Subgroups — C₅²×C₇×C₃⋊D₂₁ #36348078

Order 22050.

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

	subgroup	G/H
center, hypercenter	C₅²×C₇	C₃⋊D₂₁
commutator, cosocle, hypocenter	C₃²×C₇	C₂×C₅²×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇²	C₂
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₅²×C₇

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₇²	C₂	semidirect
3. C₅×C₇×C₃⋊D₂₁	C₅	direct (N × H)
4. C₃×C₅²×C₇²	D₃	semidirect
5. C₅²×C₃⋊D₂₁	C₇	direct (N × H)
8. C₃²×C₅×C₇²	C₂×C₅	semidirect
10. C₃²×C₅²×C₇	D₇	semidirect
11. C₃²×C₅²×C₇	C₂×C₇	semidirect
13. C₅²×C₇²	C₃⋊D₃	semidirect
16. C₇×C₃⋊D₂₁	C₅²	direct (N × H)
17. C₃×C₅×C₇²	C₅×D₃	semidirect
18. C₅×C₃⋊D₂₁	C₅×C₇	direct (N × H)
21. C₃×C₅²×C₇	D₃×C₇	semidirect
22. C₃×C₅²×C₇	D₂₁	semidirect
25. C₃²×C₇²	C₂×C₅²	semidirect
29. C₃²×C₅×C₇	C₅×D₇	semidirect
30. C₃²×C₅×C₇	C₂×C₅×C₇	semidirect
32. C₅×C₇²	C₅×C₃⋊D₃	semidirect
33. C₃²×C₅²	C₇×D₇	semidirect
37. C₅²×C₇	C₇×C₃⋊D₃	semidirect
38. C₅²×C₇	C₃⋊D₂₁	direct (N × H)
40. C₃×C₇²	C₅²×D₃	semidirect
41. C₃⋊D₂₁	C₅²×C₇	direct (N × H)
44. C₃×C₅×C₇	C₅×D₃×C₇	semidirect
45. C₃×C₅×C₇	C₅×D₂₁	semidirect
48. C₃×C₅²	C₇×D₂₁	semidirect
52. C₃²×C₇	C₅²×D₇	semidirect
53. C₃²×C₇	C₂×C₅²×C₇	semidirect
55. C₇²	C₅²×C₃⋊D₃	semidirect
56. C₃²×C₅	C₅×C₇×D₇	semidirect
60. C₅×C₇	C₅×C₇×C₃⋊D₃	semidirect
61. C₅×C₇	C₅×C₃⋊D₂₁	direct (N × H)
63. C₅²	C₇×C₃⋊D₂₁	direct (N × H)
65. C₃×C₇	C₅²×D₃×C₇	semidirect
66. C₃×C₇	C₅²×D₂₁	semidirect
68. C₃×C₅	C₅×C₇×D₂₁	semidirect
72. C₃²	C₅²×C₇×D₇	semidirect
74. C₇	C₅²×C₇×C₃⋊D₃	semidirect
75. C₇	C₅²×C₃⋊D₂₁	direct (N × H)
77. C₅	C₅×C₇×C₃⋊D₂₁	direct (N × H)
78. C₃	C₅²×C₇×D₂₁	semidirect

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×C₇×D₂₁	3
C₅×C₇×C₃⋊D₂₁	5
C₅²×C₃⋊D₂₁	7
C₅²×C₇×C₃⋊D₃	7

Subgroups — C52×C7×C3⋊D21 #36348078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₇×C₃⋊D₂₁ #36348078