Group #13509203 — subgroups

Subgroups — #13509203

Order 1792.

834 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₂×#89691	C₂	1	4	yes	maximal, complemented
2	#11739244	C₂	1	2	yes	maximal, complemented
3	C₇×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729254	C₂	1	4	yes	maximal, complemented
5	#11729067	C₂	1	2	yes	maximal, complemented
6	#11726814	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#6482	—	2	2	no	—
9	C₂×#6482	C₂²	1	4	yes	—
10	#90416	—	2	2	no	—
11	C₂×C₇×Q₃₂	C₂²	1	4	yes	—
12	C₇×#494	C₂²	1	2	yes	—
13	#89963	—	2	2	no	—
14	#89963	C₂²	1	4	yes	—
15	C₂×#6402	C₂²	1	8	yes	—
16	#89857	—	2	2	no	—
17	#89691	—	2	16	no	—
18	C₂×D₁₁₂	C₂²	1	4	yes	—
19	C₂×D₇×C₁₆	C₂²	1	2	yes	—
20	#89467	C₂²	1	2	yes	—
21	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
22	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
23	C₁₆×Dic₇	C₂²	1	2	yes	—
24	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
25	#33713	—	7	1	no	maximal, Sylow
26	C₂×#1161	—	2	4	no	—
27	C₂×C₇×Q₁₆	—	2	2	no	—
28	C₂×C₇×Q₁₆	C₂³	1	4	yes	—
29	C₇×C₄⋊Q₈	—	2	2	no	—
30	#6510	—	2	4	no	—
31	C₂×#1143	—	2	4	no	—
32	C₂×#1143	—	2	4	no	—
33	#6482	—	2	16	no	—
34	#6482	—	4	8	no	—
35	C₂×D₅₆	—	2	2	no	—
36	C₂×D₅₆	C₂³	1	4	yes	—
37	C₂×C₈×D₇	D₄	1	1	yes	characteristic
38	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
39	#6458	—	4	4	no	—
40	#6456	—	4	4	no	—
41	#6437	—	2	2	no	—
42	C₄²×D₇	D₄	1	1	yes	characteristic
43	C₇×Q₃₂	—	2	8	no	—
44	C₂×C₇×C₁₆	C₂³	1	2	yes	—
45	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
46	#6402	—	2	16	no	—
47	C₈×Dic₇	D₄	1	1	yes	characteristic
48	C₈×Dic₇	C₂³	1	1	yes	characteristic
49	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
50	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
51	D₁₁₂	—	2	8	no	—
52	D₇×C₁₆	D₄	1	8	yes	complemented
53	C₂×#508	—	7	4	no	—
54	#3147	—	7	2	no	—
55	#2249	—	7	4	no	—
56	#2248	—	7	2	no	—
57	#2247	D₇	1	1	yes	characteristic, complemented
58	#2246	—	7	1	no	—
59	C₂×C₄×C₁₆	—	7	1	no	—
60	C₂×C₇×Q₈	—	2	4	no	—
61	#1161	—	4	8	no	—
62	#1161	—	4	8	no	—
63	C₂×D₂₈	—	2	4	no	—
64	C₂×D₂₈	—	4	4	no	—
65	C₂×D₂₈	—	4	4	no	—
66	C₂×C₄×D₇	—	4	4	no	—
67	C₂×C₄×D₇	D₈	1	2	yes	—
