Group #13252889 — subgroups

Subgroups — #13252889

Order 1792.

538 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 167 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	#11730756	C₂	1	1	yes	maximal, characteristic
2	C₇×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×#89281	C₂	1	4	yes	maximal
4	#11727145	C₂	1	1	yes	maximal, characteristic
5	C₂×#6432	—	2	1	no	—
6	C₂×#6432	—	2	1	no	—
7	C₂×#6432	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₄³×C₇	C₂²	1	1	yes	characteristic
10	#89707	—	2	2	no	—
11	#89707	C₄	1	4	yes	—
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×#6377	—	2	1	no	—
14	C₇×#336	—	2	2	no	—
15	#89319	—	2	1	no	—
16	#89281	—	2	8	no	—
17	#9705	—	7	1	no	maximal, Sylow
18	C₂²×Dic₁₄	—	2	2	no	—
19	C₂²×C₇×C₈	—	2	2	no	—
20	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₇	D₄	1	2	yes	—
24	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇	—	2	4	no	—
26	C₂×C₄⋊Dic₇	—	2	4	no	—
27	C₂×C₄⋊Dic₇	—	2	2	no	—
28	C₂×C₄⋊Dic₇	C₂×C₄	1	4	yes	—
29	C₂²×C₇⋊C₈	—	2	2	no	—
30	#6432	—	2	4	no	—
31	#6432	—	2	4	no	—
32	#6432	—	4	4	no	—
33	#6432	—	4	2	no	—
34	#6432	—	4	2	no	—
35	C₇×C₄⋊C₈	—	2	4	no	—
36	C₄⋊(C₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇⋊C₈)	D₄	1	4	yes	—
38	#6377	—	4	2	no	—
39	#2868	—	7	1	no	—
40	#1831	D₇	1	1	yes	characteristic
41	#1767	—	7	1	no	—
42	C₂×#332	—	7	4	no	—
43	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₇	—	2	4	no	—
49	C₂×Dic₁₄	—	4	4	no	—
50	C₂×Dic₁₄	—	4	2	no	—
51	C₂×Dic₁₄	—	4	2	no	—
52	C₂×Dic₁₄	—	4	4	no	—
53	C₂×C₇×C₈	—	4	2	no	—
54	C₂×C₇×C₈	—	4	4	no	—
55	C₄²×C₇	—	2	4	no	—
56	C₄²×C₇	—	2	2	no	—
57	C₄²×C₇	—	2	4	no	—
58	C₄²×C₇	D₈	1	2	yes	—
59	C₄²×C₇	SD₁₆	1	2	yes	—
60	C₄²×C₇	C₂×D₄	1	2	yes	—
61	C₄²×C₇	#41	1	2	yes	—
62	C₄⋊Dic₇	—	2	8	no	—
63	C₄⋊Dic₇	—	4	8	no	—
64	C₄⋊Dic₇	—	4	8	no	—
65	C₄⋊Dic₇	—	4	2	no	—
66	C₄⋊Dic₇	—	4	2	no	—
67	C₂×C₇⋊C₈	—	2	4	no	—
68	C₂×C₇⋊C₈	—	4	2	no	—
69	C₂×C₇⋊C₈	—	4	2	no	—
70	C₂×C₄⋊Q₈	—	7	2	no	—
71	C₂×C₄⋊Q₈	—	14	1	no	—
72	C₂×C₄⋊Q₈	—	14	1	no	—
73	C₂×C₄⋊C₈	—	7	2	no	—
74	C₂×C₄⋊C₈	C₂×D₇	1	2	yes	—
75	C₂×#97	—	14	1	no	—
76	#384	—	7	4	no	—
77	#384	—	14	2	no	—
78	C₄³	C₂×D₇	1	1	yes	characteristic
79	#336	—	2	2	no	—
80	#336	—	14	1	no	—
81	#332	—	14	8	no	—
82	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	#101	1	4	yes	—
89	C₂×C₄×C₇	#102	1	4	yes	—
90	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₇	#121	1	2	yes	—
92	C₂×C₄×C₇	#124	1	1	yes	characteristic
93	C₂×C₄×C₇	#127	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₇	#137	1	4	yes	—
98	C₂×Dic₇	—	4	4	no	—
99	C₂×Dic₇	—	4	4	no	—
100	C₂×Dic₇	—	4	4	no	—
101	Dic₁₄	—	8	4	no	—
102	Dic₁₄	—	8	4	no	—
103	C₇×C₈	—	8	2	no	—
104	C₇⋊C₈	—	4	4	no	—
105	C₂²×Q₈	—	14	2	no	—
106	C₂²×C₈	—	2	2	no	—
107	C₂²×C₈	—	14	2	no	—
108	C₄⋊Q₈	—	14	4	no	—
109	C₄⋊Q₈	—	14	4	no	—
110	C₄⋊Q₈	—	28	4	no	—
