Subgroups — D₁₅×AGL(1,𝔽₈) #13132922

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, complemented
2	D₅×AGL(1,𝔽₈)	—	3	1	no	maximal, Hall
3	D₃×AGL(1,𝔽₈)	—	5	1	no	maximal, Hall
4	C₅×AGL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
5	C₂³×D₁₅	C₇	1	1	yes	maximal, characteristic, complemented, Hall
6	C₇×D₁₅	—	8	1	no	maximal
7	C₃×AGL(1,𝔽₈)	D₅	1	1	yes	characteristic, complemented
8	C₂³×C₃×C₅	C₂×C₇	1	1	yes	characteristic, complemented
9	C₂²×D₁₅	—	7	1	no	—
10	C₂²×D₁₅	—	7	1	no	—
11	C₂×AGL(1,𝔽₈)	—	15	1	no	Hall
12	C₃×C₅×C₇	—	8	1	no	Hall
13	C₂³×D₅	—	3	1	no	Hall
14	C₇×D₅	—	24	1	no	—
15	C₂²×C₃×C₅	—	7	1	no	—
16	C₂×D₁₅	—	7	1	no	—
17	C₂×D₁₅	—	7	3	no	—
18	AGL(1,𝔽₈)	D₁₅	1	1	yes	characteristic, direct product factor
19	C₂³×D₃	—	5	1	no	Hall
20	D₃×C₇	—	40	1	no	—
21	C₂³×C₅	D₃×C₇	1	1	yes	characteristic, complemented
22	C₂²×D₅	—	21	1	no	—
23	C₂²×D₅	—	21	1	no	—
24	C₅×C₇	—	8	1	no	Hall
25	C₂×C₃×C₅	—	7	1	no	—
26	D₁₅	—	7	1	no	—
27	D₁₅	AGL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
28	C₂³×C₃	C₇×D₅	1	1	yes	characteristic, complemented
29	C₂²×D₃	—	35	1	no	—
30	C₂²×D₃	—	35	1	no	—
31	C₃×C₇	—	8	1	no	Hall
32	C₂²×C₅	—	7	1	no	—
33	C₂×D₅	—	21	1	no	—
34	C₂×D₅	—	21	3	no	—
35	C₂⁴	—	15	1	no	Sylow
36	C₃×C₅	C₂×AGL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
37	C₂×C₇	—	120	1	no	—
38	C₂²×C₃	—	7	1	no	—
39	C₂×D₃	—	35	1	no	—
40	C₂×D₃	—	35	3	no	—
41	C₂×C₅	—	7	1	no	—
42	D₅	—	3	1	no	—
43	D₅	—	21	1	no	—
44	C₂³	—	105	1	no	—
45	C₂³	—	105	1	no	—
46	C₂³	C₇×D₁₅	1	1	yes	characteristic, complemented
47	C₇	—	8	1	no	Sylow
48	C₂×C₃	—	7	1	no	—
49	D₃	—	5	1	no	—
50	D₃	—	35	1	no	—
51	C₅	D₃×AGL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	7	1	no	—
53	C₂²	—	105	1	no	—
54	C₂²	—	105	3	no	—
55	C₃	D₅×AGL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	7	1	no	—
57	C₂	—	15	1	no	—
58	C₂	—	105	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂³×C₃×C₅	C₂×C₇
LCS	G ▷ C₂³×C₃×C₅
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×AGL(1,𝔽₈)	C₂	semidirect
4. C₅×AGL(1,𝔽₈)	D₃	semidirect
5. C₂³×D₁₅	C₇	semidirect
7. C₃×AGL(1,𝔽₈)	D₅	semidirect
8. C₂³×C₃×C₅	C₂×C₇	semidirect
18. AGL(1,𝔽₈)	D₁₅	direct (N × H)
21. C₂³×C₅	D₃×C₇	semidirect
27. D₁₅	AGL(1,𝔽₈)	direct (N × H)
28. C₂³×C₃	C₇×D₅	semidirect
36. C₃×C₅	C₂×AGL(1,𝔽₈)	semidirect
46. C₂³	C₇×D₁₅	semidirect
51. C₅	D₃×AGL(1,𝔽₈)	semidirect
55. C₃	D₅×AGL(1,𝔽₈)	semidirect

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
D₅×AGL(1,𝔽₈)	3
D₃×AGL(1,𝔽₈)	5
C₂³×D₁₅	7
C₇×D₁₅	8

Subgroups — D15×AGL(1,𝔽8) #13132922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₅×AGL(1,𝔽₈) #13132922