Subgroups — C₂×C₃×C₅×AGL(1,𝔽₈) #13132947

Order 1680.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	AGL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂³	C₂×C₃×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃×C₅	C₇
LCS	G ▷ C₂³
UCS	C₁ ◁ C₂×C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×AGL(1,𝔽₈)	C₂	direct (N × H)
2. C₂×C₅×AGL(1,𝔽₈)	C₃	direct (N × H)
3. C₂×C₃×AGL(1,𝔽₈)	C₅	direct (N × H)
4. C₅×AGL(1,𝔽₈)	C₂×C₃	direct (N × H)
5. C₂⁴×C₃×C₅	C₇	semidirect
7. C₃×AGL(1,𝔽₈)	C₂×C₅	direct (N × H)
10. C₂³×C₃×C₅	C₂×C₇	semidirect
11. C₂×AGL(1,𝔽₈)	C₃×C₅	direct (N × H)
13. C₂⁴×C₅	C₃×C₇	semidirect
18. AGL(1,𝔽₈)	C₂×C₃×C₅	direct (N × H)
19. C₂⁴×C₃	C₅×C₇	semidirect
23. C₂³×C₅	C₂×C₃×C₇	semidirect
27. C₂×C₃×C₅	AGL(1,𝔽₈)	direct (N × H)
30. C₂³×C₃	C₂×C₅×C₇	semidirect
35. C₂⁴	C₃×C₅×C₇	semidirect
36. C₃×C₅	C₂×AGL(1,𝔽₈)	direct (N × H)
43. C₂×C₅	C₃×AGL(1,𝔽₈)	direct (N × H)
46. C₂³	C₂×C₃×C₅×C₇	semidirect
50. C₂×C₃	C₅×AGL(1,𝔽₈)	direct (N × H)
51. C₅	C₂×C₃×AGL(1,𝔽₈)	direct (N × H)
55. C₃	C₂×C₅×AGL(1,𝔽₈)	direct (N × H)
58. C₂	C₃×C₅×AGL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×AGL(1,𝔽₈)	2
C₂×C₅×AGL(1,𝔽₈)	3
C₂×C₃×AGL(1,𝔽₈)	5
C₂⁴×C₃×C₅	7
C₂×C₃×C₅×C₇	8

Subgroups — C2×C3×C5×AGL(1,𝔽8) #13132947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×AGL(1,𝔽₈) #13132947