111	C₄⋊Q₈	—	28	2	no	—
112	C₄⋊Q₈	—	28	2	no	—
113	C₂×C₄⋊C₄	—	7	4	no	—
114	C₂×C₄⋊C₄	—	14	4	no	—
115	C₂×C₄⋊C₄	—	14	4	no	—
116	C₂×C₄⋊C₄	—	14	2	no	—
117	C₂×C₄²	C₄×D₇	1	2	yes	—
118	C₂×C₄²	D₂₈	1	1	yes	characteristic
119	C₂×C₄²	D₂₈	1	1	yes	characteristic
120	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
121	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	7	4	no	—
124	C₄⋊C₈	—	14	2	no	—
125	#97	—	28	2	no	—
126	C₂²×C₇	#390	1	1	yes	characteristic
127	C₂²×C₇	C₂×#101	1	2	yes	—
128	C₂²×C₇	C₂×#102	1	1	yes	characteristic
129	C₂²×C₇	#418	1	1	yes	characteristic
130	C₂²×C₇	#419	1	2	yes	—
131	C₄×C₇	—	2	4	no	—
132	C₄×C₇	—	2	4	no	—
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	#459	1	2	yes	—
135	C₄×C₇	#460	1	2	yes	—
136	C₄×C₇	#491	1	2	yes	—
137	C₄×C₇	#497	1	2	yes	—
138	Dic₇	—	8	4	no	—
139	C₂×Q₈	—	28	4	no	—
140	C₂×Q₈	—	28	2	no	—
141	C₂×Q₈	—	28	2	no	—
142	C₂×Q₈	—	28	4	no	—
143	C₂²×C₄	—	14	4	no	—
144	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
145	C₂²×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	4	no	—
151	C₂×C₈	—	14	4	no	—
152	C₂×C₈	—	28	2	no	—
153	C₂×C₈	—	28	2	no	—
154	C₄⋊C₄	—	14	8	no	—
155	C₄⋊C₄	—	28	8	no	—
156	C₄⋊C₄	—	28	2	no	—
157	C₄⋊C₄	—	28	2	no	—
158	C₄⋊C₄	—	28	8	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	4	no	—
162	C₄²	#1132	1	2	yes	—
163	C₄²	D₅₆	1	2	yes	—
164	C₄²	D₄×D₇	1	2	yes	—
165	C₄²	#1161	1	2	yes	—
166	C₂×C₇	#1546	1	2	yes	—
167	C₂×C₇	#1556	1	2	yes	—
168	C₂×C₇	#1960	1	1	yes	characteristic
169	C₂×C₇	#1966	1	1	yes	characteristic
170	C₂×C₇	#1982	1	1	yes	characteristic
171	C₂³	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
172	Q₈	—	56	4	no	—
173	Q₈	—	56	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	28	4	no	—
180	C₂×C₄	—	28	4	no	—
181	C₂×C₄	—	28	4	no	—
182	C₂×C₄	#6395	1	4	yes	—
183	C₂×C₄	#6398	1	4	yes	—
184	C₂×C₄	#6456	1	2	yes	—
185	C₂×C₄	#6458	1	2	yes	—
186	C₂×C₄	C₂×#1132	1	1	yes	characteristic
187	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
188	C₂×C₄	#6471	1	2	yes	—
189	C₂×C₄	#6503	1	1	yes	characteristic
190	C₂×C₄	#6510	1	1	yes	characteristic
191	C₂×C₄	#6514	1	4	yes	—
192	C₈	—	8	2	no	—
193	C₈	—	28	4	no	—
194	C₇	#9705	1	1	yes	characteristic, complemented, Sylow
195	C₂²	#89760	1	1	yes	characteristic
196	C₂²	#89776	1	2	yes	—
197	C₂²	C₂×#6395	1	1	yes	characteristic
198	C₂²	#89903	1	1	yes	characteristic
199	C₂²	C₂×#6398	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	4	no	—
203	C₄	—	56	4	no	—
204	C₄	#89615	1	2	yes	—
205	C₄	#89620	1	2	yes	—
206	C₄	#89832	1	2	yes	—
207	C₄	#89841	1	2	yes	—
208	C₂	#11727306	1	1	yes	characteristic
209	C₂	#11728211	1	1	yes	characteristic
210	C₂	#11728276	1	1	yes	characteristic
211	C₂	#11728532	1	2	yes	—
212	C₂	#11728576	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11730756

