Group #13137251 — subgroups

Subgroups — #13137251

Order 1728.

412 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₉×#875	C₂	1	1	yes	maximal, characteristic
2	C₃×#63431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88427	C₂	1	2	yes	maximal, complemented
4	C₂×#88423	C₂	1	2	yes	maximal
5	#11720516	C₂	1	1	yes	maximal, characteristic, complemented
6	#10590376	—	3	1	no	maximal
7	#10588737	—	3	1	no	maximal
8	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
9	C₂×C₉×Dic₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×Dic₁₈	C₂²	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
13	C₉×(C₂×D₃)⋊C₄	—	2	2	no	—
14	C₃×C₄⋊Dic₉	—	2	2	no	—
15	#88427	—	2	4	no	—
16	#88423	—	2	4	no	—
17	C₃×#875	D₃	1	1	yes	characteristic
18	C₃×#869	—	3	1	no	—
19	C₂×#3718	—	3	2	no	—
20	C₂×#3715	—	3	2	no	—
21	#63630	—	3	1	no	—
22	C₉×#124	—	3	1	no	—
23	C₂×#3674	—	3	2	no	—
24	C₂×#3672	—	3	2	no	—
25	#63431	D₃	1	1	yes	characteristic, complemented
26	#63362	—	3	1	no	—
27	#63362	—	6	1	no	—
28	C₂²×D₃×C₉	—	2	1	no	—
29	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₉×Dic₃	—	2	1	no	—
32	C₂×C₃×Dic₉	—	2	2	no	—
33	C₉×D₁₂	—	2	2	no	—
34	C₉×Dic₆	—	2	2	no	—
35	C₃×Dic₁₈	—	2	2	no	—
36	C₃×Dic₁₈	—	2	2	no	—
37	C₃⋊(C₉⋊C₈)	D₄	1	4	yes	—
38	#5086	—	9	1	no	—
39	#5060	—	9	1	no	—
40	C₂×C₃×D₁₂	C₂×D₃	1	1	yes	characteristic
41	C₂×C₃×Dic₆	—	3	2	no	—
42	C₂×C₃×Dic₆	C₂×D₃	1	1	yes	characteristic
43	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
45	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
46	C₃×C₄⋊Dic₃	—	6	2	no	—
47	#3718	—	6	4	no	—
48	#3715	—	6	4	no	—
49	C₂×Q₈×C₉	—	3	1	no	—
50	C₂×D₄×C₉	—	3	1	no	—
51	C₂×Dic₁₈	C₂×D₃	1	2	yes	—
52	C₉×C₂²⋊C₄	—	6	2	no	—
53	C₄²×C₉	—	2	1	no	—
54	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
55	#3674	—	6	4	no	—
56	#3672	—	6	4	no	—
57	C₄⋊Dic₉	—	2	2	no	—
58	C₂×C₉⋊C₈	—	3	2	no	—
59	C₂×C₉⋊C₈	—	6	2	no	—
60	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
61	C₂×D₃×C₉	—	4	1	no	—
62	C₂×D₃×C₉	—	4	2	no	—
63	C₃×C₄×C₉	—	2	2	no	—
64	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
65	C₉×Dic₃	—	4	1	no	—
66	C₃×Dic₉	—	4	2	no	—
67	C₃×C₄⋊Q₈	—	9	1	no	—
68	C₃×#124	—	3	1	no	—
69	C₂×#216	—	9	2	no	—
70	C₂×#216	—	9	2	no	—
71	C₂×#215	—	9	2	no	—