68	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
69	C₇×Q₁₆	—	2	8	no	—
70	C₇×Q₁₆	—	4	4	no	—
71	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
73	C₇×C₄⋊C₄	—	4	4	no	—
74	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
75	#1143	—	4	8	no	—
76	#1143	—	4	8	no	—
77	(C₂×D₇)⋊C₄	—	4	8	no	—
78	C₄×Dic₇	D₈	1	2	yes	—
79	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
82	D₅₆	—	2	8	no	—
83	D₅₆	—	4	4	no	—
84	C₈×D₇	D₈	1	4	yes	—
85	C₈×D₇	C₂×D₄	1	4	yes	—
86	C₇×C₁₆	C₂×D₄	1	4	yes	—
87	C₇⋊C₁₆	C₂×D₄	1	4	yes	—
88	C₂×#135	—	7	4	no	—
89	C₂×#135	—	14	2	no	—
90	#532	—	14	2	no	—
91	#508	—	14	16	no	—
92	C₂×Q₃₂	C₂×D₇	1	4	yes	—
93	C₂×SD₃₂	—	7	8	no	—
94	C₂×D₁₆	—	7	4	no	—
95	C₂²×C₁₆	—	7	2	no	—
96	#495	—	7	4	no	—
97	#495	—	14	2	no	—
98	#494	C₂×D₇	1	2	yes	—
99	#493	—	7	2	no	—
100	#492	—	14	2	no	—
101	C₂×C₄×C₈	—	7	1	no	—
102	C₄×C₁₆	—	7	2	no	—
103	C₄×C₁₆	—	7	1	no	—
104	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
105	C₂²×D₇	—	4	4	no	—
106	C₂²×D₇	C₂×D₈	1	1	yes	characteristic
107	C₇×Q₈	—	4	4	no	—
108	C₇×Q₈	—	4	4	no	—
109	C₂×C₄×C₇	—	4	4	no	—
110	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₇	C₂×D₈	1	2	yes	—
113	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
114	D₂₈	—	4	4	no	—
115	D₂₈	—	4	4	no	—
116	D₂₈	—	8	8	no	—
117	D₂₈	—	8	8	no	—
118	C₄×D₇	—	2	4	no	—
119	C₄×D₇	—	8	8	no	—
120	C₄×D₇	#127	1	2	yes	—
121	C₄×D₇	C₂×D₈	1	2	yes	—
122	C₇×C₈	C₂×D₈	1	2	yes	—
123	C₇×C₈	C₂²×D₄	1	2	yes	—
124	C₇⋊C₈	#127	1	2	yes	—
125	C₇⋊C₈	C₂×D₈	1	2	yes	—
126	C₂×#41	—	14	4	no	—
127	#135	—	14	16	no	—
128	#135	—	28	8	no	—
129	C₂×Q₁₆	—	2	2	no	—
130	C₂×Q₁₆	C₂²×D₇	1	4	yes	—
131	C₂×SD₁₆	—	14	4	no	—
132	C₂×SD₁₆	—	14	4	no	—
133	C₂×D₈	—	7	4	no	—
134	C₂×D₈	—	14	2	no	—
135	C₂²×C₈	—	7	1	no	—
136	C₂²×C₈	—	7	1	no	—
137	C₄⋊Q₈	—	2	2	no	—
138	#127	—	14	2	no	—
139	#124	—	14	4	no	—
140	#121	—	28	4	no	—
141	C₄×D₄	—	28	4	no	—
142	C₂×C₄²	—	7	1	no	—
143	Q₃₂	—	2	8	no	—
144	SD₃₂	—	14	16	no	—
145	D₁₆	—	14	8	no	—
146	C₂×C₁₆	—	7	2	no	—
147	C₂×C₁₆	—	7	8	no	—
148	C₂×C₁₆	C₂²×D₇	1	2	yes	—
149	C₄×C₈	—	7	1	no	—
150	C₄×C₈	—	7	1	no	—
151	C₄×C₈	—	7	1	no	—
152	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
