Subgroups — C₃×C₅²×C₃⋊(C₇⋊D₇) #36348070

Order 22050.

720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₇²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅²×C₃⋊(C₇⋊D₇)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅²×C₇⋊D₇	—	3	1	no	maximal
4	C₃×C₅×C₃⋊(C₇⋊D₇)	C₅	1	6	yes	maximal, direct product factor
5	C₃×C₅²×C₇²	—	2	1	no	—
6	C₃×C₅²×C₇²	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₅²×D₂₁	—	7	8	no	maximal
9	C₅²×C₇⋊D₇	—	3	1	no	Hall
10	C₃²×C₅×C₇²	C₂×C₅	1	6	yes	complemented
11	C₃²×C₅²×C₇	D₇	1	8	yes	complemented
12	C₅×C₃⋊(C₇⋊D₇)	C₃×C₅	1	6	yes	direct product factor
13	C₃×C₅×C₇⋊D₇	—	3	6	no	—
14	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₅²×D₂₁	—	7	8	no	—
16	C₃×C₅²×D₇	—	21	8	no	—
17	C₃×C₃⋊(C₇⋊D₇)	C₅²	1	1	yes	characteristic, direct product factor, Hall
18	C₃×C₅×C₇²	—	2	6	no	—
19	C₃×C₅×C₇²	C₅×D₃	1	6	yes	complemented
20	C₃×C₅×C₇²	C₂×C₃×C₅	1	6	yes	complemented
21	C₃×C₅×D₂₁	—	7	48	no	—
22	C₃×C₅²×C₇	—	2	8	no	—
23	C₃×C₅²×C₇	C₃×D₇	1	8	yes	complemented
24	C₃×C₅²×C₇	D₂₁	1	8	yes	complemented
25	C₅×C₇⋊D₇	—	3	6	no	—
26	C₃×C₅²×D₃	—	49	1	no	Hall
27	C₃²×C₇²	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
28	C₅²×D₇	—	21	8	no	—
29	C₃²×C₅×C₇	C₅×D₇	1	48	yes	complemented
30	C₃⋊(C₇⋊D₇)	C₃×C₅²	1	1	yes	characteristic, direct product factor
31	C₃×C₇⋊D₇	—	3	1	no	—
32	C₅×C₇²	C₃×C₅×D₃	1	6	yes	complemented
33	C₃²×C₅²	C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
34	C₅×D₂₁	—	7	48	no	—
35	C₃×C₅×D₇	—	21	48	no	—
36	C₅²×C₇	C₃×D₂₁	1	8	yes	complemented
37	C₂×C₃×C₅²	—	147	1	no	—
38	C₅²×D₃	—	49	1	no	—
39	C₃×C₇²	—	2	1	no	—
40	C₃×C₇²	C₅²×D₃	1	1	yes	characteristic, complemented
41	C₃×C₇²	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
42	C₃×D₂₁	—	7	8	no	—
43	C₃×C₅×C₇	—	2	48	no	—
44	C₃×C₅×C₇	C₃×C₅×D₇	1	48	yes	complemented
45	C₃×C₅×C₇	C₅×D₂₁	1	48	yes	complemented
46	C₇⋊D₇	—	3	1	no	Hall
47	C₃×C₅×D₃	—	49	6	no	—
48	C₃×C₅²	—	2	1	no	—
49	C₃×C₅²	C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
50	C₃×C₅²	C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor
51	C₅×D₇	—	21	48	no	—
52	C₃²×C₇	C₅²×D₇	1	8	yes	complemented
53	C₂×C₅²	—	147	1	no	Hall
54	C₇²	C₃×C₅²×D₃	1	1	yes	characteristic, complemented, Sylow
55	C₃²×C₅	C₅×C₇⋊D₇	1	6	yes	complemented
56	D₂₁	—	7	8	no	—
57	C₃×D₇	—	21	8	no	—
58	C₅×C₇	C₃×C₅×D₂₁	1	48	yes	complemented
59	C₂×C₃×C₅	—	147	6	no	—
60	C₅×D₃	—	49	6	no	—
61	C₅²	C₃×C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor, Sylow
