Group #13137141 — subgroups

Subgroups — #13137141

Order 1728.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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
2	C₉×#869	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11720298	C₂	1	4	yes	maximal
5	#10590287	—	3	1	no	maximal
6	#10588759	—	3	1	no	maximal
7	C₂×#6226	—	2	1	no	—
8	C₂×C₉×Dic₆	C₂²	1	2	yes	—
9	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
11	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
12	C₉×C₄⋊Dic₃	—	2	1	no	—
13	C₉×C₄⋊Dic₃	—	2	1	no	—
14	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
15	#64078	—	3	1	no	—
16	C₃×#869	D₃	1	1	yes	characteristic
17	C₃×C₄×C₃⋊C₈	—	3	1	no	—
18	#63566	—	3	4	no	—
19	C₉×C₄⋊Q₈	—	3	1	no	—
20	#63431	—	3	1	no	—
21	#63431	—	6	1	no	—
22	#63395	—	3	4	no	—
23	C₄×C₉⋊C₈	D₃	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₂×C₃⋊Dic₉	—	2	2	no	—
28	#6226	—	4	2	no	—
29	C₂×C₉×Dic₃	—	2	2	no	—
30	C₉×Dic₆	—	2	4	no	—
31	C₃×C₉⋊C₈	—	2	2	no	—
32	#5079	—	9	1	no	—
33	#4681	—	9	1	no	—
34	C₂×#634	—	6	1	no	—
35	C₂×C₃×Dic₆	C₂×D₃	1	2	yes	—
36	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
38	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
39	C₃×C₄⋊Dic₃	—	2	1	no	—
40	C₃×C₄⋊Dic₃	—	2	1	no	—
41	C₂×C₃×C₃⋊C₈	—	3	2	no	—
42	C₂×Q₈×C₉	—	3	2	no	—
43	C₂×Dic₁₈	—	6	2	no	—
44	C₂×Dic₁₈	—	6	1	no	—
45	C₉×C₄⋊C₄	—	6	1	no	—
46	C₉×C₄⋊C₄	—	6	1	no	—
47	C₄²×C₉	—	2	1	no	—
48	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
49	C₄⋊Dic₉	—	3	2	no	—
50	C₄⋊Dic₉	—	6	2	no	—
51	C₄⋊Dic₉	—	6	2	no	—
52	C₄⋊Dic₉	—	6	1	no	—
53	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
54	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
55	C₃×C₄×C₉	SD₁₆	1	2	yes	—
56	C₃×C₄×C₉	Q₁₆	1	2	yes	—
57	C₃×C₄×C₉	#41	1	2	yes	—
58	C₃⋊Dic₉	—	4	2	no	—
59	C₉×Dic₃	—	4	2	no	—
60	C₃×C₄⋊Q₈	—	3	1	no	—
61	#869	—	9	1	no	—
62	#869	—	9	1	no	—
63	#869	—	18	1	no	—
64	#869	D₉	1	1	yes	characteristic
65	C₃×C₄×C₈	—	9	1	no	—
66	#835	—	9	4	no	—
67	#816	—	9	4	no	—
68	C₄×C₃⋊C₈	—	3	1	no	—
69	C₂×C₃²×C₄	#67	1	1	yes	characteristic
70	C₂×C₃²×C₄	#67	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂×C₃⋊Dic₃	—	6	2	no	—
73	#634	—	12	2	no	—
74	C₂×C₃×Dic₃	—	2	2	no	—
75	C₃×Dic₆	—	2	4	no	—
76	C₃×C₃⋊C₈	—	6	2	no	—
77	Q₈×C₉	—	6	4	no	—
78	C₂×C₄×C₉	—	2	1	no	—
79	C₂×C₄×C₉	—	2	1	no	—
80	C₂×C₄×C₉	—	2	1	no	—
81	C₂×C₄×C₉	—	6	2	no	—
82	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
83	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
85	C₂×Dic₉	—	6	4	no	—
86	C₂×Dic₉	—	6	2	no	—
87	Dic₁₈	—	12	4	no	—
88	Dic₁₈	—	12	2	no	—
89	C₉⋊C₈	—	2	2	no	—
90	#487	—	27	1	no	Sylow
91	C₂×C₃×C₉	#124	1	1	yes	characteristic
92	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₃×Q₈	—	3	2	no	—
95	C₂×Dic₆	—	18	2	no	—
96	C₂×Dic₆	—	18	1	no	—
97	C₂×Dic₆	—	18	1	no	—
98	C₂×Dic₆	C₂×D₉	1	2	yes	—
99	C₂×C₃×C₈	—	9	2	no	—
100	C₃×C₄⋊C₄	—	6	1	no	—
101	C₃×C₄⋊C₄	—	6	1	no	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
104	C₃×C₄²	D₃²	1	1	yes	characteristic
105	C₄⋊Dic₃	—	2	1	no	—
106	C₄⋊Dic₃	—	2	1	no	—
107	C₄⋊Dic₃	—	9	2	no	—
108	C₄⋊Dic₃	—	9	2	no	—
109	C₄⋊Dic₃	—	18	2	no	—
110	C₄⋊Dic₃	—	18	2	no	—
111	C₄⋊Dic₃	—	18	1	no	—
112	C₄⋊Dic₃	—	18	1	no	—
113	C₂×C₃⋊C₈	—	3	2	no	—
