Subgroups — C₃×C₅×C₇×D₂₁ #17984405

Order 4410.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₇²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₇×D₂₁	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅×C₇×D₇	—	3	1	no	maximal
4	C₃×C₇×D₂₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₅×C₇²	—	2	1	no	—
6	C₃×C₅×C₇²	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₅×D₂₁	C₇	1	1	yes	maximal, characteristic, direct product factor
9	C₃×C₅×D₃×C₇	—	7	1	no	maximal
10	C₅×C₇×D₇	—	3	1	no	Hall
11	C₃²×C₇²	C₂×C₅	1	1	yes	characteristic, complemented, Hall
12	C₃²×C₅×C₇	—	2	3	no	—
13	C₃²×C₅×C₇	D₇	1	1	yes	characteristic, complemented
14	C₃²×C₅×C₇	C₂×C₇	1	1	yes	characteristic, complemented
15	C₇×D₂₁	C₃×C₅	1	1	yes	characteristic, direct product factor
16	C₃×C₇×D₇	—	3	1	no	—
17	C₅×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₅×C₇	—	21	1	no	—
19	C₅×D₂₁	C₃×C₇	1	1	yes	characteristic, direct product factor
20	C₅×D₃×C₇	—	7	1	no	—
21	C₃×C₅×D₇	—	3	1	no	—
22	C₃×C₇²	—	2	1	no	—
23	C₃×C₇²	C₅×D₃	1	1	yes	characteristic, complemented
24	C₃×C₇²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
25	C₃×D₂₁	C₅×C₇	1	1	yes	characteristic, direct product factor
26	C₃×D₃×C₇	—	7	1	no	—
27	C₃×C₅×C₇	—	2	6	no	—
28	C₃×C₅×C₇	—	2	1	no	—
29	C₃×C₅×C₇	—	2	1	no	—
30	C₃×C₅×C₇	—	2	3	no	—
31	C₃×C₅×C₇	—	2	3	no	—
32	C₃×C₅×C₇	D₃×C₇	1	1	yes	characteristic, complemented
33	C₃×C₅×C₇	C₃×D₇	1	1	yes	characteristic, complemented
34	C₃×C₅×C₇	D₂₁	1	1	yes	characteristic, direct product factor
35	C₃×C₅×C₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
36	C₇×D₇	—	3	1	no	Hall
37	C₃×C₅×D₃	—	7	1	no	Hall
38	C₂×C₅×C₇	—	21	1	no	—
39	C₅×D₇	—	3	1	no	—
40	C₃²×C₇	—	2	3	no	—
41	C₃²×C₇	C₅×D₇	1	1	yes	characteristic, complemented
42	C₃²×C₇	C₂×C₅×C₇	1	1	yes	characteristic, complemented
43	C₇²	C₃×C₅×D₃	1	1	yes	characteristic, complemented, Sylow
44	C₃²×C₅	C₇×D₇	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₇	—	21	1	no	—
46	D₂₁	C₃×C₅×C₇	1	1	yes	characteristic, direct product factor
47	C₃×D₇	—	3	1	no	—
48	D₃×C₇	—	7	1	no	—
49	C₅×C₇	—	2	3	no	—
50	C₅×C₇	C₃×D₃×C₇	1	1	yes	characteristic, complemented
51	C₅×C₇	C₃×D₂₁	1	1	yes	characteristic, direct product factor
52	C₂×C₃×C₅	—	21	1	no	—
53	C₅×D₃	—	7	1	no	—
54	C₃×C₇	—	2	6	no	—
55	C₃×C₇	—	2	1	no	—
56	C₃×C₇	—	2	1	no	—
57	C₃×C₇	—	2	3	no	—
58	C₃×C₇	—	2	3	no	—
59	C₃×C₇	C₃×C₅×D₇	1	1	yes	characteristic, complemented
60	C₃×C₇	C₅×D₃×C₇	1	1	yes	characteristic, complemented
61	C₃×C₇	C₅×D₂₁	1	1	yes	characteristic, direct product factor
62	C₃×C₇	C₂×C₃×C₅×C₇	1	1	yes	characteristic, complemented
63	C₃×D₃	—	7	1	no	Hall
64	C₃×C₅	—	2	1	no	—
65	C₃×C₅	C₃×C₇×D₇	1	1	yes	characteristic, complemented
66	C₃×C₅	C₇×D₂₁	1	1	yes	characteristic, direct product factor
67	C₂×C₇	—	21	1	no	—
68	D₇	—	3	1	no	—
69	C₂×C₅	—	21	1	no	Hall
70	C₃²	C₅×C₇×D₇	1	1	yes	characteristic, complemented, Sylow
71	C₇	—	2	3	no	—
72	C₇	C₃×C₅×D₃×C₇	1	1	yes	characteristic, complemented
73	C₇	C₃×C₅×D₂₁	1	1	yes	characteristic, direct product factor
74	C₂×C₃	—	21	1	no	—
75	D₃	—	7	1	no	—
76	C₅	C₃×C₇×D₂₁	1	1	yes	characteristic, direct product factor, Sylow
77	C₃	—	2	1	no	—
78	C₃	C₃×C₅×C₇×D₇	1	1	yes	characteristic, complemented
79	C₃	C₅×C₇×D₂₁	1	1	yes	characteristic, direct product factor
80	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅×C₇	D₂₁
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₃×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇²	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₃×C₅×C₇