72	C₂×#214	—	9	2	no	—
73	#875	D₉	1	1	yes	characteristic
74	#869	—	3	1	no	—
75	#803	—	9	1	no	—
76	#803	—	9	1	no	—
77	#803	—	18	1	no	—
78	C₂²×C₃×D₃	—	2	1	no	—
79	C₂×C₃²×C₄	#67	1	2	yes	—
80	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂×C₃×Dic₃	—	2	1	no	—
82	C₂×C₃×Dic₃	—	6	2	no	—
83	C₃×D₁₂	—	2	2	no	—
84	C₃×Dic₆	—	2	2	no	—
85	C₃×Dic₆	—	6	2	no	—
86	C₃×Dic₆	—	6	2	no	—
87	C₂³×C₉	—	6	1	no	—
88	C₃⋊(C₃⋊C₈)	—	3	4	no	—
89	Q₈×C₉	—	6	2	no	—
90	D₄×C₉	—	6	2	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	6	1	no	—
94	C₂×C₄×C₉	#67	1	2	yes	—
95	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
96	C₂×Dic₉	—	2	2	no	—
97	Dic₁₈	—	2	2	no	—
98	Dic₁₈	—	2	2	no	—
99	C₉⋊C₈	—	3	4	no	—
100	C₉⋊C₈	—	6	4	no	—
101	#497	—	27	1	no	Sylow
102	C₂×C₃×C₉	#127	1	1	yes	characteristic
103	C₂×C₃×C₉	#137	1	2	yes	—
104	D₃×C₉	—	8	1	no	—
105	C₂×C₃×Q₈	—	3	1	no	—
106	C₂×C₃×Q₈	—	9	2	no	—
107	C₂×C₃×D₄	—	3	1	no	—
108	C₂×D₁₂	C₂×D₉	1	1	yes	characteristic
109	C₂×Dic₆	—	3	2	no	—
110	C₂×Dic₆	C₂×D₉	1	1	yes	characteristic
111	C₃×C₄⋊C₄	—	18	2	no	—
112	C₃×C₂²⋊C₄	—	6	2	no	—
113	C₃×C₄²	—	2	1	no	—
114	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
115	C₃×C₄²	D₃²	1	1	yes	characteristic
116	#216	—	18	4	no	—
117	#216	—	18	4	no	—
118	#215	—	18	4	no	—
119	#214	—	18	4	no	—
120	(C₂×D₃)⋊C₄	—	2	2	no	—
121	C₄⋊Dic₃	—	6	2	no	—
122	C₂×C₃⋊C₈	—	9	2	no	—
123	C₂×C₃⋊C₈	—	9	2	no	—
124	C₂×C₃⋊C₈	—	18	2	no	—
125	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
126	C₂×C₃×D₃	—	4	1	no	—
127	C₂×C₃×D₃	—	4	2	no	—
128	C₃²×C₄	—	2	2	no	—
129	C₃²×C₄	D₃×D₄	1	2	yes	—
130	C₃×Dic₃	—	4	1	no	—
131	C₃×Dic₃	—	12	2	no	—
132	C₂²×C₉	—	2	1	no	—
133	C₂²×C₉	—	12	1	no	—
134	C₂²×C₉	—	12	2	no	—
135	C₂²×C₉	C₂×#67	1	1	yes	characteristic
136	C₄×C₉	—	2	2	no	—
137	C₄×C₉	—	2	2	no	—
138	C₄×C₉	—	4	2	no	—
139	C₄×C₉	—	12	1	no	—
140	C₄×C₉	D₃×D₄	1	2	yes	—
141	Dic₉	—	4	2	no	—
142	C₂×Q₁₆	—	27	2	no	—
143	C₂×SD₁₆	—	27	2	no	—
144	C₄⋊Q₈	—	9	1	no	—
145	#124	—	3	1	no	—
146	#97	—	27	1	no	—
147	C₃×C₉	#497	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₃	—	6	1	no	—
149	C₂²×D₃	—	2	1	no	—
150	C₃×Q₈	—	6	2	no	—
151	C₃×Q₈	—	18	2	no	—
152	C₃×Q₈	—	18	2	no	—
153	C₃×D₄	—	6	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	6	1	no	—
