Subgroups — C₂²×C₃×C₅×D₇ #11719877

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×C₃×C₅×D₇	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅×D₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₅×C₇	C₂²	1	3	yes	complemented
5	C₃×C₅×D₇	C₂²	1	4	yes	direct product factor
6	C₂²×C₃×D₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂×C₅×D₇	C₂×C₃	1	6	yes	direct product factor
9	C₂³×C₃×C₅	—	7	1	no	maximal, 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×C₃×D₇	C₂×C₅	1	6	yes	direct product factor
13	C₂×C₅×C₇	C₂²×C₃	1	3	yes	complemented
14	C₅×D₇	C₂²×C₃	1	4	yes	direct product factor
15	C₂²×C₃×C₅	—	7	6	no	—
16	C₂²×C₃×C₅	D₇	1	1	yes	characteristic, direct product factor
17	C₂²×D₇	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₇	C₂²×C₅	1	3	yes	complemented
19	C₃×D₇	C₂²×C₅	1	4	yes	direct product factor
20	C₂³×C₅	—	7	1	no	Hall
21	C₅×C₇	C₂³×C₃	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₅	—	7	4	no	—
23	C₂×C₃×C₅	C₂×D₇	1	3	yes	direct product factor
24	C₂²×C₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
25	C₂×D₇	C₂×C₃×C₅	1	6	yes	direct product factor
26	C₂³×C₃	—	7	1	no	Hall
27	C₃×C₇	C₂³×C₅	1	1	yes	characteristic, complemented, Hall
28	C₂²×C₅	—	7	6	no	—
29	C₂²×C₅	C₃×D₇	1	1	yes	characteristic, direct product factor
30	C₃×C₅	C₂²×D₇	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₇	C₂²×C₃×C₅	1	3	yes	complemented
32	D₇	C₂²×C₃×C₅	1	4	yes	direct product factor
33	C₂²×C₃	—	7	6	no	—
34	C₂²×C₃	C₅×D₇	1	1	yes	characteristic, direct product factor
35	C₂×C₅	—	7	4	no	—
36	C₂×C₅	C₂×C₃×D₇	1	3	yes	direct product factor
37	C₂³	—	7	1	no	Sylow
38	C₇	C₂³×C₃×C₅	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃	—	7	4	no	—
40	C₂×C₃	C₂×C₅×D₇	1	3	yes	direct product factor
41	C₅	C₂²×C₃×D₇	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²	—	7	6	no	—
43	C₂²	C₃×C₅×D₇	1	1	yes	characteristic, direct product factor
44	C₃	C₂²×C₅×D₇	1	1	yes	characteristic, direct product factor, Sylow
45	C₂	—	7	4	no	—
46	C₂	C₂×C₃×C₅×D₇	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₅	D₇
commutator, cosocle, hypocenter	C₇	C₂³×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅×C₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₃×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×C₃×C₅×D₇	C₂	direct (N × H)
3. C₂²×C₅×D₇	C₃	direct (N × H)
4. C₂×C₃×C₅×C₇	C₂²	semidirect
5. C₃×C₅×D₇	C₂²	direct (N × H)
6. C₂²×C₃×D₇	C₅	direct (N × H)
7. C₂²×C₅×C₇	C₂×C₃	semidirect
8. C₂×C₅×D₇	C₂×C₃	direct (N × H)
10. C₃×C₅×C₇	C₂³	semidirect
11. C₂²×C₃×C₇	C₂×C₅	semidirect
12. C₂×C₃×D₇	C₂×C₅	direct (N × H)
13. C₂×C₅×C₇	C₂²×C₃	semidirect
14. C₅×D₇	C₂²×C₃	direct (N × H)
16. C₂²×C₃×C₅	D₇	direct (N × H)
17. C₂²×D₇	C₃×C₅	direct (N × H)
18. C₂×C₃×C₇	C₂²×C₅	semidirect
19. C₃×D₇	C₂²×C₅	direct (N × H)
21. C₅×C₇	C₂³×C₃	semidirect
23. C₂×C₃×C₅	C₂×D₇	direct (N × H)
24. C₂²×C₇	C₂×C₃×C₅	semidirect
25. C₂×D₇	C₂×C₃×C₅	direct (N × H)
27. C₃×C₇	C₂³×C₅	semidirect
29. C₂²×C₅	C₃×D₇	direct (N × H)
30. C₃×C₅	C₂²×D₇	direct (N × H)
31. C₂×C₇	C₂²×C₃×C₅	semidirect
32. D₇	C₂²×C₃×C₅	direct (N × H)
34. C₂²×C₃	C₅×D₇	direct (N × H)
36. C₂×C₅	C₂×C₃×D₇	direct (N × H)
38. C₇	C₂³×C₃×C₅	semidirect
40. C₂×C₃	C₂×C₅×D₇	direct (N × H)
41. C₅	C₂²×C₃×D₇	direct (N × H)
43. C₂²	C₃×C₅×D₇	direct (N × H)
44. C₃	C₂²×C₅×D₇	direct (N × H)
46. C₂	C₂×C₃×C₅×D₇	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×C₃×C₅×D₇	2
C₂²×C₅×D₇	3
C₂²×C₃×D₇	5
C₂³×C₃×C₅	7

Subgroups — C22×C3×C5×D7 #11719877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅×D₇ #11719877