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₇²	C₂	semidirect
2. C₅×C₇×D₂₁	C₃	direct (N × H)
4. C₃×C₇×D₂₁	C₅	direct (N × H)
6. C₃×C₅×C₇²	D₃	semidirect
7. C₃×C₅×C₇²	C₂×C₃	semidirect
8. C₃×C₅×D₂₁	C₇	direct (N × H)
11. C₃²×C₇²	C₂×C₅	semidirect
13. C₃²×C₅×C₇	D₇	semidirect
14. C₃²×C₅×C₇	C₂×C₇	semidirect
15. C₇×D₂₁	C₃×C₅	direct (N × H)
17. C₅×C₇²	C₃×D₃	semidirect
19. C₅×D₂₁	C₃×C₇	direct (N × H)
23. C₃×C₇²	C₅×D₃	semidirect
24. C₃×C₇²	C₂×C₃×C₅	semidirect
25. C₃×D₂₁	C₅×C₇	direct (N × H)
32. C₃×C₅×C₇	D₃×C₇	semidirect
33. C₃×C₅×C₇	C₃×D₇	semidirect
34. C₃×C₅×C₇	D₂₁	direct (N × H)
35. C₃×C₅×C₇	C₂×C₃×C₇	semidirect
41. C₃²×C₇	C₅×D₇	semidirect
42. C₃²×C₇	C₂×C₅×C₇	semidirect
43. C₇²	C₃×C₅×D₃	semidirect
44. C₃²×C₅	C₇×D₇	semidirect
46. D₂₁	C₃×C₅×C₇	direct (N × H)
50. C₅×C₇	C₃×D₃×C₇	semidirect
51. C₅×C₇	C₃×D₂₁	direct (N × H)
59. C₃×C₇	C₃×C₅×D₇	semidirect
60. C₃×C₇	C₅×D₃×C₇	semidirect
61. C₃×C₇	C₅×D₂₁	direct (N × H)
62. C₃×C₇	C₂×C₃×C₅×C₇	semidirect
65. C₃×C₅	C₃×C₇×D₇	semidirect
66. C₃×C₅	C₇×D₂₁	direct (N × H)
70. C₃²	C₅×C₇×D₇	semidirect
72. C₇	C₃×C₅×D₃×C₇	semidirect
73. C₇	C₃×C₅×D₂₁	direct (N × H)
76. C₅	C₃×C₇×D₂₁	direct (N × H)
78. C₃	C₃×C₅×C₇×D₇	semidirect
79. C₃	C₅×C₇×D₂₁	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×C₅×C₇²	2
C₅×C₇×D₂₁	3
C₃×C₅×C₇×D₇	3
C₃×C₇×D₂₁	5
C₃×C₅×D₂₁	7
C₃×C₅×D₃×C₇	7

Subgroups — C3×C5×C7×D21 #17984405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₇×D₂₁ #17984405