Group #13256422 — subgroups

Subgroups — #13256422

Order 1792.

510 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 139 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×#89622	C₂	1	2	yes	maximal
2	C₂×#89454	C₂	1	1	yes	maximal, characteristic
3	C₂×#89276	C₂	1	2	yes	maximal
4	#11727542	C₂	1	1	yes	maximal, characteristic
5	C₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×#6432	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₄	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6392	—	2	2	no	—
11	C₂×#6392	—	2	2	no	—
12	C₂×#6390	—	2	2	no	—
13	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
14	#89622	—	2	4	no	—
15	#89622	—	2	4	no	—
16	#89454	—	2	4	no	—
17	#89345	—	2	2	no	—
18	#89276	—	2	4	no	—
19	#10279	—	7	1	no	maximal, Sylow
20	C₂²×Dic₁₄	—	2	2	no	—
21	C₂²×Dic₁₄	D₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	—	2	2	no	—
23	C₂²×C₇×C₈	D₄	1	2	yes	—
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇	—	2	2	no	—
26	C₂×C₄⋊Dic₇	—	2	2	no	—
27	C₂×C₄⋊Dic₇	D₄	1	1	yes	characteristic
28	C₂×Dic₇⋊C₄	—	2	1	no	—
29	C₂×C₄×Dic₇	—	2	1	no	—
30	C₂²×C₇⋊C₈	—	2	1	no	—
31	#6432	—	2	2	no	—
32	#6432	—	2	2	no	—
33	#6432	—	2	2	no	—
34	#6432	D₄	1	4	yes	—
35	C₄×Dic₁₄	—	2	2	no	—
36	C₄×Dic₁₄	—	2	2	no	—
37	C₇×C₄⋊C₈	—	2	2	no	—
38	C₇×C₄⋊C₈	D₄	1	4	yes	—
39	C₄×C₇×C₈	—	2	2	no	—
40	#6392	—	2	8	no	—
41	#6392	—	2	8	no	—
42	#6390	—	2	8	no	—
43	C₄⋊(C₇⋊C₈)	—	2	2	no	—
44	C₂×#500	—	7	1	no	—
45	C₂×#477	—	7	2	no	—
46	#1984	—	7	1	no	—
47	C₂×#333	—	7	2	no	—
48	#1277	D₇	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	#41	1	1	yes	characteristic
52	C₂²×Dic₇	—	2	1	no	—
53	C₂²×Dic₇	—	2	1	no	—
54	C₂²×Dic₇	—	4	2	no	—
55	C₂×Dic₁₄	—	2	2	no	—
56	C₂×Dic₁₄	—	2	2	no	—
57	C₂×Dic₁₄	—	2	1	no	—
58	C₂×Dic₁₄	—	2	4	no	—
59	C₂×Dic₁₄	—	4	2	no	—
60	C₂×Dic₁₄	—	8	4	no	—
61	C₂×Dic₁₄	Q₁₆	1	2	yes	—
62	C₂×C₇×C₈	—	2	2	no	—
63	C₂×C₇×C₈	—	2	2	no	—
64	C₂×C₇×C₈	—	2	4	no	—
65	C₂×C₇×C₈	—	4	2	no	—
66	C₂×C₇×C₈	—	4	2	no	—
67	C₂×C₇×C₈	Q₁₆	1	4	yes	—
68	C₄²×C₇	C₂×D₄	1	2	yes	—
69	C₄²×C₇	C₂×D₄	1	2	yes	—
70	C₄⋊Dic₇	—	2	2	no	—
71	C₄⋊Dic₇	—	2	4	no	—
72	C₄⋊Dic₇	—	2	4	no	—
73	C₄⋊Dic₇	—	4	2	no	—
74	C₄⋊Dic₇	—	4	2	no	—
75	Dic₇⋊C₄	—	4	2	no	—
