Subgroups — ΓL(1,𝔽₈)×D₄₅₅ #34062171

Order 19110.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×C₁₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₄₅₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	ΓL(1,𝔽₈)×D₉₁	—	5	1	no	maximal, Hall
4	C₅×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	ΓL(1,𝔽₈)×D₆₅	—	7	1	no	maximal
6	C₃×D₄₅₅	—	7	1	no	maximal
7	C₇×C₁₃×ΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented, Hall
8	ΓL(1,𝔽₈)×D₃₅	—	13	1	no	maximal, Hall
9	C₃×C₅×C₇×C₁₃	—	7	1	no	—
10	C₅×C₁₃×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic, complemented
11	C₇×D₉₁	—	5	1	no	Hall
12	D₄₅₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
13	C₇×D₆₅	—	7	1	no	—
14	C₅×C₇×ΓL(1,𝔽₈)	D₁₃	1	1	yes	characteristic, complemented, Hall
15	C₇²×C₁₃	C₃×D₅	1	1	yes	characteristic, complemented, Hall
16	C₃×D₉₁	—	35	1	no	—
17	ΓL(1,𝔽₈)×D₁₃	—	35	1	no	—
18	C₇×D₃₅	—	13	1	no	Hall
19	C₅×C₇×C₁₃	—	6	1	no	—
20	C₅×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₅×C₇×C₁₃	C₃×D₇	1	1	yes	characteristic, complemented
22	C₃×D₆₅	—	49	1	no	Hall
23	D₇×ΓL(1,𝔽₈)	—	65	1	no	Hall
24	C₃×C₇×C₁₃	—	7	1	no	—
25	C₁₃×ΓL(1,𝔽₈)	D₃₅	1	1	yes	characteristic, complemented
26	C₅×C₇²	C₃×D₁₃	1	1	yes	characteristic, complemented, Hall
27	C₃×D₃₅	—	91	1	no	—
28	D₅×ΓL(1,𝔽₈)	—	91	1	no	—
29	C₃×C₅×C₁₃	—	7	1	no	Hall
30	D₉₁	—	5	1	no	—
31	C₇×D₁₃	—	35	1	no	—
32	C₇×ΓL(1,𝔽₈)	D₆₅	1	1	yes	characteristic, complemented, Hall
33	D₆₅	—	7	1	no	Hall
34	C₃×C₅×C₇	—	7	1	no	—
35	C₅×ΓL(1,𝔽₈)	D₉₁	1	1	yes	characteristic, complemented
36	C₇×D₇	—	65	1	no	Hall
37	C₇×C₁₃	—	6	1	no	—
38	C₇×C₁₃	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
39	C₇×C₁₃	C₃×D₃₅	1	1	yes	characteristic, complemented
40	C₃×D₁₃	—	245	1	no	Hall
41	D₃₅	—	13	1	no	—
42	C₇×D₅	—	91	1	no	—
43	C₅×C₁₃	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
44	C₇²	C₃×D₆₅	1	1	yes	characteristic, complemented, Sylow
45	C₃×D₇	—	455	1	no	—
46	C₂×ΓL(1,𝔽₈)	—	455	1	no	—
47	C₃×C₁₃	—	7	1	no	Hall
48	C₅×C₇	—	6	1	no	—
49	C₅×C₇	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
50	C₅×C₇	C₃×D₉₁	1	1	yes	characteristic, complemented
51	C₃×D₅	—	637	1	no	Hall
52	D₁₃	—	35	1	no	Hall
53	C₃×C₇	—	7	1	no	—
54	ΓL(1,𝔽₈)	D₄₅₅	1	1	yes	characteristic, direct product factor
55	C₃×C₅	—	7	1	no	Hall
56	C₂×C₇	—	455	1	no	—
57	D₇	—	65	1	no	—
58	C₁₃	ΓL(1,𝔽₈)×D₃₅	1	1	yes	characteristic, complemented, Sylow
59	D₅	—	91	1	no	Hall
60	C₇	—	6	1	no	—
61	C₇	C₃×D₄₅₅	1	1	yes	characteristic, complemented
62	C₇	ΓL(1,𝔽₈)×D₆₅	1	1	yes	characteristic, complemented
63	C₂×C₃	—	3185	1	no	Hall
64	C₅	ΓL(1,𝔽₈)×D₉₁	1	1	yes	characteristic, complemented, Sylow
65	C₃	—	7	1	no	Sylow
66	C₂	—	455	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₅×C₇²×C₁₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×C₁₃×ΓL(1,𝔽₈)	C₂	semidirect
2. C₇×D₄₅₅	C₃	semidirect
4. C₅×C₇²×C₁₃	C₂×C₃	semidirect
7. C₇×C₁₃×ΓL(1,𝔽₈)	D₅	semidirect
10. C₅×C₁₃×ΓL(1,𝔽₈)	D₇	semidirect
12. D₄₅₅	ΓL(1,𝔽₈)	direct (N × H)
14. C₅×C₇×ΓL(1,𝔽₈)	D₁₃	semidirect
15. C₇²×C₁₃	C₃×D₅	semidirect
20. C₅×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	semidirect
21. C₅×C₇×C₁₃	C₃×D₇	semidirect
25. C₁₃×ΓL(1,𝔽₈)	D₃₅	semidirect
26. C₅×C₇²	C₃×D₁₃	semidirect
32. C₇×ΓL(1,𝔽₈)	D₆₅	semidirect
35. C₅×ΓL(1,𝔽₈)	D₉₁	semidirect
38. C₇×C₁₃	D₅×ΓL(1,𝔽₈)	semidirect
39. C₇×C₁₃	C₃×D₃₅	semidirect
43. C₅×C₁₃	D₇×ΓL(1,𝔽₈)	semidirect
44. C₇²	C₃×D₆₅	semidirect
49. C₅×C₇	ΓL(1,𝔽₈)×D₁₃	semidirect
50. C₅×C₇	C₃×D₉₁	semidirect
54. ΓL(1,𝔽₈)	D₄₅₅	direct (N × H)
58. C₁₃	ΓL(1,𝔽₈)×D₃₅	semidirect
61. C₇	C₃×D₄₅₅	semidirect
62. C₇	ΓL(1,𝔽₈)×D₆₅	semidirect
64. C₅	ΓL(1,𝔽₈)×D₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₇×C₁₃×ΓL(1,𝔽₈)	2
C₇×D₄₅₅	3
ΓL(1,𝔽₈)×D₉₁	5
ΓL(1,𝔽₈)×D₆₅	7
C₃×D₄₅₅	7
ΓL(1,𝔽₈)×D₃₅	13

Subgroups — ΓL(1,𝔽8)×D455 #34062171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₄₅₅ #34062171