Subgroups — C₂²×C₅×D₃×C₇ #11719880

Order 840.

128 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	6	yes	maximal, direct product factor
3	C₂³×C₅×C₇	—	3	1	no	maximal, Hall
4	C₂×C₃×C₅×C₇	C₂²	1	3	yes	complemented
5	C₅×D₃×C₇	C₂²	1	4	yes	direct product factor
6	C₂²×D₃×C₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₅×C₇	—	3	6	no	—
8	C₂²×C₅×C₇	D₃	1	1	yes	characteristic, direct product factor
9	C₂²×C₅×D₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×C₅×C₇	C₂³	1	1	yes	characteristic, complemented, Hall
11	C₂²×C₃×C₇	C₂×C₅	1	1	yes	characteristic, complemented
12	C₂×D₃×C₇	C₂×C₅	1	6	yes	direct product factor
13	C₂×C₅×C₇	—	3	4	no	—
14	C₂×C₅×C₇	C₂×D₃	1	3	yes	direct product factor
15	C₂²×C₃×C₅	C₂×C₇	1	1	yes	characteristic, complemented
16	C₂×C₅×D₃	C₂×C₇	1	6	yes	direct product factor
17	C₂³×C₇	—	3	1	no	Hall
18	C₂×C₃×C₇	C₂²×C₅	1	3	yes	complemented
19	D₃×C₇	C₂²×C₅	1	4	yes	direct product factor
20	C₂³×C₅	—	3	1	no	Hall
21	C₅×C₇	C₂²×D₃	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₃×C₅	C₂²×C₇	1	3	yes	complemented
23	C₅×D₃	C₂²×C₇	1	4	yes	direct product factor
24	C₂²×C₇	—	3	6	no	—
25	C₂²×C₇	C₅×D₃	1	1	yes	characteristic, direct product factor
26	C₂²×D₃	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
27	C₃×C₇	C₂³×C₅	1	1	yes	characteristic, complemented, Hall
28	C₂²×C₅	—	3	6	no	—
29	C₂²×C₅	D₃×C₇	1	1	yes	characteristic, direct product factor
30	C₃×C₅	C₂³×C₇	1	1	yes	characteristic, complemented, Hall
31	C₂×C₇	—	3	4	no	—
32	C₂×C₇	C₂×C₅×D₃	1	3	yes	direct product factor
33	C₂²×C₃	C₂×C₅×C₇	1	1	yes	characteristic, complemented
34	C₂×D₃	C₂×C₅×C₇	1	6	yes	direct product factor
35	C₂×C₅	—	3	4	no	—
36	C₂×C₅	C₂×D₃×C₇	1	3	yes	direct product factor
37	C₂³	—	3	1	no	Sylow
38	C₇	C₂²×C₅×D₃	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₃	C₂²×C₅×C₇	1	3	yes	complemented
40	D₃	C₂²×C₅×C₇	1	4	yes	direct product factor
41	C₅	C₂²×D₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²	—	3	6	no	—
43	C₂²	C₅×D₃×C₇	1	1	yes	characteristic, direct product factor
44	C₃	C₂³×C₅×C₇	1	1	yes	characteristic, complemented, Sylow
45	C₂	—	3	4	no	—
46	C₂	C₂×C₅×D₃×C₇	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₅×C₇	D₃
commutator, cosocle, hypocenter	C₃	C₂³×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅×C₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ 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	direct (N × H)
4. C₂×C₃×C₅×C₇	C₂²	semidirect
5. C₅×D₃×C₇	C₂²	direct (N × H)
6. C₂²×D₃×C₇	C₅	direct (N × H)
8. C₂²×C₅×C₇	D₃	direct (N × H)
9. C₂²×C₅×D₃	C₇	direct (N × H)
10. C₃×C₅×C₇	C₂³	semidirect
11. C₂²×C₃×C₇	C₂×C₅	semidirect
12. C₂×D₃×C₇	C₂×C₅	direct (N × H)
14. C₂×C₅×C₇	C₂×D₃	direct (N × H)
15. C₂²×C₃×C₅	C₂×C₇	semidirect
16. C₂×C₅×D₃	C₂×C₇	direct (N × H)
18. C₂×C₃×C₇	C₂²×C₅	semidirect
19. D₃×C₇	C₂²×C₅	direct (N × H)
21. C₅×C₇	C₂²×D₃	direct (N × H)
22. C₂×C₃×C₅	C₂²×C₇	semidirect
23. C₅×D₃	C₂²×C₇	direct (N × H)
25. C₂²×C₇	C₅×D₃	direct (N × H)
26. C₂²×D₃	C₅×C₇	direct (N × H)
27. C₃×C₇	C₂³×C₅	semidirect
29. C₂²×C₅	D₃×C₇	direct (N × H)
30. C₃×C₅	C₂³×C₇	semidirect
32. C₂×C₇	C₂×C₅×D₃	direct (N × H)
33. C₂²×C₃	C₂×C₅×C₇	semidirect
34. C₂×D₃	C₂×C₅×C₇	direct (N × H)
36. C₂×C₅	C₂×D₃×C₇	direct (N × H)
38. C₇	C₂²×C₅×D₃	direct (N × H)
39. C₂×C₃	C₂²×C₅×C₇	semidirect
40. D₃	C₂²×C₅×C₇	direct (N × H)
41. C₅	C₂²×D₃×C₇	direct (N × H)
43. C₂²	C₅×D₃×C₇	direct (N × H)
44. C₃	C₂³×C₅×C₇	semidirect
46. C₂	C₂×C₅×D₃×C₇	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×C₃×C₅×C₇	2
C₂×C₅×D₃×C₇	2
C₂³×C₅×C₇	3
C₂²×D₃×C₇	5
C₂²×C₅×D₃	7

Subgroups — C22×C5×D3×C7 #11719880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×D₃×C₇ #11719880