Subgroups — C₅×D₃×D₂₂₁ #27210563

Order 13260.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₆₆₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₃×C₁₃×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×D₂₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×D₂₂₁	—	3	1	no	maximal, Hall
5	C₃×C₅×C₁₃×C₁₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₃×D₂₂₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×D₂₂₁	—	3	1	no	—
8	C₅×D₂₂₁	D₃	1	1	yes	characteristic, direct product factor
9	C₂×C₅×C₁₃×C₁₇	—	3	1	no	—
10	D₆₆₃	C₂×C₅	1	1	yes	characteristic, complemented
11	D₃×C₁₃×C₁₇	C₂×C₅	1	1	yes	characteristic, complemented
12	C₃×D₂₂₁	C₂×C₅	1	1	yes	characteristic, complemented
13	C₅×C₁₃×C₁₇	C₂×D₃	1	1	yes	characteristic, complemented, Hall
14	C₅×D₃×D₁₇	—	13	1	no	maximal, Hall
15	C₂×D₂₂₁	—	3	1	no	Hall
16	C₅×D₃×D₁₃	—	17	1	no	maximal, Hall
17	C₃×C₁₃×C₁₇	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
18	C₅×D₅₁	—	13	1	no	—
19	C₅×D₃×C₁₇	D₁₃	1	1	yes	characteristic, complemented
20	C₃×C₅×D₁₇	—	13	1	no	—
21	C₂×C₁₃×C₁₇	—	3	1	no	—
22	D₂₂₁	—	3	1	no	—
23	D₂₂₁	C₅×D₃	1	1	yes	characteristic, direct product factor
24	C₅×D₃₉	—	17	1	no	—
25	C₅×D₃×C₁₃	D₁₇	1	1	yes	characteristic, complemented
26	C₃×C₅×D₁₃	—	17	1	no	—
27	C₂×C₅×D₁₇	—	39	1	no	Hall
28	C₂×C₅×D₁₃	—	51	1	no	Hall
29	C₃×C₅×C₁₇	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
30	C₁₃×C₁₇	C₂×C₅×D₃	1	1	yes	characteristic, complemented, Hall
31	D₃×D₁₇	—	13	1	no	Hall
32	C₃×C₅×C₁₃	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
33	C₂×C₅×C₁₇	—	3	1	no	—
34	C₅×D₁₇	—	13	1	no	—
35	C₅×D₁₇	—	39	1	no	—
36	D₃×D₁₃	—	17	1	no	Hall
37	C₂×C₅×C₁₃	—	3	1	no	—
38	C₅×D₁₃	—	17	1	no	—
39	C₅×D₁₃	—	51	1	no	—
40	D₅₁	—	13	1	no	—
41	C₃×D₁₇	—	13	1	no	—
42	D₃×C₁₇	C₅×D₁₃	1	1	yes	characteristic, complemented
43	C₅×C₁₇	D₃×D₁₃	1	1	yes	characteristic, complemented, Hall
44	D₃₉	—	17	1	no	—
45	C₃×D₁₃	—	17	1	no	—
46	D₃×C₁₃	C₅×D₁₇	1	1	yes	characteristic, complemented
47	C₂×D₁₇	—	39	1	no	Hall
48	C₅×C₁₃	D₃×D₁₇	1	1	yes	characteristic, complemented, Hall
49	C₂×C₅×D₃	—	221	1	no	Hall
50	C₂×D₁₃	—	51	1	no	Hall
51	C₃×C₁₇	C₂×C₅×D₁₃	1	1	yes	characteristic, complemented, Hall
52	C₃×C₁₃	C₂×C₅×D₁₇	1	1	yes	characteristic, complemented, Hall
53	C₂×C₁₇	—	3	1	no	—
54	D₁₇	—	13	1	no	—
55	D₁₇	—	39	1	no	—
56	C₂×C₃×C₅	—	221	1	no	—
57	C₅×D₃	—	221	1	no	—
58	C₅×D₃	D₂₂₁	1	1	yes	characteristic, direct product factor
59	C₂×C₁₃	—	3	1	no	—
60	D₁₃	—	17	1	no	—
61	D₁₃	—	51	1	no	—
62	C₂²×C₅	—	663	1	no	Hall
63	C₁₇	C₅×D₃×D₁₃	1	1	yes	characteristic, complemented, Sylow
64	C₃×C₅	C₂×D₂₂₁	1	1	yes	characteristic, complemented, Hall
65	C₁₃	C₅×D₃×D₁₇	1	1	yes	characteristic, complemented, Sylow
66	C₂×D₃	—	221	1	no	Hall
67	C₂×C₅	—	3	1	no	—
68	C₂×C₅	—	663	1	no	—
69	C₂×C₅	—	221	1	no	—
70	C₂×C₃	—	221	1	no	—
71	D₃	—	221	1	no	—
72	D₃	C₅×D₂₂₁	1	1	yes	characteristic, direct product factor
73	C₅	D₃×D₂₂₁	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²	—	663	1	no	Sylow
75	C₃	C₂×C₅×D₂₂₁	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	3	1	no	—
77	C₂	—	663	1	no	—
78	C₂	—	221	1	no	—

