Subgroups — ΓL(1,𝔽₈)×C₂₃×D₃₅ #46798949

Order 33810.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×ΓL(1,𝔽₈)×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₂₃×D₃₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	D₇×ΓL(1,𝔽₈)×C₂₃	—	5	1	no	maximal, Hall
4	C₅×C₇²×C₂₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	D₅×ΓL(1,𝔽₈)×C₂₃	—	7	1	no	maximal
6	C₃×C₂₃×D₃₅	—	7	1	no	maximal
7	C₇×ΓL(1,𝔽₈)×C₂₃	D₅	1	1	yes	characteristic, complemented, Hall
8	C₅×ΓL(1,𝔽₈)×C₂₃	D₇	1	1	yes	characteristic, complemented
9	C₃×C₅×C₇×C₂₃	—	7	1	no	—
10	C₇×D₇×C₂₃	—	5	1	no	Hall
11	C₂₃×D₃₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
12	C₇×D₅×C₂₃	—	7	1	no	—
13	ΓL(1,𝔽₈)×D₃₅	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₇²×C₂₃	C₃×D₅	1	1	yes	characteristic, complemented, Hall
15	C₃×D₇×C₂₃	—	35	1	no	—
16	C₂×ΓL(1,𝔽₈)×C₂₃	—	35	1	no	—
17	C₅×C₇×C₂₃	—	6	1	no	—
18	C₅×C₇×C₂₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
19	C₅×C₇×C₂₃	C₃×D₇	1	1	yes	characteristic, complemented
20	C₅×C₇×ΓL(1,𝔽₈)	C₂×C₂₃	1	1	yes	characteristic, complemented, Hall
21	C₃×D₅×C₂₃	—	49	1	no	Hall
22	C₇×D₃₅	C₃×C₂₃	1	1	yes	characteristic, complemented, Hall
23	C₃×C₇×C₂₃	—	7	1	no	—
24	ΓL(1,𝔽₈)×C₂₃	D₃₅	1	1	yes	characteristic, direct product factor
25	C₃×C₅×C₂₃	—	7	1	no	Hall
26	C₂×C₇×C₂₃	—	35	1	no	—
27	D₇×C₂₃	—	5	1	no	—
28	D₇×ΓL(1,𝔽₈)	—	5	1	no	Hall
29	C₅×C₇²	C₂×C₃×C₂₃	1	1	yes	characteristic, complemented, Hall
30	D₅×C₂₃	—	7	1	no	Hall
31	C₃×D₃₅	—	7	1	no	—
32	D₅×ΓL(1,𝔽₈)	—	7	1	no	—
33	C₇×C₂₃	—	6	1	no	—
34	C₇×C₂₃	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
35	C₇×C₂₃	C₃×D₃₅	1	1	yes	characteristic, complemented
36	C₇×ΓL(1,𝔽₈)	D₅×C₂₃	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₂₃	—	245	1	no	Hall
38	C₅×C₂₃	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
39	C₃×C₅×C₇	—	7	1	no	—
40	C₅×ΓL(1,𝔽₈)	D₇×C₂₃	1	1	yes	characteristic, complemented
41	C₇×D₇	—	5	1	no	Hall
42	D₃₅	ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, direct product factor
43	C₇×D₅	—	7	1	no	—
44	C₃×C₂₃	—	7	1	no	Hall
45	C₇²	C₃×D₅×C₂₃	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₂₃	—	35	1	no	Hall
47	C₃×D₇	—	35	1	no	—
48	C₂×ΓL(1,𝔽₈)	—	35	1	no	—
49	C₅×C₇	—	6	1	no	—
50	C₅×C₇	C₂×ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, complemented
51	C₅×C₇	C₃×D₇×C₂₃	1	1	yes	characteristic, complemented
52	C₃×D₅	—	49	1	no	Hall
53	C₂₃	ΓL(1,𝔽₈)×D₃₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₃×C₇	—	7	1	no	—
55	ΓL(1,𝔽₈)	C₂₃×D₃₅	1	1	yes	characteristic, direct product factor
56	C₃×C₅	—	7	1	no	Hall
57	C₂×C₇	—	35	1	no	—
58	D₇	—	5	1	no	—
59	D₅	—	7	1	no	Hall
60	C₇	—	6	1	no	—
61	C₇	C₃×C₂₃×D₃₅	1	1	yes	characteristic, complemented
62	C₇	D₅×ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, complemented
63	C₂×C₃	—	245	1	no	Hall
64	C₅	D₇×ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, complemented, Sylow
65	C₃	—	7	1	no	Sylow
66	C₂	—	35	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₂₃	ΓL(1,𝔽₈)×D₃₅
commutator, cosocle, hypocenter	C₅×C₇²	C₂×C₃×C₂₃
Frattini	C₁	G
Fitting, socle	C₅×C₇²×C₂₃	C₂×C₃
LCS	G ▷ C₅×C₇²
UCS	C₁ ◁ C₂₃

