Group #13091480 — subgroups

Subgroups — #13091480

Order 1458.

296 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 52 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⋊D₃×3₋¹⁺²	—	3	1	no	maximal
3	#6895⋊C₂	C₃	1	2	yes	maximal, complemented
4	#92302	C₃	1	1	yes	maximal, characteristic
5	#92295	C₃	1	1	yes	maximal, characteristic, complemented
6	#92265	—	3	3	no	maximal
7	C₃²×3₋¹⁺²	D₃	1	1	yes	characteristic
8	#6900	—	2	3	no	—
9	#6900	D₃	1	3	yes	complemented
10	#6899	—	2	6	no	—
11	#6899	—	2	3	no	—
12	#6895	—	2	2	no	—
13	#6895	C₂×C₃	1	2	yes	complemented
14	#6891	—	2	1	no	—
15	#6891	C₂×C₃	1	1	yes	characteristic
16	#6887	—	2	1	no	—
17	#6887	C₂×C₃	1	1	yes	characteristic, complemented
18	C₃²×C₃⋊D₃	—	3	1	no	—
19	C₉×C₃⋊D₃	—	3	1	no	—
20	C₉×C₃⋊D₃	—	3	2	no	—
21	D₃×3₋¹⁺²	—	9	3	no	—
22	D₃×3₋¹⁺²	—	9	1	no	—
23	C₃×AGL(1,ℤ/9)	—	3	3	no	—
24	#4181	C₃²	1	1	yes	characteristic
25	D₉⋊C₉	—	3	6	no	—
26	C₉×D₉	—	3	3	no	—
27	C₃⁴	C₃×D₃	1	1	yes	characteristic
28	C₃×3₋¹⁺²	—	2	1	no	—
29	C₃×3₋¹⁺²	—	2	3	no	—
30	C₃×3₋¹⁺²	—	2	2	no	—
31	C₃×3₋¹⁺²	—	2	1	no	—
32	C₃×3₋¹⁺²	—	3	3	no	—
33	C₃×3₋¹⁺²	—	6	3	no	—
34	C₃×3₋¹⁺²	C₃×D₃	1	3	yes	—
35	C₃×3₋¹⁺²	C₃⋊D₃	1	1	yes	characteristic
36	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
37	C₃²×C₉	C₃×D₃	1	2	yes	—
38	C₉⋊C₉	—	2	3	no	—
39	C₉⋊C₉	—	2	12	no	—
40	C₉⋊C₉	—	2	3	no	—
41	C₉⋊C₉	—	2	6	no	—
42	C₉⋊C₉	—	2	3	no	—
43	C₉⋊C₉	—	2	6	no	—
44	C₉⋊C₉	C₃×D₃	1	6	yes	complemented
45	C₃²⋊C₉	—	2	6	no	—
46	C₃²⋊C₉	—	2	2	no	—
47	C₃²⋊C₉	—	2	1	no	—
48	C₃²⋊C₉	—	2	1	no	—
49	C₃²⋊C₉	—	2	3	no	—
50	C₃²⋊C₉	C₂×C₃²	1	1	yes	characteristic
51	C₉²	—	2	3	no	—
52	C₉²	C₃×D₃	1	3	yes	complemented
53	C₃×C₃⋊D₃	—	3	1	no	—
54	C₃×C₃⋊D₃	—	9	1	no	—
55	C₃²×D₃	—	9	3	no	—
56	C₃²×D₃	—	9	1	no	—
57	C₂×3₋¹⁺²	—	27	1	no	—
58	AGL(1,ℤ/9)	—	9	3	no	—
59	D₃×C₉	—	9	6	no	—
60	D₃×C₉	—	9	3	no	—
61	D₃×C₉	—	9	1	no	—
62	D₃×C₉	—	9	2	no	—
63	C₃×D₉	—	3	3	no	—
64	C₃³	—	2	1	no	—
65	C₃³	—	3	3	no	—
66	C₃³	—	9	1	no	—
67	C₃³	—	18	1	no	—
68	C₃³	C₃²×D₃	1	1	yes	characteristic
69	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
70	3₋¹⁺²	—	3	1	no	—
71	3₋¹⁺²	—	6	1	no	—
72	3₋¹⁺²	—	6	6	no	—
73	3₋¹⁺²	—	6	3	no	—
74	3₋¹⁺²	—	6	2	no	—
75	3₋¹⁺²	—	6	1	no	—
76	3₋¹⁺²	—	9	3	no	—
77	3₋¹⁺²	—	18	3	no	—
78	C₃×C₉	—	2	3	no	—
79	C₃×C₉	—	2	1	no	—
80	C₃×C₉	—	2	3	no	—
81	C₃×C₉	—	2	6	no	—
82	C₃×C₉	—	2	3	no	—
83	C₃×C₉	—	2	2	no	—
84	C₃×C₉	—	3	6	no	—
85	C₃×C₉	AGL(1,ℤ/9)	1	3	yes	complemented
86	C₃×C₉	C₃²×D₃	1	3	yes	—
87	C₃×C₉	C₃×C₃⋊D₃	1	1	yes	characteristic
88	C₃×C₉	C₃×C₃⋊D₃	1	2	yes	—
89	C₂×C₃²	—	27	1	no	—
90	C₃⋊D₃	—	3	1	no	—
91	C₃×D₃	—	9	3	no	—
92	C₃×D₃	—	9	1	no	—
93	C₃×D₃	—	27	1	no	—
94	C₃×D₃	—	27	3	no	—
95	C₂×C₉	—	27	2	no	—
96	C₂×C₉	—	27	1	no	—
97	D₉	—	3	3	no	—
98	C₃²	—	3	1	no	—
99	C₃²	—	3	1	no	—
100	C₃²	—	3	1	no	—
101	C₃²	—	6	1	no	—
102	C₃²	—	6	1	no	—
103	C₃²	—	6	2	no	—
104	C₃²	—	6	1	no	—
105	C₃²	—	6	1	no	—
106	C₃²	—	9	3	no	—
107	C₃²	—	18	3	no	—
108	C₃²	C₃×AGL(1,ℤ/9)	1	3	yes	—
109	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
110	C₉	—	2	3	no	—
111	C₉	—	2	1	no	—
112	C₉	—	3	2	no	—
113	C₉	—	3	3	no	—
114	C₉	—	6	6	no	—
115	C₉	—	6	3	no	—
116	C₉	—	6	2	no	—
117	C₉	—	6	3	no	—
118	C₉	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
119	C₂×C₃	—	27	1	no	—
120	C₂×C₃	—	81	1	no	—
121	D₃	—	9	3	no	—
122	D₃	—	9	1	no	—
123	C₃	—	2	1	no	—
124	C₃	—	3	3	no	—
125	C₃	—	9	1	no	—
126	C₃	—	18	1	no	—
127	C₃	C₃×C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
128	C₃	#92394	1	1	yes	characteristic
129	C₂	—	27	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624884

