Group #13091484 — subgroups

Subgroups — #13091484

Order 1458.

314 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624885	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×C₃⋊AGL(1,ℤ/9)	—	3	1	no	maximal
3	#92309	C₃	1	3	yes	maximal, complemented
4	#92296	C₃	1	1	yes	maximal, characteristic, complemented
5	#4169⋊C₃	—	3	3	no	maximal
6	C₃²×3₋¹⁺²	D₃	1	1	yes	characteristic
7	#6896	—	2	3	no	—
8	#6896	—	2	3	no	—
9	#6896	—	2	6	no	—
10	#6896	C₂×C₃	1	3	yes	complemented
11	#6895	—	2	3	no	—
12	#6891	D₃	1	3	yes	—
13	#6887	—	2	1	no	—
14	#6887	C₂×C₃	1	1	yes	characteristic, complemented
15	C₃²×C₃⋊D₃	—	9	1	no	—
16	C₃⋊AGL(1,ℤ/9)	—	3	3	no	—
17	C₃⋊AGL(1,ℤ/9)	—	3	3	no	—
18	C₃×C₃⋊D₉	—	3	1	no	—
19	C₃×AGL(1,ℤ/9)	—	9	3	no	—
20	C₉⋊D₉	C₃²	1	1	yes	characteristic, complemented
21	#4169	—	3	3	no	—
22	#4169	—	3	6	no	—
23	C₃⁴	C₃⋊D₃	1	1	yes	characteristic
24	C₃×3₋¹⁺²	—	2	3	no	—
25	C₃×3₋¹⁺²	—	2	3	no	—
26	C₃×3₋¹⁺²	—	3	3	no	—
27	C₃×3₋¹⁺²	—	6	3	no	—
28	C₃×3₋¹⁺²	C₃×D₃	1	3	yes	—
29	C₃×3₋¹⁺²	C₃×D₃	1	3	yes	—
30	C₃²×C₉	—	2	1	no	—
31	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
32	C₉⋊C₉	—	2	3	no	—
33	C₉⋊C₉	—	2	3	no	—
34	C₉⋊C₉	—	2	6	no	—
35	C₉⋊C₉	—	2	9	no	—
36	C₉⋊C₉	—	2	3	no	—
37	C₉⋊C₉	—	2	3	no	—
38	C₉⋊C₉	—	2	6	no	—
39	C₉⋊C₉	—	2	3	no	—
40	C₃²⋊C₉	—	2	3	no	—
41	C₃²⋊C₉	—	2	6	no	—
42	C₃²⋊C₉	C₃×D₃	1	3	yes	—
43	C₃²⋊C₉	C₃×D₃	1	6	yes	—
44	C₉²	—	2	1	no	—
45	C₉²	—	2	2	no	—
46	C₉²	—	2	1	no	—
47	C₉²	C₂×C₃²	1	1	yes	characteristic, complemented
48	C₃×C₃⋊D₃	—	9	1	no	—
49	C₃×C₃⋊D₃	—	9	3	no	—
50	C₃²×D₃	—	27	1	no	—
51	C₃²×D₃	—	27	3	no	—
52	C₃⋊D₉	—	3	1	no	—
53	C₃⋊D₉	—	3	3	no	—
54	AGL(1,ℤ/9)	—	9	9	no	—
55	AGL(1,ℤ/9)	—	27	3	no	—
56	C₃×D₉	—	9	3	no	—
57	C₃³	—	3	1	no	—
58	C₃³	—	6	1	no	—
59	C₃³	—	9	3	no	—
60	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
61	C₃³	C₃×C₃⋊D₃	1	3	yes	—
62	3₋¹⁺²	—	3	9	no	—
63	3₋¹⁺²	—	6	9	no	—
64	3₋¹⁺²	—	9	3	no	—
65	3₋¹⁺²	—	18	3	no	—
66	C₃×C₉	—	2	1	no	—
67	C₃×C₉	—	2	3	no	—
68	C₃×C₉	—	2	3	no	—
69	C₃×C₉	—	2	6	no	—
70	C₃×C₉	—	2	3	no	—
71	C₃×C₉	—	2	3	no	—
72	C₃×C₉	—	6	2	no	—
73	C₃×C₉	—	6	1	no	—
74	C₃×C₉	AGL(1,ℤ/9)	1	3	yes	complemented
75	C₃×C₉	C₃²×D₃	1	3	yes	—
76	C₃×C₉	C₃²×D₃	1	1	yes	characteristic
77	C₂×C₃²	—	81	1	no	—
78	C₃⋊D₃	—	9	1	no	—
79	C₃×D₃	—	27	3	no	—
80	C₃×D₃	—	27	3	no	—
81	C₃×D₃	—	27	3	no	—
82	C₃×D₃	—	27	6	no	—
83	C₃×D₃	—	27	1	no	—
84	D₉	—	9	3	no	—
85	D₉	—	27	3	no	—
86	C₃²	—	2	1	no	—
87	C₃²	—	3	3	no	—
88	C₃²	—	3	3	no	—
89	C₃²	—	3	3	no	—
90	C₃²	—	3	6	no	—
91	C₃²	—	9	1	no	—
92	C₃²	—	18	1	no	—
93	C₃²	—	18	3	no	—
94	C₃²	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
95	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
96	C₉	—	2	3	no	—
97	C₉	—	3	3	no	—
98	C₉	—	6	3	no	—
99	C₉	—	6	3	no	—
100	C₉	—	6	6	no	—
101	C₉	—	6	3	no	—
102	C₉	C₃×AGL(1,ℤ/9)	1	3	yes	—
103	C₂×C₃	—	81	1	no	—
104	C₂×C₃	—	81	3	no	—
105	D₃	—	27	1	no	—
106	D₃	—	27	3	no	—
107	C₃	—	3	1	no	—
108	C₃	—	6	1	no	—
109	C₃	—	9	3	no	—
110	C₃	C₃×C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
111	C₃	#4207⋊C₃	1	3	yes	—
112	C₂	—	81	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624885