	subgroup	G/H
center, hypercenter	C₅	D₃×D₂₂₁
commutator, cosocle, hypocenter	C₃×C₁₃×C₁₇	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃×C₁₇	C₂²
LCS	G ▷ C₃×C₁₃×C₁₇
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₅×D₆₆₃	C₂	semidirect
2. C₅×D₃×C₁₃×C₁₇	C₂	semidirect
3. C₃×C₅×D₂₂₁	C₂	semidirect
5. C₃×C₅×C₁₃×C₁₇	C₂²	semidirect
6. D₃×D₂₂₁	C₅	direct (N × H)
8. C₅×D₂₂₁	D₃	direct (N × H)
10. D₆₆₃	C₂×C₅	semidirect
11. D₃×C₁₃×C₁₇	C₂×C₅	semidirect
12. C₃×D₂₂₁	C₂×C₅	semidirect
13. C₅×C₁₃×C₁₇	C₂×D₃	semidirect
17. C₃×C₁₃×C₁₇	C₂²×C₅	semidirect
19. C₅×D₃×C₁₇	D₁₃	semidirect
23. D₂₂₁	C₅×D₃	direct (N × H)
25. C₅×D₃×C₁₃	D₁₇	semidirect
29. C₃×C₅×C₁₇	C₂×D₁₃	semidirect
30. C₁₃×C₁₇	C₂×C₅×D₃	semidirect
32. C₃×C₅×C₁₃	C₂×D₁₇	semidirect
42. D₃×C₁₇	C₅×D₁₃	semidirect
43. C₅×C₁₇	D₃×D₁₃	semidirect
46. D₃×C₁₃	C₅×D₁₇	semidirect
48. C₅×C₁₃	D₃×D₁₇	semidirect
51. C₃×C₁₇	C₂×C₅×D₁₃	semidirect
52. C₃×C₁₃	C₂×C₅×D₁₇	semidirect
58. C₅×D₃	D₂₂₁	direct (N × H)
63. C₁₇	C₅×D₃×D₁₃	semidirect
64. C₃×C₅	C₂×D₂₂₁	semidirect
65. C₁₃	C₅×D₃×D₁₇	semidirect
72. D₃	C₅×D₂₂₁	direct (N × H)
73. C₅	D₃×D₂₂₁	direct (N × H)
75. C₃	C₂×C₅×D₂₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₆₆₃	2
C₅×D₃×C₁₃×C₁₇	2
C₃×C₅×D₂₂₁	2
C₂×C₅×D₂₂₁	3
D₃×D₂₂₁	5
C₅×D₃×D₁₇	13
C₅×D₃×D₁₃	17

Subgroups — C5×D3×D221 #27210563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₃×D₂₂₁ #27210563