Group #13258253 — subgroups

Subgroups — #13258253

Order 1792.

406 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 127 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	#11731787	C₂	1	1	yes	maximal, characteristic
2	C₇×#1844	C₂	1	1	yes	maximal, characteristic
3	#11727321	C₂	1	2	yes	maximal
4	#11725461	C₂	1	2	yes	maximal
5	#11725440	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×#106	—	2	2	no	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×#384	C₂²	1	1	yes	characteristic
9	C₇×#384	C₂²	1	1	yes	characteristic
10	#89758	—	2	2	no	—
11	#89757	—	2	2	no	—
12	#89749	—	2	2	no	—
13	C₂×#6381	—	2	2	no	—
14	C₄×C₄⋊Dic₇	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
16	#89324	—	2	2	no	—
17	#10577	—	7	1	no	maximal, Sylow
18	C₂²×C₇×C₈	—	2	1	no	—
19	C₂×C₇×C₄⋊C₄	—	2	2	no	—
20	C₂×C₇×C₄⋊C₄	—	2	2	no	—
21	C₂×C₇×C₄⋊C₄	—	2	1	no	—
22	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
23	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₇	—	2	2	no	—
25	C₂×C₄⋊Dic₇	D₄	1	2	yes	—
26	C₂×C₄⋊Dic₇	D₄	1	2	yes	—
27	C₂×C₄×Dic₇	—	2	2	no	—
28	C₂²×C₇⋊C₈	—	2	2	no	—
29	C₇×#106	—	4	4	no	—
30	C₄×C₇×C₈	—	2	1	no	—
31	C₄×C₇×C₈	Q₈	1	2	yes	—
32	C₇⋊#95	—	2	2	no	—
33	C₇⋊#95	—	2	4	no	—
34	C₇⋊#95	—	2	2	no	—
35	#6381	—	4	4	no	—
36	C₄⋊(C₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	—
38	#2571	—	7	1	no	—
39	#1947	—	7	2	no	—
40	#1844	D₇	1	1	yes	characteristic
41	#1296	—	7	1	no	—
42	#1279	—	7	2	no	—
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	—	4	1	no	—
45	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
47	C₂²×Dic₇	—	2	2	no	—
48	C₂²×Dic₇	—	2	2	no	—
49	C₂×C₇×C₈	—	2	2	no	—
50	C₂×C₇×C₈	—	2	2	no	—
51	C₂×C₇×C₈	—	2	2	no	—
52	C₇×C₄⋊C₄	—	4	2	no	—
53	C₇×C₄⋊C₄	—	4	2	no	—
54	C₇×C₄⋊C₄	—	4	2	no	—
55	C₇×C₄⋊C₄	—	4	2	no	—
56	C₇×C₄⋊C₄	—	4	2	no	—
57	C₇×C₄⋊C₄	—	4	2	no	—
58	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇	C₂×Q₈	1	2	yes	—
60	C₄²×C₇	#41	1	1	yes	characteristic
61	C₄⋊Dic₇	—	2	2	no	—
62	C₄⋊Dic₇	—	2	2	no	—
63	C₄⋊Dic₇	—	2	2	no	—
64	C₄⋊Dic₇	—	2	2	no	—
65	C₄⋊Dic₇	—	8	2	no	—
66	C₄×Dic₇	—	4	2	no	—
67	C₄×Dic₇	—	4	2	no	—
68	C₂×C₇⋊C₈	—	2	4	no	—
69	C₂×C₇⋊C₈	—	4	2	no	—
70	C₂×C₇⋊C₈	—	4	2	no	—
71	C₂×#107	—	14	2	no	—
72	C₂×#106	—	2	2	no	—
73	C₂×C₄⋊C₈	—	7	2	no	—
74	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
75	#400	—	14	2	no	—
76	#389	—	14	2	no	—
77	#384	C₂×D₇	1	1	yes	characteristic
78	#384	C₂×D₇	1	1	yes	characteristic
79	#382	—	14	2	no	—
80	C₄×C₄⋊C₄	—	7	2	no	—
81	#340	—	14	2	no	—
82	C₂³×C₇	#120	1	1	yes	characteristic
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	4	4	no	—
86	C₂×C₄×C₇	—	4	2	no	—
87	C₂×C₄×C₇	—	8	2	no	—
88	C₂×C₄×C₇	—	8	1	no	—
89	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
92	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₇	#135	1	2	yes	—
94	C₂×C₄×C₇	#135	1	2	yes	—
95	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₇	#137	1	2	yes	—
97	C₂×Dic₇	—	4	2	no	—
