Group #13137249 — subgroups

Subgroups — #13137249

Order 1728.

494 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 121 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	#11721706	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#874	C₂	1	1	yes	maximal, characteristic
3	C₂×#88424	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#10590369	—	3	1	no	maximal
6	#10588753	—	3	1	no	maximal
7	C₂×#6226	C₂²	1	2	yes	—
8	C₂×C₉×D₁₂	—	2	2	no	—
9	C₂×C₉×D₁₂	C₂²	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊Dic₉)	—	2	2	no	—
12	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
13	#88424	—	2	8	no	—
14	#64078	—	3	1	no	—
15	C₃×#874	D₃	1	1	yes	characteristic
16	C₂×#3716	—	3	4	no	—
17	C₃×C₄×C₃⋊C₈	—	3	1	no	—
18	C₉×#127	—	3	1	no	—
19	C₂×#3672	—	3	4	no	—
20	#63431	—	3	1	no	—
21	#63431	—	6	1	no	—
22	C₄×C₉⋊C₈	D₃	1	1	yes	characteristic, complemented
23	C₂²×D₃×C₉	—	2	2	no	—
24	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
25	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
26	C₂×C₃⋊Dic₉	—	2	2	no	—
27	#6226	—	2	4	no	—
28	C₉×D₁₂	—	2	4	no	—
29	C₉×D₁₂	—	4	4	no	—
30	C₃×C₉⋊C₈	D₄	1	4	yes	complemented
31	#5076	—	9	1	no	—
32	#4693	—	9	1	no	—
33	C₂×#634	—	3	2	no	—
34	C₂×C₃×D₁₂	—	2	2	no	—
35	C₂×C₃×D₁₂	C₂×D₃	1	2	yes	—
36	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₄⋊(C₃⋊Dic₃)	—	6	2	no	—
38	C₂×C₃×C₃⋊C₈	—	3	2	no	—
39	#3716	—	6	8	no	—
40	C₂×D₄×C₉	—	3	2	no	—
41	C₂×D₄×C₉	—	6	2	no	—
42	C₂×Dic₁₈	—	3	2	no	—
43	C₂×Dic₁₈	—	6	2	no	—
44	C₄²×C₉	—	2	1	no	—
45	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
46	#3672	—	6	8	no	—
47	C₄⋊Dic₉	—	6	4	no	—
48	C₄⋊Dic₉	—	6	2	no	—
49	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
50	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
51	C₂×D₃×C₉	—	4	2	no	—
52	C₂×D₃×C₉	—	4	4	no	—
53	C₃×C₄×C₉	SD₁₆	1	4	yes	complemented
54	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
55	C₃⋊Dic₉	—	4	2	no	—
56	C₃×#127	—	3	1	no	—
57	C₂×#214	—	9	4	no	—
58	C₂×#204	—	9	4	no	—
59	#874	D₉	1	1	yes	characteristic
60	#869	—	9	1	no	—
61	#869	—	9	1	no	—
62	#869	—	18	1	no	—
63	C₃×C₄×C₈	—	9	1	no	—
64	C₄×C₃⋊C₈	—	3	1	no	—
65	C₂²×C₃×D₃	—	2	2	no	—
66	C₂×C₃²×C₄	#67	1	2	yes	—
67	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₂×C₃⋊Dic₃	—	6	2	no	—
69	#634	—	6	4	no	—
70	C₃×D₁₂	—	2	4	no	—
71	C₃×D₁₂	—	4	4	no	—
72	C₂³×C₉	—	6	2	no	—
73	C₃×C₃⋊C₈	—	3	4	no	—
74	D₄×C₉	—	6	4	no	—
75	D₄×C₉	—	12	4	no	—
76	C₂×C₄×C₉	—	2	2	no	—
77	C₂×C₄×C₉	—	2	1	no	—
78	C₂×C₄×C₉	D₁₂	1	2	yes	—
79	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
80	C₂×Dic₉	—	6	4	no	—
81	C₂×Dic₉	—	6	2	no	—
82	Dic₁₈	—	6	4	no	—
83	Dic₁₈	—	12	4	no	—
84	C₉⋊C₈	D₁₂	1	4	yes	complemented
85	#492	—	27	1	no	Sylow
86	C₂×C₃×C₉	#127	1	1	yes	characteristic
87	C₂×C₃×C₉	C₂×SD₁₆	1	2	yes	—
88	D₃×C₉	—	8	2	no	—
89	C₂×C₃×D₄	—	3	2	no	—
90	C₂×C₃×D₄	—	6	2	no	—
91	C₂×D₁₂	—	2	2	no	—
92	C₂×D₁₂	C₂×D₉	1	2	yes	—
93	C₂×Dic₆	—	9	2	no	—
94	C₂×Dic₆	—	9	2	no	—
95	C₂×Dic₆	—	18	2	no	—
96	C₂×C₃×C₈	—	9	2	no	—
97	C₃×C₄²	—	2	1	no	—
98	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
99	C₃×C₄²	D₃²	1	1	yes	characteristic
