Subgroups — C₂³×C₃×C₅×D₇ #13132978

Order 1680.

536 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₅×D₇	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₅×D₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₅×C₇	C₂²	1	7	yes	complemented
5	C₂×C₃×C₅×D₇	C₂²	1	28	yes	direct product factor
6	C₂³×C₃×D₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂²×C₅×D₇	C₂×C₃	1	14	yes	direct product factor
9	C₂⁴×C₃×C₅	—	7	1	no	maximal, Hall
10	C₂×C₃×C₅×C₇	C₂³	1	7	yes	complemented
11	C₃×C₅×D₇	C₂³	1	8	yes	direct product factor
12	C₂³×C₃×C₇	C₂×C₅	1	1	yes	characteristic, complemented
13	C₂²×C₃×D₇	C₂×C₅	1	14	yes	direct product factor
14	C₂²×C₅×C₇	C₂²×C₃	1	7	yes	complemented
15	C₂×C₅×D₇	C₂²×C₃	1	28	yes	direct product factor
16	C₂³×C₃×C₅	—	7	14	no	—
17	C₂³×C₃×C₅	D₇	1	1	yes	characteristic, direct product factor
18	C₂³×D₇	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
19	C₃×C₅×C₇	C₂⁴	1	1	yes	characteristic, complemented, Hall
20	C₂²×C₃×C₇	C₂²×C₅	1	7	yes	complemented
21	C₂×C₃×D₇	C₂²×C₅	1	28	yes	direct product factor
22	C₂⁴×C₅	—	7	1	no	Hall
23	C₂×C₅×C₇	C₂³×C₃	1	7	yes	complemented
24	C₅×D₇	C₂³×C₃	1	8	yes	direct product factor
25	C₂²×C₃×C₅	—	7	28	no	—
26	C₂²×C₃×C₅	C₂×D₇	1	7	yes	direct product factor
27	C₂³×C₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
28	C₂²×D₇	C₂×C₃×C₅	1	14	yes	direct product factor
29	C₂⁴×C₃	—	7	1	no	Hall
30	C₂×C₃×C₇	C₂³×C₅	1	7	yes	complemented
31	C₃×D₇	C₂³×C₅	1	8	yes	direct product factor
32	C₂³×C₅	—	7	14	no	—
33	C₂³×C₅	C₃×D₇	1	1	yes	characteristic, direct product factor
34	C₅×C₇	C₂⁴×C₃	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₅	—	7	8	no	—
36	C₂×C₃×C₅	C₂²×D₇	1	7	yes	direct product factor
37	C₂²×C₇	C₂²×C₃×C₅	1	7	yes	complemented
38	C₂×D₇	C₂²×C₃×C₅	1	28	yes	direct product factor
39	C₂³×C₃	—	7	14	no	—
40	C₂³×C₃	C₅×D₇	1	1	yes	characteristic, direct product factor
41	C₃×C₇	C₂⁴×C₅	1	1	yes	characteristic, complemented, Hall
42	C₂²×C₅	—	7	28	no	—
43	C₂²×C₅	C₂×C₃×D₇	1	7	yes	direct product factor
44	C₂⁴	—	7	1	no	Sylow
45	C₃×C₅	C₂³×D₇	1	1	yes	characteristic, direct product factor, Hall
46	C₂×C₇	C₂³×C₃×C₅	1	7	yes	complemented
47	D₇	C₂³×C₃×C₅	1	8	yes	direct product factor
48	C₂²×C₃	—	7	28	no	—
49	C₂²×C₃	C₂×C₅×D₇	1	7	yes	direct product factor
50	C₂×C₅	—	7	8	no	—
51	C₂×C₅	C₂²×C₃×D₇	1	7	yes	direct product factor
52	C₂³	—	7	14	no	—
53	C₂³	C₃×C₅×D₇	1	1	yes	characteristic, direct product factor
