Group #13091481 — subgroups

Subgroups — #13091481

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	#10624884	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×C₃⋊AGL(1,ℤ/9)	—	3	1	no	maximal
3	#92314	C₃	1	1	yes	maximal, characteristic, complemented
4	#6899⋊C₂	C₃	1	2	yes	maximal, complemented
5	#92296	C₃	1	1	yes	maximal, characteristic, complemented
6	#4169⋊C₃	—	3	3	no	maximal
7	C₃²×3₋¹⁺²	D₃	1	1	yes	characteristic
8	#6900	—	2	3	no	—
9	#6900	—	2	1	no	—
10	#6900	C₂×C₃	1	1	yes	characteristic, complemented
11	#6899	—	2	2	no	—
12	#6899	—	2	6	no	—
13	#6899	C₂×C₃	1	2	yes	complemented
14	#6895	—	2	3	no	—
15	#6891	D₃	1	3	yes	—
16	#6887	—	2	1	no	—
17	#6887	C₂×C₃	1	1	yes	characteristic, complemented
18	C₃²×C₃⋊D₃	—	9	1	no	—
19	C₃⋊AGL(1,ℤ/9)	—	3	2	no	—
20	C₃⋊AGL(1,ℤ/9)	—	3	1	no	—
21	C₃⋊AGL(1,ℤ/9)	—	3	3	no	—
22	C₃×C₃⋊D₉	—	3	1	no	—
23	C₃×AGL(1,ℤ/9)	—	9	3	no	—
24	C₉⋊D₉	C₃²	1	1	yes	characteristic, complemented
25	#4169	—	3	3	no	—
26	#4169	—	3	6	no	—
27	C₃⁴	C₃⋊D₃	1	1	yes	characteristic
28	C₃×3₋¹⁺²	—	2	2	no	—
29	C₃×3₋¹⁺²	—	2	1	no	—
30	C₃×3₋¹⁺²	—	2	3	no	—
31	C₃×3₋¹⁺²	—	3	3	no	—
32	C₃×3₋¹⁺²	—	6	3	no	—
33	C₃×3₋¹⁺²	C₃×D₃	1	2	yes	—
34	C₃×3₋¹⁺²	C₃×D₃	1	1	yes	characteristic
35	C₃×3₋¹⁺²	C₃×D₃	1	3	yes	—
36	C₃²×C₉	—	2	1	no	—
37	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
38	C₉⋊C₉	—	2	3	no	—
39	C₉⋊C₉	—	2	12	no	—
40	C₉⋊C₉	—	2	6	no	—
41	C₉⋊C₉	—	2	1	no	—
42	C₉⋊C₉	—	2	2	no	—
43	C₉⋊C₉	—	2	3	no	—
44	C₉⋊C₉	—	2	3	no	—
45	C₉⋊C₉	—	2	6	no	—
46	C₃²⋊C₉	—	2	3	no	—
47	C₃²⋊C₉	—	2	6	no	—
48	C₃²⋊C₉	C₃×D₃	1	3	yes	—
49	C₃²⋊C₉	C₃×D₃	1	6	yes	—
50	C₉²	—	2	3	no	—
51	C₉²	—	2	1	no	—
52	C₉²	C₂×C₃²	1	1	yes	characteristic, complemented
53	C₃×C₃⋊D₃	—	9	1	no	—
54	C₃×C₃⋊D₃	—	9	2	no	—
55	C₃×C₃⋊D₃	—	9	1	no	—
56	C₃²×D₃	—	27	1	no	—
57	C₃²×D₃	—	27	3	no	—
58	C₃⋊D₉	—	3	1	no	—
59	C₃⋊D₉	—	3	3	no	—
60	AGL(1,ℤ/9)	—	9	6	no	—
61	AGL(1,ℤ/9)	—	9	3	no	—
62	AGL(1,ℤ/9)	—	27	3	no	—
63	C₃×D₉	—	9	3	no	—
64	C₃³	—	3	1	no	—
65	C₃³	—	6	1	no	—
66	C₃³	—	9	3	no	—
67	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
68	C₃³	C₃×C₃⋊D₃	1	2	yes	—
69	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
70	3₋¹⁺²	—	3	6	no	—
71	3₋¹⁺²	—	3	3	no	—
72	3₋¹⁺²	—	6	6	no	—
73	3₋¹⁺²	—	6	3	no	—
74	3₋¹⁺²	—	9	3	no	—
75	3₋¹⁺²	—	18	3	no	—
76	C₃×C₉	—	2	1	no	—
77	C₃×C₉	—	2	3	no	—
78	C₃×C₉	—	2	3	no	—
79	C₃×C₉	—	2	6	no	—
80	C₃×C₉	—	2	3	no	—
81	C₃×C₉	—	2	1	no	—
82	C₃×C₉	—	2	2	no	—
83	C₃×C₉	—	6	2	no	—
84	C₃×C₉	—	6	1	no	—
85	C₃×C₉	AGL(1,ℤ/9)	1	3	yes	complemented
86	C₃×C₉	C₃²×D₃	1	3	yes	—
87	C₃×C₉	C₃²×D₃	1	1	yes	characteristic
88	C₂×C₃²	—	81	1	no	—
89	C₃⋊D₃	—	9	1	no	—
90	C₃×D₃	—	27	2	no	—
91	C₃×D₃	—	27	1	no	—
92	C₃×D₃	—	27	3	no	—
93	C₃×D₃	—	27	3	no	—
94	C₃×D₃	—	27	6	no	—
95	C₃×D₃	—	27	1	no	—
96	D₉	—	9	3	no	—
97	D₉	—	27	3	no	—
98	C₃²	—	2	1	no	—
99	C₃²	—	3	3	no	—
100	C₃²	—	3	2	no	—
101	C₃²	—	3	1	no	—
102	C₃²	—	3	3	no	—
103	C₃²	—	3	6	no	—
104	C₃²	—	9	1	no	—
105	C₃²	—	18	1	no	—
106	C₃²	—	18	3	no	—
107	C₃²	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
108	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
109	C₉	—	2	3	no	—
110	C₉	—	3	3	no	—
111	C₉	—	6	1	no	—
112	C₉	—	6	2	no	—
113	C₉	—	6	3	no	—
114	C₉	—	6	6	no	—
115	C₉	—	6	3	no	—
116	C₉	C₃×AGL(1,ℤ/9)	1	3	yes	—
117	C₂×C₃	—	81	1	no	—
118	C₂×C₃	—	81	1	no	—
119	C₂×C₃	—	81	2	no	—
120	D₃	—	27	1	no	—
121	D₃	—	27	3	no	—
122	C₃	—	3	1	no	—
123	C₃	—	6	1	no	—
124	C₃	—	9	1	no	—
125	C₃	—	9	2	no	—
126	C₃	C₃×C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
127	C₃	#4207⋊C₃	1	3	yes	—
128	C₂	—	81	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624884