100	#214	—	18	8	no	—
101	C₄⋊Dic₃	—	18	4	no	—
102	C₄⋊Dic₃	—	18	2	no	—
103	C₄⋊Dic₃	—	18	2	no	—
104	C₂×C₃⋊C₈	—	3	2	no	—
105	#204	—	18	8	no	—
106	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
107	C₂×C₃×D₃	—	4	4	no	—
108	C₂×C₃×D₃	—	4	2	no	—
109	C₃²×C₄	#214	1	4	yes	—
110	C₃²×C₄	D₃×D₄	1	2	yes	—
111	C₃⋊Dic₃	—	12	2	no	—
112	C₂²×C₉	—	2	1	no	—
113	C₂²×C₉	—	12	2	no	—
114	C₂²×C₉	—	12	4	no	—
115	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
116	C₄×C₉	—	2	4	no	—
117	C₄×C₉	—	2	2	no	—
118	C₄×C₉	#204	1	4	yes	complemented
119	C₄×C₉	C₂×D₁₂	1	2	yes	—
120	Dic₉	—	12	4	no	—
121	Dic₉	—	12	2	no	—
122	C₂×SD₁₆	—	27	4	no	—
123	C₄⋊Q₈	—	27	1	no	—
124	#127	—	3	1	no	—
125	C₄×C₈	—	9	1	no	—
126	C₃×C₉	#492	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₃	—	6	2	no	—
128	C₂²×D₃	—	2	2	no	—
129	C₃×D₄	—	6	4	no	—
130	C₃×D₄	—	12	4	no	—
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	#611	1	2	yes	—
134	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
135	C₂×C₃×C₄	#626	1	2	yes	—
136	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
137	C₂×Dic₃	—	18	4	no	—
138	C₂×Dic₃	—	18	2	no	—
139	C₂×Dic₃	—	18	2	no	—
140	D₁₂	—	2	4	no	—
141	D₁₂	—	4	4	no	—
142	Dic₆	—	18	4	no	—
143	Dic₆	—	18	4	no	—
144	Dic₆	—	36	4	no	—
145	C₃×C₈	—	9	4	no	—
146	C₃⋊C₈	—	3	4	no	—
147	C₂×C₃²	C₂×#214	1	2	yes	—
148	C₂×C₃²	#940	1	1	yes	characteristic
149	C₃×D₃	—	8	2	no	—
150	C₂×C₉	—	2	2	no	—
151	C₂×C₉	—	2	1	no	—
152	C₂×C₉	—	24	2	no	—
153	C₂×C₉	#874	1	1	yes	characteristic
154	C₂×C₉	C₂×#204	1	2	yes	—
155	C₂×Q₈	—	27	2	no	—
156	C₂×D₄	—	3	2	no	—
157	C₂×D₄	—	6	2	no	—
158	SD₁₆	—	54	8	no	—
159	C₂×C₈	—	9	2	no	—
160	C₄⋊C₄	—	54	2	no	—
161	C₄²	D₃×D₉	1	1	yes	characteristic
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	12	4	no	—
164	C₂²×C₃	—	12	2	no	—
165	C₂²×C₃	C₂×#611	1	1	yes	characteristic
166	C₂²×C₃	C₂×#626	1	1	yes	characteristic
167	C₂×D₃	—	4	4	no	—
168	C₂×D₃	—	4	2	no	—
169	C₃×C₄	—	2	2	no	—
170	C₃×C₄	—	2	4	no	—
171	C₃×C₄	#3672	1	4	yes	complemented
172	C₃×C₄	D₄×D₉	1	2	yes	—
173	C₃×C₄	#3716	1	4	yes	—
174	C₃×C₄	D₃×D₁₂	1	2	yes	—
175	Dic₃	—	36	4	no	—
176	Dic₃	—	36	2	no	—
177	Dic₃	—	36	2	no	—
178	C₃²	#5076	1	1	yes	characteristic
179	C₉	—	2	1	no	—
180	C₉	#4693	1	1	yes	characteristic, complemented
181	C₂³	—	6	2	no	—
182	Q₈	—	54	4	no	—
183	D₄	—	6	4	no	—
184	D₄	—	12	4	no	—
185	C₂×C₄	—	54	2	no	—
186	C₂×C₄	#6195	1	2	yes	—
187	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
188	C₈	—	9	4	no	—
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	2	2	no	—
191	C₂×C₃	—	24	2	no	—
192	C₂×C₃	C₂×#3672	1	2	yes	—
193	C₂×C₃	#63502	1	1	yes	characteristic
194	C₂×C₃	C₂×#3716	1	2	yes	—
195	C₂×C₃	#63911	1	1	yes	characteristic
196	D₃	—	8	2	no	—
197	C₂²	—	12	2	no	—
198	C₂²	—	12	4	no	—
199	C₂²	C₂×#6195	1	1	yes	characteristic
200	C₄	—	108	2	no	—
201	C₄	#88424	1	4	yes	complemented
202	C₄	D₉×D₁₂	1	2	yes	—
203	C₃	—	2	1	no	—
204	C₃	#10588753	1	1	yes	characteristic, complemented
205	C₃	#10590369	1	1	yes	characteristic
206	C₂	—	24	2	no	—
207	C₂	C₂×#88424	1	2	yes	—
208	C₂	#11721040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11721706