114	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
115	C₃²×C₄	#215	1	2	yes	—
116	C₃²×C₄	#216	1	2	yes	—
117	C₃²×C₄	#237	1	2	yes	—
118	C₃⋊Dic₃	—	12	2	no	—
119	C₃×Dic₃	—	4	2	no	—
120	C₂²×C₉	—	2	1	no	—
121	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
122	C₄×C₉	—	2	2	no	—
123	C₄×C₉	—	2	2	no	—
124	C₄×C₉	—	2	2	no	—
125	C₄×C₉	—	12	2	no	—
126	C₄×C₉	#204	1	2	yes	—
127	C₄×C₉	Dic₁₂	1	2	yes	—
128	C₄×C₉	#235	1	2	yes	—
129	Dic₉	—	12	4	no	—
130	Dic₉	—	12	2	no	—
131	C₄⋊Q₈	—	3	1	no	—
132	C₄⋊Q₈	—	27	1	no	—
133	#103	—	27	4	no	—
134	C₄×C₈	—	9	1	no	—
135	C₃×C₉	#487	1	1	yes	characteristic, complemented, Sylow
136	C₃×Q₈	—	6	4	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	6	2	no	—
141	C₂×C₃×C₄	#611	1	1	yes	characteristic
142	C₂×C₃×C₄	#611	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
144	C₂×C₃×C₄	#626	1	1	yes	characteristic
145	C₂×C₃×C₄	#626	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
147	C₂×Dic₃	—	2	2	no	—
148	C₂×Dic₃	—	18	4	no	—
149	C₂×Dic₃	—	18	2	no	—
150	C₂×Dic₃	—	18	2	no	—
151	Dic₆	—	2	4	no	—
152	Dic₆	—	36	4	no	—
153	Dic₆	—	36	2	no	—
154	Dic₆	—	36	2	no	—
155	C₃×C₈	—	18	2	no	—
156	C₃⋊C₈	—	6	2	no	—
157	C₂×C₃²	#936	1	1	yes	characteristic
158	C₂×C₃²	C₂×#215	1	1	yes	characteristic
159	C₂×C₃²	C₂×#216	1	1	yes	characteristic
160	C₂×C₉	—	2	1	no	—
161	C₂×C₉	—	2	1	no	—
162	C₂×C₉	—	2	1	no	—
163	C₂×C₉	#875	1	1	yes	characteristic
164	C₂×C₉	C₂×#204	1	1	yes	characteristic
165	C₂×C₉	C₂×Dic₁₂	1	1	yes	characteristic
166	C₂×Q₈	—	3	2	no	—
167	C₂×Q₈	—	54	1	no	—
168	C₂×C₈	—	9	2	no	—
169	C₄⋊C₄	—	6	1	no	—
170	C₄⋊C₄	—	6	1	no	—
171	C₄⋊C₄	—	27	2	no	—
172	C₄⋊C₄	—	54	1	no	—
173	C₄²	D₃×D₉	1	1	yes	characteristic
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	C₂×#611	1	1	yes	characteristic
176	C₂²×C₃	C₂×#626	1	1	yes	characteristic
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	12	2	no	—
181	C₃×C₄	#3674	1	2	yes	—
182	C₃×C₄	#3675	1	2	yes	—
183	C₃×C₄	#3699	1	2	yes	—
184	C₃×C₄	#3717	1	2	yes	—
185	C₃×C₄	#3719	1	2	yes	—
186	C₃×C₄	#3795	1	2	yes	—
187	Dic₃	—	4	2	no	—
188	Dic₃	—	36	4	no	—
189	Dic₃	—	36	2	no	—
190	Dic₃	—	36	2	no	—
191	C₃²	#5079	1	1	yes	characteristic
192	C₉	—	2	1	no	—
193	C₉	#4681	1	1	yes	characteristic, complemented
194	Q₈	—	6	4	no	—
195	Q₈	—	108	2	no	—
196	C₂×C₄	—	6	2	no	—
197	C₂×C₄	—	54	2	no	—
198	C₂×C₄	#6195	1	1	yes	characteristic
199	C₂×C₄	#6195	1	1	yes	characteristic
200	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
201	C₈	—	18	2	no	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	#63498	1	1	yes	characteristic
206	C₂×C₃	C₂×#3674	1	1	yes	characteristic
207	C₂×C₃	C₂×#3675	1	1	yes	characteristic
208	C₂×C₃	C₂×#3717	1	1	yes	characteristic
209	C₂×C₃	C₂×#3719	1	1	yes	characteristic
210	C₂×C₃	#63860	1	1	yes	characteristic
211	C₂²	C₂×#6195	1	1	yes	characteristic
212	C₄	—	12	2	no	—
213	C₄	—	108	2	no	—
214	C₄	#88426	1	2	yes	—
215	C₄	#88428	1	2	yes	—
216	C₄	#88638	1	2	yes	—
217	C₃	—	2	1	no	—
218	C₃	#10588759	1	1	yes	characteristic, complemented
219	C₃	#10590287	1	1	yes	characteristic
220	C₂	C₂×#88426	1	1	yes	characteristic
221	C₂	C₂×#88428	1	1	yes	characteristic
222	C₂	#11720964	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11721706