157	C₂×C₃×C₄	—	18	2	no	—
158	C₂×C₃×C₄	#611	1	2	yes	—
159	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
160	C₂×C₃×C₄	#625	1	2	yes	—
161	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
162	C₂×Dic₃	—	2	1	no	—
163	C₂×Dic₃	—	6	2	no	—
164	D₁₂	—	2	2	no	—
165	Dic₆	—	2	2	no	—
166	Dic₆	—	6	2	no	—
167	Dic₆	—	6	2	no	—
168	C₃⋊C₈	—	9	4	no	—
169	C₃⋊C₈	—	9	4	no	—
170	C₃⋊C₈	—	18	4	no	—
171	C₂×C₃²	#940	1	1	yes	characteristic
172	C₂×C₃²	#951	1	2	yes	—
173	C₃×D₃	—	8	1	no	—
174	C₂×C₉	—	2	2	no	—
175	C₂×C₉	—	2	1	no	—
176	C₂×C₉	—	24	1	no	—
177	C₂×C₉	#940	1	1	yes	characteristic
178	C₂×C₉	#942	1	2	yes	—
179	C₂×Q₈	—	3	1	no	—
180	C₂×Q₈	—	9	2	no	—
181	C₂×D₄	—	3	1	no	—
182	Q₁₆	—	54	4	no	—
183	SD₁₆	—	54	4	no	—
184	C₂×C₈	—	27	2	no	—
185	C₄⋊C₄	—	18	2	no	—
186	C₂²⋊C₄	—	6	2	no	—
187	C₄²	D₃×D₉	1	1	yes	characteristic
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	12	2	no	—
190	C₂²×C₃	—	12	1	no	—
191	C₂²×C₃	C₂×#611	1	1	yes	characteristic
192	C₂²×C₃	C₂×#625	1	1	yes	characteristic
193	C₂×D₃	—	4	1	no	—
194	C₂×D₃	—	4	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	4	2	no	—
199	C₃×C₄	—	12	1	no	—
200	C₃×C₄	—	36	2	no	—
201	C₃×C₄	D₄×D₉	1	2	yes	—
202	C₃×C₄	#3802	1	2	yes	—
203	Dic₃	—	4	1	no	—
204	Dic₃	—	12	2	no	—
205	C₃²	#5060	1	1	yes	characteristic
206	C₉	—	2	1	no	—
207	C₉	#5086	1	1	yes	characteristic, complemented
208	C₂³	—	6	1	no	—
209	Q₈	—	6	2	no	—
210	Q₈	—	18	2	no	—
211	Q₈	—	18	2	no	—
212	D₄	—	6	2	no	—
213	C₂×C₄	—	6	1	no	—
214	C₂×C₄	—	18	2	no	—
215	C₂×C₄	#6194	1	2	yes	—
216	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
217	C₈	—	27	4	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	—	24	1	no	—
221	C₂×C₃	#63502	1	1	yes	characteristic
222	C₂×C₃	#63511	1	2	yes	—
223	C₂×C₃	#63827	1	2	yes	—
224	C₂×C₃	#63914	1	1	yes	characteristic
225	D₃	—	8	1	no	—
226	C₂²	—	12	1	no	—
227	C₂²	—	12	2	no	—
228	C₂²	C₂×#6194	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	12	1	no	—
231	C₄	—	36	2	no	—
232	C₄	#88646	1	2	yes	—
233	C₃	—	2	1	no	—
234	C₃	#10588737	1	1	yes	characteristic, complemented
235	C₃	#10590376	1	1	yes	characteristic
236	C₂	—	24	1	no	—
237	C₂	#11720887	1	2	yes	—
238	C₂	#11721043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#875

