Group #13091495 — subgroups

Subgroups — #13091495

Order 1458.

266 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 48 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	#10624890	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	#6899⋊C₂	C₃	1	2	yes	maximal, complemented
3	#92312	C₃	1	1	yes	maximal, characteristic, complemented
4	#92299	C₃	1	1	yes	maximal, characteristic, complemented
5	#4169⋊C₃	—	3	2	no	maximal
6	#4169⋊C₃	—	3	1	no	maximal
7	C₃×#4169	—	3	1	no	maximal
8	#6899	—	2	2	no	—
9	#6899	—	2	2	no	—
10	#6899	—	2	2	no	—
11	#6899	—	2	2	no	—
12	#6899	C₂×C₃	1	2	yes	complemented
13	#6898	—	2	1	no	—
14	#6898	—	2	2	no	—
15	#6898	—	2	1	no	—
16	#6898	C₂×C₃	1	1	yes	characteristic, complemented
17	#6894	—	2	1	no	—
18	#6894	—	2	2	no	—
19	#6891	D₃	1	2	yes	—
20	#6891	D₃	1	1	yes	characteristic
21	C₉×3₋¹⁺²	—	2	1	no	—
22	C₉×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×C₃²⋊C₉	D₃	1	1	yes	characteristic
24	C₃²×C₃⋊D₃	—	9	1	no	—
25	C₃⋊AGL(1,ℤ/9)	—	3	1	no	—
26	C₃⋊AGL(1,ℤ/9)	—	3	2	no	—
27	C₃×C₃⋊D₉	—	3	1	no	—
28	C₉⋊D₉	C₃²	1	1	yes	characteristic, complemented
29	#4169	—	3	1	no	—
30	#4169	—	3	2	no	—
31	#4169	—	3	2	no	—
32	#4169	—	3	2	no	—
33	#4169	—	3	1	no	—
34	#4169	—	3	2	no	—
35	#4169	—	3	2	no	—
36	C₃⁴	C₃⋊D₃	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	2	1	no	—
38	C₃×3₋¹⁺²	—	2	2	no	—
39	C₃×3₋¹⁺²	C₃×D₃	1	1	yes	characteristic
40	C₃×3₋¹⁺²	C₃×D₃	1	2	yes	—
41	C₃²×C₉	—	2	1	no	—
42	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
43	C₉⋊C₉	—	2	2	no	—
44	C₉⋊C₉	—	2	1	no	—
45	C₉⋊C₉	—	2	1	no	—
46	C₉⋊C₉	—	2	2	no	—
47	C₉⋊C₉	—	2	1	no	—
48	C₉⋊C₉	—	2	2	no	—
49	C₉⋊C₉	—	2	2	no	—
50	C₉⋊C₉	—	2	2	no	—
51	C₉⋊C₉	—	2	2	no	—
52	C₉⋊C₉	—	2	4	no	—
53	C₉⋊C₉	—	2	2	no	—
54	C₉⋊C₉	—	2	2	no	—
55	C₉⋊C₉	—	2	1	no	—
56	C₉⋊C₉	—	2	2	no	—
57	C₉⋊C₉	—	2	1	no	—
58	C₉⋊C₉	—	2	2	no	—
59	C₉⋊C₉	—	2	2	no	—
60	C₉⋊C₉	—	2	2	no	—
61	C₉⋊C₉	—	2	1	no	—
62	C₉⋊C₉	—	2	2	no	—
63	C₃²⋊C₉	—	2	1	no	—
64	C₃²⋊C₉	—	2	2	no	—
65	C₃²⋊C₉	—	2	2	no	—
66	C₃²⋊C₉	—	2	1	no	—
67	C₃²⋊C₉	—	2	2	no	—
68	C₃²⋊C₉	—	2	2	no	—
69	C₃²⋊C₉	—	2	2	no	—
70	C₃²⋊C₉	C₃×D₃	1	1	yes	characteristic
71	C₃²⋊C₉	C₃×D₃	1	2	yes	—
72	C₃²⋊C₉	C₃×D₃	1	2	yes	—
73	C₃²⋊C₉	C₃×D₃	1	1	yes	characteristic
74	C₃²⋊C₉	C₃×D₃	1	2	yes	—
75	C₃²⋊C₉	C₃×D₃	1	2	yes	—
76	C₃²⋊C₉	C₃×D₃	1	2	yes	—
77	C₉²	—	2	1	no	—
78	C₉²	—	2	2	no	—
79	C₉²	—	2	1	no	—