C₂

yes

maximal, characteristic, complemented

C₉×#874

C₂

yes

maximal, characteristic

C₂×#88424

C₂

yes

maximal, complemented

C₃×C₄×C₉⋊C₈

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

—

C₂×C₉×D₁₂

—

C₂×C₉×D₁₂

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₉)

—

C₂×C₃×C₉⋊C₈

C₂²

yes

—

#88424

—

#64078

—

C₃×#874

D₃

yes

characteristic

C₂×#3716

—

C₃×C₄×C₃⋊C₈

—

—

—

—

—

C₄×C₉⋊C₈

D₃

yes

characteristic, complemented

C₂²×D₃×C₉

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₃⋊Dic₉

—

#6226

—

C₉×D₁₂

—

C₉×D₁₂

—

C₃×C₉⋊C₈

D₄

yes

—

—

—

C₂×C₃×D₁₂

—

C₂×C₃×D₁₂

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃×C₃⋊C₈

—

#3716

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×Dic₁₈

—

C₂×Dic₁₈

—

C₄²×C₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

#3672

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₂×C₉⋊C₈

C₂×D₃

yes

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₂×D₃×C₉

—

C₂×D₃×C₉

—

C₃×C₄×C₉

SD₁₆

yes

complemented

C₃×C₄×C₉

C₂×D₄

yes

—

C₃⋊Dic₉

—

—

—

—

D₉

yes

—

—

—

C₃×C₄×C₈

—

C₄×C₃⋊C₈

—

C₂²×C₃×D₃

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₃×D₁₂

—

C₃×D₁₂

—

C₂³×C₉

—

C₃×C₃⋊C₈

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

D₁₂

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₉⋊C₈

D₁₂

yes

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

C₂×SD₁₆

yes

—

D₃×C₉

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×D₁₂

—

C₂×D₁₂

C₂×D₉

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

C₂×D₉

yes

characteristic

C₃×C₄²

D₃²

yes

characteristic

100

#214

—

101

C₄⋊Dic₃

—

102

C₄⋊Dic₃

—

103

C₄⋊Dic₃

—

104

C₂×C₃⋊C₈

—

105

#204

—

106

C₂²×C₃²

C₂×#67

yes

characteristic

107

C₂×C₃×D₃

—

108

C₂×C₃×D₃

—

109

C₃²×C₄

#214

yes

—

110

C₃²×C₄

D₃×D₄

yes

—

111

C₃⋊Dic₃

—

112

C₂²×C₉

—

113

C₂²×C₉

—

114

C₂²×C₉

—

115

C₂²×C₉

C₂×D₁₂

yes

characteristic

116

C₄×C₉

—

117

C₄×C₉

—

118

C₄×C₉

#204

yes

complemented

119

C₄×C₉

C₂×D₁₂

yes

—

120

Dic₉

—

121

Dic₉

—

122

C₂×SD₁₆

—

123

C₄⋊Q₈

—

124

#127

—

125

C₄×C₈

—

126

C₃×C₉

#492

yes

characteristic, complemented, Sylow

127

C₂³×C₃

—

128

C₂²×D₃

—

129

C₃×D₄

—

130

C₃×D₄

—

131

C₂×C₃×C₄

—

132

C₂×C₃×C₄

—

133

C₂×C₃×C₄

#611

yes

—

134

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

135

C₂×C₃×C₄

#626

yes

—

136

C₂×C₃×C₄

C₂×D₃²

yes

characteristic

137

C₂×Dic₃

—

138

C₂×Dic₃

—

139

C₂×Dic₃

—

140

D₁₂

—

141

D₁₂

—

142

Dic₆

—

143

Dic₆

—

144

Dic₆

—

145

C₃×C₈

—

146

C₃⋊C₈

—

147

C₂×C₃²

C₂×#214

yes

—

148

C₂×C₃²

#940

yes

characteristic

149

C₃×D₃

—

150

C₂×C₉

—

151

C₂×C₉

—

152

C₂×C₉

—

153

C₂×C₉

#874

yes

characteristic

154

C₂×C₉

C₂×#204

yes

—