C₂

yes

maximal, characteristic

C₉×#869

C₂

yes

maximal, characteristic

C₃×C₄×C₉⋊C₈

C₂

yes

maximal, characteristic

#11720298

C₂

yes

—

—

—

C₂×C₉×Dic₆

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₉)

—

C₄⋊(C₃⋊Dic₉)

C₂²

yes

—

C₉×C₄⋊Dic₃

—

C₉×C₄⋊Dic₃

—

C₂×C₃×C₉⋊C₈

C₂²

yes

—

#64078

—

C₃×#869

D₃

yes

characteristic

C₃×C₄×C₃⋊C₈

—

#63566

—

C₉×C₄⋊Q₈

—

#63431

—

#63431

—

#63395

—

C₄×C₉⋊C₈

D₃

yes

characteristic

C₂×C₃×C₄×C₉

D₄

yes

characteristic

C₂×C₃×C₄×C₉

D₄

yes

characteristic

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₃⋊Dic₉

—

#6226

—

C₂×C₉×Dic₃

—

C₉×Dic₆

—

C₃×C₉⋊C₈

—

#5079

—

#4681

—

C₂×#634

—

C₂×C₃×Dic₆

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

—

C₂×C₃×C₃⋊C₈

—

C₂×Q₈×C₉

—

C₂×Dic₁₈

—

C₂×Dic₁₈

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₂×C₉⋊C₈

C₂×D₃

yes

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₃×C₄×C₉

SD₁₆

yes

—

C₃×C₄×C₉

Q₁₆

yes

—

C₃×C₄×C₉

#41

yes

—

C₃⋊Dic₉

—

C₉×Dic₃

—

C₃×C₄⋊Q₈

—

—

—

—

D₉

yes

characteristic

C₃×C₄×C₈

—

#835

—

#816

—

C₄×C₃⋊C₈

—

C₂×C₃²×C₄

yes

C₂×C₃²×C₄

yes

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

D₁₂

yes

characteristic

C₂×C₄×C₉

D₁₂

yes

characteristic

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₉⋊C₈

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

C₂×SD₁₆

yes

characteristic

C₂×C₃×C₉

C₂×Q₁₆

yes

characteristic

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

C₂×D₉

yes

—

C₂×C₃×C₈

—

100

C₃×C₄⋊C₄

—

101

C₃×C₄⋊C₄

—

102

C₃×C₄²

—

103

C₃×C₄²

C₂×D₉

yes

characteristic

104

C₃×C₄²

D₃²

yes

characteristic

105

C₄⋊Dic₃

—

106

C₄⋊Dic₃

—

107

C₄⋊Dic₃

—

108

C₄⋊Dic₃

—

109

C₄⋊Dic₃

—

110

C₄⋊Dic₃

—

111

C₄⋊Dic₃

—

112

C₄⋊Dic₃

—

113

C₂×C₃⋊C₈

—

114

C₂²×C₃²

C₂×#67

yes

characteristic

115

C₃²×C₄

#215

yes

—

116

C₃²×C₄

#216

yes

—

117

C₃²×C₄

#237

yes

—

118

C₃⋊Dic₃

—

119

C₃×Dic₃

—

120

C₂²×C₉

—

121

C₂²×C₉

C₂×D₁₂

yes

characteristic

122

C₄×C₉

—

123

C₄×C₉

—

124

C₄×C₉

—

125

C₄×C₉

—

126

C₄×C₉

#204

yes

—

127

C₄×C₉

Dic₁₂

yes

—

128

C₄×C₉

#235

yes

—

129

Dic₉

—

130

Dic₉

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

133

#103

—

134

C₄×C₈

—

135

C₃×C₉

#487

yes

characteristic, complemented, Sylow

136

C₃×Q₈

—

137

C₂×C₃×C₄

—

138

C₂×C₃×C₄

—

139

C₂×C₃×C₄

—

140

C₂×C₃×C₄

—

141

C₂×C₃×C₄

#611

yes

characteristic

142

C₂×C₃×C₄

#611

yes

characteristic

143

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

144

C₂×C₃×C₄

#626

yes

characteristic

145

C₂×C₃×C₄

#626

yes

characteristic

146

C₂×C₃×C₄

C₂×D₃²

yes

characteristic

147