80	C₉²	C₂×C₃²	1	1	yes	characteristic, complemented
81	C₃×C₃⋊D₃	—	9	1	no	—
82	C₃×C₃⋊D₃	—	9	2	no	—
83	C₃×C₃⋊D₃	—	9	1	no	—
84	C₃²×D₃	—	27	1	no	—
85	C₃²×D₃	—	27	1	no	—
86	C₃²×D₃	—	27	2	no	—
87	C₃⋊D₉	—	3	1	no	—
88	C₃⋊D₉	—	3	2	no	—
89	C₃⋊D₉	—	3	1	no	—
90	AGL(1,ℤ/9)	—	27	1	no	—
91	AGL(1,ℤ/9)	—	27	2	no	—
92	C₃×D₉	—	27	1	no	—
93	C₃³	—	3	1	no	—
94	C₃³	—	6	1	no	—
95	C₃³	—	9	1	no	—
96	C₃³	—	9	2	no	—
97	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
98	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
99	C₃³	C₃×C₃⋊D₃	1	2	yes	—
100	3₋¹⁺²	—	3	1	no	—
101	3₋¹⁺²	—	6	1	no	—
102	3₋¹⁺²	—	6	2	no	—
103	3₋¹⁺²	—	6	1	no	—
104	3₋¹⁺²	—	9	2	no	—
105	3₋¹⁺²	—	18	2	no	—
106	C₃×C₉	—	2	1	no	—
107	C₃×C₉	—	2	1	no	—
108	C₃×C₉	—	2	2	no	—
109	C₃×C₉	—	2	2	no	—
110	C₃×C₉	—	2	1	no	—
111	C₃×C₉	—	2	2	no	—
112	C₃×C₉	—	2	2	no	—
113	C₃×C₉	—	2	2	no	—
114	C₃×C₉	—	2	2	no	—
115	C₃×C₉	—	2	1	no	—
116	C₃×C₉	—	9	1	no	—
117	C₃×C₉	—	18	1	no	—
118	C₃×C₉	C₃²×D₃	1	1	yes	characteristic
119	C₃×C₉	C₃²×D₃	1	1	yes	characteristic
120	C₃×C₉	C₃²×D₃	1	2	yes	—
121	C₂×C₃²	—	81	1	no	—
122	C₃⋊D₃	—	9	1	no	—
123	C₃×D₃	—	27	1	no	—
124	C₃×D₃	—	27	2	no	—
125	C₃×D₃	—	27	1	no	—
126	C₃×D₃	—	27	2	no	—
127	C₃×D₃	—	27	2	no	—
128	C₃×D₃	—	27	1	no	—
129	C₃×D₃	—	27	2	no	—
130	C₃×D₃	—	27	2	no	—
131	C₃×D₃	—	27	2	no	—
132	C₃×D₃	—	27	1	no	—
133	D₉	—	27	1	no	—
134	D₉	—	27	1	no	—
135	D₉	—	27	2	no	—
136	C₃²	—	2	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	2	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	2	no	—
141	C₃²	—	3	2	no	—
142	C₃²	—	3	1	no	—
143	C₃²	—	3	2	no	—
144	C₃²	—	3	2	no	—
145	C₃²	—	3	2	no	—
146	C₃²	—	9	1	no	—
147	C₃²	—	18	1	no	—
148	C₃²	—	18	3	no	—
149	C₃²	#4207	1	1	yes	characteristic
150	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
151	C₉	—	3	1	no	—
152	C₉	—	3	1	no	—
153	C₉	—	3	2	no	—
154	C₉	—	6	1	no	—
155	C₉	—	6	2	no	—
156	C₉	—	6	1	no	—
157	C₉	—	6	1	no	—
158	C₉	—	6	2	no	—
159	C₉	—	6	1	no	—
160	C₉	—	6	2	no	—
161	C₉	—	6	2	no	—
162	C₉	—	6	2	no	—
163	C₉	—	6	2	no	—
164	C₂×C₃	—	81	1	no	—
165	C₂×C₃	—	81	2	no	—
166	C₂×C₃	—	81	1	no	—
167	D₃	—	27	1	no	—
168	D₃	—	27	1	no	—
169	D₃	—	27	2	no	—
170	C₃	—	3	1	no	—
171	C₃	—	6	1	no	—
172	C₃	—	9	2	no	—
173	C₃	—	9	1	no	—
174	C₃	C₃×#4207	1	1	yes	characteristic
175	C₃	#4207⋊C₃	1	1	yes	characteristic
176	C₃	#4207⋊C₃	1	2	yes	—
177	C₂	—	81	1	no	Sylow