155

C₂×Q₈

—

156

C₂×D₄

—

157

C₂×D₄

—

158

SD₁₆

—

159

C₂×C₈

—

160

C₄⋊C₄

—

161

C₄²

D₃×D₉

yes

characteristic

162

C₂²×C₃

—

163

C₂²×C₃

—

164

C₂²×C₃

—

165

C₂²×C₃

C₂×#611

yes

characteristic

166

C₂²×C₃

C₂×#626

yes

characteristic

167

C₂×D₃

—

168

C₂×D₃

—

169

C₃×C₄

—

170

C₃×C₄

—

171

C₃×C₄

#3672

yes

complemented

172

C₃×C₄

D₄×D₉

yes

—

173

C₃×C₄

#3716

yes

—

174

C₃×C₄

D₃×D₁₂

yes

—

175

Dic₃

—

176

Dic₃

—

177

Dic₃

—

178

C₃²

#5076

yes

characteristic

179

C₉

—

180

C₉

#4693

yes

characteristic, complemented

181

C₂³

—

182

Q₈

—

183

D₄

—

184

D₄

—

185

C₂×C₄

—

186

C₂×C₄

#6195

yes

—

187

C₂×C₄

C₂×D₃×D₉

yes

characteristic

188

C₈

—

189

C₂×C₃

—

190

C₂×C₃

—

191

C₂×C₃

—

192

C₂×C₃

C₂×#3672

yes

—

193

C₂×C₃

#63502

yes

characteristic

194

C₂×C₃

C₂×#3716

yes

—

195

C₂×C₃

#63911

yes

characteristic

196

D₃

—

197

C₂²

—

198

C₂²

—

199

C₂²

C₂×#6195

yes

characteristic

200

C₄

—

108

—

201

C₄

#88424

yes

complemented

202

C₄

D₉×D₁₂

yes

—

203

C₃

—

204

C₃

#10588753

yes

characteristic, complemented

205

C₃

#10590369

yes

characteristic

206

C₂

—

207

C₂

C₂×#88424

yes

—

208

C₂

#11721040

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6195
commutator, cosocle	C₂×C₃×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₃×C₄²×C₉	C₂²
socle	C₂²×C₃²	C₂×#67
prime radical	C₃×#874	D₃
hypercenter	C₄²	D₃×D₉
hypocenter	C₃×C₉	#492
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#6195

commutator, cosocle

C₂×C₃×C₄×C₉

C₂³

Frattini

C₂×C₃×C₄

C₂×D₃²

Fitting

C₃×C₄²×C₉

C₂²

socle

C₂²×C₃²

C₂×#67

prime radical

C₃×#874

D₃

hypercenter

C₄²

D₃×D₉

hypocenter

C₃×C₉

#492

LCS

G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
1. #11721706	C₂	semidirect
3. C₂×#88424	C₂	semidirect
4. C₃×C₄×C₉⋊C₈	C₂	semidirect
22. C₄×C₉⋊C₈	D₃	semidirect
30. C₃×C₉⋊C₈	D₄	semidirect
53. C₃×C₄×C₉	SD₁₆	semidirect
84. C₉⋊C₈	D₁₂	semidirect
118. C₄×C₉	#204	semidirect
126. C₃×C₉	#492	semidirect
171. C₃×C₄	#3672	semidirect
180. C₉	#4693	semidirect
201. C₄	#88424	semidirect
204. C₃	#10588753	semidirect

H ≅ G/N

type

1. #11721706

C₂

semidirect

3. C₂×#88424

C₂

semidirect

4. C₃×C₄×C₉⋊C₈

C₂

semidirect

22. C₄×C₉⋊C₈

D₃

semidirect

30. C₃×C₉⋊C₈

D₄

semidirect

53. C₃×C₄×C₉

SD₁₆

semidirect

84. C₉⋊C₈

D₁₂

semidirect

118. C₄×C₉

#204

semidirect

126. C₃×C₉

#492

semidirect

171. C₃×C₄

#3672

semidirect

180. C₉

#4693

semidirect

201. C₄

#88424

semidirect

204. C₃

#10588753

semidirect

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

subgroup	index
#11721706	2
C₉×#874	2
C₂×#88424	2
C₃×C₄×C₉⋊C₈	2
#10590369	3
#10588753	3

subgroup

index

C₃×C₄×C₉⋊C₈

Subgroups — #13137249

Conjugacy classes of subgroups

Complements

Maximal subgroups