62	C₃×C₇	—	2	8	no	—
63	C₃×C₇	C₃×C₅²×D₇	1	8	yes	complemented
64	C₃×C₇	C₅²×D₂₁	1	8	yes	complemented
65	C₃×D₃	—	49	1	no	Hall
66	C₃×C₅	—	2	6	no	—
67	C₃×C₅	C₃×C₅×C₇⋊D₇	1	6	yes	complemented
68	C₃×C₅	C₅×C₃⋊(C₇⋊D₇)	1	6	yes	direct product factor
69	D₇	—	21	8	no	—
70	C₂×C₅	—	147	6	no	—
71	C₃²	C₅²×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
72	C₇	C₃×C₅²×D₂₁	1	8	yes	complemented
73	C₂×C₃	—	147	1	no	—
74	D₃	—	49	1	no	—
75	C₅	C₃×C₅×C₃⋊(C₇⋊D₇)	1	6	yes	direct product factor
76	C₃	—	2	1	no	—
77	C₃	C₃×C₅²×C₇⋊D₇	1	1	yes	characteristic, complemented
78	C₃	C₅²×C₃⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor
79	C₂	—	147	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅²	C₃⋊(C₇⋊D₇)
commutator, cosocle, hypocenter	C₃×C₇²	C₂×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇²	C₂
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₇²	C₂	semidirect
2. C₅²×C₃⋊(C₇⋊D₇)	C₃	direct (N × H)
4. C₃×C₅×C₃⋊(C₇⋊D₇)	C₅	direct (N × H)
6. C₃×C₅²×C₇²	D₃	semidirect
7. C₃×C₅²×C₇²	C₂×C₃	semidirect
10. C₃²×C₅×C₇²	C₂×C₅	semidirect
11. C₃²×C₅²×C₇	D₇	semidirect
12. C₅×C₃⋊(C₇⋊D₇)	C₃×C₅	direct (N × H)
14. C₅²×C₇²	C₃×D₃	semidirect
17. C₃×C₃⋊(C₇⋊D₇)	C₅²	direct (N × H)
19. C₃×C₅×C₇²	C₅×D₃	semidirect
20. C₃×C₅×C₇²	C₂×C₃×C₅	semidirect
23. C₃×C₅²×C₇	C₃×D₇	semidirect
24. C₃×C₅²×C₇	D₂₁	semidirect
27. C₃²×C₇²	C₂×C₅²	semidirect
29. C₃²×C₅×C₇	C₅×D₇	semidirect
30. C₃⋊(C₇⋊D₇)	C₃×C₅²	direct (N × H)
32. C₅×C₇²	C₃×C₅×D₃	semidirect
33. C₃²×C₅²	C₇⋊D₇	semidirect
36. C₅²×C₇	C₃×D₂₁	semidirect
40. C₃×C₇²	C₅²×D₃	semidirect
41. C₃×C₇²	C₂×C₃×C₅²	semidirect
44. C₃×C₅×C₇	C₃×C₅×D₇	semidirect
45. C₃×C₅×C₇	C₅×D₂₁	semidirect
49. C₃×C₅²	C₃×C₇⋊D₇	semidirect
50. C₃×C₅²	C₃⋊(C₇⋊D₇)	direct (N × H)
52. C₃²×C₇	C₅²×D₇	semidirect
54. C₇²	C₃×C₅²×D₃	semidirect
55. C₃²×C₅	C₅×C₇⋊D₇	semidirect
58. C₅×C₇	C₃×C₅×D₂₁	semidirect
61. C₅²	C₃×C₃⋊(C₇⋊D₇)	direct (N × H)
63. C₃×C₇	C₃×C₅²×D₇	semidirect
64. C₃×C₇	C₅²×D₂₁	semidirect
67. C₃×C₅	C₃×C₅×C₇⋊D₇	semidirect
68. C₃×C₅	C₅×C₃⋊(C₇⋊D₇)	direct (N × H)
71. C₃²	C₅²×C₇⋊D₇	semidirect
72. C₇	C₃×C₅²×D₂₁	semidirect
75. C₅	C₃×C₅×C₃⋊(C₇⋊D₇)	direct (N × H)
77. C₃	C₃×C₅²×C₇⋊D₇	semidirect
78. C₃	C₅²×C₃⋊(C₇⋊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
C₃²×C₅²×C₇²	2
C₅²×C₃⋊(C₇⋊D₇)	3
C₃×C₅²×C₇⋊D₇	3
C₃×C₅×C₃⋊(C₇⋊D₇)	5
C₃×C₅²×D₂₁	7

Subgroups — C3×C52×C3⋊(C7⋊D7) #36348070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×C₃⋊(C₇⋊D₇) #36348070