Group #13258546 — subgroups

Subgroups — #13258546

Order 1792.

350 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₇×#1834	C₂	1	1	yes	maximal, characteristic
2	#11727542	C₂	1	1	yes	maximal, characteristic
3	#11727342	C₂	1	2	yes	maximal
4	#11725460	C₂	1	1	yes	maximal, characteristic
5	#11725440	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₁₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×#384	C₂²	1	2	yes	—
9	C₂×#6384	—	2	2	no	—
10	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
12	C₇×#340	—	2	2	no	—
13	#89345	—	2	2	no	—
14	#89324	—	2	2	no	—
15	#10624	—	7	1	no	maximal, Sylow
16	C₂²×Dic₁₄	D₄	1	1	yes	characteristic
17	C₂²×C₇×C₈	D₄	1	2	yes	—
18	C₂×C₇×C₄⋊C₄	—	2	2	no	—
19	C₂×C₇×C₄⋊C₄	—	2	2	no	—
20	C₂×C₇×C₄⋊C₄	—	2	2	no	—
21	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₇	D₄	1	1	yes	characteristic
23	C₂×Dic₇⋊C₄	—	2	1	no	—
24	C₂×C₄×Dic₇	—	2	1	no	—
25	C₂²×C₇⋊C₈	—	2	1	no	—
26	C₂²×C₇⋊C₈	—	2	2	no	—
27	C₄×Dic₁₄	—	2	2	no	—
28	C₄×Dic₁₄	—	2	2	no	—
29	C₄×C₇×C₈	—	2	2	no	—
30	#6384	—	4	4	no	—
31	C₄⋊(C₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇⋊C₈)	—	2	2	no	—
34	#1984	—	7	1	no	—
35	#1834	D₇	1	1	yes	characteristic
36	#1803	—	7	2	no	—
37	#1296	—	7	2	no	—
38	#1277	—	7	1	no	—
39	C₂²×C₄×C₇	—	4	2	no	—
40	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇	#41	1	1	yes	characteristic
43	C₂²×Dic₇	—	2	1	no	—
44	C₂²×Dic₇	—	2	1	no	—
45	C₂×Dic₁₄	—	2	2	no	—
46	C₂×Dic₁₄	—	2	2	no	—
47	C₂×Dic₁₄	—	2	1	no	—
48	C₂×Dic₁₄	Q₁₆	1	2	yes	—
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	—	8	2	no	—
57	C₄²×C₇	#41	1	2	yes	—
58	C₄²×C₇	#41	1	2	yes	—
59	C₄⋊Dic₇	—	2	2	no	—
60	Dic₇⋊C₄	—	4	2	no	—
61	C₄×Dic₇	—	4	1	no	—
62	C₄×Dic₇	—	4	1	no	—
63	C₂×C₇⋊C₈	—	4	2	no	—
64	C₂×C₇⋊C₈	—	4	1	no	—
65	C₂×C₇⋊C₈	—	4	2	no	—
66	C₂×C₇⋊C₈	—	4	2	no	—
67	C₂×C₇⋊C₈	—	4	2	no	—
68	C₂×C₄×Q₈	—	7	1	no	—
69	C₂×C₄⋊C₈	—	7	2	no	—
70	C₂×C₄⋊C₈	—	7	1	no	—
71	C₂×#103	—	14	2	no	—
72	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
73	#384	C₂×D₇	1	2	yes	—
74	#340	—	2	2	no	—
75	#340	—	14	2	no	—
76	#336	—	14	2	no	—
77	C₂³×C₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	8	2	no	—
80	C₂×C₄×C₇	—	8	2	no	—
81	C₂×C₄×C₇	—	8	2	no	—
82	C₂×C₄×C₇	#123	1	2	yes	—
83	C₂×C₄×C₇	#123	1	1	yes	characteristic
84	C₂×C₄×C₇	#124	1	1	yes	characteristic
85	C₂×C₄×C₇	#126	1	2	yes	—
86	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇	#135	1	1	yes	characteristic
88	C₂×C₄×C₇	#135	1	2	yes	—
89	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₇	#137	1	2	yes	—
91	C₂×Dic₇	—	2	2	no	—
92	C₂×Dic₇	—	4	1	no	—
93	C₂×Dic₇	—	4	1	no	—
94	C₂×Dic₇	—	4	1	no	—
95	C₂×Dic₇	—	4	2	no	—
96	Dic₁₄	—	2	4	no	—
97	Dic₁₄	—	4	2	no	—
98	C₇×C₈	—	4	2	no	—
99	C₇⋊C₈	—	8	2	no	—
100	C₇⋊C₈	—	8	1	no	—
101	C₂²×Q₈	—	7	1	no	—
102	C₂²×C₈	—	14	2	no	—
103	C₂²×C₈	—	14	1	no	—
104	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
105	C₄×Q₈	—	14	2	no	—
106	C₄×Q₈	—	14	2	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	7	1	no	—
111	C₂×C₄⋊C₄	—	14	1	no	—
112	C₂×C₄²	—	14	1	no	—
113	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
114	C₄⋊C₈	—	14	2	no	—
115	C₄⋊C₈	—	14	2	no	—
116	C₄⋊C₈	—	14	2	no	—
117	#103	—	28	4	no	—
