Subgroups — C₂×C₃×D₅×D₇ #11719867

Order 840.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₇×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₅×D₇	C₂	1	4	yes	maximal, direct product factor
5	C₂×D₅×D₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×C₇	C₂²	1	1	yes	characteristic, complemented
7	C₃×D₃₅	C₂²	1	2	yes	complemented
8	C₃×C₇×D₅	C₂²	1	2	yes	complemented
9	C₃×C₅×D₇	C₂²	1	2	yes	complemented
10	C₂²×C₃×D₇	—	5	1	no	maximal, Hall
11	C₂×D₃₅	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂×C₇×D₅	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×C₅×D₇	C₂×C₃	1	1	yes	characteristic, complemented
14	D₅×D₇	C₂×C₃	1	4	yes	direct product factor
15	C₂²×C₃×D₅	—	7	1	no	maximal, Hall
16	C₃×C₅×C₇	C₂³	1	1	yes	characteristic, complemented, Hall
17	C₂²×C₃×C₇	—	5	1	no	—
18	C₂×C₃×D₇	—	5	1	no	—
19	C₂×C₃×D₇	—	5	4	no	—
20	C₂×C₃×D₇	D₅	1	1	yes	characteristic, direct product factor
21	C₂×C₅×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
22	D₃₅	C₂²×C₃	1	2	yes	complemented
23	C₅×D₇	C₂²×C₃	1	2	yes	complemented
24	C₇×D₅	C₂²×C₃	1	2	yes	complemented
25	C₂²×C₃×C₅	—	7	1	no	—
26	C₂×C₃×D₅	—	7	1	no	—
27	C₂×C₃×D₅	—	7	4	no	—
28	C₂×C₃×D₅	D₇	1	1	yes	characteristic, direct product factor
29	C₂²×D₇	—	5	1	no	Hall
30	C₂×C₃×C₇	—	5	2	no	—
31	C₂×C₃×C₇	C₂×D₅	1	1	yes	characteristic, complemented
32	C₃×D₇	—	5	2	no	—
33	C₃×D₇	C₂×D₅	1	2	yes	direct product factor
34	C₂²×D₅	—	7	1	no	Hall
35	C₅×C₇	C₂³×C₃	1	1	yes	characteristic, complemented, Hall
36	C₂×C₃×C₅	—	7	2	no	—
37	C₂×C₃×C₅	C₂×D₇	1	1	yes	characteristic, complemented
38	C₃×D₅	—	7	2	no	—
39	C₃×D₅	C₂×D₇	1	2	yes	direct product factor
40	C₂²×C₇	—	5	1	no	—
41	C₂×D₇	—	5	1	no	—
42	C₂×D₇	—	5	4	no	—
43	C₂×D₇	C₃×D₅	1	1	yes	characteristic, direct product factor
44	C₂³×C₃	—	35	1	no	Hall
45	C₃×C₇	C₂²×D₅	1	1	yes	characteristic, complemented, Hall
46	C₂²×C₅	—	7	1	no	—
47	C₂×D₅	—	7	1	no	—
48	C₂×D₅	—	7	4	no	—
49	C₂×D₅	C₃×D₇	1	1	yes	characteristic, direct product factor
50	C₃×C₅	C₂²×D₇	1	1	yes	characteristic, complemented, Hall
51	C₂×C₇	—	5	2	no	—
52	C₂×C₇	C₂×C₃×D₅	1	1	yes	characteristic, complemented
53	D₇	—	5	2	no	—
54	D₇	C₂×C₃×D₅	1	2	yes	direct product factor
55	C₂²×C₃	—	5	1	no	—
56	C₂²×C₃	—	7	1	no	—
57	C₂²×C₃	—	35	4	no	—