76	C₄×Dic₇	—	4	1	no	—
77	C₄×Dic₇	—	4	1	no	—
78	C₂×C₇⋊C₈	—	4	1	no	—
79	C₂×C₇⋊C₈	—	4	2	no	—
80	C₂×C₄⋊Q₈	—	7	2	no	—
81	C₂×C₄×Q₈	—	7	1	no	—
82	#500	—	14	4	no	—
83	#477	—	14	4	no	—
84	#477	—	14	4	no	—
85	C₂×#107	—	14	2	no	—
86	C₂×#107	—	14	2	no	—
87	C₂×C₄⋊C₈	—	7	1	no	—
88	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
89	C₂×#103	—	14	2	no	—
90	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
91	#336	—	14	2	no	—
92	#333	—	14	4	no	—
93	C₂³×C₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	#120	1	1	yes	characteristic
96	C₂×C₄×C₇	#120	1	2	yes	—
97	C₂×C₄×C₇	#121	1	2	yes	—
98	C₂×C₄×C₇	#127	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₇	#135	1	1	yes	characteristic
102	C₂×C₄×C₇	#137	1	2	yes	—
103	C₂×C₄×C₇	#137	1	2	yes	—
104	C₂×Dic₇	—	2	2	no	—
105	C₂×Dic₇	—	4	1	no	—
106	C₂×Dic₇	—	4	1	no	—
107	C₂×Dic₇	—	4	1	no	—
108	C₂×Dic₇	—	4	4	no	—
109	C₂×Dic₇	—	4	2	no	—
110	C₂×Dic₇	—	8	4	no	—
111	Dic₁₄	—	2	4	no	—
112	Dic₁₄	—	4	2	no	—
113	Dic₁₄	—	8	4	no	—
114	Dic₁₄	—	8	4	no	—
115	C₇×C₈	—	2	4	no	—
116	C₇×C₈	—	4	4	no	—
117	C₇⋊C₈	—	8	1	no	—
118	C₂²×Q₈	—	7	1	no	—
119	C₂²×Q₈	—	14	2	no	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	14	1	no	—
122	C₂²×C₈	D₂₈	1	2	yes	—
123	C₄⋊Q₈	—	7	4	no	—
124	C₄⋊Q₈	—	14	2	no	—
125	C₄⋊Q₈	—	14	2	no	—
126	C₄⋊Q₈	—	14	2	no	—
127	C₄×Q₈	—	14	2	no	—
128	C₄×Q₈	—	14	2	no	—
129	C₂×C₄⋊C₄	—	7	1	no	—
130	C₂×C₄⋊C₄	—	14	2	no	—
131	C₂×C₄⋊C₄	—	14	2	no	—
132	C₂×C₄⋊C₄	—	14	1	no	—
133	C₂×C₄²	—	14	1	no	—
134	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
135	#107	—	14	8	no	—
136	#107	—	14	8	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	14	2	no	—
139	C₄⋊C₈	D₂₈	1	4	yes	—
140	#103	—	14	8	no	—
141	C₄×C₈	—	2	2	no	—
142	C₂²×C₇	#392	1	1	yes	characteristic
143	C₂²×C₇	#462	1	2	yes	—
144	C₂²×C₇	#467	1	1	yes	characteristic
145	C₂²×C₇	#470	1	1	yes	characteristic
146	C₂²×C₇	#472	1	2	yes	—
147	C₄×C₇	—	2	2	no	—
148	C₄×C₇	#451	1	2	yes	—
149	C₄×C₇	#451	1	2	yes	—
150	C₄×C₇	#452	1	2	yes	—
151	C₄×C₇	#494	1	1	yes	characteristic
152	C₄×C₇	#497	1	1	yes	characteristic
153	Dic₇	—	4	2	no	—
154	Dic₇	—	8	4	no	—
155	Dic₇	—	8	1	no	—
156	C₂×Q₈	—	7	2	no	—
157	C₂×Q₈	—	14	1	no	—
158	C₂×Q₈	—	14	2	no	—