98	C₂×Dic₇	—	4	2	no	—
99	C₂×Dic₇	—	4	2	no	—
100	C₂×Dic₇	—	4	2	no	—
101	C₂×Dic₇	—	4	2	no	—
102	C₂×Dic₇	—	4	2	no	—
103	C₇×C₈	—	2	4	no	—
104	C₇⋊C₈	—	4	4	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	14	2	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₄⋊C₄	—	7	2	no	—
111	C₂×C₄⋊C₄	—	7	2	no	—
112	C₂×C₄⋊C₄	—	14	2	no	—
113	C₂×C₄⋊C₄	C₂²⋊D₇	1	2	yes	—
114	C₂×C₄²	—	14	2	no	—
115	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
116	#107	—	28	4	no	—
117	#106	—	4	4	no	—
118	C₄⋊C₈	—	7	4	no	—
119	C₄⋊C₈	—	14	2	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	Dic₁₄	1	2	yes	—
122	#95	—	14	2	no	—
123	#95	—	14	4	no	—
124	#95	—	14	2	no	—
125	C₂²×C₇	#395	1	1	yes	characteristic
126	C₂²×C₇	#449	1	2	yes	—
127	C₂²×C₇	#452	1	2	yes	—
128	C₂²×C₇	#453	1	1	yes	characteristic
129	C₂²×C₇	#456	1	1	yes	characteristic
130	C₄×C₇	—	2	1	no	—
131	C₄×C₇	—	2	1	no	—
132	C₄×C₇	—	8	2	no	—
133	C₄×C₇	—	16	1	no	—
134	C₄×C₇	#478	1	2	yes	—
135	C₄×C₇	#479	1	2	yes	—
136	C₄×C₇	#499	1	2	yes	—
137	C₄×C₇	#501	1	2	yes	—
138	Dic₇	—	8	2	no	—
139	Dic₇	—	8	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	14	2	no	—
143	C₂²×C₄	—	14	2	no	—
144	C₂²×C₄	D₄×D₇	1	2	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	14	4	no	—
150	C₂×C₈	—	28	2	no	—
151	C₂×C₈	—	28	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	14	2	no	—
159	C₄⋊C₄	—	14	2	no	—
160	C₄⋊C₄	—	14	2	no	—
161	C₄⋊C₄	—	14	2	no	—
162	C₄⋊C₄	—	56	2	no	—
163	C₄²	—	28	2	no	—
164	C₄²	—	28	2	no	—
165	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
166	C₄²	#1157	1	1	yes	characteristic
167	C₄²	Q₈×D₇	1	2	yes	—
168	C₂×C₇	#1624	1	1	yes	characteristic
169	C₂×C₇	#1630	1	1	yes	characteristic
170	C₂×C₇	#1652	1	2	yes	—
171	C₂×C₇	#2059	1	2	yes	—
172	C₂×C₇	#2060	1	1	yes	characteristic
173	C₂³	#6500	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	1	no	—
180	C₂×C₄	—	28	2	no	—
181	C₂×C₄	—	28	2	no	—
182	C₂×C₄	—	28	2	no	—
183	C₂×C₄	—	28	2	no	—
184	C₂×C₄	—	28	2	no	—
185	C₂×C₄	—	28	2	no	—
186	C₂×C₄	#6451	1	2	yes	—
187	C₂×C₄	#6459	1	2	yes	—
188	C₂×C₄	#6477	1	2	yes	—
189	C₂×C₄	#6478	1	2	yes	—
190	C₂×C₄	#6479	1	2	yes	—
191	C₂×C₄	#6479	1	2	yes	—
192	C₂×C₄	#6487	1	1	yes	characteristic
193	C₂×C₄	#6507	1	1	yes	characteristic
194	C₈	—	2	4	no	—
195	C₈	—	28	4	no	—
196	C₇	#10577	1	1	yes	characteristic, complemented, Sylow
197	C₂²	#89751	1	1	yes	characteristic
198	C₂²	#89849	1	1	yes	characteristic
199	C₂²	#89867	1	1	yes	characteristic
200	C₂²	#89944	1	2	yes	—
201	C₂²	#89945	1	2	yes	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	16	1	no	—
206	C₄	—	56	2	no	—
207	C₄	—	56	2	no	—
208	C₄	#89628	1	2	yes	—
209	C₄	#89629	1	2	yes	—
210	C₄	#89630	1	2	yes	—
211	C₄	#89643	1	2	yes	—
212	C₂	#11727327	1	1	yes	characteristic
213	C₂	#11728320	1	2	yes	—
214	C₂	#11729019	1	1	yes	characteristic
215	C₂	#11729024	1	1	yes	characteristic
216	C₂	#11729100	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11731787

