	characteristic
50	D₄	—	21	1	no	Sylow
51	C₇	C₅×#67	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	C₂×C₅×D₇	1	1	yes	characteristic
54	D₃	—	14	1	no	—
55	C₅	#4268	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²	—	21	1	no	—
57	C₂²	C₅×D₂₁	1	1	yes	characteristic, complemented
58	C₄	—	21	1	no	—
59	C₃	C₅×C₂²⋊D₇	1	1	yes	characteristic, complemented, Sylow
60	C₂	—	2	1	no	—
61	C₂	—	42	1	no	—
62	C₂	C₂×C₅×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₂₁
commutator, cosocle	C₂×C₃×C₇	C₂²×C₅
Frattini	C₂	C₂×C₅×D₂₁
Fitting	C₂²×C₃×C₅×C₇	C₂
socle	C₂×C₃×C₅×C₇	C₂²
hypercenter	C₂²×C₅	D₂₁
hypocenter	C₃×C₇	C₅×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅

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×D₂₁	C₂	semidirect
3. C₅×Dic₂₁	C₂	semidirect
8. #4268	C₅	direct (N × H)
9. C₂²×C₅×C₇	D₃	semidirect
13. C₃×C₅×C₇	D₄	semidirect
14. C₂²×C₃×C₇	C₂×C₅	semidirect
15. C₂×D₂₁	C₂×C₅	semidirect
16. Dic₂₁	C₂×C₅	semidirect
20. C₂²×C₃×C₅	D₇	semidirect
28. C₅×C₇	#67	semidirect
32. C₂²×C₇	C₅×D₃	semidirect
36. C₃×C₇	C₅×D₄	semidirect
38. C₂²×C₅	D₂₁	semidirect
40. C₃×C₅	C₂²⋊D₇	semidirect
44. C₂²×C₃	C₅×D₇	semidirect
51. C₇	C₅×#67	semidirect
55. C₅	#4268	direct (N × H)
57. C₂²	C₅×D₂₁	semidirect
59. C₃	C₅×C₂²⋊D₇	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×D₂₁	2
C₅×Dic₂₁	2
C₅×C₂²⋊D₇	3
#4268	5
C₅×#67	7

Subgroups — C5×#4268 #11719818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#4268 #11719818