Subgroups — C₃⋊ΓL(1,𝔽₆₄) #11765744

Order 1134.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4428	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃⋊D₃×ΓL(1,𝔽₈)	—	3	1	no	maximal
3	C₇×C₃⋊D₉	C₃	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₆₄)	—	3	3	no	maximal
5	C₃²×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic
6	C₃×C₇×C₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃×C₇⋊C₉	—	2	1	no	—
8	#4428	—	2	3	no	—
9	#4428	D₃	1	3	yes	complemented
10	C₃⋊AGL(1,ℤ/9)	—	7	1	no	maximal, Hall
11	C₇×C₃⋊D₃	—	3	1	no	—
12	D₃×ΓL(1,𝔽₈)	—	9	3	no	—
13	D₃×ΓL(1,𝔽₈)	—	9	1	no	—
14	C₇×D₉	—	3	3	no	—
15	C₃×3₋¹⁺²	—	7	1	no	Sylow
16	C₃²×C₇	C₃×D₃	1	1	yes	characteristic
17	C₃×ΓL(1,𝔽₈)	—	2	1	no	—
18	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
19	C₃×ΓL(1,𝔽₈)	C₃⋊D₃	1	1	yes	characteristic
20	C₇×C₉	C₃×D₃	1	3	yes	complemented
21	C₇⋊C₉	—	2	3	no	—
22	C₃×C₃⋊D₃	—	21	1	no	—
23	C₃⋊D₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	AGL(1,ℤ/9)	—	21	3	no	—
25	D₃×C₇	—	9	3	no	—
26	D₃×C₇	—	9	1	no	—
27	C₂×ΓL(1,𝔽₈)	—	27	1	no	—
28	C₃³	—	7	1	no	—
29	3₋¹⁺²	—	7	3	no	—
30	3₋¹⁺²	—	14	3	no	—
31	C₃×C₉	—	14	1	no	—
32	C₃×C₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
33	C₃×C₇	AGL(1,ℤ/9)	1	3	yes	complemented
34	C₃×C₇	C₃×C₃⋊D₃	1	1	yes	characteristic
35	ΓL(1,𝔽₈)	—	3	1	no	—
36	ΓL(1,𝔽₈)	—	6	1	no	—
37	C₃⋊D₃	—	3	1	no	—
38	C₃×D₃	—	63	1	no	—
39	C₃×D₃	—	63	3	no	—
40	D₉	—	3	3	no	—
41	C₂×C₇	—	27	1	no	Hall
42	C₃²	—	7	1	no	—
43	C₃²	—	14	1	no	—
44	C₃²	—	21	3	no	—
45	C₃²	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
46	C₉	—	14	3	no	—
47	C₉	D₃×ΓL(1,𝔽₈)	1	3	yes	complemented
48	C₇	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃	—	189	1	no	—
50	D₃	—	9	3	no	—
51	D₃	—	9	1	no	—
52	C₃	—	21	1	no	—
53	C₃	—	42	1	no	—
54	C₃	ΓL(1,𝔽₆₄)	1	3	yes	complemented
55	C₃	C₃⋊D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
56	C₂	—	27	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₇×C₉	C₂×C₃
Frattini	C₃	C₃⋊D₃×ΓL(1,𝔽₈)
socle	C₃²×C₇	C₃×D₃
LCS	G ▷ C₃×C₇×C₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×#4428	C₂	semidirect
3. C₇×C₃⋊D₉	C₃	semidirect
6. C₃×C₇×C₉	C₂×C₃	semidirect
9. #4428	D₃	semidirect
20. C₇×C₉	C₃×D₃	semidirect
23. C₃⋊D₉	ΓL(1,𝔽₈)	semidirect
32. C₃×C₉	C₂×ΓL(1,𝔽₈)	semidirect
33. C₃×C₇	AGL(1,ℤ/9)	semidirect
47. C₉	D₃×ΓL(1,𝔽₈)	semidirect
48. C₇	C₃⋊AGL(1,ℤ/9)	semidirect
54. C₃	ΓL(1,𝔽₆₄)	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4428	2
C₃⋊D₃×ΓL(1,𝔽₈)	3
C₇×C₃⋊D₉	3
ΓL(1,𝔽₆₄)	3
C₃⋊AGL(1,ℤ/9)	7

Subgroups — C3⋊ΓL(1,𝔽64) #11765744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊ΓL(1,𝔽₆₄) #11765744