58	C₂²×C₃	—	35	1	no	—
59	C₂×C₅	—	7	2	no	—
60	C₂×C₅	C₂×C₃×D₇	1	1	yes	characteristic, complemented
61	D₅	—	7	2	no	—
62	D₅	C₂×C₃×D₇	1	2	yes	direct product factor
63	C₂³	—	35	1	no	Sylow
64	C₇	C₂²×C₃×D₅	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	5	2	no	—
66	C₂×C₃	—	7	2	no	—
67	C₂×C₃	—	35	2	no	—
68	C₂×C₃	D₅×D₇	1	1	yes	characteristic, direct product factor
69	C₅	C₂²×C₃×D₇	1	1	yes	characteristic, complemented, Sylow
70	C₂²	—	5	1	no	—
71	C₂²	—	7	1	no	—
72	C₂²	—	35	4	no	—
73	C₂²	—	35	1	no	—
74	C₃	C₂×D₅×D₇	1	1	yes	characteristic, direct product factor, Sylow
75	C₂	—	5	2	no	—
76	C₂	—	7	2	no	—
77	C₂	—	35	2	no	—
78	C₂	C₃×D₅×D₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	D₅×D₇
commutator, cosocle, hypocenter	C₅×C₇	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₇	C₂²
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×D₃₅	C₂	semidirect
2. C₂×C₃×C₇×D₅	C₂	semidirect
3. C₂×C₃×C₅×D₇	C₂	semidirect
4. C₃×D₅×D₇	C₂	direct (N × H)
5. C₂×D₅×D₇	C₃	direct (N × H)
6. C₂×C₃×C₅×C₇	C₂²	semidirect
7. C₃×D₃₅	C₂²	semidirect
8. C₃×C₇×D₅	C₂²	semidirect
9. C₃×C₅×D₇	C₂²	semidirect
11. C₂×D₃₅	C₂×C₃	semidirect
12. C₂×C₇×D₅	C₂×C₃	semidirect
13. C₂×C₅×D₇	C₂×C₃	semidirect
14. D₅×D₇	C₂×C₃	direct (N × H)
16. C₃×C₅×C₇	C₂³	semidirect
20. C₂×C₃×D₇	D₅	direct (N × H)
21. C₂×C₅×C₇	C₂²×C₃	semidirect
22. D₃₅	C₂²×C₃	semidirect
23. C₅×D₇	C₂²×C₃	semidirect
24. C₇×D₅	C₂²×C₃	semidirect
28. C₂×C₃×D₅	D₇	direct (N × H)
31. C₂×C₃×C₇	C₂×D₅	semidirect
33. C₃×D₇	C₂×D₅	direct (N × H)
35. C₅×C₇	C₂³×C₃	semidirect
37. C₂×C₃×C₅	C₂×D₇	semidirect
39. C₃×D₅	C₂×D₇	direct (N × H)
43. C₂×D₇	C₃×D₅	direct (N × H)
45. C₃×C₇	C₂²×D₅	semidirect
49. C₂×D₅	C₃×D₇	direct (N × H)
50. C₃×C₅	C₂²×D₇	semidirect
52. C₂×C₇	C₂×C₃×D₅	semidirect
54. D₇	C₂×C₃×D₅	direct (N × H)
60. C₂×C₅	C₂×C₃×D₇	semidirect
62. D₅	C₂×C₃×D₇	direct (N × H)
64. C₇	C₂²×C₃×D₅	semidirect
68. C₂×C₃	D₅×D₇	direct (N × H)
69. C₅	C₂²×C₃×D₇	semidirect
74. C₃	C₂×D₅×D₇	direct (N × H)
78. C₂	C₃×D₅×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
C₂×C₃×D₃₅	2
C₂×C₃×C₇×D₅	2
C₂×C₃×C₅×D₇	2
C₃×D₅×D₇	2
C₂×D₅×D₇	3
C₂²×C₃×D₇	5
C₂²×C₃×D₅	7

Subgroups — C2×C3×D5×D7 #11719867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₅×D₇ #11719867