Subgroups — C₁₇×D₁₅⋊D₂₃ #37553337

Order 23460.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×C₂₃×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₁₇×D₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₁₇×D₁₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×C₁₇×D₂₃	—	3	1	no	maximal, Hall
5	C₃×C₅×C₁₇×C₂₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₃×C₁₇×D₂₃	—	5	1	no	maximal, Hall
7	C₁₇×D₁₁₅	D₃	1	1	yes	characteristic, complemented
8	D₅×C₁₇×C₂₃	—	3	1	no	—
9	C₅×C₁₇×D₂₃	—	3	1	no	—
10	C₁₇×D₆₉	D₅	1	1	yes	characteristic, complemented
11	D₃×C₁₇×C₂₃	—	5	1	no	—
12	C₃×C₁₇×D₂₃	—	5	1	no	—
13	C₅×C₁₇×C₂₃	C₂×D₃	1	1	yes	characteristic, complemented, Hall
14	C₂×C₁₇×D₂₃	—	15	1	no	Hall
15	D₁₅⋊D₂₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₃×C₁₇×C₂₃	C₂×D₅	1	1	yes	characteristic, complemented, Hall
17	D₃×D₅×C₁₇	—	23	1	no	maximal, Hall
18	C₂×C₁₇×C₂₃	—	15	1	no	—
19	C₁₇×D₂₃	—	3	1	no	—
20	C₁₇×D₂₃	—	5	1	no	—
21	C₂₃×D₁₅	C₂×C₁₇	1	1	yes	characteristic, complemented
22	C₅×D₆₉	C₂×C₁₇	1	1	yes	characteristic, complemented
23	C₃×D₁₁₅	C₂×C₁₇	1	1	yes	characteristic, complemented
24	C₁₇×D₁₅	D₂₃	1	1	yes	characteristic, complemented
25	C₅×D₃×C₁₇	—	23	1	no	—
26	C₃×D₅×C₁₇	—	23	1	no	—
27	D₅×D₂₃	—	3	1	no	Hall
28	C₁₇×C₂₃	D₃×D₅	1	1	yes	characteristic, complemented, Hall
29	C₃×C₅×C₂₃	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
30	C₂×D₅×C₁₇	—	69	1	no	Hall
31	D₃×D₂₃	—	5	1	no	Hall
32	C₃×C₅×C₁₇	C₂×D₂₃	1	1	yes	characteristic, complemented, Hall
33	D₁₁₅	D₃×C₁₇	1	1	yes	characteristic, complemented
34	C₅×D₂₃	—	3	1	no	—
35	D₅×C₂₃	—	3	1	no	—
36	C₂×D₃×C₁₇	—	115	1	no	Hall
37	C₂×C₅×C₁₇	—	69	1	no	—
38	D₅×C₁₇	—	3	1	no	—
39	D₅×C₁₇	—	23	1	no	—
40	D₆₉	D₅×C₁₇	1	1	yes	characteristic, complemented
41	C₃×D₂₃	—	5	1	no	—
42	D₃×C₂₃	—	5	1	no	—
43	C₅×C₂₃	C₂×D₃×C₁₇	1	1	yes	characteristic, complemented, Hall
44	C₂×C₃×C₁₇	—	115	1	no	—
45	D₃×C₁₇	—	5	1	no	—
46	D₃×C₁₇	—	23	1	no	—
47	C₂×D₂₃	—	15	1	no	Hall
48	C₅×C₁₇	D₃×D₂₃	1	1	yes	characteristic, complemented, Hall
49	C₃×C₂₃	C₂×D₅×C₁₇	1	1	yes	characteristic, complemented, Hall
50	C₂²×C₁₇	—	345	1	no	Hall
51	D₃×D₅	—	23	1	no	Hall
52	C₃×C₁₇	D₅×D₂₃	1	1	yes	characteristic, complemented, Hall
53	C₂×C₂₃	—	15	1	no	—
54	D₂₃	—	3	1	no	—
55	D₂₃	—	5	1	no	—
56	C₂×C₁₇	—	15	1	no	—
57	C₂×C₁₇	—	69	1	no	—
58	C₂×C₁₇	—	115	1	no	—
59	D₁₅	C₁₇×D₂₃	1	1	yes	characteristic, complemented
60	C₃×D₅	—	23	1	no	—
61	C₅×D₃	—	23	1	no	—
62	C₂₃	D₃×D₅×C₁₇	1	1	yes	characteristic, complemented, Sylow
63	C₂×D₅	—	69	1	no	Hall
64	C₁₇	D₁₅⋊D₂₃	1	1	yes	characteristic, direct product factor, Sylow
65	C₃×C₅	C₂×C₁₇×D₂₃	1	1	yes	characteristic, complemented, Hall
66	C₂×D₃	—	115	1	no	Hall
67	C₂×C₅	—	69	1	no	—
68	D₅	—	3	1	no	—
69	D₅	—	23	1	no	—
70	C₂×C₃	—	115	1	no	—
71	D₃	—	5	1	no	—
72	D₃	—	23	1	no	—
73	C₅	D₃×C₁₇×D₂₃	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	345	1	no	Sylow
75	C₃	D₅×C₁₇×D₂₃	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	15	1	no	—
77	C₂	—	69	1	no	—
78	C₂	—	115	1	no	—