153	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₇	—	2	2	no	—
155	C₂×D₇	—	8	4	no	—
156	C₂×D₇	—	8	8	no	—
157	C₂×D₇	#493	1	2	yes	—
158	C₄×C₇	—	8	4	no	—
159	C₄×C₇	#508	1	4	yes	complemented
160	C₄×C₇	C₂×#127	1	1	yes	characteristic
161	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
162	Dic₇	—	2	2	no	—
163	Dic₇	#493	1	2	yes	—
164	#41	—	28	8	no	—
165	#41	—	28	8	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×D₄	—	14	4	no	—
168	C₂×D₄	—	28	4	no	—
169	C₂×D₄	—	28	4	no	—
170	C₂²×C₄	—	7	1	no	—
171	C₂²×C₄	—	7	2	no	—
172	C₂²×C₄	—	28	4	no	—
173	Q₁₆	—	2	8	no	—
174	Q₁₆	—	4	4	no	—
175	SD₁₆	—	28	8	no	—
176	SD₁₆	—	28	8	no	—
177	D₈	—	14	8	no	—
178	D₈	—	28	4	no	—
179	C₂×C₈	—	7	1	no	—
180	C₂×C₈	—	7	1	no	—
181	C₂×C₈	—	7	4	no	—
182	C₂×C₈	—	7	4	no	—
183	C₂×C₈	D₄×D₇	1	1	yes	characteristic
184	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
185	C₄⋊C₄	—	4	4	no	—
186	C₂²⋊C₄	—	28	8	no	—
187	C₄²	—	7	2	no	—
188	C₄²	—	7	1	no	—
189	C₄²	D₄×D₇	1	1	yes	characteristic
190	C₁₆	—	7	4	no	—
191	C₁₆	D₄×D₇	1	4	yes	—
192	C₂×C₇	C₂×#493	1	1	yes	characteristic
193	C₂×C₇	C₂×#508	1	2	yes	—
194	D₇	—	2	2	no	—
195	D₇	—	16	4	no	—
196	C₂³	—	7	1	no	—
197	C₂³	—	28	4	no	—
198	Q₈	—	4	4	no	—
199	Q₈	—	4	4	no	—
200	D₄	—	28	4	no	—
201	D₄	—	28	4	no	—
202	D₄	—	56	8	no	—
203	D₄	—	56	8	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	7	1	no	—
206	C₂×C₄	—	7	2	no	—
207	C₂×C₄	—	7	2	no	—
208	C₂×C₄	—	7	2	no	—
209	C₂×C₄	—	14	4	no	—
210	C₂×C₄	—	56	8	no	—
211	C₂×C₄	D₇×D₈	1	2	yes	—
212	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
213	C₈	—	7	2	no	—
214	C₈	—	7	2	no	—
215	C₈	D₇×D₈	1	2	yes	—
216	C₈	C₂×D₄×D₇	1	2	yes	—
217	C₇	#33713	1	1	yes	characteristic, complemented, Sylow
218	C₂²	—	7	2	no	—
219	C₂²	—	14	2	no	—
220	C₂²	—	56	8	no	—
221	C₂²	—	56	4	no	—
222	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
223	C₄	—	7	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	14	2	no	—
226	C₄	#89691	1	4	yes	complemented
227	C₄	D₇×#127	1	1	yes	characteristic
228	C₄	C₂×D₇×D₈	1	1	yes	characteristic
229	C₂	—	14	2	no	—
230	C₂	—	112	4	no	—
231	C₂	D₇×#493	1	1	yes	characteristic
232	C₂	C₂×#89691	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#89691

