Subgroups — C₅×D₂₃×D₅₁ #37553313

Order 23460.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₁₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₂₃×D₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×C₁₇×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₁₇×D₂₃	—	3	1	no	maximal, Hall
5	C₃×C₅×C₁₇×C₂₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₂₃×D₅₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₅×D₃₉₁	—	3	1	no	—
8	C₅×C₂₃×D₁₇	—	3	1	no	—
9	C₅×C₁₇×D₂₃	D₃	1	1	yes	characteristic, complemented
10	D₁₁₇₃	C₂×C₅	1	1	yes	characteristic, complemented
11	C₂₃×D₅₁	C₂×C₅	1	1	yes	characteristic, complemented
12	C₃×C₁₇×D₂₃	C₂×C₅	1	1	yes	characteristic, complemented
13	C₅×C₁₇×C₂₃	C₂×D₃	1	1	yes	characteristic, complemented, Hall
14	D₁₇×D₂₃	—	3	1	no	Hall
15	C₅×D₃×D₂₃	—	17	1	no	maximal, Hall
16	C₃×C₁₇×C₂₃	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
17	C₂×C₅×D₅₁	—	23	1	no	maximal, Hall
18	D₃₉₁	—	3	1	no	—
19	C₁₇×D₂₃	C₅×D₃	1	1	yes	characteristic, complemented
20	C₂₃×D₁₇	—	3	1	no	—
21	C₅×D₆₉	—	17	1	no	—
22	C₅×D₃×C₂₃	—	17	1	no	—
23	C₃×C₅×D₂₃	D₁₇	1	1	yes	characteristic, complemented
24	C₂×C₃×C₅×C₁₇	—	23	1	no	—
25	C₅×D₅₁	—	23	1	no	—
26	C₅×D₅₁	D₂₃	1	1	yes	characteristic, direct product factor
27	C₂×C₅×D₂₃	—	51	1	no	Hall
28	C₁₇×C₂₃	C₂×C₅×D₃	1	1	yes	characteristic, complemented, Hall
29	C₃×C₅×C₂₃	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
30	C₂×C₅×D₁₇	—	69	1	no	Hall
31	D₃×D₂₃	—	17	1	no	Hall
32	C₃×C₅×C₁₇	C₂×D₂₃	1	1	yes	characteristic, complemented, Hall
33	C₂×C₅×C₂₃	—	51	1	no	—
34	C₅×D₂₃	—	51	1	no	—
35	C₅×D₂₃	D₅₁	1	1	yes	characteristic, direct product factor
36	C₂×D₅₁	—	23	1	no	Hall
37	C₂×C₅×C₁₇	—	23	1	no	—
38	C₅×D₁₇	—	3	1	no	—
39	C₅×D₁₇	—	69	1	no	—
40	D₆₉	—	17	1	no	—
41	C₃×D₂₃	C₅×D₁₇	1	1	yes	characteristic, complemented
42	D₃×C₂₃	—	17	1	no	—
43	C₅×C₂₃	C₂×D₅₁	1	1	yes	characteristic, complemented, Hall
44	C₂×C₃×C₁₇	—	23	1	no	—
45	D₅₁	—	23	1	no	—
46	D₅₁	C₅×D₂₃	1	1	yes	characteristic, direct product factor
47	C₂×D₂₃	—	51	1	no	Hall
48	C₅×C₁₇	D₃×D₂₃	1	1	yes	characteristic, complemented, Hall
49	C₃×C₂₃	C₂×C₅×D₁₇	1	1	yes	characteristic, complemented, Hall
50	C₂×D₁₇	—	69	1	no	Hall
51	C₂×C₅×D₃	—	391	1	no	Hall
52	C₃×C₁₇	C₂×C₅×D₂₃	1	1	yes	characteristic, complemented, Hall
53	C₂×C₂₃	—	51	1	no	—
54	D₂₃	—	51	1	no	—
55	D₂₃	C₅×D₅₁	1	1	yes	characteristic, direct product factor
56	C₂×C₁₇	—	23	1	no	—
57	D₁₇	—	3	1	no	—
58	D₁₇	—	69	1	no	—
59	C₂×C₃×C₅	—	23	1	no	—
60	C₅×D₃	—	17	1	no	—
61	C₅×D₃	—	391	1	no	—
62	C₂₃	C₂×C₅×D₅₁	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₅	—	1173	1	no	Hall
64	C₁₇	C₅×D₃×D₂₃	1	1	yes	characteristic, complemented, Sylow
65	C₃×C₅	D₁₇×D₂₃	1	1	yes	characteristic, complemented, Hall
66	C₂×D₃	—	391	1	no	Hall
67	C₂×C₅	—	23	1	no	—
68	C₂×C₅	—	51	1	no	—
69	C₂×C₅	—	1173	1	no	—
70	C₂×C₃	—	23	1	no	—
71	D₃	—	17	1	no	—
72	D₃	—	391	1	no	—
73	C₅	D₂₃×D₅₁	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²	—	1173	1	no	Sylow
75	C₃	C₅×D₁₇×D₂₃	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	23	1	no	—
77	C₂	—	51	1	no	—
78	C₂	—	1173	1	no	—