	subgroup	G/H
center, hypercenter	C₁₇	D₁₅⋊D₂₃
commutator, cosocle, hypocenter	C₃×C₅×C₂₃	C₂²×C₁₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₇×C₂₃	C₂²
LCS	G ▷ C₃×C₅×C₂₃
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
1. C₁₇×C₂₃×D₁₅	C₂	semidirect
2. C₅×C₁₇×D₆₉	C₂	semidirect
3. C₃×C₁₇×D₁₁₅	C₂	semidirect
5. C₃×C₅×C₁₇×C₂₃	C₂²	semidirect
7. C₁₇×D₁₁₅	D₃	semidirect
10. C₁₇×D₆₉	D₅	semidirect
13. C₅×C₁₇×C₂₃	C₂×D₃	semidirect
15. D₁₅⋊D₂₃	C₁₇	direct (N × H)
16. C₃×C₁₇×C₂₃	C₂×D₅	semidirect
21. C₂₃×D₁₅	C₂×C₁₇	semidirect
22. C₅×D₆₉	C₂×C₁₇	semidirect
23. C₃×D₁₁₅	C₂×C₁₇	semidirect
24. C₁₇×D₁₅	D₂₃	semidirect
28. C₁₇×C₂₃	D₃×D₅	semidirect
29. C₃×C₅×C₂₃	C₂²×C₁₇	semidirect
32. C₃×C₅×C₁₇	C₂×D₂₃	semidirect
33. D₁₁₅	D₃×C₁₇	semidirect
40. D₆₉	D₅×C₁₇	semidirect
43. C₅×C₂₃	C₂×D₃×C₁₇	semidirect
48. C₅×C₁₇	D₃×D₂₃	semidirect
49. C₃×C₂₃	C₂×D₅×C₁₇	semidirect
52. C₃×C₁₇	D₅×D₂₃	semidirect
59. D₁₅	C₁₇×D₂₃	semidirect
62. C₂₃	D₃×D₅×C₁₇	semidirect
64. C₁₇	D₁₅⋊D₂₃	direct (N × H)
65. C₃×C₅	C₂×C₁₇×D₂₃	semidirect
73. C₅	D₃×C₁₇×D₂₃	semidirect
75. C₃	D₅×C₁₇×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×C₂₃×D₁₅	2
C₅×C₁₇×D₆₉	2
C₃×C₁₇×D₁₁₅	2
D₅×C₁₇×D₂₃	3
D₃×C₁₇×D₂₃	5
D₁₅⋊D₂₃	17
D₃×D₅×C₁₇	23

Subgroups — C17×D15⋊D23 #37553337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×D₁₅⋊D₂₃ #37553337