C₂

yes

maximal, characteristic

C₃×#63431

C₂

yes

maximal, characteristic, complemented

C₂×#88427

C₂

yes

maximal, complemented

C₂×#88423

C₂

yes

maximal

#11720516

C₂

yes

maximal, characteristic, complemented

—

—

C₂×C₉×D₁₂

C₂²

yes

characteristic

C₂×C₉×Dic₆

C₂²

yes

characteristic

C₂×C₃×Dic₁₈

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₂×C₃⋊(C₉⋊C₈)

C₂²

yes

—

C₉×(C₂×D₃)⋊C₄

—

C₃×C₄⋊Dic₉

—

#88427

—

#88423

—

C₃×#875

D₃

yes

—

—

—

—

—

—

—

D₃

yes

characteristic, complemented

#63362

—

#63362

—

C₂²×D₃×C₉

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₉×Dic₃

—

C₂×C₃×Dic₉

—

C₉×D₁₂

—

C₉×Dic₆

—

C₃×Dic₁₈

—

C₃×Dic₁₈

—

C₃⋊(C₉⋊C₈)

D₄

yes

—

#5086

—

#5060

—

C₂×C₃×D₁₂

C₂×D₃

yes

characteristic

C₂×C₃×Dic₆

—

C₂×C₃×Dic₆

C₂×D₃

yes

characteristic

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

—

C₃×(C₂×D₃)⋊C₄

—

C₃×C₄⋊Dic₃

—

#3718

—

#3715

—

C₂×Q₈×C₉

—

C₂×D₄×C₉

—

C₂×Dic₁₈

C₂×D₃

yes

—

C₉×C₂²⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

#3674

—

#3672

—

C₄⋊Dic₉

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₂×D₃×C₉

—

C₂×D₃×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×D₄

yes

—

C₉×Dic₃

—

C₃×Dic₉

—

C₃×C₄⋊Q₈

—

—

—

—

—

—

D₉

yes

—

—

—

—

C₂²×C₃×D₃

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×D₁₂

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃×Dic₆

—

C₂³×C₉

—

C₃⋊(C₃⋊C₈)