C₂

yes

maximal, complemented

#11739244

C₂

yes

maximal, complemented

C₇×#2247

C₂

yes

maximal, characteristic, complemented

#11729254

C₂

yes

maximal, complemented

#11729067

C₂

yes

maximal, complemented

#11726814

C₂

yes

maximal, characteristic, complemented

C₄×D₇×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#6482

—

C₂×#6482

C₂²

yes

—

#90416

—

C₂×C₇×Q₃₂

C₂²

yes

—

C₇×#494

C₂²

yes

—

#89963

—

#89963

C₂²

yes

—

C₂×#6402

C₂²

yes

—

#89857

—

#89691

—

C₂×D₁₁₂

C₂²

yes

—

C₂×D₇×C₁₆

C₂²

yes

—

#89467

C₂²

yes

—

C₄×C₈×D₇

C₂²

yes

characteristic

C₄×C₇×C₁₆

C₂²

yes

characteristic

C₁₆×Dic₇

C₂²

yes

—

C₄×C₇⋊C₁₆

C₂²

yes

—

—

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

C₂³

yes

—

C₇×C₄⋊Q₈

—

—

—

—

—

—

C₂×D₅₆

—

C₂×D₅₆

C₂³

yes

—

C₂×C₈×D₇

D₄

yes

characteristic

C₂×C₈×D₇

C₂³

yes

—

—

—

C₄²×D₇

D₄

yes

characteristic

C₇×Q₃₂

—

C₂×C₇×C₁₆

C₂³

yes

—

C₄×C₇×C₈

C₂³

yes

characteristic

#6402

—

C₈×Dic₇

D₄

yes

characteristic

C₈×Dic₇

C₂³

yes

characteristic

C₂×C₇⋊C₁₆

C₂³

yes

—

C₄×C₇⋊C₈

D₄

yes

characteristic

D₁₁₂

—

D₇×C₁₆

D₄

yes

—

—

—

—

D₇

yes

characteristic, complemented

#2246

—

C₂×C₄×C₁₆

—

C₂×C₇×Q₈

—

#1161

—

#1161

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×C₄×D₇

—

C₂×C₄×D₇

D₈

yes

—

C₂×C₄×D₇

C₂×D₄

yes

characteristic

C₇×Q₁₆

—

C₇×Q₁₆

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂⁴

yes

characteristic

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

characteristic

#1143

—

#1143

—

(C₂×D₇)⋊C₄

—

C₄×Dic₇

D₈

yes

—

C₄×Dic₇

C₂×D₄

yes

characteristic

C₂×C₇⋊C₈

C₂×D₄

yes

characteristic

C₂×C₇⋊C₈

C₂×D₄

yes

characteristic

D₅₆

—

D₅₆

—

C₈×D₇

D₈

yes

—

C₈×D₇

C₂×D₄

yes

—

C₇×C₁₆

C₂×D₄

yes

—

C₇⋊C₁₆

C₂×D₄

yes

—

—

—

—

—

C₂×Q₃₂

C₂×D₇

yes

—

C₂×SD₃₂

—

C₂×D₁₆

—

C₂²×C₁₆

—

#495

—

#495

—

#494

C₂×D₇

yes

—

#493

—

100

#492

—

101

C₂×C₄×C₈

—

102

C₄×C₁₆

—

103

C₄×C₁₆

—

104

C₄×C₁₆

C₂×D₇

yes

characteristic

105

C₂²×D₇

—

106

C₂²×D₇

C₂×D₈

yes

characteristic

107

C₇×Q₈

—

108

C₇×Q₈

—

109

C₂×C₄×C₇

—

110

C₂×C₄×C₇

C₂×D₈

yes

—

111

C₂×C₄×C₇

C₂²×D₄

yes

characteristic

112

C₂×Dic₇

C₂×D₈

yes

—

113

C₂×Dic₇

C₂×D₈

yes

characteristic

114

D₂₈

—

115

D₂₈

—

116

D₂₈

—

117

D₂₈

—

118

C₄×D₇

—

119

C₄×D₇

—

120

C₄×D₇

#127

yes

—

121

C₄×D₇

C₂×D₈

yes

—

122

C₇×C₈

C₂×D₈

yes

—

123

C₇×C₈

C₂²×D₄

yes

—

124

C₇⋊C₈

#127

yes

—

125

C₇⋊C₈

C₂×D₈

yes

—

126

C₂×#41

—

127

#135

—

128

#135

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

C₂²×D₇

yes

—

131

C₂×SD₁₆

—

132

C₂×SD₁₆

—

133

C₂×D₈

—

134

C₂×D₈

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₄⋊Q₈

—

138

#127

—

139

#124

—

140

#121

—

141

C₄×D₄

—

142

C₂×C₄²

—

143

Q₃₂

—

144

SD₃₂

—

145

D₁₆

—

146

C₂×C₁₆

—

147

C₂×C₁₆

—

148

C₂×C₁₆

C₂²×D₇

yes