C₂

yes

maximal, characteristic, complemented, Sylow

C₃×C₃⋊AGL(1,ℤ/9)

—

maximal

#92314

C₃

yes

maximal, characteristic, complemented

#6899⋊C₂

C₃

yes

maximal, complemented

#92296

C₃

yes

maximal, characteristic, complemented

#4169⋊C₃

—

maximal

C₃²×3₋¹⁺²

D₃

yes

—

—

C₂×C₃

yes

characteristic, complemented

#6899

—

#6899

—

#6899

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⋊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

—

C₃×3₋¹⁺²

C₃×D₃

yes

—

C₃×3₋¹⁺²

C₃×D₃

yes

characteristic

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

C₂×C₃²

yes

characteristic, complemented

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃⋊D₉

—

C₃⋊D₉

—

AGL(1,ℤ/9)

—

AGL(1,ℤ/9)

—

AGL(1,ℤ/9)

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₃⋊D₃

yes

characteristic

C₃³

C₃×C₃⋊D₃

yes

—

C₃³

C₃×C₃⋊D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

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

—

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

—

C₃×D₃

—

D₉

—

D₉

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

C₃⋊AGL(1,ℤ/9)

yes

characteristic

108

C₃²

C₃²×C₃⋊D₃

yes

characteristic

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

C₃×AGL(1,ℤ/9)

yes

—

117

C₂×C₃

—

118

C₂×C₃

—

119

C₂×C₃

—

120

D₃

—

121

D₃

—

122

C₃

—

123

C₃

—

124

C₃

—

125

C₃

—

126

C₃

C₃×C₃⋊AGL(1,ℤ/9)

yes

characteristic

127

C₃

#4207⋊C₃

yes

—

128

C₂

—

Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉²	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊D₃
Fitting	#10624884	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. #10624884	C₂	semidirect
3. #92314	C₃	semidirect
4. #6899⋊C₂	C₃	semidirect
5. #92296	C₃	semidirect
10. #6900	C₂×C₃	semidirect
13. #6899	C₂×C₃	semidirect
17. #6887	C₂×C₃	semidirect
24. C₉⋊D₉	C₃²	semidirect
52. C₉²	C₂×C₃²	semidirect
85. C₃×C₉	AGL(1,ℤ/9)	semidirect

H ≅ G/N

type

1. #10624884

C₂

semidirect

3. #92314

C₃

semidirect

4. #6899⋊C₂

C₃

semidirect

5. #92296

C₃

semidirect

10. #6900

C₂×C₃

semidirect

13. #6899

C₂×C₃

semidirect

17. #6887

C₂×C₃

semidirect

24. C₉⋊D₉

C₃²

semidirect

52. C₉²

C₂×C₃²

semidirect

85. 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
#10624884	2
C₃×C₃⋊AGL(1,ℤ/9)	3
#92314	3
#6899⋊C₂	3
#92296	3
#4169⋊C₃	3

subgroup

index

Subgroups — #13091481

Conjugacy classes of subgroups

Complements

Maximal subgroups