Subgroups — #11765607

Order 1134.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₃×ΓL(1,𝔽₈))⋊C₉	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	C₃²×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic
5	C₃×C₇×C₉	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×C₇⋊C₉	—	2	1	no	—
7	#4169	—	7	1	no	maximal, Hall
8	C₇×C₃⋊D₃	—	3	1	no	—
9	D₃×ΓL(1,𝔽₈)	—	9	1	no	—
10	D₃×ΓL(1,𝔽₈)	—	9	1	no	—
11	D₃×ΓL(1,𝔽₈)	—	9	1	no	—
12	D₃×ΓL(1,𝔽₈)	—	9	1	no	—
13	C₇×D₉	—	9	1	no	—
14	C₃²⋊C₉	—	7	1	no	Sylow
15	C₃²×C₇	C₃×D₃	1	1	yes	characteristic
16	C₃×ΓL(1,𝔽₈)	—	2	1	no	—
17	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
18	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
19	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
20	C₃×ΓL(1,𝔽₈)	D₉	1	1	yes	characteristic, complemented
21	C₇×C₉	—	3	1	no	—
22	C₇⋊C₉	—	6	1	no	—
23	C₃×C₃⋊D₃	—	21	1	no	—
24	C₃⋊D₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
25	D₃×C₇	—	9	1	no	—
26	D₃×C₇	—	9	1	no	—
27	D₃×C₇	—	9	1	no	—
28	D₃×C₇	—	9	1	no	—
29	C₂×ΓL(1,𝔽₈)	—	27	1	no	—
30	C₃³	—	7	1	no	—
31	C₃×C₉	—	14	1	no	—
32	C₃×C₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
33	C₃×C₇	C₃×D₉	1	1	yes	characteristic
34	C₃×C₇	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
35	C₃×C₇	AGL(1,ℤ/9)	1	1	yes	characteristic
36	C₃×C₇	AGL(1,ℤ/9)	1	1	yes	characteristic
37	ΓL(1,𝔽₈)	—	3	1	no	—
38	ΓL(1,𝔽₈)	—	6	1	no	—
39	C₃⋊D₃	—	3	1	no	—
40	C₃×D₃	—	63	1	no	—
41	C₃×D₃	—	63	1	no	—
42	C₃×D₃	—	63	1	no	—
43	C₃×D₃	—	63	1	no	—
44	D₉	—	9	1	no	—
45	C₂×C₇	—	27	1	no	Hall
46	C₃²	—	7	1	no	—
47	C₃²	—	14	1	no	—
48	C₃²	—	21	1	no	—
49	C₃²	—	21	1	no	—
50	C₃²	—	21	1	no	—
51	C₃²	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
52	C₉	—	3	1	no	—
53	C₉	—	42	1	no	—
54	C₇	#4169	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₃	—	189	1	no	—
56	D₃	—	9	1	no	—
57	D₃	—	9	1	no	—
58	D₃	—	9	1	no	—
59	D₃	—	9	1	no	—
60	C₃	—	21	1	no	—
61	C₃	—	42	1	no	—
62	C₃	ΓL(1,𝔽₆₄)	1	1	yes	characteristic
63	C₃	#67361	1	1	yes	characteristic
64	C₃	D₉×ΓL(1,𝔽₈)	1	1	yes	characteristic
65	C₃	(C₃⋊D₃)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
66	C₂	—	27	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₇×C₉	C₂×C₃
Frattini	C₃²	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×ΓL(1,𝔽₈))⋊C₉	C₂	semidirect
3. C₇×C₃⋊D₉	C₃	semidirect
5. C₃×C₇×C₉	C₂×C₃	semidirect
20. C₃×ΓL(1,𝔽₈)	D₉	semidirect
24. C₃⋊D₉	ΓL(1,𝔽₈)	semidirect
32. C₃×C₉	C₂×ΓL(1,𝔽₈)	semidirect
54. C₇	#4169	semidirect

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))⋊C₉	2
C₃⋊D₃×ΓL(1,𝔽₈)	3
C₇×C₃⋊D₉	3
#4169	7