C₂

yes

maximal, characteristic

C₇×#1831

C₂

yes

maximal, characteristic

C₂×#89281

C₂

yes

maximal

#11727145

C₂

yes

maximal, characteristic

C₂×#6432

—

C₂×#6432

—

C₂×#6432

C₂²

yes

—

C₂×C₇×C₄⋊C₈

C₂²

yes

—

C₄³×C₇

C₂²

yes

characteristic

#89707

—

#89707

C₄

yes

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

—

—

—

—

—

C₂²×Dic₁₄

—

C₂²×C₇×C₈

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

C₂×C₄

yes

—

C₂²×C₇⋊C₈

—

—

—

—

—

—

C₇×C₄⋊C₈

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

D₄

yes

—

#6377

—

#2868

—

#1831

D₇

yes

characteristic

#1767

—

C₂×#332

—

C₂²×C₄×C₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

D₈

yes

—

C₄²×C₇

SD₁₆

yes

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

#41

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₇

yes

—

C₂×#97

—

#384

—

#384

—

C₄³

C₂×D₇

yes

—

—

—

C₂³×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#101

yes

—

C₂×C₄×C₇

#102

yes

—

C₂×C₄×C₇

C₄×D₄

yes

—

C₂×C₄×C₇

#121

yes

—

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇

#137

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

100

C₂×Dic₇

—

101

Dic₁₄

—

102

Dic₁₄

—

103

C₇×C₈

—

104

C₇⋊C₈

—

105

C₂²×Q₈

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄²

C₄×D₇

yes

—

118

C₂×C₄²

D₂₈

yes

characteristic

119

C₂×C₄²

D₂₈

yes

characteristic

120

C₂×C₄²

C₂²⋊D₇

yes

—

121

C₂×C₄²

C₂²×D₇

yes

characteristic

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

#97

—

126

C₂²×C₇

#390

yes

characteristic

127

C₂²×C₇

C₂×#101

yes

—

128

C₂²×C₇

C₂×#102

yes

characteristic

129

C₂²×C₇

#418

yes

characteristic

130

C₂²×C₇

#419

yes

—

131

C₄×C₇

—

132

C₄×C₇

—

133

C₄×C₇

—

134

C₄×C₇

#459

yes

—

135

C₄×C₇

#460

yes

—

136

C₄×C₇

#491

yes

—

137

C₄×C₇

#497

yes

—

138

Dic₇

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

(C₂×D₇)⋊C₄

yes

—

145

C₂²×C₄

(C₂×D₇)⋊C₄

yes

—

146

C₂²×C₄

C₂×C₄×D₇

yes

characteristic

147

C₂²×C₄

C₂×D₂₈

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

#1132

yes

—

163

C₄²

D₅₆

yes

—

164

C₄²

D₄×D₇

yes

—

165

C₄²

#1161

yes

—

166

C₂×C₇

#1546

yes

—

167

C₂×C₇

#1556

yes

—

168

C₂×C₇

#1960

yes

characteristic

169

C₂×C₇

#1966

yes

characteristic

170

C₂×C₇

#1982

yes

characteristic

171

C₂³

C₂×(C₂×D₇)⋊C₄

yes

characteristic

172

Q₈

—

173

Q₈

—

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

#6395

yes

—

183

C₂×C₄

#6398

yes

—

184

C₂×C₄

#6456

yes

—

185

C₂×C₄

#6458

yes

—

186

C₂×C₄

C₂×#1132

yes

characteristic

187

C₂×C₄

C₂×D₅₆

yes

characteristic

188

C₂×C₄

#6471

yes

—

189

C₂×C₄

#6503

yes

characteristic

190

C₂×C₄

#6510

yes

characteristic

191

C₂×C₄

#6514

yes

—

192

C₈

—

193

C₈

—

194

C₇

#9705

yes

characteristic, complemented, Sylow

195

C₂²

#89760

yes

characteristic

196

C₂²

#89776

yes

—

197

C₂²

C₂×#6395

yes

characteristic

198

C₂²

#89903

yes

characteristic

199

C₂²

C₂×#6398

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

#89615

yes

—

205

C₄

#89620

yes

—

206

C₄

#89832

yes

—

207

C₄

#89841

yes

—

208

C₂

#11727306

yes

characteristic

209

C₂

#11728211

yes

characteristic

210

C₂

#11728276

yes

characteristic

211

C₂

#11728532

yes

—

212

C₂

#11728576

yes

—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇)⋊C₄
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1831	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1831	D₇
hypocenter	C₇	#9705
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×(C₂×D₇)⋊C₄

commutator

C₂²×C₄×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

Fitting

C₇×#1831

C₂

socle, prime radical

C₂³×C₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
194. C₇	#9705	semidirect

H ≅ G/N

type

194. C₇

#9705

semidirect

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

subgroup	index
#11730756	2
C₇×#1831	2
C₂×#89281	2
#11727145	2
#9705	7

subgroup

index

Subgroups — #13252889

Conjugacy classes of subgroups

Complements

Maximal subgroups