C₂

yes

maximal, characteristic

C₇×#1844

C₂

yes

maximal, characteristic

#11727321

C₂

yes

maximal

#11725461

C₂

yes

maximal

#11725440

C₂

yes

maximal, characteristic

C₂×C₇×#106

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#384

C₂²

yes

characteristic

C₇×#384

C₂²

yes

—

—

—

—

C₄×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

#89324

—

#10577

—

maximal, Sylow

C₂²×C₇×C₈

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

D₄

yes

—

C₂×C₄⋊Dic₇

D₄

yes

—

C₂×C₄×Dic₇

—

C₂²×C₇⋊C₈

—

C₇×#106

—

C₄×C₇×C₈

—

C₄×C₇×C₈

Q₈

yes

—

—

—

—

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

Q₈

yes

—

#2571

—

#1947

—

#1844

D₇

yes

characteristic

#1296

—

#1279

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂²×Dic₇

—

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×Q₈

yes

characteristic

C₄²×C₇

C₂×Q₈

yes

—

C₄²×C₇

#41

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄×Dic₇

—

C₄×Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#107

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

C₂×D₇

yes

characteristic

#384

C₂×D₇

yes

characteristic

#382

—

C₄×C₄⋊C₄

—

#340

—

C₂³×C₇

#120

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇

C₄⋊Q₈

yes

—

C₂×C₄×C₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇

#135

yes

—

C₂×C₄×C₇

#135

yes

—

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇

#137

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

100

C₂×Dic₇

—

101

C₂×Dic₇

—

102

C₂×Dic₇

—

103

C₇×C₈

—

104

C₇⋊C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂²⋊D₇

yes

—

114

C₂×C₄²

—

115

C₂×C₄²

C₂²×D₇

yes

characteristic

116

#107

—

117

#106

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄×C₈

—

121

C₄×C₈

Dic₁₄

yes

—

122

#95

—

123

#95

—

124

#95

—

125

C₂²×C₇

#395

yes

characteristic

126

C₂²×C₇

#449

yes

—

127

C₂²×C₇

#452

yes

—

128

C₂²×C₇

#453

yes

characteristic

129

C₂²×C₇

#456

yes

characteristic

130

C₄×C₇

—

131

C₄×C₇

—

132

C₄×C₇

—

133

C₄×C₇

—

134

C₄×C₇

#478

yes

—

135

C₄×C₇

#479

yes

—

136

C₄×C₇

#499

yes

—

137

C₄×C₇

#501

yes

—

138

Dic₇

—

139

Dic₇

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

D₄×D₇

yes

—

145

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₄²

C₂×Dic₁₄

yes

characteristic

166

C₄²

#1157

yes

characteristic

167

C₄²

Q₈×D₇

yes

—

168

C₂×C₇

#1624

yes

characteristic

169

C₂×C₇

#1630

yes

characteristic

170

C₂×C₇

#1652

yes

—

171

C₂×C₇

#2059

yes

—

172

C₂×C₇

#2060

yes

characteristic

173

C₂³

#6500

yes

characteristic

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

#6451

yes

—

187

C₂×C₄

#6459

yes

—

188

C₂×C₄

#6477

yes

—

189

C₂×C₄

#6478

yes

—

190

C₂×C₄

#6479

yes

—

191

C₂×C₄

#6479

yes

—

192

C₂×C₄

#6487

yes

characteristic

193

C₂×C₄

#6507

yes

characteristic

194

C₈

—

195

C₈

—

196

C₇

#10577

yes

characteristic, complemented, Sylow

197

C₂²

#89751

yes

characteristic

198

C₂²

#89849

yes

characteristic

199

C₂²

#89867

yes

characteristic

200

C₂²

#89944

yes

—

201

C₂²

#89945

yes

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

#89628

yes

—

209

C₄

#89629

yes

—

210

C₄

#89630

yes

—

211

C₄

#89643

yes

—

212

C₂

#11727327

yes

characteristic

213

C₂

#11728320

yes

—

214

C₂

#11729019

yes

characteristic

215

C₂

#11729024

yes

characteristic

216

C₂

#11729100

yes

—

	subgroup	G/H
center	C₂³	#6500
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1844	C₂
socle, prime radical	C₂³×C₇	#120
hypercenter	#1844	D₇
hypocenter	C₇	#10577
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#6500

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 ◁ #1844

N	H ≅ G/N	type
196. C₇	#10577	semidirect

H ≅ G/N

type

196. C₇

#10577

semidirect

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

subgroup	index
#11731787	2
C₇×#1844	2
#11727321	2
#11725461	2
#11725440	2
#10577	7

subgroup

index

Subgroups — #13258253

Conjugacy classes of subgroups

Complements

Maximal subgroups