Subgroups — C₄²×C₅×ΓL(1,𝔽₈) #13132247

Order 1680.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal
2	C₄²×C₅×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²×C₅×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
4	C₄×C₅×ΓL(1,𝔽₈)	C₄	1	6	yes	direct product factor
5	C₄²×ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₄×C₅×C₇	C₂×C₃	1	3	yes	—
7	C₃×C₄²×C₅	—	7	1	no	maximal, Hall
8	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₄	1	3	yes	—
9	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₅	1	3	yes	—
10	C₂²×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
11	C₄×C₅×C₇	C₃×C₄	1	6	yes	complemented
12	C₂×C₃×C₄×C₅	—	7	3	no	—
13	C₄²×C₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
14	C₅×ΓL(1,𝔽₈)	C₄²	1	1	yes	characteristic, direct product factor, Hall
15	C₂²×ΓL(1,𝔽₈)	C₂²×C₅	1	1	yes	characteristic
16	C₄×ΓL(1,𝔽₈)	C₄×C₅	1	6	yes	direct product factor
17	C₄²×C₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₅×C₇	C₂×C₃×C₄	1	3	yes	—
19	C₂²×C₃×C₅	—	7	1	no	—
20	C₃×C₄×C₅	—	7	6	no	—
21	C₂×C₄×C₇	C₂×C₃×C₅	1	3	yes	—
22	C₃×C₄²	—	7	1	no	Hall
23	C₂×ΓL(1,𝔽₈)	C₂×C₄×C₅	1	3	yes	—
24	C₂×C₄×C₅	C₂×ΓL(1,𝔽₈)	1	3	yes	—
25	C₅×C₇	C₃×C₄²	1	1	yes	characteristic, complemented, Hall
26	C₂×C₃×C₅	—	7	3	no	—
27	C₂²×C₇	C₂²×C₃×C₅	1	1	yes	characteristic
28	C₄×C₇	C₃×C₄×C₅	1	6	yes	complemented
29	C₂×C₃×C₄	—	7	3	no	—
30	ΓL(1,𝔽₈)	C₄²×C₅	1	1	yes	characteristic, direct product factor, Hall
31	C₂²×C₅	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
32	C₄×C₅	C₄×ΓL(1,𝔽₈)	1	6	yes	direct product factor
33	C₄²	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
34	C₃×C₅	—	7	1	no	Hall
35	C₂×C₇	C₂×C₃×C₄×C₅	1	3	yes	—
36	C₂²×C₃	—	7	1	no	—
37	C₃×C₄	—	7	6	no	—
38	C₂×C₅	C₂×C₄×ΓL(1,𝔽₈)	1	3	yes	—
39	C₂×C₄	C₂×C₅×ΓL(1,𝔽₈)	1	3	yes	—
40	C₇	C₃×C₄²×C₅	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	7	3	no	—
42	C₅	C₄²×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²	C₂²×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
44	C₄	C₄×C₅×ΓL(1,𝔽₈)	1	6	yes	direct product factor
45	C₃	—	7	1	no	Sylow
46	C₂	C₂×C₄×C₅×ΓL(1,𝔽₈)	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²×C₅	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₃×C₄²×C₅
Frattini	C₂²	C₂²×C₅×ΓL(1,𝔽₈)
Fitting	C₄²×C₅×C₇	C₃
socle	C₂²×C₅×C₇	C₂²×C₃
prime radical	C₂²×C₅×ΓL(1,𝔽₈)	C₂²
cosocle	C₂²×C₇	C₂²×C₃×C₅
LCS	G ▷ C₇
UCS	C₁ ◁ C₄²×C₅

Complements

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

N	H ≅ G/N	type
2. C₄²×C₅×C₇	C₃	semidirect
4. C₄×C₅×ΓL(1,𝔽₈)	C₄	direct (N × H)
5. C₄²×ΓL(1,𝔽₈)	C₅	direct (N × H)
11. C₄×C₅×C₇	C₃×C₄	semidirect
13. C₄²×C₇	C₃×C₅	semidirect
14. C₅×ΓL(1,𝔽₈)	C₄²	direct (N × H)
16. C₄×ΓL(1,𝔽₈)	C₄×C₅	direct (N × H)
17. C₄²×C₅	ΓL(1,𝔽₈)	direct (N × H)
25. C₅×C₇	C₃×C₄²	semidirect
28. C₄×C₇	C₃×C₄×C₅	semidirect
30. ΓL(1,𝔽₈)	C₄²×C₅	direct (N × H)
32. C₄×C₅	C₄×ΓL(1,𝔽₈)	direct (N × H)
33. C₄²	C₅×ΓL(1,𝔽₈)	direct (N × H)
40. C₇	C₃×C₄²×C₅	semidirect
42. C₅	C₄²×ΓL(1,𝔽₈)	direct (N × H)
44. C₄	C₄×C₅×ΓL(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
C₂×C₄×C₅×ΓL(1,𝔽₈)	2
C₄²×C₅×C₇	3
C₄²×ΓL(1,𝔽₈)	5
C₃×C₄²×C₅	7

Subgroups — C42×C5×ΓL(1,𝔽8) #13132247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₅×ΓL(1,𝔽₈) #13132247