Subgroups — ΓL(1,𝔽₈)×D₁₇×D₂₃ #45659271

Order 32844.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₃₉₁	C₂	1	1	yes	maximal, characteristic, complemented
2	ΓL(1,𝔽₈)×C₂₃×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×ΓL(1,𝔽₈)×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₁₇×D₂₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₁₇×ΓL(1,𝔽₈)×C₂₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₇×D₃₉₁	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×C₂₃×D₁₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×C₁₇×D₂₃	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×D₁₇×D₂₃	—	7	1	no	maximal, Hall
10	C₇×C₁₇×C₂₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
11	C₃×D₃₉₁	—	7	1	no	—
12	C₃×C₂₃×D₁₇	—	7	1	no	—
13	C₃×C₁₇×D₂₃	—	7	1	no	—
14	C₂×ΓL(1,𝔽₈)×D₂₃	—	17	1	no	maximal, Hall
15	D₁₇×D₂₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
16	C₂×ΓL(1,𝔽₈)×D₁₇	—	23	1	no	maximal, Hall
17	C₃×C₁₇×C₂₃	—	7	1	no	Hall
18	C₂×ΓL(1,𝔽₈)×C₂₃	—	17	1	no	—
19	ΓL(1,𝔽₈)×D₂₃	—	17	1	no	—
20	ΓL(1,𝔽₈)×D₂₃	D₁₇	1	1	yes	characteristic, direct product factor
21	D₃₉₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₁₇×D₂₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₂₃×D₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₂×C₁₇×ΓL(1,𝔽₈)	—	23	1	no	—
25	ΓL(1,𝔽₈)×D₁₇	—	23	1	no	—
26	ΓL(1,𝔽₈)×D₁₇	D₂₃	1	1	yes	characteristic, direct product factor
27	C₂×C₇×D₂₃	—	17	1	no	Hall
28	ΓL(1,𝔽₈)×C₂₃	C₂×D₁₇	1	1	yes	characteristic, complemented, Hall
29	C₂×C₇×D₁₇	—	23	1	no	Hall
30	C₁₇×C₂₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
31	C₁₇×ΓL(1,𝔽₈)	C₂×D₂₃	1	1	yes	characteristic, complemented, Hall
32	C₂×C₇×C₂₃	—	17	1	no	—
33	C₇×D₂₃	—	17	1	no	—
34	C₇×D₂₃	C₃×D₁₇	1	1	yes	characteristic, complemented
35	C₂×C₃×D₂₃	—	119	1	no	Hall
36	C₂×C₇×C₁₇	—	23	1	no	—
37	C₇×D₁₇	—	23	1	no	—
38	C₇×D₁₇	C₃×D₂₃	1	1	yes	characteristic, complemented
39	C₂×C₃×D₁₇	—	161	1	no	Hall
40	C₇×C₂₃	C₂×C₃×D₁₇	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₂₃	—	119	1	no	—
42	C₃×D₂₃	—	7	1	no	—
43	C₃×D₂₃	—	119	1	no	—
44	C₇×C₁₇	C₂×C₃×D₂₃	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₁₇	—	161	1	no	—
46	C₃×D₁₇	—	7	1	no	—
47	C₃×D₁₇	—	161	1	no	—
48	C₂×D₂₃	—	17	1	no	Hall
49	C₂²×ΓL(1,𝔽₈)	—	391	1	no	Hall
50	C₃×C₂₃	—	7	1	no	Hall