159	C₂×Q₈	—	14	4	no	—
160	C₂×Q₈	—	14	2	no	—
161	C₂×Q₈	—	28	2	no	—
162	C₂×Q₈	—	56	4	no	—
163	C₂²×C₄	—	14	1	no	—
164	C₂²×C₄	—	14	1	no	—
165	C₂²×C₄	—	28	2	no	—
166	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
167	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
168	C₂²×C₄	#1161	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	28	1	no	—
175	C₂×C₈	—	28	2	no	—
176	C₂×C₈	Dic₂₈	1	4	yes	—
177	C₄⋊C₄	—	14	4	no	—
178	C₄⋊C₄	—	14	2	no	—
179	C₄⋊C₄	—	14	4	no	—
180	C₄⋊C₄	—	28	2	no	—
181	C₄⋊C₄	—	28	2	no	—
182	C₄⋊C₄	—	28	2	no	—
183	C₄²	—	28	1	no	—
184	C₄²	—	28	1	no	—
185	C₄²	C₂×D₂₈	1	2	yes	—
186	C₄²	D₄×D₇	1	2	yes	—
187	C₂×C₇	#1676	1	2	yes	—
188	C₂×C₇	#1696	1	2	yes	—
189	C₂×C₇	#2001	1	1	yes	characteristic
190	C₂×C₇	#2016	1	1	yes	characteristic
191	C₂×C₇	#2018	1	1	yes	characteristic
192	C₂³	#6458	1	1	yes	characteristic
193	Q₈	—	14	4	no	—
194	Q₈	—	28	2	no	—
195	Q₈	—	56	4	no	—
196	Q₈	—	56	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	14	2	no	—
199	C₂×C₄	—	28	1	no	—
200	C₂×C₄	—	28	1	no	—
201	C₂×C₄	—	28	2	no	—
202	C₂×C₄	—	28	1	no	—
203	C₂×C₄	—	28	4	no	—
204	C₂×C₄	—	56	4	no	—
205	C₂×C₄	#6437	1	1	yes	characteristic
206	C₂×C₄	#6445	1	1	yes	characteristic
207	C₂×C₄	#6445	1	2	yes	—
208	C₂×C₄	#6458	1	2	yes	—
209	C₂×C₄	C₂×Dic₂₈	1	2	yes	—
210	C₂×C₄	#6472	1	2	yes	—
211	C₂×C₄	#6475	1	1	yes	characteristic
212	C₂×C₄	#6478	1	2	yes	—
213	C₂×C₄	D₇×Q₁₆	1	1	yes	characteristic
214	C₈	—	2	4	no	—
215	C₈	—	4	4	no	—
216	C₈	—	56	1	no	—
217	C₇	#10279	1	1	yes	characteristic, complemented, Sylow
218	C₂²	#89443	1	1	yes	characteristic
219	C₂²	#89621	1	2	yes	—
220	C₂²	#89640	1	1	yes	characteristic
221	C₂²	#89659	1	1	yes	characteristic
222	C₂²	#89669	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	28	2	no	—
225	C₄	—	56	1	no	—
226	C₄	—	56	4	no	—
227	C₄	#89471	1	1	yes	characteristic
228	C₄	#89487	1	1	yes	characteristic
229	C₄	#89511	1	2	yes	—
230	C₄	#89555	1	2	yes	—
231	C₄	#89580	1	2	yes	—
232	C₂	#11726307	1	2	yes	—
233	C₂	#11726343	1	2	yes	—
234	C₂	#11726442	1	1	yes	characteristic
235	C₂	#11726642	1	1	yes	characteristic
236	C₂	#11726696	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#89622

