Subgroups — C₃×ΓL(1,𝔽₈)×Dic₁₂₇ #44544736

Order 32004.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	ΓL(1,𝔽₈)×Dic₁₂₇	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₇×Dic₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×ΓL(1,𝔽₈)×C₁₂₇	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×ΓL(1,𝔽₈)×C₁₂₇	C₂×C₃	1	3	yes	—
6	C₂×C₃×C₇×C₁₂₇	C₂×C₃	1	1	yes	characteristic
7	C₃²×Dic₁₂₇	—	7	1	no	maximal, Hall
8	C₇×Dic₁₂₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇×C₁₂₇	C₃×C₄	1	1	yes	characteristic, complemented
10	ΓL(1,𝔽₈)×C₁₂₇	C₃×C₄	1	3	yes	complemented
11	C₂×C₃²×C₁₂₇	—	7	1	no	—
12	C₂×C₇×C₁₂₇	C₂×C₃²	1	1	yes	characteristic
13	C₃×Dic₁₂₇	—	7	3	no	—
14	C₃×Dic₁₂₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₃²×C₁₂₇	—	7	1	no	Hall
16	C₇×C₁₂₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₁₂₇	—	7	3	no	—
18	C₂×C₃×C₁₂₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
19	Dic₁₂₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
20	C₃×C₁₂₇	—	7	3	no	—
21	C₃×C₁₂₇	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₂×C₁₂₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
23	C₃×C₄×ΓL(1,𝔽₈)	—	127	1	no	maximal, Hall
24	C₁₂₇	C₃×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃×ΓL(1,𝔽₈)	D₁₂₇	1	1	yes	characteristic
26	C₃×C₄×C₇	—	127	1	no	—
27	C₄×ΓL(1,𝔽₈)	—	127	3	no	—
28	C₃×ΓL(1,𝔽₈)	Dic₁₂₇	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₃×C₇	C₃×D₁₂₇	1	1	yes	characteristic
30	C₂×ΓL(1,𝔽₈)	C₃×D₁₂₇	1	3	yes	—
31	C₃²×C₄	—	889	1	no	Hall
32	C₄×C₇	—	127	1	no	Hall
33	C₃×C₇	C₃×Dic₁₂₇	1	1	yes	characteristic, complemented
34	ΓL(1,𝔽₈)	C₃×Dic₁₂₇	1	3	yes	direct product factor
35	C₂×C₃²	—	7	1	no	—
36	C₂×C₇	C₃²×D₁₂₇	1	1	yes	characteristic
37	C₃×C₄	—	889	3	no	—
38	C₃×C₄	—	127	1	no	—
39	C₃²	—	7	1	no	Sylow
40	C₇	C₃²×Dic₁₂₇	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	7	3	no	—
42	C₂×C₃	ΓL(1,𝔽₈)×D₁₂₇	1	1	yes	characteristic
43	C₄	—	127	1	no	Sylow
44	C₃	—	7	3	no	—
45	C₃	ΓL(1,𝔽₈)×Dic₁₂₇	1	1	yes	characteristic, direct product factor
46	C₂	C₃×ΓL(1,𝔽₈)×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	ΓL(1,𝔽₈)×D₁₂₇
commutator, hypocenter	C₇×C₁₂₇	C₃²×C₄
Frattini	C₂	C₃×ΓL(1,𝔽₈)×D₁₂₇
Fitting, socle	C₂×C₃×C₇×C₁₂₇	C₂×C₃
prime radical	C₂×C₃×ΓL(1,𝔽₈)×C₁₂₇	C₂
cosocle	C₂×C₇×C₁₂₇	C₂×C₃²
LCS	G ▷ C₇×C₁₂₇
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
2. ΓL(1,𝔽₈)×Dic₁₂₇	C₃	direct (N × H)
3. C₃×C₇×Dic₁₂₇	C₃	semidirect
4. C₃×ΓL(1,𝔽₈)×C₁₂₇	C₄	semidirect
8. C₇×Dic₁₂₇	C₃²	semidirect
9. C₃×C₇×C₁₂₇	C₃×C₄	semidirect
10. ΓL(1,𝔽₈)×C₁₂₇	C₃×C₄	semidirect
14. C₃×Dic₁₂₇	ΓL(1,𝔽₈)	direct (N × H)
16. C₇×C₁₂₇	C₃²×C₄	semidirect
19. Dic₁₂₇	C₃×ΓL(1,𝔽₈)	direct (N × H)
21. C₃×C₁₂₇	C₄×ΓL(1,𝔽₈)	semidirect
24. C₁₂₇	C₃×C₄×ΓL(1,𝔽₈)	semidirect
28. C₃×ΓL(1,𝔽₈)	Dic₁₂₇	direct (N × H)
33. C₃×C₇	C₃×Dic₁₂₇	semidirect
34. ΓL(1,𝔽₈)	C₃×Dic₁₂₇	direct (N × H)
40. C₇	C₃²×Dic₁₂₇	semidirect
45. C₃	ΓL(1,𝔽₈)×Dic₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×ΓL(1,𝔽₈)×C₁₂₇	2
ΓL(1,𝔽₈)×Dic₁₂₇	3
C₃×C₇×Dic₁₂₇	3
C₃²×Dic₁₂₇	7
C₃×C₄×ΓL(1,𝔽₈)	127

Subgroups — C3×ΓL(1,𝔽8)×Dic127 #44544736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₈)×Dic₁₂₇ #44544736