51	C₂×D₁₇	—	23	1	no	Hall
52	C₃×C₁₇	—	7	1	no	Hall
53	C₂×C₂₃	—	17	1	no	—
54	D₂₃	—	17	1	no	—
55	D₂₃	ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, direct product factor
56	C₂×ΓL(1,𝔽₈)	—	17	1	no	—
57	C₂×ΓL(1,𝔽₈)	—	23	1	no	—
58	C₂×ΓL(1,𝔽₈)	—	391	1	no	—
59	C₂×C₁₇	—	23	1	no	—
60	D₁₇	—	23	1	no	—
61	D₁₇	ΓL(1,𝔽₈)×D₂₃	1	1	yes	characteristic, direct product factor
62	C₂²×C₇	—	391	1	no	Hall
63	C₂₃	C₂×ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, complemented, Sylow
64	ΓL(1,𝔽₈)	D₁₇×D₂₃	1	1	yes	characteristic, direct product factor, Hall
65	C₁₇	C₂×ΓL(1,𝔽₈)×D₂₃	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₇	—	17	1	no	—
67	C₂×C₇	—	23	1	no	—
68	C₂×C₇	—	391	1	no	—
69	C₂²×C₃	—	2737	1	no	Hall
70	C₇	C₃×D₁₇×D₂₃	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	—	119	1	no	—
72	C₂×C₃	—	161	1	no	—
73	C₂×C₃	—	2737	1	no	—
74	C₂²	—	391	1	no	Sylow
75	C₃	—	7	1	no	Sylow
76	C₂	—	17	1	no	—
77	C₂	—	23	1	no	—
78	C₂	—	391	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₇×C₂₃	C₂²×C₃
LCS	G ▷ C₇×C₁₇×C₂₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₃₉₁	C₂	semidirect
2. ΓL(1,𝔽₈)×C₂₃×D₁₇	C₂	semidirect
3. C₁₇×ΓL(1,𝔽₈)×D₂₃	C₂	semidirect
4. C₇×D₁₇×D₂₃	C₃	semidirect
5. C₁₇×ΓL(1,𝔽₈)×C₂₃	C₂²	semidirect
6. C₇×D₃₉₁	C₂×C₃	semidirect
7. C₇×C₂₃×D₁₇	C₂×C₃	semidirect
8. C₇×C₁₇×D₂₃	C₂×C₃	semidirect
10. C₇×C₁₇×C₂₃	C₂²×C₃	semidirect
15. D₁₇×D₂₃	ΓL(1,𝔽₈)	direct (N × H)
20. ΓL(1,𝔽₈)×D₂₃	D₁₇	direct (N × H)
21. D₃₉₁	C₂×ΓL(1,𝔽₈)	semidirect
22. C₁₇×D₂₃	C₂×ΓL(1,𝔽₈)	semidirect
23. C₂₃×D₁₇	C₂×ΓL(1,𝔽₈)	semidirect
26. ΓL(1,𝔽₈)×D₁₇	D₂₃	direct (N × H)
28. ΓL(1,𝔽₈)×C₂₃	C₂×D₁₇	semidirect
30. C₁₇×C₂₃	C₂²×ΓL(1,𝔽₈)	semidirect
31. C₁₇×ΓL(1,𝔽₈)	C₂×D₂₃	semidirect
34. C₇×D₂₃	C₃×D₁₇	semidirect
38. C₇×D₁₇	C₃×D₂₃	semidirect
40. C₇×C₂₃	C₂×C₃×D₁₇	semidirect
44. C₇×C₁₇	C₂×C₃×D₂₃	semidirect
55. D₂₃	ΓL(1,𝔽₈)×D₁₇	direct (N × H)
61. D₁₇	ΓL(1,𝔽₈)×D₂₃	direct (N × H)
63. C₂₃	C₂×ΓL(1,𝔽₈)×D₁₇	semidirect
64. ΓL(1,𝔽₈)	D₁₇×D₂₃	direct (N × H)
65. C₁₇	C₂×ΓL(1,𝔽₈)×D₂₃	semidirect
70. C₇	C₃×D₁₇×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×D₃₉₁	2
ΓL(1,𝔽₈)×C₂₃×D₁₇	2
C₁₇×ΓL(1,𝔽₈)×D₂₃	2
C₇×D₁₇×D₂₃	3
C₃×D₁₇×D₂₃	7
C₂×ΓL(1,𝔽₈)×D₂₃	17
C₂×ΓL(1,𝔽₈)×D₁₇	23

Subgroups — ΓL(1,𝔽8)×D17×D23 #45659271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₁₇×D₂₃ #45659271