Subgroups — C₅×C₂₉×C₂²⋊ΓL(1,𝔽₈) #27131812

Order 12180.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₇×C₂₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×ΓL(1,𝔽₈)×C₂₉	—	4	1	no	maximal, Hall
3	C₂₉×C₂²⋊ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₅×C₇×C₂₉	—	3	1	no	—
5	C₅×A₄×C₂₉	—	7	1	no	maximal, Hall
6	C₅×C₇×C₂₉	A₄	1	1	yes	characteristic, complemented, Hall
7	C₂²×C₇×C₂₉	C₃×C₅	1	1	yes	characteristic, complemented, Hall
8	ΓL(1,𝔽₈)×C₂₉	—	4	1	no	Hall
9	C₂²×C₅×C₂₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
10	C₃×C₅×C₂₉	—	28	1	no	Hall
11	C₅×C₂²⋊ΓL(1,𝔽₈)	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₇×C₂₉	—	3	1	no	—
13	A₄×C₂₉	—	7	1	no	Hall
14	C₂×C₅×C₂₉	—	3	1	no	—
15	C₇×C₂₉	C₅×A₄	1	1	yes	characteristic, complemented, Hall
16	C₅×C₂₉	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
17	C₂²×C₅×C₇	C₃×C₂₉	1	1	yes	characteristic, complemented, Hall
18	C₂²×C₂₉	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
19	C₅×ΓL(1,𝔽₈)	—	4	1	no	Hall
20	C₃×C₂₉	—	28	1	no	Hall
21	C₂²⋊ΓL(1,𝔽₈)	C₅×C₂₉	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₅×C₇	—	3	1	no	—
23	C₅×A₄	—	7	1	no	Hall
24	C₂×C₂₉	—	3	1	no	—
25	C₅×C₇	A₄×C₂₉	1	1	yes	characteristic, complemented, Hall
26	C₂₉	C₅×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
27	C₂²×C₇	C₃×C₅×C₂₉	1	1	yes	characteristic, complemented, Hall
28	ΓL(1,𝔽₈)	—	4	1	no	Hall
29	C₂²×C₅	ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, complemented, Hall
30	C₃×C₅	—	28	1	no	Hall
31	C₂×C₇	—	3	1	no	—
32	A₄	—	7	1	no	Hall
33	C₂×C₅	—	3	1	no	—
34	C₇	C₅×A₄×C₂₉	1	1	yes	characteristic, complemented, Sylow
35	C₅	C₂₉×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²	C₅×ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, complemented, Sylow
37	C₃	—	28	1	no	Sylow
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅×C₂₉	C₂²⋊ΓL(1,𝔽₈)
commutator, hypocenter	C₂²×C₇	C₃×C₅×C₂₉
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₇×C₂₉	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₅×C₂₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×C₇×C₂₉	C₃	semidirect
3. C₂₉×C₂²⋊ΓL(1,𝔽₈)	C₅	direct (N × H)
6. C₅×C₇×C₂₉	A₄	semidirect
7. C₂²×C₇×C₂₉	C₃×C₅	semidirect
9. C₂²×C₅×C₂₉	ΓL(1,𝔽₈)	semidirect
11. C₅×C₂²⋊ΓL(1,𝔽₈)	C₂₉	direct (N × H)
15. C₇×C₂₉	C₅×A₄	semidirect
16. C₅×C₂₉	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
17. C₂²×C₅×C₇	C₃×C₂₉	semidirect
18. C₂²×C₂₉	C₅×ΓL(1,𝔽₈)	semidirect
21. C₂²⋊ΓL(1,𝔽₈)	C₅×C₂₉	direct (N × H)
25. C₅×C₇	A₄×C₂₉	semidirect
26. C₂₉	C₅×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
27. C₂²×C₇	C₃×C₅×C₂₉	semidirect
29. C₂²×C₅	ΓL(1,𝔽₈)×C₂₉	semidirect
34. C₇	C₅×A₄×C₂₉	semidirect
35. C₅	C₂₉×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
36. C₂²	C₅×ΓL(1,𝔽₈)×C₂₉	semidirect

Subgroups — C5×C29×C22⋊ΓL(1,𝔽8) #27131812

Conjugacy classes of subgroups

Complements

Subgroups — C₅×C₂₉×C₂²⋊ΓL(1,𝔽₈) #27131812