	subgroup	G/H
center, hypercenter	C₅	D₂₃×D₅₁
commutator, cosocle, hypocenter	C₃×C₁₇×C₂₃	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₇×C₂₃	C₂²
LCS	G ▷ C₃×C₁₇×C₂₃
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₅×D₁₁₇₃	C₂	semidirect
2. C₅×C₂₃×D₅₁	C₂	semidirect
3. C₃×C₅×C₁₇×D₂₃	C₂	semidirect
5. C₃×C₅×C₁₇×C₂₃	C₂²	semidirect
6. D₂₃×D₅₁	C₅	direct (N × H)
9. C₅×C₁₇×D₂₃	D₃	semidirect
10. D₁₁₇₃	C₂×C₅	semidirect
11. C₂₃×D₅₁	C₂×C₅	semidirect
12. C₃×C₁₇×D₂₃	C₂×C₅	semidirect
13. C₅×C₁₇×C₂₃	C₂×D₃	semidirect
16. C₃×C₁₇×C₂₃	C₂²×C₅	semidirect
19. C₁₇×D₂₃	C₅×D₃	semidirect
23. C₃×C₅×D₂₃	D₁₇	semidirect
26. C₅×D₅₁	D₂₃	direct (N × H)
28. C₁₇×C₂₃	C₂×C₅×D₃	semidirect
29. C₃×C₅×C₂₃	C₂×D₁₇	semidirect
32. C₃×C₅×C₁₇	C₂×D₂₃	semidirect
35. C₅×D₂₃	D₅₁	direct (N × H)
41. C₃×D₂₃	C₅×D₁₇	semidirect
43. C₅×C₂₃	C₂×D₅₁	semidirect
46. D₅₁	C₅×D₂₃	direct (N × H)
48. C₅×C₁₇	D₃×D₂₃	semidirect
49. C₃×C₂₃	C₂×C₅×D₁₇	semidirect
52. C₃×C₁₇	C₂×C₅×D₂₃	semidirect
55. D₂₃	C₅×D₅₁	direct (N × H)
62. C₂₃	C₂×C₅×D₅₁	semidirect
64. C₁₇	C₅×D₃×D₂₃	semidirect
65. C₃×C₅	D₁₇×D₂₃	semidirect
73. C₅	D₂₃×D₅₁	direct (N × H)
75. C₃	C₅×D₁₇×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₁₁₇₃	2
C₅×C₂₃×D₅₁	2
C₃×C₅×C₁₇×D₂₃	2
C₅×D₁₇×D₂₃	3
D₂₃×D₅₁	5
C₅×D₃×D₂₃	17
C₂×C₅×D₅₁	23

Subgroups — C5×D23×D51 #37553313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₂₃×D₅₁ #37553313