Subgroups — C₁₇×AGL(1,𝔽₇)×D₂₃ #45659261

Order 32844.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×ΓL(1,𝔽₈)⋊D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×C₂₃×AGL(1,𝔽₇)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×ΓL(1,𝔽₈)×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×C₁₇×D₂₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₁₇×ΓL(1,𝔽₈)×C₂₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₁₇×D₁₆₁	C₂×C₃	1	1	yes	characteristic, complemented
7	D₇×C₁₇×C₂₃	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×C₁₇×D₂₃	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×C₃×C₁₇×D₂₃	—	7	1	no	maximal, Hall
10	C₇×C₁₇×C₂₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
11	C₃×C₁₇×D₂₃	—	7	1	no	—
12	C₃×C₁₇×D₂₃	—	7	1	no	—
13	C₂×C₃×C₁₇×C₂₃	—	7	1	no	—
14	AGL(1,𝔽₇)×D₂₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₁₇×D₂₃	—	7	1	no	Hall
16	C₂×C₁₇×AGL(1,𝔽₇)	—	23	1	no	maximal, Hall
17	C₃×C₁₇×C₂₃	—	7	1	no	Hall
18	ΓL(1,𝔽₈)⋊D₂₃	C₂×C₁₇	1	1	yes	characteristic, complemented
19	C₂₃×AGL(1,𝔽₇)	C₂×C₁₇	1	1	yes	characteristic, complemented
20	ΓL(1,𝔽₈)×D₂₃	C₂×C₁₇	1	1	yes	characteristic, complemented
21	C₂×C₁₇×C₂₃	—	7	1	no	—
22	C₁₇×D₂₃	—	7	1	no	—
23	C₁₇×D₂₃	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
24	C₂×C₁₇×ΓL(1,𝔽₈)	—	23	1	no	—
25	C₁₇×AGL(1,𝔽₇)	—	23	1	no	—
26	C₁₇×AGL(1,𝔽₇)	D₂₃	1	1	yes	characteristic, direct product factor
27	D₇×D₂₃	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
28	ΓL(1,𝔽₈)×C₂₃	C₂²×C₁₇	1	1	yes	characteristic, complemented, Hall
29	C₂×D₇×C₁₇	—	23	1	no	Hall
30	C₁₇×C₂₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
31	C₁₇×ΓL(1,𝔽₈)	C₂×D₂₃	1	1	yes	characteristic, complemented, Hall
32	D₁₆₁	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
33	C₇×D₂₃	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
34	D₇×C₂₃	C₂×C₃×C₁₇	1	1	yes	characteristic, complemented
35	C₂×C₃×D₂₃	—	7	1	no	Hall
36	C₂×C₇×C₁₇	—	23	1	no	—
37	D₇×C₁₇	—	23	1	no	—
38	D₇×C₁₇	C₃×D₂₃	1	1	yes	characteristic, complemented
39	C₂²×C₃×C₁₇	—	161	1	no	Hall
40	C₇×C₂₃	C₂²×C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₂₃	—	7	1	no	—
42	C₃×D₂₃	—	7	1	no	—
43	C₃×D₂₃	—	7	1	no	—
44	C₇×C₁₇	C₂×C₃×D₂₃	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₁₇	—	7	1	no	—
46	C₂×C₃×C₁₇	—	161	1	no	—
47	C₂×C₃×C₁₇	—	161	1	no	—
48	C₂×D₂₃	—	7	1	no	Hall
49	C₂×AGL(1,𝔽₇)	—	23	1	no	Hall
50	C₃×C₂₃	—	7	1	no	Hall
51	C₂²×C₁₇	—	161	1	no	Hall
52	C₃×C₁₇	—	7	1	no	Hall
53	C₂×C₂₃	—	7	1	no	—
54	D₂₃	—	7	1	no	—
55	D₂₃	C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
56	C₂×ΓL(1,𝔽₈)	—	23	1	no	—
57	AGL(1,𝔽₇)	—	23	1	no	—
58	AGL(1,𝔽₇)	C₁₇×D₂₃	1	1	yes	characteristic, direct product factor
59	C₂×C₁₇	—	7	1	no	—
60	C₂×C₁₇	—	23	1	no	—
61	C₂×C₁₇	—	161	1	no	—
62	C₂×D₇	—	23	1	no	Hall
63	C₂₃	C₂×C₁₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
64	ΓL(1,𝔽₈)	C₂×C₁₇×D₂₃	1	1	yes	characteristic, complemented, Hall
65	C₁₇	AGL(1,𝔽₇)×D₂₃	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₇	—	23	1	no	—
67	D₇	—	23	1	no	—
68	D₇	C₃×C₁₇×D₂₃	1	1	yes	characteristic, complemented
69	C₂²×C₃	—	161	1	no	Hall
70	C₇	C₂×C₃×C₁₇×D₂₃	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	7	1	no	—
72	C₂×C₃	—	161	1	no	—
73	C₂×C₃	—	161	1	no	—
74	C₂²	—	161	1	no	Sylow
75	C₃	—	7	1	no	Sylow
76	C₂	—	7	1	no	—
77	C₂	—	23	1	no	—
78	C₂	—	161	1	no	—