C₂

yes

maximal, characteristic, complemented, Sylow

C₃×C₃⋊AGL(1,ℤ/9)

—

maximal

#92309

C₃

yes

maximal, complemented

#92296

C₃

yes

maximal, characteristic, complemented

#4169⋊C₃

—

maximal

C₃²×3₋¹⁺²

D₃

yes

—

—

—

C₂×C₃

yes

complemented

#6895

—

#6891

D₃

yes

—

#6887

—

#6887

C₂×C₃

yes

characteristic, complemented

C₃²×C₃⋊D₃

—

C₃⋊AGL(1,ℤ/9)

—

C₃⋊AGL(1,ℤ/9)

—

C₃×C₃⋊D₉

—

C₃×AGL(1,ℤ/9)

—

C₉⋊D₉

C₃²

yes

characteristic, complemented

#4169

—

#4169

—

C₃⁴

C₃⋊D₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×D₃

yes

—

C₃×3₋¹⁺²

C₃×D₃

yes

—

C₃²×C₉

—

C₃²×C₉

C₃×D₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×D₃

yes

—

C₃²⋊C₉

C₃×D₃

yes

—

C₉²

—

C₉²

—

C₉²

—

C₉²

C₂×C₃²

yes

characteristic, complemented

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃⋊D₉

—

C₃⋊D₉

—

AGL(1,ℤ/9)

—

AGL(1,ℤ/9)

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₃⋊D₃

yes

characteristic

C₃³

C₃×C₃⋊D₃

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

AGL(1,ℤ/9)

yes

complemented

C₃×C₉

C₃²×D₃

yes

—

C₃×C₉

C₃²×D₃

yes

characteristic

C₂×C₃²

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

D₉

—

D₉

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃⋊AGL(1,ℤ/9)

yes

characteristic

C₃²

C₃²×C₃⋊D₃

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

100

C₉

—

101

C₉

—

102

C₉

C₃×AGL(1,ℤ/9)

yes

—

103

C₂×C₃

—

104

C₂×C₃

—

105

D₃

—

106

D₃

—

107

C₃

—

108

C₃

—

109

C₃

—

110

C₃

C₃×C₃⋊AGL(1,ℤ/9)

yes

characteristic

111

C₃

#4207⋊C₃

yes

—

112

C₂

—

Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉²	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊D₃
Fitting	#10624885	C₂
LCS	G ▷ C₉²
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, cosocle, hypocenter

C₉²

C₂×C₃²

Frattini, socle

C₃²

C₃²×C₃⋊D₃

C₂

G ▷ C₉²

C₁

N	H ≅ G/N	type
1. #10624885	C₂	semidirect
3. #92309	C₃	semidirect
4. #92296	C₃	semidirect
10. #6896	C₂×C₃	semidirect
14. #6887	C₂×C₃	semidirect
20. C₉⋊D₉	C₃²	semidirect
47. C₉²	C₂×C₃²	semidirect
74. C₃×C₉	AGL(1,ℤ/9)	semidirect

H ≅ G/N

type

1. #10624885

C₂

semidirect

3. #92309

C₃

semidirect

4. #92296

C₃

semidirect

10. #6896

C₂×C₃

semidirect

14. #6887

C₂×C₃

semidirect

20. C₉⋊D₉

C₃²

semidirect

47. C₉²

C₂×C₃²

semidirect

74. C₃×C₉

AGL(1,ℤ/9)

semidirect

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

subgroup	index
#10624885	2
C₃×C₃⋊AGL(1,ℤ/9)	3
#92309	3
#92296	3
#4169⋊C₃	3

subgroup

index

Subgroups — #13091484

Conjugacy classes of subgroups

Complements

Maximal subgroups