Subgroups — C₃×C₇×C₁₇×D₂₁ #28418437

Order 14994.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₇²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₁₇×D₂₁	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₇×D₇×C₁₇	—	3	1	no	maximal
4	C₃×C₇²×C₁₇	—	2	1	no	—
5	C₃×C₇²×C₁₇	D₃	1	1	yes	characteristic, complemented
6	C₃×C₇²×C₁₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₁₇×D₂₁	C₇	1	1	yes	maximal, characteristic, direct product factor
8	C₃×D₃×C₇×C₁₇	—	7	1	no	maximal
9	C₇×D₇×C₁₇	—	3	1	no	Hall
10	C₃²×C₇×C₁₇	—	2	3	no	—
11	C₃²×C₇×C₁₇	D₇	1	1	yes	characteristic, complemented
12	C₃²×C₇×C₁₇	C₂×C₇	1	1	yes	characteristic, complemented
13	C₃×C₇×D₂₁	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₇²×C₁₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃×C₇×C₁₇	—	21	1	no	—
16	C₁₇×D₂₁	C₃×C₇	1	1	yes	characteristic, direct product factor
17	D₃×C₇×C₁₇	—	7	1	no	—
18	C₃×D₇×C₁₇	—	3	1	no	—
19	C₃²×C₇²	C₂×C₁₇	1	1	yes	characteristic, complemented, Hall
20	C₃×C₇×C₁₇	—	2	6	no	—
21	C₃×C₇×C₁₇	—	2	1	no	—
22	C₃×C₇×C₁₇	—	2	1	no	—
23	C₃×C₇×C₁₇	—	2	3	no	—
24	C₃×C₇×C₁₇	—	2	3	no	—
25	C₃×C₇×C₁₇	D₃×C₇	1	1	yes	characteristic, complemented
26	C₃×C₇×C₁₇	C₃×D₇	1	1	yes	characteristic, complemented
27	C₃×C₇×C₁₇	D₂₁	1	1	yes	characteristic, direct product factor
28	C₃×C₇×C₁₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
29	C₃×D₃×C₁₇	—	7	1	no	Hall
30	C₇×D₂₁	C₃×C₁₇	1	1	yes	characteristic, direct product factor
31	C₃×C₇×D₇	—	3	1	no	—
32	C₂×C₇×C₁₇	—	21	1	no	—
33	D₇×C₁₇	—	3	1	no	—
34	C₃²×C₁₇	C₇×D₇	1	1	yes	characteristic, complemented, Hall
35	C₃×C₇²	—	2	1	no	—
36	C₃×C₇²	D₃×C₁₇	1	1	yes	characteristic, complemented
37	C₃×C₇²	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
38	C₃×D₂₁	C₇×C₁₇	1	1	yes	characteristic, direct product factor
39	C₃×D₃×C₇	—	7	1	no	—
40	C₇×C₁₇	—	2	3	no	—
41	C₇×C₁₇	C₃×D₃×C₇	1	1	yes	characteristic, complemented
42	C₇×C₁₇	C₃×D₂₁	1	1	yes	characteristic, direct product factor
43	C₂×C₃×C₁₇	—	21	1	no	—
44	D₃×C₁₇	—	7	1	no	—
45	C₇×D₇	—	3	1	no	Hall
46	C₃²×C₇	—	2	3	no	—
47	C₃²×C₇	D₇×C₁₇	1	1	yes	characteristic, complemented
48	C₃²×C₇	C₂×C₇×C₁₇	1	1	yes	characteristic, complemented
49	C₃×C₁₇	—	2	1	no	—
50	C₃×C₁₇	C₃×C₇×D₇	1	1	yes	characteristic, complemented
51	C₃×C₁₇	C₇×D₂₁	1	1	yes	characteristic, direct product factor
52	C₇²	C₃×D₃×C₁₇	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃×C₇	—	21	1	no	—
54	D₂₁	C₃×C₇×C₁₇	1	1	yes	characteristic, direct product factor
55	C₃×D₇	—	3	1	no	—
56	D₃×C₇	—	7	1	no	—
57	C₂×C₁₇	—	21	1	no	Hall
58	C₃×C₇	—	2	6	no	—
59	C₃×C₇	—	2	1	no	—