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

C₂²×D₇

yes

characteristic

153

C₂²×C₇

C₂²×D₈

yes

characteristic

154

C₂×D₇

—

155

C₂×D₇

—

156

C₂×D₇

—

157

C₂×D₇

#493

yes

—

158

C₄×C₇

—

159

C₄×C₇

#508

yes

complemented

160

C₄×C₇

C₂×#127

yes

characteristic

161

C₄×C₇

C₂²×D₈

yes

characteristic

162

Dic₇

—

163

Dic₇

#493

yes

—

164

#41

—

165

#41

—

166

C₂×Q₈

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂×D₄

—

170

C₂²×C₄

—

171

C₂²×C₄

—

172

C₂²×C₄

—

173

Q₁₆

—

174

Q₁₆

—

175

SD₁₆

—

176

SD₁₆

—

177

D₈

—

178

D₈

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

D₄×D₇

yes

characteristic

184

C₂×C₈

C₂³×D₇

yes

characteristic

185

C₄⋊C₄

—

186

C₂²⋊C₄

—

187

C₄²

—

188

C₄²

—

189

C₄²

D₄×D₇

yes

characteristic

190

C₁₆

—

191

C₁₆

D₄×D₇

yes

—

192

C₂×C₇

C₂×#493

yes

characteristic

193

C₂×C₇

C₂×#508

yes

—

194

D₇

—

195

D₇

—

196

C₂³

—

197

C₂³

—

198

Q₈

—

199

Q₈

—

200

D₄

—

201

D₄

—

202

D₄

—

203

D₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

D₇×D₈

yes

—

212

C₂×C₄

C₂×D₄×D₇

yes

characteristic

213

C₈

—

214

C₈

—

215

C₈

D₇×D₈

yes

—

216

C₈

C₂×D₄×D₇

yes

—

217

C₇

#33713

yes

characteristic, complemented, Sylow

218

C₂²

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

222

C₂²

C₂×D₇×D₈

yes

characteristic

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

#89691

yes

complemented

227

C₄

D₇×#127

yes

characteristic

228

C₄

C₂×D₇×D₈

yes

characteristic

229

C₂

—

230

C₂

—

112

—

231

C₂

D₇×#493

yes

characteristic

232

C₂

C₂×#89691

yes

—

	subgroup	G/H
center	C₂²	C₂×D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2247	C₂
socle	C₂²×C₇	C₂²×D₈
hypercenter	#2247	D₇
hypocenter	C₇	#33713
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

C₂×D₇×D₈

commutator, cosocle

C₂×C₇×C₈

C₂⁴

Frattini

C₂×C₈

C₂³×D₇

Fitting

C₇×#2247

C₂

socle

C₂²×C₇

C₂²×D₈

D₇

C₇

G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
1. C₂×#89691	C₂	semidirect
2. #11739244	C₂	semidirect
3. C₇×#2247	C₂	semidirect
4. #11729254	C₂	semidirect
5. #11729067	C₂	semidirect
6. #11726814	C₂	semidirect
7. C₄×D₇×C₁₆	C₂	semidirect
52. D₇×C₁₆	D₄	semidirect
57. #2247	D₇	semidirect
159. C₄×C₇	#508	semidirect
217. C₇	#33713	semidirect
226. C₄	#89691	semidirect

H ≅ G/N

type

1. C₂×#89691

C₂

semidirect

2. #11739244

C₂

semidirect

3. C₇×#2247

C₂

semidirect

4. #11729254

C₂

semidirect

5. #11729067

C₂

semidirect

6. #11726814

C₂

semidirect

7. C₄×D₇×C₁₆

C₂

semidirect

52. D₇×C₁₆

D₄

semidirect

57. #2247

D₇

semidirect

159. C₄×C₇

#508

semidirect

217. C₇

#33713

semidirect

226. C₄

#89691

semidirect

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

subgroup	index
C₂×#89691	2
#11739244	2
C₇×#2247	2
#11729254	2
#11729067	2
#11726814	2
C₄×D₇×C₁₆	2
#33713	7

subgroup

index

C₄×D₇×C₁₆

Subgroups — #13509203

Conjugacy classes of subgroups

Complements

Maximal subgroups