Subgroups — C₅×ΓL(1,𝔽₈)⋊C₁₆ #13131981

Order 1680.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₅×C₇⋊C₁₆	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄×C₅×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
4	ΓL(1,𝔽₈)⋊C₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×C₇×C₈	C₂×C₃	1	1	yes	characteristic
6	C₃×C₅×C₁₆	—	7	1	no	maximal, Hall
7	C₂×C₅×ΓL(1,𝔽₈)	C₈	1	1	yes	characteristic
8	C₈×ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic
9	C₄×C₅×C₇	C₃×C₄	1	1	yes	characteristic
10	C₃×C₅×C₈	—	7	1	no	—
11	C₇⋊C₁₆	C₃×C₅	1	1	yes	characteristic, complemented, Hall
12	C₅×ΓL(1,𝔽₈)	C₁₆	1	1	yes	characteristic, complemented, Hall
13	C₄×ΓL(1,𝔽₈)	C₄×C₅	1	1	yes	characteristic
14	C₅×C₁₆	—	7	1	no	Hall
15	C₂×C₅×C₇	C₃×C₈	1	1	yes	characteristic
16	C₃×C₄×C₅	—	7	1	no	—
17	C₇×C₈	C₂×C₃×C₅	1	1	yes	characteristic
18	C₃×C₁₆	—	7	1	no	Hall
19	C₂×ΓL(1,𝔽₈)	C₅×C₈	1	1	yes	characteristic
20	C₅×C₈	AGL(1,𝔽₇)	1	1	yes	characteristic
21	C₅×C₇	C₃×C₁₆	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₅	—	7	1	no	—
23	C₄×C₇	C₃×C₄×C₅	1	1	yes	characteristic
24	C₃×C₈	—	7	1	no	—
25	ΓL(1,𝔽₈)	C₅×C₁₆	1	1	yes	characteristic, complemented, Hall
26	C₄×C₅	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
27	C₁₆	—	7	1	no	Sylow
28	C₃×C₅	—	7	1	no	Hall
29	C₂×C₇	C₃×C₅×C₈	1	1	yes	characteristic
30	C₃×C₄	—	7	1	no	—
31	C₂×C₅	ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic
32	C₈	C₅×AGL(1,𝔽₇)	1	1	yes	characteristic
33	C₇	C₃×C₅×C₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃	—	7	1	no	—
35	C₅	ΓL(1,𝔽₈)⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
36	C₄	C₅×ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic
37	C₃	—	7	1	no	Sylow
38	C₂	C₅×ΓL(1,𝔽₈)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₅×C₈	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₃×C₅×C₁₆
Frattini	C₈	C₅×AGL(1,𝔽₇)
Fitting	C₅×C₇×C₈	C₂×C₃
socle	C₂×C₅×C₇	C₃×C₈
prime radical	C₂×C₅×ΓL(1,𝔽₈)	C₈
cosocle	C₇×C₈	C₂×C₃×C₅
LCS	G ▷ C₇
UCS	C₁ ◁ C₅×C₈

Complements

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

N	H ≅ G/N	type
2. C₅×C₇⋊C₁₆	C₃	semidirect
4. ΓL(1,𝔽₈)⋊C₁₆	C₅	direct (N × H)
11. C₇⋊C₁₆	C₃×C₅	semidirect
12. C₅×ΓL(1,𝔽₈)	C₁₆	semidirect
21. C₅×C₇	C₃×C₁₆	semidirect
25. ΓL(1,𝔽₈)	C₅×C₁₆	semidirect
33. C₇	C₃×C₅×C₁₆	semidirect
35. C₅	ΓL(1,𝔽₈)⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₈×ΓL(1,𝔽₈)	2
C₅×C₇⋊C₁₆	3
ΓL(1,𝔽₈)⋊C₁₆	5
C₃×C₅×C₁₆	7

Subgroups — C5×ΓL(1,𝔽8)⋊C16 #13131981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×ΓL(1,𝔽₈)⋊C₁₆ #13131981