Subgroups — C₅×C₁₃×ΓL(1,𝔽₈)×C₂₃ #44536032

Order 31395.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×C₁₃×C₂₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₃×ΓL(1,𝔽₈)×C₂₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₁₃×C₂₃	—	7	1	no	maximal, Hall
4	C₅×ΓL(1,𝔽₈)×C₂₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₇×C₁₃×C₂₃	C₃×C₅	1	1	yes	characteristic, complemented, Hall
6	C₅×C₁₃×C₂₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
7	C₅×C₁₃×ΓL(1,𝔽₈)	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₁₃×C₂₃	—	7	1	no	Hall
9	C₅×C₇×C₂₃	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
10	ΓL(1,𝔽₈)×C₂₃	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
11	C₅×C₇×C₁₃	C₃×C₂₃	1	1	yes	characteristic, complemented, Hall
12	C₃×C₅×C₂₃	—	7	1	no	Hall
13	C₁₃×C₂₃	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
14	C₁₃×ΓL(1,𝔽₈)	C₅×C₂₃	1	1	yes	characteristic, direct product factor, Hall
15	C₃×C₅×C₁₃	—	7	1	no	Hall
16	C₇×C₂₃	C₃×C₅×C₁₃	1	1	yes	characteristic, complemented, Hall
17	C₅×C₂₃	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
18	C₅×ΓL(1,𝔽₈)	C₁₃×C₂₃	1	1	yes	characteristic, direct product factor, Hall
19	C₇×C₁₃	C₃×C₅×C₂₃	1	1	yes	characteristic, complemented, Hall
20	C₃×C₂₃	—	7	1	no	Hall
21	C₅×C₁₃	ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, direct product factor, Hall
22	C₃×C₁₃	—	7	1	no	Hall
23	C₅×C₇	C₃×C₁₃×C₂₃	1	1	yes	characteristic, complemented, Hall
24	C₂₃	C₅×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
25	ΓL(1,𝔽₈)	C₅×C₁₃×C₂₃	1	1	yes	characteristic, direct product factor, Hall
26	C₃×C₅	—	7	1	no	Hall
27	C₁₃	C₅×ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
28	C₇	C₃×C₅×C₁₃×C₂₃	1	1	yes	characteristic, complemented, Sylow
29	C₅	C₁₃×ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
30	C₃	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₁₃×C₂₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₅×C₁₃×C₂₃
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₁₃×C₂₃	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₅×C₁₃×C₂₃

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×C₁₃×C₂₃	C₃	semidirect
2. C₁₃×ΓL(1,𝔽₈)×C₂₃	C₅	direct (N × H)
4. C₅×ΓL(1,𝔽₈)×C₂₃	C₁₃	direct (N × H)
5. C₇×C₁₃×C₂₃	C₃×C₅	semidirect
6. C₅×C₁₃×C₂₃	ΓL(1,𝔽₈)	direct (N × H)
7. C₅×C₁₃×ΓL(1,𝔽₈)	C₂₃	direct (N × H)
9. C₅×C₇×C₂₃	C₃×C₁₃	semidirect
10. ΓL(1,𝔽₈)×C₂₃	C₅×C₁₃	direct (N × H)
11. C₅×C₇×C₁₃	C₃×C₂₃	semidirect
13. C₁₃×C₂₃	C₅×ΓL(1,𝔽₈)	direct (N × H)
14. C₁₃×ΓL(1,𝔽₈)	C₅×C₂₃	direct (N × H)
16. C₇×C₂₃	C₃×C₅×C₁₃	semidirect
17. C₅×C₂₃	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
18. C₅×ΓL(1,𝔽₈)	C₁₃×C₂₃	direct (N × H)
19. C₇×C₁₃	C₃×C₅×C₂₃	semidirect
21. C₅×C₁₃	ΓL(1,𝔽₈)×C₂₃	direct (N × H)
23. C₅×C₇	C₃×C₁₃×C₂₃	semidirect
24. C₂₃	C₅×C₁₃×ΓL(1,𝔽₈)	direct (N × H)
25. ΓL(1,𝔽₈)	C₅×C₁₃×C₂₃	direct (N × H)
27. C₁₃	C₅×ΓL(1,𝔽₈)×C₂₃	direct (N × H)
28. C₇	C₃×C₅×C₁₃×C₂₃	semidirect
29. C₅	C₁₃×ΓL(1,𝔽₈)×C₂₃	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×C₇×C₁₃×C₂₃	3
C₁₃×ΓL(1,𝔽₈)×C₂₃	5
C₃×C₅×C₁₃×C₂₃	7
C₅×ΓL(1,𝔽₈)×C₂₃	13
C₅×C₁₃×ΓL(1,𝔽₈)	23

Subgroups — C5×C13×ΓL(1,𝔽8)×C23 #44536032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₁₃×ΓL(1,𝔽₈)×C₂₃ #44536032