C₂

yes

maximal

C₂×#89454

C₂

yes

maximal, characteristic

C₂×#89276

C₂

yes

maximal

#11727542

C₂

yes

maximal, characteristic

C₇×#1277

C₂

yes

maximal, characteristic

C₂×#6432

C₂²

yes

—

C₂×C₄×Dic₁₄

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

—

—

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

—

—

—

—

—

C₂²×Dic₁₄

—

C₂²×Dic₁₄

D₄

yes

characteristic

C₂²×C₇×C₈

—

C₂²×C₇×C₈

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

D₄

yes

characteristic

C₂×Dic₇⋊C₄

—

C₂×C₄×Dic₇

—

C₂²×C₇⋊C₈

—

—

—

—

D₄

yes

—

C₄×Dic₁₄

—

C₄×Dic₁₄

—

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

D₄

yes

—

C₄×C₇×C₈

—

#6392

—

#6392

—

#6390

—

C₄⋊(C₇⋊C₈)

—

—

—

—

—

D₇

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

characteristic

C₂²×Dic₇

—

C₂²×Dic₇

—

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

Q₁₆

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

Q₁₆

yes

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×D₄

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₇

yes

—

C₂×#103

—

C₂×C₄×C₈

C₂×D₇

yes

characteristic

#336

—

#333

—

C₂³×C₇

#121

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

#120

yes

characteristic

C₂×C₄×C₇

#120

yes

—

C₂×C₄×C₇

#121

yes

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×Q₁₆

yes

—

100

C₂×C₄×C₇

C₂×Q₁₆

yes

characteristic

101

C₂×C₄×C₇

#135

yes

characteristic

102

C₂×C₄×C₇

#137

yes

—

103

C₂×C₄×C₇

#137

yes

—

104

C₂×Dic₇

—

105

C₂×Dic₇

—

106

C₂×Dic₇

—

107

C₂×Dic₇

—

108

C₂×Dic₇

—

109

C₂×Dic₇

—

110

C₂×Dic₇

—

111

Dic₁₄

—

112

Dic₁₄

—

113

Dic₁₄

—

114

Dic₁₄

—

115

C₇×C₈

—

116

C₇×C₈

—

117

C₇⋊C₈

—

118

C₂²×Q₈

—

119

C₂²×Q₈

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

D₂₈

yes

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄×Q₈

—

128

C₄×Q₈

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

—

134

C₂×C₄²

C₂²×D₇

yes

characteristic

135

#107

—

136

#107

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

D₂₈

yes

—

140

#103

—

141

C₄×C₈

—

142

C₂²×C₇

#392

yes

characteristic

143

C₂²×C₇

#462

yes

—

144

C₂²×C₇

#467

yes

characteristic

145

C₂²×C₇

#470

yes

characteristic

146

C₂²×C₇

#472

yes

—

147

C₄×C₇

—

148

C₄×C₇

#451

yes

—

149

C₄×C₇

#451

yes

—

150

C₄×C₇

#452

yes

—

151

C₄×C₇

#494

yes

characteristic

152

C₄×C₇

#497

yes

characteristic

153

Dic₇

—

154

Dic₇

—

155

Dic₇

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂²×C₄

—

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

C₂×D₂₈

yes

characteristic

167

C₂²×C₄

D₄×D₇

yes

characteristic

168

C₂²×C₄

#1161

yes

characteristic

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

Dic₂₈

yes

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄²

—

184

C₄²

—

185

C₄²

C₂×D₂₈

yes

—

186

C₄²

D₄×D₇

yes

—

187

C₂×C₇

#1676

yes

—

188

C₂×C₇

#1696

yes

—

189

C₂×C₇

#2001

yes

characteristic

190

C₂×C₇

#2016

yes

characteristic

191

C₂×C₇

#2018

yes

characteristic

192

C₂³

#6458

yes

characteristic

193

Q₈

—

194

Q₈

—

195

Q₈

—

196

Q₈

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

#6437

yes

characteristic

206

C₂×C₄

#6445

yes

characteristic

207

C₂×C₄

#6445

yes

—

208

C₂×C₄

#6458

yes

—

209

C₂×C₄

C₂×Dic₂₈

yes

—

210

C₂×C₄

#6472

yes

—

211

C₂×C₄

#6475

yes

characteristic

212

C₂×C₄

#6478

yes

—

213

C₂×C₄

D₇×Q₁₆

yes

characteristic

214

C₈

—

215

C₈

—

216

C₈

—

217

C₇

#10279

yes

characteristic, complemented, Sylow

218

C₂²

#89443

yes

characteristic

219

C₂²

#89621

yes

—

220

C₂²

#89640

yes

characteristic

221

C₂²

#89659

yes

characteristic

222

C₂²

#89669

yes

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

#89471

yes

characteristic

228

C₄

#89487

yes

characteristic

229

C₄

#89511

yes

—

230

C₄

#89555

yes

—

231

C₄

#89580

yes

—

232

C₂

#11726307

yes

—

233

C₂

#11726343

yes

—

234

C₂

#11726442

yes

characteristic

235

C₂

#11726642

yes

characteristic

236

C₂

#11726696

yes

characteristic

	subgroup	G/H
center	C₂³	#6458
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1277	C₂
socle, prime radical	C₂³×C₇	#121
hypercenter	#1277	D₇
hypocenter	C₇	#10279
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂³

#6458

commutator, cosocle

C₂×C₄²×C₇

C₂³

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂³×C₇

D₇

C₇

G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
217. C₇	#10279	semidirect

H ≅ G/N

type

217. C₇

#10279

semidirect

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

subgroup	index
C₂×#89622	2
C₂×#89454	2
C₂×#89276	2
#11727542	2
C₇×#1277	2
#10279	7

subgroup

index

Subgroups — #13256422

Conjugacy classes of subgroups

Complements

Maximal subgroups