C₂×Dic₃

—

148

C₂×Dic₃

—

149

C₂×Dic₃

—

150

C₂×Dic₃

—

151

Dic₆

—

152

Dic₆

—

153

Dic₆

—

154

Dic₆

—

155

C₃×C₈

—

156

C₃⋊C₈

—

157

C₂×C₃²

#936

yes

characteristic

158

C₂×C₃²

C₂×#215

yes

characteristic

159

C₂×C₃²

C₂×#216

yes

characteristic

160

C₂×C₉

—

161

C₂×C₉

—

162

C₂×C₉

—

163

C₂×C₉

#875

yes

characteristic

164

C₂×C₉

C₂×#204

yes

characteristic

165

C₂×C₉

C₂×Dic₁₂

yes

characteristic

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂×C₈

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄²

D₃×D₉

yes

characteristic

174

C₂²×C₃

—

175

C₂²×C₃

C₂×#611

yes

characteristic

176

C₂²×C₃

C₂×#626

yes

characteristic

177

C₃×C₄

—

178

C₃×C₄

—

179

C₃×C₄

—

180

C₃×C₄

—

181

C₃×C₄

#3674

yes

—

182

C₃×C₄

#3675

yes

—

183

C₃×C₄

#3699

yes

—

184

C₃×C₄

#3717

yes

—

185

C₃×C₄

#3719

yes

—

186

C₃×C₄

#3795

yes

—

187

Dic₃

—

188

Dic₃

—

189

Dic₃

—

190

Dic₃

—

191

C₃²

#5079

yes

characteristic

192

C₉

—

193

C₉

#4681

yes

characteristic, complemented

194

Q₈

—

195

Q₈

—

108

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

#6195

yes

characteristic

199

C₂×C₄

#6195

yes

characteristic

200

C₂×C₄

C₂×D₃×D₉

yes

characteristic

201

C₈

—

202

C₂×C₃

—

203

C₂×C₃

—

204

C₂×C₃

—

205

C₂×C₃

#63498

yes

characteristic

206

C₂×C₃

C₂×#3674

yes

characteristic

207

C₂×C₃

C₂×#3675

yes

characteristic

208

C₂×C₃

C₂×#3717

yes

characteristic

209

C₂×C₃

C₂×#3719

yes

characteristic

210

C₂×C₃

#63860

yes

characteristic

211

C₂²

C₂×#6195

yes

characteristic

212

C₄

—

213

C₄

—

108

—

214

C₄

#88426

yes

—

215

C₄

#88428

yes

—

216

C₄

#88638

yes

—

217

C₃

—

218

C₃

#10588759

yes

characteristic, complemented

219

C₃

#10590287

yes

characteristic

220

C₂

C₂×#88426

yes

characteristic

221

C₂

C₂×#88428

yes

characteristic

222

C₂

#11720964

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, prime radical	C₂²×C₃²	C₂×#67
hypercenter	C₄²	D₃×D₉
hypocenter	C₃×C₉	#487
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, prime radical

C₂²×C₃²

C₂×#67

hypercenter

C₄²

D₃×D₉

hypocenter

C₃×C₉

#487

LCS

G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
135. C₃×C₉	#487	semidirect
193. C₉	#4681	semidirect
218. C₃	#10588759	semidirect

H ≅ G/N

type

135. C₃×C₉

#487

semidirect

193. C₉

#4681

semidirect

218. C₃

#10588759

semidirect

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

subgroup	index
#11721706	2
C₉×#869	2
C₃×C₄×C₉⋊C₈	2
#11720298	2
#10590287	3
#10588759	3

subgroup

index

C₃×C₄×C₉⋊C₈

Subgroups — #13137141

Conjugacy classes of subgroups

Complements

Maximal subgroups