	subgroup	G/H
center, hypercenter	C₁₇	AGL(1,𝔽₇)×D₂₃
commutator, cosocle, hypocenter	C₇×C₂₃	C₂²×C₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₂₃	C₂²×C₃
LCS	G ▷ C₇×C₂₃
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
1. C₁₇×ΓL(1,𝔽₈)⋊D₂₃	C₂	semidirect
2. C₁₇×C₂₃×AGL(1,𝔽₇)	C₂	semidirect
3. C₁₇×ΓL(1,𝔽₈)×D₂₃	C₂	semidirect
4. D₇×C₁₇×D₂₃	C₃	semidirect
5. C₁₇×ΓL(1,𝔽₈)×C₂₃	C₂²	semidirect
6. C₁₇×D₁₆₁	C₂×C₃	semidirect
7. D₇×C₁₇×C₂₃	C₂×C₃	semidirect
8. C₇×C₁₇×D₂₃	C₂×C₃	semidirect
10. C₇×C₁₇×C₂₃	C₂²×C₃	semidirect
14. AGL(1,𝔽₇)×D₂₃	C₁₇	direct (N × H)
18. ΓL(1,𝔽₈)⋊D₂₃	C₂×C₁₇	semidirect
19. C₂₃×AGL(1,𝔽₇)	C₂×C₁₇	semidirect
20. ΓL(1,𝔽₈)×D₂₃	C₂×C₁₇	semidirect
23. C₁₇×D₂₃	AGL(1,𝔽₇)	direct (N × H)
26. C₁₇×AGL(1,𝔽₇)	D₂₃	direct (N × H)
27. D₇×D₂₃	C₃×C₁₇	semidirect
28. ΓL(1,𝔽₈)×C₂₃	C₂²×C₁₇	semidirect
30. C₁₇×C₂₃	C₂×AGL(1,𝔽₇)	semidirect
31. C₁₇×ΓL(1,𝔽₈)	C₂×D₂₃	semidirect
32. D₁₆₁	C₂×C₃×C₁₇	semidirect
33. C₇×D₂₃	C₂×C₃×C₁₇	semidirect
34. D₇×C₂₃	C₂×C₃×C₁₇	semidirect
38. D₇×C₁₇	C₃×D₂₃	semidirect
40. C₇×C₂₃	C₂²×C₃×C₁₇	semidirect
44. C₇×C₁₇	C₂×C₃×D₂₃	semidirect
55. D₂₃	C₁₇×AGL(1,𝔽₇)	direct (N × H)
58. AGL(1,𝔽₇)	C₁₇×D₂₃	direct (N × H)
63. C₂₃	C₂×C₁₇×AGL(1,𝔽₇)	semidirect
64. ΓL(1,𝔽₈)	C₂×C₁₇×D₂₃	semidirect
65. C₁₇	AGL(1,𝔽₇)×D₂₃	direct (N × H)
68. D₇	C₃×C₁₇×D₂₃	semidirect
70. C₇	C₂×C₃×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×ΓL(1,𝔽₈)⋊D₂₃	2
C₁₇×C₂₃×AGL(1,𝔽₇)	2
C₁₇×ΓL(1,𝔽₈)×D₂₃	2
D₇×C₁₇×D₂₃	3
C₂×C₃×C₁₇×D₂₃	7
AGL(1,𝔽₇)×D₂₃	17
C₂×C₁₇×AGL(1,𝔽₇)	23

Subgroups — C17×AGL(1,𝔽7)×D23 #45659261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×AGL(1,𝔽₇)×D₂₃ #45659261