subgroup

G/H

class size

multiplicity

normal?

properties

#10624890

C₂

yes

maximal, characteristic, complemented, Sylow

#6899⋊C₂

C₃

yes

maximal, complemented

#92312

C₃

yes

maximal, characteristic, complemented

#92299

C₃

yes

maximal, characteristic, complemented

—

—

—

—

—

—

—

C₂×C₃

yes

—

—

—

C₂×C₃

yes

characteristic, complemented

#6894

—

#6894

—

#6891

D₃

yes

—

#6891

D₃

yes

characteristic

C₉×3₋¹⁺²

—

C₉×3₋¹⁺²

C₂×C₃

yes

characteristic, complemented

C₃×C₃²⋊C₉

D₃

yes

characteristic

C₃²×C₃⋊D₃

—

C₃⋊AGL(1,ℤ/9)

—

C₃⋊AGL(1,ℤ/9)

—

C₃×C₃⋊D₉

—

C₉⋊D₉

C₃²

yes

characteristic, complemented

—

—

—

—

—

—

—

C₃⁴

C₃⋊D₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

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

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×D₃

yes

characteristic

C₃²⋊C₉

C₃×D₃

yes

—

C₃²⋊C₉

C₃×D₃

yes

—

C₃²⋊C₉

C₃×D₃

yes

characteristic

C₃²⋊C₉

C₃×D₃

yes

—

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

—

C₃×C₃⋊D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃²×D₃

—

C₃⋊D₉

—

C₃⋊D₉

—

C₃⋊D₉

—

AGL(1,ℤ/9)

—

AGL(1,ℤ/9)

—

C₃×D₉

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₃⋊D₃

yes

characteristic

C₃³

C₃×C₃⋊D₃

yes

characteristic

C₃³

C₃×C₃⋊D₃

yes

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

—

103

3₋¹⁺²

—

104

3₋¹⁺²

—

105

3₋¹⁺²

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₃×C₉

—

118

C₃×C₉

C₃²×D₃

yes

characteristic

119

C₃×C₉

C₃²×D₃

yes

characteristic

120

C₃×C₉

C₃²×D₃

yes

—

121

C₂×C₃²

—

122

C₃⋊D₃

—

123

C₃×D₃

—

124

C₃×D₃

—

125

C₃×D₃

—

126

C₃×D₃

—

127

C₃×D₃

—

128

C₃×D₃

—

129

C₃×D₃

—

130

C₃×D₃

—

131

C₃×D₃

—

132

C₃×D₃

—

133

D₉

—

134

D₉

—

135

D₉

—

136

C₃²

—

137

C₃²

—

138

C₃²

—

139

C₃²

—

140

C₃²

—

141

C₃²

—

142

C₃²

—

143

C₃²

—

144

C₃²

—

145

C₃²

—

146

C₃²

—

147

C₃²

—

148

C₃²

—

149

C₃²

#4207

yes

characteristic

150

C₃²

C₃²×C₃⋊D₃

yes

characteristic

151

C₉

—

152

C₉

—

153

C₉

—

154

C₉

—

155

C₉

—

156

C₉

—

157

C₉

—

158

C₉

—

159

C₉

—

160

C₉

—

161

C₉

—

162

C₉

—

163

C₉

—

164

C₂×C₃

—

165

C₂×C₃

—

166

C₂×C₃

—

167

D₃

—

168

D₃

—

169

D₃

—

170

C₃

—

171

C₃

—

172

C₃

—

173

C₃

—

174

C₃

C₃×#4207

yes

characteristic

175

C₃

#4207⋊C₃

yes

characteristic

176

C₃

#4207⋊C₃

yes

—

177

C₂

—

Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉²	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊D₃
Fitting	#10624890	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. #10624890	C₂	semidirect
2. #6899⋊C₂	C₃	semidirect
3. #92312	C₃	semidirect
4. #92299	C₃	semidirect
12. #6899	C₂×C₃	semidirect
16. #6898	C₂×C₃	semidirect
22. C₉×3₋¹⁺²	C₂×C₃	semidirect
28. C₉⋊D₉	C₃²	semidirect
80. C₉²	C₂×C₃²	semidirect

H ≅ G/N

type

1. #10624890

C₂

semidirect

2. #6899⋊C₂

C₃

semidirect

3. #92312

C₃

semidirect

4. #92299

C₃

semidirect

12. #6899

C₂×C₃

semidirect

16. #6898

C₂×C₃

semidirect

22. C₉×3₋¹⁺²

C₂×C₃

semidirect

28. C₉⋊D₉

C₃²

semidirect

80. C₉²

C₂×C₃²

semidirect

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

subgroup	index
#10624890	2
#6899⋊C₂	3
#92312	3
#92299	3
#4169⋊C₃	3
C₃×#4169	3

subgroup

index

Subgroups — #13091495

Conjugacy classes of subgroups

Complements

Maximal subgroups