Subgroups — C₂×C₁₃×ΓL(1,𝔽₈)×C₅₉ #44567249

Order 32214.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×ΓL(1,𝔽₈)×C₅₉	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₇×C₁₃×C₅₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₃×C₅₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₁₃×C₅₉	—	7	1	no	maximal, Hall
5	C₂×ΓL(1,𝔽₈)×C₅₉	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₁₃×C₅₉	—	7	1	no	Hall
7	C₂×C₁₃×C₅₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
8	ΓL(1,𝔽₈)×C₅₉	C₂×C₁₃	1	1	yes	characteristic, direct product factor, Hall
9	C₂×C₇×C₅₉	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
10	C₁₃×C₅₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₁₃×ΓL(1,𝔽₈)	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₇×C₅₉	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×C₅₉	—	7	1	no	Hall
14	C₁₃×ΓL(1,𝔽₈)	C₂×C₅₉	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₇×C₁₃	C₃×C₅₉	1	1	yes	characteristic, complemented, Hall
16	C₃×C₅₉	—	7	1	no	Hall
17	C₂×C₅₉	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
18	C₇×C₁₃	C₂×C₃×C₅₉	1	1	yes	characteristic, complemented, Hall
19	C₂×C₃×C₁₃	—	7	1	no	Hall
20	C₅₉	C₂×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
21	C₂×ΓL(1,𝔽₈)	C₁₃×C₅₉	1	1	yes	characteristic, direct product factor, Hall
22	C₃×C₁₃	—	7	1	no	Hall
23	C₂×C₁₃	ΓL(1,𝔽₈)×C₅₉	1	1	yes	characteristic, direct product factor, Hall
24	ΓL(1,𝔽₈)	C₂×C₁₃×C₅₉	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₇	C₃×C₁₃×C₅₉	1	1	yes	characteristic, complemented, Hall
26	C₁₃	C₂×ΓL(1,𝔽₈)×C₅₉	1	1	yes	characteristic, direct product factor, Sylow
27	C₇	C₂×C₃×C₁₃×C₅₉	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃	—	7	1	no	Hall
29	C₃	—	7	1	no	Sylow
30	C₂	C₁₃×ΓL(1,𝔽₈)×C₅₉	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₁₃×C₅₉	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₁₃×C₅₉
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₃×C₅₉	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₃×C₅₉

Complements

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

N	H ≅ G/N	type
1. C₁₃×ΓL(1,𝔽₈)×C₅₉	C₂	direct (N × H)
2. C₂×C₇×C₁₃×C₅₉	C₃	semidirect
3. C₇×C₁₃×C₅₉	C₂×C₃	semidirect
5. C₂×ΓL(1,𝔽₈)×C₅₉	C₁₃	direct (N × H)
7. C₂×C₁₃×C₅₉	ΓL(1,𝔽₈)	direct (N × H)
8. ΓL(1,𝔽₈)×C₅₉	C₂×C₁₃	direct (N × H)
9. C₂×C₇×C₅₉	C₃×C₁₃	semidirect
10. C₁₃×C₅₉	C₂×ΓL(1,𝔽₈)	direct (N × H)
11. C₂×C₁₃×ΓL(1,𝔽₈)	C₅₉	direct (N × H)
12. C₇×C₅₉	C₂×C₃×C₁₃	semidirect
14. C₁₃×ΓL(1,𝔽₈)	C₂×C₅₉	direct (N × H)
15. C₂×C₇×C₁₃	C₃×C₅₉	semidirect
17. C₂×C₅₉	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
18. C₇×C₁₃	C₂×C₃×C₅₉	semidirect
20. C₅₉	C₂×C₁₃×ΓL(1,𝔽₈)	direct (N × H)
21. C₂×ΓL(1,𝔽₈)	C₁₃×C₅₉	direct (N × H)
23. C₂×C₁₃	ΓL(1,𝔽₈)×C₅₉	direct (N × H)
24. ΓL(1,𝔽₈)	C₂×C₁₃×C₅₉	direct (N × H)
25. C₂×C₇	C₃×C₁₃×C₅₉	semidirect
26. C₁₃	C₂×ΓL(1,𝔽₈)×C₅₉	direct (N × H)
27. C₇	C₂×C₃×C₁₃×C₅₉	semidirect
30. C₂	C₁₃×ΓL(1,𝔽₈)×C₅₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×ΓL(1,𝔽₈)×C₅₉	2
C₂×C₇×C₁₃×C₅₉	3
C₂×C₃×C₁₃×C₅₉	7
C₂×ΓL(1,𝔽₈)×C₅₉	13
C₂×C₁₃×ΓL(1,𝔽₈)	59

Subgroups — C2×C13×ΓL(1,𝔽8)×C59 #44567249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×ΓL(1,𝔽₈)×C₅₉ #44567249