C₂

yes

maximal, characteristic, complemented, Sylow

C₃⋊D₃×3₋¹⁺²

—

maximal

#6895⋊C₂

C₃

yes

maximal, complemented

#92302

C₃

yes

maximal, characteristic

#92295

C₃

yes

maximal, characteristic, complemented

#92265

—

maximal

C₃²×3₋¹⁺²

D₃

yes

characteristic

#6900

—

#6900

D₃

yes

—

—

—

C₂×C₃

yes

complemented

#6891

—

#6891

C₂×C₃

yes

characteristic

#6887

—

#6887

C₂×C₃

yes

characteristic, complemented

C₃²×C₃⋊D₃

—

C₉×C₃⋊D₃

—

C₉×C₃⋊D₃

—

D₃×3₋¹⁺²

—

D₃×3₋¹⁺²

—

C₃×AGL(1,ℤ/9)

—

#4181

C₃²

yes

characteristic

D₉⋊C₉

—

C₉×D₉

—

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

—

C₃×3₋¹⁺²

C₃×D₃

yes

—

C₃×3₋¹⁺²

C₃⋊D₃

yes

characteristic

C₃²×C₉

C₃×D₃

yes

characteristic

C₃²×C₉

C₃×D₃

yes

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

C₃×D₃

yes

complemented

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₂×C₃²

yes

characteristic

C₉²

—

C₉²

C₃×D₃

yes

complemented

C₃×C₃⋊D₃

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

AGL(1,ℤ/9)

—

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

D₃×C₉

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×D₃

yes

characteristic

C₃³

C₃×C₃⋊D₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

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

AGL(1,ℤ/9)

yes

complemented

C₃×C₉

C₃²×D₃

yes

—

C₃×C₉

C₃×C₃⋊D₃

yes

characteristic

C₃×C₉

C₃×C₃⋊D₃

yes

—

C₂×C₃²

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

D₉

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

C₃×AGL(1,ℤ/9)

yes

—

109

C₃²

C₃²×C₃⋊D₃

yes

characteristic

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

—

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₉

C₃⋊AGL(1,ℤ/9)

yes

characteristic

119

C₂×C₃

—

120

C₂×C₃

—

121

D₃

—

122

D₃

—

123

C₃

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

C₃×C₃⋊AGL(1,ℤ/9)

yes

characteristic

128

C₃

#92394

yes

characteristic

129

C₂

—

Sylow

	subgroup	G/H
center	C₃	C₃×C₃⋊AGL(1,ℤ/9)
commutator, cosocle, hypocenter	C₃²⋊C₉	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊D₃
Fitting	#10624884	C₂
prime radical	#92302	C₃
hypercenter	C₉	C₃⋊AGL(1,ℤ/9)
LCS	G ▷ C₃²⋊C₉
UCS	C₁ ◁ C₃ ◁ C₉

subgroup

G/H

center

C₃

C₃×C₃⋊AGL(1,ℤ/9)

commutator, cosocle, hypocenter

C₃²⋊C₉

C₂×C₃²

Frattini, socle

C₃²

C₃²×C₃⋊D₃

C₂

C₃

C₉

G ▷ C₃²⋊C₉

C₁ ◁ C₃ ◁ C₉

N	H ≅ G/N	type
1. #10624884	C₂	semidirect
3. #6895⋊C₂	C₃	semidirect
5. #92295	C₃	semidirect
9. #6900	D₃	semidirect
13. #6895	C₂×C₃	semidirect
17. #6887	C₂×C₃	semidirect
44. C₉⋊C₉	C₃×D₃	semidirect
52. C₉²	C₃×D₃	semidirect
85. C₃×C₉	AGL(1,ℤ/9)	semidirect

H ≅ G/N

type

1. #10624884

C₂

semidirect

3. #6895⋊C₂

C₃

semidirect

5. #92295

C₃

semidirect

9. #6900

D₃

semidirect

13. #6895

C₂×C₃

semidirect

17. #6887

C₂×C₃

semidirect

44. C₉⋊C₉

C₃×D₃

semidirect

52. C₉²

C₃×D₃

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⋊D₃×3₋¹⁺²	3
#6895⋊C₂	3
#92302	3
#92295	3
#92265	3

subgroup

index

C₃⋊D₃×3₋¹⁺²

Subgroups — #13091480

Conjugacy classes of subgroups

Complements

Maximal subgroups