60	C₃×C₇	—	2	1	no	—
61	C₃×C₇	—	2	3	no	—
62	C₃×C₇	—	2	3	no	—
63	C₃×C₇	C₃×D₇×C₁₇	1	1	yes	characteristic, complemented
64	C₃×C₇	D₃×C₇×C₁₇	1	1	yes	characteristic, complemented
65	C₃×C₇	C₁₇×D₂₁	1	1	yes	characteristic, direct product factor
66	C₃×C₇	C₂×C₃×C₇×C₁₇	1	1	yes	characteristic, complemented
67	C₃×D₃	—	7	1	no	Hall
68	C₁₇	C₃×C₇×D₂₁	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₇	—	21	1	no	—
70	D₇	—	3	1	no	—
71	C₃²	C₇×D₇×C₁₇	1	1	yes	characteristic, complemented, Sylow
72	C₇	—	2	3	no	—
73	C₇	C₃×D₃×C₇×C₁₇	1	1	yes	characteristic, complemented
74	C₇	C₃×C₁₇×D₂₁	1	1	yes	characteristic, direct product factor
75	C₂×C₃	—	21	1	no	—
76	D₃	—	7	1	no	—
77	C₃	—	2	1	no	—
78	C₃	C₃×C₇×D₇×C₁₇	1	1	yes	characteristic, complemented
79	C₃	C₇×C₁₇×D₂₁	1	1	yes	characteristic, direct product factor
80	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₇×C₁₇	D₂₁
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₃×C₇×C₁₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇²×C₁₇	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₃×C₇×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₃²×C₇²×C₁₇	C₂	semidirect
2. C₇×C₁₇×D₂₁	C₃	direct (N × H)
5. C₃×C₇²×C₁₇	D₃	semidirect
6. C₃×C₇²×C₁₇	C₂×C₃	semidirect
7. C₃×C₁₇×D₂₁	C₇	direct (N × H)
11. C₃²×C₇×C₁₇	D₇	semidirect
12. C₃²×C₇×C₁₇	C₂×C₇	semidirect
13. C₃×C₇×D₂₁	C₁₇	direct (N × H)
14. C₇²×C₁₇	C₃×D₃	semidirect
16. C₁₇×D₂₁	C₃×C₇	direct (N × H)
19. C₃²×C₇²	C₂×C₁₇	semidirect
25. C₃×C₇×C₁₇	D₃×C₇	semidirect
26. C₃×C₇×C₁₇	C₃×D₇	semidirect
27. C₃×C₇×C₁₇	D₂₁	direct (N × H)
28. C₃×C₇×C₁₇	C₂×C₃×C₇	semidirect
30. C₇×D₂₁	C₃×C₁₇	direct (N × H)
34. C₃²×C₁₇	C₇×D₇	semidirect
36. C₃×C₇²	D₃×C₁₇	semidirect
37. C₃×C₇²	C₂×C₃×C₁₇	semidirect
38. C₃×D₂₁	C₇×C₁₇	direct (N × H)
41. C₇×C₁₇	C₃×D₃×C₇	semidirect
42. C₇×C₁₇	C₃×D₂₁	direct (N × H)
47. C₃²×C₇	D₇×C₁₇	semidirect
48. C₃²×C₇	C₂×C₇×C₁₇	semidirect
50. C₃×C₁₇	C₃×C₇×D₇	semidirect
51. C₃×C₁₇	C₇×D₂₁	direct (N × H)
52. C₇²	C₃×D₃×C₁₇	semidirect
54. D₂₁	C₃×C₇×C₁₇	direct (N × H)
63. C₃×C₇	C₃×D₇×C₁₇	semidirect
64. C₃×C₇	D₃×C₇×C₁₇	semidirect
65. C₃×C₇	C₁₇×D₂₁	direct (N × H)
66. C₃×C₇	C₂×C₃×C₇×C₁₇	semidirect
68. C₁₇	C₃×C₇×D₂₁	direct (N × H)
71. C₃²	C₇×D₇×C₁₇	semidirect
73. C₇	C₃×D₃×C₇×C₁₇	semidirect
74. C₇	C₃×C₁₇×D₂₁	direct (N × H)
78. C₃	C₃×C₇×D₇×C₁₇	semidirect
79. C₃	C₇×C₁₇×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×C₇²×C₁₇	2
C₇×C₁₇×D₂₁	3
C₃×C₇×D₇×C₁₇	3
C₃×C₁₇×D₂₁	7
C₃×D₃×C₇×C₁₇	7
C₃×C₇×D₂₁	17

Subgroups — C3×C7×C17×D21 #28418437

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×C₁₇×D₂₁ #28418437