Subgroups — C₃⋊((C₃⋊D₃)⋊ΓL(1,𝔽₈)) #11765749

Order 1134.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₃⋊(C₃⋊D₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃⋊D₃×ΓL(1,𝔽₈)	—	3	1	no	maximal
4	(C₃⋊D₃)⋊ΓL(1,𝔽₈)	—	3	3	no	maximal
5	C₃³×C₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃²×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₃²×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, complemented
8	C₇⋊H(3,𝔽₃)	—	2	3	no	—
9	C₇⋊H(3,𝔽₃)	D₃	1	3	yes	complemented
10	C₃⋊((C₃⋊D₃)⋊C₃)	—	7	1	no	maximal, Hall
11	C₇×C₃⋊D₃	—	3	1	no	—
12	C₇×C₃⋊D₃	—	3	3	no	—
13	C₇×C₃⋊D₃	—	9	3	no	—
14	D₃×ΓL(1,𝔽₈)	—	9	3	no	—
15	D₃×ΓL(1,𝔽₈)	—	9	1	no	—
16	C₃×H(3,𝔽₃)	—	7	1	no	Sylow
17	C₃²×C₇	—	3	3	no	—
18	C₃²×C₇	C₃×D₃	1	1	yes	characteristic
19	C₃²×C₇	C₃×D₃	1	3	yes	complemented
20	C₃×ΓL(1,𝔽₈)	—	2	1	no	—
21	C₃×ΓL(1,𝔽₈)	—	2	3	no	—
22	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
23	C₃×ΓL(1,𝔽₈)	—	6	3	no	—
24	C₃×ΓL(1,𝔽₈)	C₃⋊D₃	1	1	yes	characteristic, complemented
25	C₃⋊(C₃⋊D₃)	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
26	C₃×C₃⋊D₃	—	21	1	no	—
27	(C₃⋊D₃)⋊C₃	—	21	3	no	—
28	D₃×C₇	—	9	3	no	—
29	D₃×C₇	—	9	1	no	—
30	D₃×C₇	—	27	3	no	—
31	C₂×ΓL(1,𝔽₈)	—	27	1	no	—
32	C₃³	—	7	1	no	—
33	C₃³	—	14	1	no	—
34	C₃³	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
35	H(3,𝔽₃)	—	7	3	no	—
36	H(3,𝔽₃)	—	14	3	no	—
37	C₃×C₇	—	3	3	no	—
38	C₃×C₇	(C₃⋊D₃)⋊C₃	1	3	yes	complemented
39	C₃×C₇	C₃×C₃⋊D₃	1	1	yes	characteristic
40	ΓL(1,𝔽₈)	—	3	1	no	—
41	ΓL(1,𝔽₈)	—	6	1	no	—
42	ΓL(1,𝔽₈)	—	6	3	no	—
43	C₃⋊D₃	—	3	1	no	—
44	C₃⋊D₃	—	3	3	no	—
45	C₃⋊D₃	—	9	3	no	—
46	C₃×D₃	—	63	1	no	—
47	C₃×D₃	—	63	3	no	—
48	C₂×C₇	—	27	1	no	Hall
49	C₃²	—	3	3	no	—
50	C₃²	—	7	1	no	—
51	C₃²	—	14	3	no	—
52	C₃²	—	14	1	no	—
53	C₃²	—	21	3	no	—
54	C₃²	—	42	3	no	—
55	C₃²	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
56	C₃²	D₃×ΓL(1,𝔽₈)	1	3	yes	complemented
57	C₇	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₃	—	189	1	no	—
59	D₃	—	9	3	no	—
60	D₃	—	9	1	no	—
61	D₃	—	27	3	no	—
62	C₃	—	3	3	no	—
63	C₃	—	21	1	no	—
64	C₃	—	42	1	no	—
65	C₃	—	42	3	no	—
66	C₃	(C₃⋊D₃)⋊ΓL(1,𝔽₈)	1	3	yes	complemented
67	C₃	C₃⋊D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
68	C₂	—	27	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃³×C₇	C₂×C₃
Frattini	C₃	C₃⋊D₃×ΓL(1,𝔽₈)
socle	C₃²×C₇	C₃×D₃
LCS	G ▷ C₃³×C₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₇⋊H(3,𝔽₃)	C₂	semidirect
2. C₇×C₃⋊(C₃⋊D₃)	C₃	semidirect
5. C₃³×C₇	C₂×C₃	semidirect
7. C₃²×ΓL(1,𝔽₈)	D₃	semidirect
9. C₇⋊H(3,𝔽₃)	D₃	semidirect
19. C₃²×C₇	C₃×D₃	semidirect
24. C₃×ΓL(1,𝔽₈)	C₃⋊D₃	semidirect
25. C₃⋊(C₃⋊D₃)	ΓL(1,𝔽₈)	semidirect
34. C₃³	C₂×ΓL(1,𝔽₈)	semidirect
38. C₃×C₇	(C₃⋊D₃)⋊C₃	semidirect
56. C₃²	D₃×ΓL(1,𝔽₈)	semidirect
57. C₇	C₃⋊((C₃⋊D₃)⋊C₃)	semidirect
66. C₃	(C₃⋊D₃)⋊ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇⋊H(3,𝔽₃)	2
C₇×C₃⋊(C₃⋊D₃)	3
C₃⋊D₃×ΓL(1,𝔽₈)	3
(C₃⋊D₃)⋊ΓL(1,𝔽₈)	3
C₃⋊((C₃⋊D₃)⋊C₃)	7

Subgroups — C3⋊((C3⋊D3)⋊ΓL(1,𝔽8)) #11765749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊((C₃⋊D₃)⋊ΓL(1,𝔽₈)) #11765749