Subgroups — C₂×C₃×ΓL(1,𝔽₈)×C₁₅₇ #34089989

Order 19782.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×ΓL(1,𝔽₈)×C₁₅₇	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×ΓL(1,𝔽₈)×C₁₅₇	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×C₇×C₁₅₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×C₁₅₇	C₂×C₃	1	1	yes	characteristic, complemented
5	ΓL(1,𝔽₈)×C₁₅₇	C₂×C₃	1	3	yes	direct product factor
6	C₂×C₃²×C₁₅₇	—	7	1	no	maximal, Hall
7	C₂×C₇×C₁₅₇	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃²×C₁₅₇	—	7	1	no	Hall
9	C₇×C₁₅₇	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
10	C₂×C₃×C₁₅₇	—	7	3	no	—
11	C₂×C₃×C₁₅₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
12	C₃×C₁₅₇	—	7	3	no	—
13	C₃×C₁₅₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
14	C₂×C₁₅₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
15	C₁₅₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
16	C₂×C₃×ΓL(1,𝔽₈)	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₃×ΓL(1,𝔽₈)	C₂×C₁₅₇	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₇	C₃×C₁₅₇	1	1	yes	characteristic, complemented
19	C₂×ΓL(1,𝔽₈)	C₃×C₁₅₇	1	3	yes	direct product factor
20	C₃×C₇	C₂×C₃×C₁₅₇	1	1	yes	characteristic, complemented
21	ΓL(1,𝔽₈)	C₂×C₃×C₁₅₇	1	3	yes	direct product factor
22	C₂×C₃²	—	7	1	no	Hall
23	C₂×C₇	C₃²×C₁₅₇	1	1	yes	characteristic, complemented, Hall
24	C₃²	—	7	1	no	Sylow
25	C₇	C₂×C₃²×C₁₅₇	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₃	—	7	3	no	—
27	C₂×C₃	ΓL(1,𝔽₈)×C₁₅₇	1	1	yes	characteristic, direct product factor
28	C₃	—	7	3	no	—
29	C₃	C₂×ΓL(1,𝔽₈)×C₁₅₇	1	1	yes	characteristic, direct product factor
30	C₂	C₃×ΓL(1,𝔽₈)×C₁₅₇	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₅₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃²×C₁₅₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇×C₁₅₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₁₅₇

Complements

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

N	H ≅ G/N	type
1. C₃×ΓL(1,𝔽₈)×C₁₅₇	C₂	direct (N × H)
2. C₂×ΓL(1,𝔽₈)×C₁₅₇	C₃	direct (N × H)
3. C₂×C₃×C₇×C₁₅₇	C₃	semidirect
4. C₃×C₇×C₁₅₇	C₂×C₃	semidirect
5. ΓL(1,𝔽₈)×C₁₅₇	C₂×C₃	direct (N × H)
7. C₂×C₇×C₁₅₇	C₃²	semidirect
9. C₇×C₁₅₇	C₂×C₃²	semidirect
11. C₂×C₃×C₁₅₇	ΓL(1,𝔽₈)	direct (N × H)
13. C₃×C₁₅₇	C₂×ΓL(1,𝔽₈)	direct (N × H)
14. C₂×C₁₅₇	C₃×ΓL(1,𝔽₈)	direct (N × H)
15. C₁₅₇	C₂×C₃×ΓL(1,𝔽₈)	direct (N × H)
16. C₂×C₃×ΓL(1,𝔽₈)	C₁₅₇	direct (N × H)
17. C₃×ΓL(1,𝔽₈)	C₂×C₁₅₇	direct (N × H)
18. C₂×C₃×C₇	C₃×C₁₅₇	semidirect
19. C₂×ΓL(1,𝔽₈)	C₃×C₁₅₇	direct (N × H)
20. C₃×C₇	C₂×C₃×C₁₅₇	semidirect
21. ΓL(1,𝔽₈)	C₂×C₃×C₁₅₇	direct (N × H)
23. C₂×C₇	C₃²×C₁₅₇	semidirect
25. C₇	C₂×C₃²×C₁₅₇	semidirect
27. C₂×C₃	ΓL(1,𝔽₈)×C₁₅₇	direct (N × H)
29. C₃	C₂×ΓL(1,𝔽₈)×C₁₅₇	direct (N × H)
30. C₂	C₃×ΓL(1,𝔽₈)×C₁₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×ΓL(1,𝔽₈)×C₁₅₇	2
C₂×ΓL(1,𝔽₈)×C₁₅₇	3
C₂×C₃×C₇×C₁₅₇	3
C₂×C₃²×C₁₅₇	7
C₂×C₃×ΓL(1,𝔽₈)	157

Subgroups — C2×C3×ΓL(1,𝔽8)×C157 #34089989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×ΓL(1,𝔽₈)×C₁₅₇ #34089989