Subgroups — D₅×AΓL(1,𝔽₈) #13132913

Order 1680.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×AΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×AGL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×AΓL(1,𝔽₈)	—	5	1	no	maximal, Hall
4	C₅×AGL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic, complemented
5	C₂×D₅×A₄	—	7	1	no	maximal, Hall
6	D₅×ΓL(1,𝔽₈)	—	8	1	no	maximal
7	AΓL(1,𝔽₈)	D₅	1	1	yes	characteristic, direct product factor
8	C₂×C₅×A₄	—	7	1	no	—
9	D₅×A₄	—	7	1	no	—
10	D₅×A₄	—	7	1	no	—
11	C₂×AGL(1,𝔽₈)	—	5	1	no	Hall
12	C₅×ΓL(1,𝔽₈)	—	8	1	no	Hall
13	C₂³×D₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
14	C₇×D₅	—	8	1	no	—
15	C₂×C₃×D₅	—	28	1	no	—
16	C₅×A₄	—	7	1	no	—
17	AGL(1,𝔽₈)	C₃×D₅	1	1	yes	characteristic, complemented
18	C₂²×A₄	—	35	1	no	Hall
19	C₂×ΓL(1,𝔽₈)	—	40	1	no	—
20	C₂³×C₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₂²×D₅	—	7	1	no	—
22	C₂²×D₅	—	7	1	no	—
23	C₅×C₇	—	8	1	no	Hall
24	C₂×C₃×C₅	—	28	1	no	—
25	C₃×D₅	—	28	1	no	—
26	C₃×D₅	—	28	1	no	—
27	C₂×A₄	—	7	1	no	—
28	C₂×A₄	—	35	1	no	—
29	C₂×A₄	—	35	1	no	—
30	ΓL(1,𝔽₈)	—	8	1	no	Hall
31	C₂²×C₅	—	7	1	no	—
32	C₂×D₅	—	7	1	no	—
33	C₂×D₅	—	21	1	no	—
34	C₂⁴	—	5	1	no	Sylow
35	C₃×C₅	—	28	1	no	Hall
36	C₂×C₇	—	40	1	no	—
37	C₂²×C₃	—	140	1	no	—
38	A₄	—	7	1	no	—
39	C₂×C₅	—	7	1	no	—
40	D₅	—	7	1	no	—
41	D₅	AΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
42	C₂³	—	35	1	no	—
43	C₂³	—	35	1	no	—
44	C₂³	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
45	C₇	—	8	1	no	Sylow
46	C₂×C₃	—	28	1	no	—
47	C₂×C₃	—	140	1	no	—
48	C₂×C₃	—	140	1	no	—
49	C₅	C₂×AΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	7	1	no	—
51	C₂²	—	35	1	no	—
52	C₂²	—	105	1	no	—
53	C₃	—	28	1	no	Sylow
54	C₂	—	5	1	no	—
55	C₂	—	7	1	no	—
56	C₂	—	35	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₅×AΓL(1,𝔽₈)	C₂	semidirect
2. D₅×AGL(1,𝔽₈)	C₃	semidirect
4. C₅×AGL(1,𝔽₈)	C₂×C₃	semidirect
7. AΓL(1,𝔽₈)	D₅	direct (N × H)
13. C₂³×D₅	ΓL(1,𝔽₈)	semidirect
17. AGL(1,𝔽₈)	C₃×D₅	semidirect
20. C₂³×C₅	C₂×ΓL(1,𝔽₈)	semidirect
41. D₅	AΓL(1,𝔽₈)	direct (N × H)
44. C₂³	D₅×ΓL(1,𝔽₈)	semidirect
49. C₅	C₂×AΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
C₅×AΓL(1,𝔽₈)	2
D₅×AGL(1,𝔽₈)	3
C₂×AΓL(1,𝔽₈)	5
C₂×D₅×A₄	7
D₅×ΓL(1,𝔽₈)	8

Subgroups — D5×AΓL(1,𝔽8) #13132913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×AΓL(1,𝔽₈) #13132913