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×ΓL(1,𝔽₈)×C₂₃	C₂	semidirect
2. C₇×C₂₃×D₃₅	C₃	semidirect
4. C₅×C₇²×C₂₃	C₂×C₃	semidirect
7. C₇×ΓL(1,𝔽₈)×C₂₃	D₅	semidirect
8. C₅×ΓL(1,𝔽₈)×C₂₃	D₇	semidirect
11. C₂₃×D₃₅	ΓL(1,𝔽₈)	direct (N × H)
13. ΓL(1,𝔽₈)×D₃₅	C₂₃	direct (N × H)
14. C₇²×C₂₃	C₃×D₅	semidirect
18. C₅×C₇×C₂₃	C₂×ΓL(1,𝔽₈)	semidirect
19. C₅×C₇×C₂₃	C₃×D₇	semidirect
20. C₅×C₇×ΓL(1,𝔽₈)	C₂×C₂₃	semidirect
22. C₇×D₃₅	C₃×C₂₃	semidirect
24. ΓL(1,𝔽₈)×C₂₃	D₃₅	direct (N × H)
29. C₅×C₇²	C₂×C₃×C₂₃	semidirect
34. C₇×C₂₃	D₅×ΓL(1,𝔽₈)	semidirect
35. C₇×C₂₃	C₃×D₃₅	semidirect
36. C₇×ΓL(1,𝔽₈)	D₅×C₂₃	semidirect
38. C₅×C₂₃	D₇×ΓL(1,𝔽₈)	semidirect
40. C₅×ΓL(1,𝔽₈)	D₇×C₂₃	semidirect
42. D₃₅	ΓL(1,𝔽₈)×C₂₃	direct (N × H)
45. C₇²	C₃×D₅×C₂₃	semidirect
50. C₅×C₇	C₂×ΓL(1,𝔽₈)×C₂₃	semidirect
51. C₅×C₇	C₃×D₇×C₂₃	semidirect
53. C₂₃	ΓL(1,𝔽₈)×D₃₅	direct (N × H)
55. ΓL(1,𝔽₈)	C₂₃×D₃₅	direct (N × H)
61. C₇	C₃×C₂₃×D₃₅	semidirect
62. C₇	D₅×ΓL(1,𝔽₈)×C₂₃	semidirect
64. C₅	D₇×ΓL(1,𝔽₈)×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₇×ΓL(1,𝔽₈)×C₂₃	2
C₇×C₂₃×D₃₅	3
D₇×ΓL(1,𝔽₈)×C₂₃	5
D₅×ΓL(1,𝔽₈)×C₂₃	7
C₃×C₂₃×D₃₅	7
ΓL(1,𝔽₈)×D₃₅	23

Subgroups — ΓL(1,𝔽8)×C23×D35 #46798949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₂₃×D₃₅ #46798949