—

Q₈×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

#67

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₉⋊C₈

—

100

C₉⋊C₈

—

101

#497

—

Sylow

102

C₂×C₃×C₉

#127

yes

characteristic

103

C₂×C₃×C₉

#137

yes

—

104

D₃×C₉

—

105

C₂×C₃×Q₈

—

106

C₂×C₃×Q₈

—

107

C₂×C₃×D₄

—

108

C₂×D₁₂

C₂×D₉

yes

characteristic

109

C₂×Dic₆

—

110

C₂×Dic₆

C₂×D₉

yes

characteristic

111

C₃×C₄⋊C₄

—

112

C₃×C₂²⋊C₄

—

113

C₃×C₄²

—

114

C₃×C₄²

C₂×D₉

yes

characteristic

115

C₃×C₄²

D₃²

yes

characteristic

116

#216

—

117

#216

—

118

#215

—

119

#214

—

120

(C₂×D₃)⋊C₄

—

121

C₄⋊Dic₃

—

122

C₂×C₃⋊C₈

—

123

C₂×C₃⋊C₈

—

124

C₂×C₃⋊C₈

—

125

C₂²×C₃²

C₂×#67

yes

characteristic

126

C₂×C₃×D₃

—

127

C₂×C₃×D₃

—

128

C₃²×C₄

—

129

C₃²×C₄

D₃×D₄

yes

—

130

C₃×Dic₃

—

131

C₃×Dic₃

—

132

C₂²×C₉

—

133

C₂²×C₉

—

134

C₂²×C₉

—

135

C₂²×C₉

C₂×#67

yes

characteristic

136

C₄×C₉

—

137

C₄×C₉

—

138

C₄×C₉

—

139

C₄×C₉

—

140

C₄×C₉

D₃×D₄

yes

—

141

Dic₉

—

142

C₂×Q₁₆

—

143

C₂×SD₁₆

—

144

C₄⋊Q₈

—

145

#124

—

146

#97

—

147

C₃×C₉

#497

yes

characteristic, complemented, Sylow

148

C₂³×C₃

—

149

C₂²×D₃

—

150

C₃×Q₈

—

151

C₃×Q₈

—

152

C₃×Q₈

—

153

C₃×D₄

—

154

C₂×C₃×C₄

—

155

C₂×C₃×C₄

—

156

C₂×C₃×C₄

—

157

C₂×C₃×C₄

—

158

C₂×C₃×C₄

#611

yes

—

159

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

160

C₂×C₃×C₄

#625

yes

—

161

C₂×C₃×C₄

C₂×D₃²

yes

characteristic

162

C₂×Dic₃

—

163

C₂×Dic₃

—

164

D₁₂

—

165

Dic₆

—

166

Dic₆

—

167

Dic₆

—

168

C₃⋊C₈

—

169

C₃⋊C₈

—

170

C₃⋊C₈

—

171

C₂×C₃²

#940

yes

characteristic

172

C₂×C₃²

#951

yes

—

173

C₃×D₃

—

174

C₂×C₉

—

175

C₂×C₉

—

176

C₂×C₉

—

177

C₂×C₉

#940

yes

characteristic

178

C₂×C₉

#942

yes

—

179

C₂×Q₈

—

180

C₂×Q₈

—

181

C₂×D₄

—

182

Q₁₆

—

183

SD₁₆

—

184

C₂×C₈

—

185

C₄⋊C₄

—

186

C₂²⋊C₄

—

187

C₄²

D₃×D₉

yes

characteristic

188

C₂²×C₃

—

189

C₂²×C₃

—

190

C₂²×C₃

—

191

C₂²×C₃

C₂×#611

yes

characteristic

192

C₂²×C₃

C₂×#625

yes

characteristic

193

C₂×D₃

—

194

C₂×D₃

—

195

C₃×C₄

—

196

C₃×C₄

—

197

C₃×C₄

—

198

C₃×C₄

—

199

C₃×C₄

—

200

C₃×C₄

—

201

C₃×C₄

D₄×D₉

yes

—

202

C₃×C₄

#3802

yes

—

203

Dic₃

—

204

Dic₃

—

205

C₃²

#5060

yes

characteristic

206

C₉

—

207

C₉

#5086

yes

characteristic, complemented

208

C₂³

—

209

Q₈

—

210

Q₈

—

211

Q₈

—

212

D₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

#6194

yes

—

216

C₂×C₄

C₂×D₃×D₉

yes

characteristic

217

C₈

—

218

C₂×C₃

—

219

C₂×C₃

—

220

C₂×C₃

—

221

C₂×C₃

#63502

yes

characteristic

222

C₂×C₃

#63511

yes

—

223

C₂×C₃

#63827

yes

—

224

C₂×C₃

#63914

yes

characteristic

225

D₃

—

226

C₂²

—

227

C₂²

—

228

C₂²

C₂×#6194

yes

characteristic

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

#88646

yes

—

233

C₃

—

234

C₃

#10588737

yes

characteristic, complemented

235

C₃

#10590376

yes

characteristic

236

C₂

—

237

C₂

#11720887

yes

—

238

C₂

#11721043

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6194
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	C₂×C₃×D₁₂	C₂×D₃
hypercenter	C₄²	D₃×D₉
hypocenter	C₃×C₉	#497
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#6194

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

C₂×C₃×D₁₂

C₂×D₃

hypercenter

C₄²

D₃×D₉

hypocenter

C₃×C₉

#497

LCS

G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
2. C₃×#63431	C₂	semidirect
3. C₂×#88427	C₂	semidirect
5. #11720516	C₂	semidirect
25. #63431	D₃	semidirect
147. C₃×C₉	#497	semidirect
207. C₉	#5086	semidirect
234. C₃	#10588737	semidirect

H ≅ G/N

type

2. C₃×#63431

C₂

semidirect

3. C₂×#88427

C₂

semidirect

5. #11720516

C₂

semidirect

25. #63431

D₃

semidirect

147. C₃×C₉

#497

semidirect

207. C₉

#5086

semidirect

234. C₃

#10588737

semidirect

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

subgroup	index
C₉×#875	2
C₃×#63431	2
C₂×#88427	2
C₂×#88423	2
#11720516	2
#10590376	3
#10588737	3

subgroup

index

Subgroups — #13137251

Conjugacy classes of subgroups

Complements

Maximal subgroups