Subgroups — C₂⁴×C₅×ΓL(1,𝔽₈) #13132973

Order 1680.

536 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×ΓL(1,𝔽₈)	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×C₅×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²×C₅×ΓL(1,𝔽₈)	C₂²	1	35	yes	direct product factor
4	C₂⁴×ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₅×C₇	C₂×C₃	1	15	yes	complemented
6	C₂⁴×C₃×C₅	—	7	1	no	maximal, Hall
7	C₂×C₅×ΓL(1,𝔽₈)	C₂³	1	15	yes	direct product factor
8	C₂³×ΓL(1,𝔽₈)	C₂×C₅	1	15	yes	direct product factor
9	C₂²×C₅×C₇	C₂²×C₃	1	35	yes	complemented
10	C₂³×C₃×C₅	—	7	15	no	—
11	C₂⁴×C₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
12	C₅×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic, direct product factor, Hall
13	C₂²×ΓL(1,𝔽₈)	C₂²×C₅	1	35	yes	direct product factor
14	C₂⁴×C₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₅×C₇	C₂³×C₃	1	15	yes	complemented
16	C₂²×C₃×C₅	—	7	35	no	—
17	C₂³×C₇	C₂×C₃×C₅	1	15	yes	complemented
18	C₂⁴×C₃	—	7	1	no	Hall
19	C₂×ΓL(1,𝔽₈)	C₂³×C₅	1	15	yes	direct product factor
20	C₂³×C₅	C₂×ΓL(1,𝔽₈)	1	15	yes	direct product factor
21	C₅×C₇	C₂⁴×C₃	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₅	—	7	15	no	—
23	C₂²×C₇	C₂²×C₃×C₅	1	35	yes	complemented
24	C₂³×C₃	—	7	15	no	—
25	ΓL(1,𝔽₈)	C₂⁴×C₅	1	1	yes	characteristic, direct product factor, Hall
26	C₂²×C₅	C₂²×ΓL(1,𝔽₈)	1	35	yes	direct product factor
27	C₂⁴	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
28	C₃×C₅	—	7	1	no	Hall
29	C₂×C₇	C₂³×C₃×C₅	1	15	yes	complemented
30	C₂²×C₃	—	7	35	no	—
31	C₂×C₅	C₂³×ΓL(1,𝔽₈)	1	15	yes	direct product factor
32	C₂³	C₂×C₅×ΓL(1,𝔽₈)	1	15	yes	direct product factor
33	C₇	C₂⁴×C₃×C₅	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃	—	7	15	no	—
35	C₅	C₂⁴×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²	C₂²×C₅×ΓL(1,𝔽₈)	1	35	yes	direct product factor
37	C₃	—	7	1	no	Sylow
38	C₂	C₂³×C₅×ΓL(1,𝔽₈)	1	15	yes	direct product factor

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

Complements

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

N	H ≅ G/N	type
1. C₂³×C₅×ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂⁴×C₅×C₇	C₃	semidirect
3. C₂²×C₅×ΓL(1,𝔽₈)	C₂²	direct (N × H)
4. C₂⁴×ΓL(1,𝔽₈)	C₅	direct (N × H)
5. C₂³×C₅×C₇	C₂×C₃	semidirect
7. C₂×C₅×ΓL(1,𝔽₈)	C₂³	direct (N × H)
8. C₂³×ΓL(1,𝔽₈)	C₂×C₅	direct (N × H)
9. C₂²×C₅×C₇	C₂²×C₃	semidirect
11. C₂⁴×C₇	C₃×C₅	semidirect
12. C₅×ΓL(1,𝔽₈)	C₂⁴	direct (N × H)
13. C₂²×ΓL(1,𝔽₈)	C₂²×C₅	direct (N × H)
14. C₂⁴×C₅	ΓL(1,𝔽₈)	direct (N × H)
15. C₂×C₅×C₇	C₂³×C₃	semidirect
17. C₂³×C₇	C₂×C₃×C₅	semidirect
19. C₂×ΓL(1,𝔽₈)	C₂³×C₅	direct (N × H)
20. C₂³×C₅	C₂×ΓL(1,𝔽₈)	direct (N × H)
21. C₅×C₇	C₂⁴×C₃	semidirect
23. C₂²×C₇	C₂²×C₃×C₅	semidirect
25. ΓL(1,𝔽₈)	C₂⁴×C₅	direct (N × H)
26. C₂²×C₅	C₂²×ΓL(1,𝔽₈)	direct (N × H)
27. C₂⁴	C₅×ΓL(1,𝔽₈)	direct (N × H)
29. C₂×C₇	C₂³×C₃×C₅	semidirect
31. C₂×C₅	C₂³×ΓL(1,𝔽₈)	direct (N × H)
32. C₂³	C₂×C₅×ΓL(1,𝔽₈)	direct (N × H)
33. C₇	C₂⁴×C₃×C₅	semidirect
35. C₅	C₂⁴×ΓL(1,𝔽₈)	direct (N × H)
36. C₂²	C₂²×C₅×ΓL(1,𝔽₈)	direct (N × H)
38. C₂	C₂³×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
C₂³×C₅×ΓL(1,𝔽₈)	2
C₂⁴×C₅×C₇	3
C₂⁴×ΓL(1,𝔽₈)	5
C₂⁴×C₃×C₅	7

Subgroups — C24×C5×ΓL(1,𝔽8) #13132973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₅×ΓL(1,𝔽₈) #13132973