Subgroups — C₃×C₁₇×ΓL(1,𝔽₈)×C₂₃ #37590149

Order 24633.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×ΓL(1,𝔽₈)×C₂₃	C₃	1	3	yes	maximal, direct product factor
2	C₃×C₇×C₁₇×C₂₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₁₇×C₂₃	—	7	1	no	maximal, Hall
4	C₇×C₁₇×C₂₃	C₃²	1	1	yes	characteristic, complemented, Hall
5	C₃×ΓL(1,𝔽₈)×C₂₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₁₇×C₂₃	—	7	3	no	—
7	C₃×C₁₇×C₂₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
8	C₃×C₁₇×ΓL(1,𝔽₈)	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₇×C₂₃	C₃×C₁₇	1	1	yes	characteristic, complemented
10	ΓL(1,𝔽₈)×C₂₃	C₃×C₁₇	1	3	yes	direct product factor
11	C₁₇×C₂₃	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
12	C₃×C₇×C₁₇	C₃×C₂₃	1	1	yes	characteristic, complemented
13	C₁₇×ΓL(1,𝔽₈)	C₃×C₂₃	1	3	yes	direct product factor
14	C₃²×C₂₃	—	7	1	no	Hall
15	C₇×C₂₃	C₃²×C₁₇	1	1	yes	characteristic, complemented, Hall
16	C₃²×C₁₇	—	7	1	no	Hall
17	C₇×C₁₇	C₃²×C₂₃	1	1	yes	characteristic, complemented, Hall
18	C₃×C₂₃	—	7	3	no	—
19	C₃×C₂₃	C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
20	C₃×ΓL(1,𝔽₈)	C₁₇×C₂₃	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₁₇	—	7	3	no	—
22	C₃×C₁₇	ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, direct product factor
23	C₂₃	C₃×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
24	C₃×C₇	C₃×C₁₇×C₂₃	1	1	yes	characteristic, complemented
25	ΓL(1,𝔽₈)	C₃×C₁₇×C₂₃	1	3	yes	direct product factor
26	C₁₇	C₃×ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
27	C₃²	—	7	1	no	Sylow
28	C₇	C₃²×C₁₇×C₂₃	1	1	yes	characteristic, complemented, Sylow
29	C₃	—	7	3	no	—
30	C₃	C₁₇×ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₃×C₁₇×C₂₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃²×C₁₇×C₂₃
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₇×C₂₃	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₁₇×C₂₃

Complements

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

N	H ≅ G/N	type
1. C₁₇×ΓL(1,𝔽₈)×C₂₃	C₃	direct (N × H)
2. C₃×C₇×C₁₇×C₂₃	C₃	semidirect
4. C₇×C₁₇×C₂₃	C₃²	semidirect
5. C₃×ΓL(1,𝔽₈)×C₂₃	C₁₇	direct (N × H)
7. C₃×C₁₇×C₂₃	ΓL(1,𝔽₈)	direct (N × H)
8. C₃×C₁₇×ΓL(1,𝔽₈)	C₂₃	direct (N × H)
9. C₃×C₇×C₂₃	C₃×C₁₇	semidirect
10. ΓL(1,𝔽₈)×C₂₃	C₃×C₁₇	direct (N × H)
11. C₁₇×C₂₃	C₃×ΓL(1,𝔽₈)	direct (N × H)
12. C₃×C₇×C₁₇	C₃×C₂₃	semidirect
13. C₁₇×ΓL(1,𝔽₈)	C₃×C₂₃	direct (N × H)
15. C₇×C₂₃	C₃²×C₁₇	semidirect
17. C₇×C₁₇	C₃²×C₂₃	semidirect
19. C₃×C₂₃	C₁₇×ΓL(1,𝔽₈)	direct (N × H)
20. C₃×ΓL(1,𝔽₈)	C₁₇×C₂₃	direct (N × H)
22. C₃×C₁₇	ΓL(1,𝔽₈)×C₂₃	direct (N × H)
23. C₂₃	C₃×C₁₇×ΓL(1,𝔽₈)	direct (N × H)
24. C₃×C₇	C₃×C₁₇×C₂₃	semidirect
25. ΓL(1,𝔽₈)	C₃×C₁₇×C₂₃	direct (N × H)
26. C₁₇	C₃×ΓL(1,𝔽₈)×C₂₃	direct (N × H)
28. C₇	C₃²×C₁₇×C₂₃	semidirect
30. C₃	C₁₇×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
C₁₇×ΓL(1,𝔽₈)×C₂₃	3
C₃×C₇×C₁₇×C₂₃	3
C₃²×C₁₇×C₂₃	7
C₃×ΓL(1,𝔽₈)×C₂₃	17
C₃×C₁₇×ΓL(1,𝔽₈)	23

Subgroups — C3×C17×ΓL(1,𝔽8)×C23 #37590149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₁₇×ΓL(1,𝔽₈)×C₂₃ #37590149