54	C₇	C₂⁴×C₃×C₅	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	—	7	8	no	—
56	C₂×C₃	C₂²×C₅×D₇	1	7	yes	direct product factor
57	C₅	C₂³×C₃×D₇	1	1	yes	characteristic, direct product factor, Sylow
58	C₂²	—	7	28	no	—
59	C₂²	C₂×C₃×C₅×D₇	1	7	yes	direct product factor
60	C₃	C₂³×C₅×D₇	1	1	yes	characteristic, direct product factor, Sylow
61	C₂	—	7	8	no	—
62	C₂	C₂²×C₃×C₅×D₇	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₃×C₅	D₇
commutator, cosocle, hypocenter	C₇	C₂⁴×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂³×C₃×C₅×C₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³×C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×C₇	C₂	semidirect
2. C₂²×C₃×C₅×D₇	C₂	direct (N × H)
3. C₂³×C₅×D₇	C₃	direct (N × H)
4. C₂²×C₃×C₅×C₇	C₂²	semidirect
5. C₂×C₃×C₅×D₇	C₂²	direct (N × H)
6. C₂³×C₃×D₇	C₅	direct (N × H)
7. C₂³×C₅×C₇	C₂×C₃	semidirect
8. C₂²×C₅×D₇	C₂×C₃	direct (N × H)
10. C₂×C₃×C₅×C₇	C₂³	semidirect
11. C₃×C₅×D₇	C₂³	direct (N × H)
12. C₂³×C₃×C₇	C₂×C₅	semidirect
13. C₂²×C₃×D₇	C₂×C₅	direct (N × H)
14. C₂²×C₅×C₇	C₂²×C₃	semidirect
15. C₂×C₅×D₇	C₂²×C₃	direct (N × H)
17. C₂³×C₃×C₅	D₇	direct (N × H)
18. C₂³×D₇	C₃×C₅	direct (N × H)
19. C₃×C₅×C₇	C₂⁴	semidirect
20. C₂²×C₃×C₇	C₂²×C₅	semidirect
21. C₂×C₃×D₇	C₂²×C₅	direct (N × H)
23. C₂×C₅×C₇	C₂³×C₃	semidirect
24. C₅×D₇	C₂³×C₃	direct (N × H)
26. C₂²×C₃×C₅	C₂×D₇	direct (N × H)
27. C₂³×C₇	C₂×C₃×C₅	semidirect
28. C₂²×D₇	C₂×C₃×C₅	direct (N × H)
30. C₂×C₃×C₇	C₂³×C₅	semidirect
31. C₃×D₇	C₂³×C₅	direct (N × H)
33. C₂³×C₅	C₃×D₇	direct (N × H)
34. C₅×C₇	C₂⁴×C₃	semidirect
36. C₂×C₃×C₅	C₂²×D₇	direct (N × H)
37. C₂²×C₇	C₂²×C₃×C₅	semidirect
38. C₂×D₇	C₂²×C₃×C₅	direct (N × H)
40. C₂³×C₃	C₅×D₇	direct (N × H)
41. C₃×C₇	C₂⁴×C₅	semidirect
43. C₂²×C₅	C₂×C₃×D₇	direct (N × H)
45. C₃×C₅	C₂³×D₇	direct (N × H)
46. C₂×C₇	C₂³×C₃×C₅	semidirect
47. D₇	C₂³×C₃×C₅	direct (N × H)
49. C₂²×C₃	C₂×C₅×D₇	direct (N × H)
51. C₂×C₅	C₂²×C₃×D₇	direct (N × H)
53. C₂³	C₃×C₅×D₇	direct (N × H)
54. C₇	C₂⁴×C₃×C₅	semidirect
56. C₂×C₃	C₂²×C₅×D₇	direct (N × H)
57. C₅	C₂³×C₃×D₇	direct (N × H)
59. C₂²	C₂×C₃×C₅×D₇	direct (N × H)
60. C₃	C₂³×C₅×D₇	direct (N × H)
62. C₂	C₂²×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
C₂³×C₃×C₅×C₇	2
C₂²×C₃×C₅×D₇	2
C₂³×C₅×D₇	3
C₂³×C₃×D₇	5
C₂⁴×C₃×C₅	7

Subgroups — C23×C3×C5×D7 #13132978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×C₅×D₇ #13132978