118	C₄×C₈	—	2	2	no	—
119	C₂²×C₇	#397	1	1	yes	characteristic
120	C₂²×C₇	#464	1	2	yes	—
121	C₂²×C₇	#467	1	1	yes	characteristic
122	C₂²×C₇	#469	1	1	yes	characteristic
123	C₂²×C₇	#471	1	2	yes	—
124	C₄×C₇	—	2	2	no	—
125	C₄×C₇	—	16	2	no	—
126	C₄×C₇	#482	1	2	yes	—
127	C₄×C₇	#484	1	2	yes	—
128	C₄×C₇	#485	1	2	yes	—
129	C₄×C₇	#488	1	1	yes	characteristic
130	C₄×C₇	#491	1	1	yes	characteristic
131	Dic₇	—	4	2	no	—
132	Dic₇	—	8	1	no	—
133	C₂×Q₈	—	7	2	no	—
134	C₂×Q₈	—	14	1	no	—
135	C₂×Q₈	—	14	2	no	—
136	C₂×Q₈	—	14	2	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	14	1	no	—
139	C₂²×C₄	—	14	1	no	—
140	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
141	C₂²×C₄	#1159	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	28	2	no	—
147	C₂×C₈	—	28	2	no	—
148	C₂×C₈	—	28	2	no	—
149	C₂×C₈	—	28	1	no	—
150	C₂×C₈	—	28	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	8	2	no	—
156	C₄⋊C₄	—	14	2	no	—
157	C₄⋊C₄	—	28	2	no	—
158	C₄²	—	28	1	no	—
159	C₄²	—	28	1	no	—
160	C₄²	#1157	1	2	yes	—
161	C₄²	#1161	1	2	yes	—
162	C₂×C₇	#1684	1	2	yes	—
163	C₂×C₇	#1702	1	2	yes	—
164	C₂×C₇	#2074	1	1	yes	characteristic
165	C₂×C₇	#2076	1	1	yes	characteristic
166	C₂×C₇	#2079	1	1	yes	characteristic
167	C₂³	#6501	1	1	yes	characteristic
168	Q₈	—	14	4	no	—
169	Q₈	—	28	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	14	2	no	—
175	C₂×C₄	—	28	1	no	—
176	C₂×C₄	—	28	1	no	—
177	C₂×C₄	—	28	2	no	—
178	C₂×C₄	—	28	1	no	—
179	C₂×C₄	#6457	1	2	yes	—
180	C₂×C₄	#6461	1	2	yes	—
181	C₂×C₄	#6474	1	2	yes	—
182	C₂×C₄	D₇×Q₁₆	1	1	yes	characteristic
183	C₂×C₄	#6482	1	1	yes	characteristic
184	C₂×C₄	#6492	1	1	yes	characteristic
185	C₂×C₄	#6510	1	1	yes	characteristic
186	C₂×C₄	#6513	1	2	yes	—
187	C₂×C₄	#6514	1	2	yes	—
188	C₈	—	4	2	no	—
189	C₈	—	56	2	no	—
190	C₈	—	56	1	no	—
191	C₇	#10624	1	1	yes	characteristic, complemented, Sylow
192	C₂²	#89765	1	1	yes	characteristic
193	C₂²	#89826	1	2	yes	—
194	C₂²	#89931	1	1	yes	characteristic
195	C₂²	#89949	1	2	yes	—
196	C₂²	#89960	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	16	2	no	—
199	C₄	—	28	2	no	—
200	C₄	—	56	1	no	—
201	C₄	#89656	1	2	yes	—
202	C₄	#89657	1	2	yes	—
203	C₄	#89768	1	2	yes	—
204	C₄	#89840	1	1	yes	characteristic
205	C₄	#89841	1	1	yes	characteristic
206	C₂	#11727314	1	1	yes	characteristic
207	C₂	#11728347	1	1	yes	characteristic
208	C₂	#11728391	1	1	yes	characteristic
209	C₂	#11728965	1	2	yes	—
210	C₂	#11728995	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1834

C₂

yes

maximal, characteristic

#11727542

C₂

yes

maximal, characteristic

#11727342

C₂

yes

maximal

#11725460

C₂

yes

maximal, characteristic

#11725440

C₂

yes

maximal

C₂×C₄×Dic₁₄

C₂²

yes

characteristic

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#384

C₂²

yes

—

C₂×#6384

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

characteristic

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

—

—

—

—

C₂²×Dic₁₄

D₄

yes

characteristic

C₂²×C₇×C₈

D₄

yes

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

D₄

yes

characteristic

C₂×Dic₇⋊C₄

—

C₂×C₄×Dic₇

—

C₂²×C₇⋊C₈

—

C₂²×C₇⋊C₈

—

C₄×Dic₁₄

—

C₄×Dic₁₄

—

C₄×C₇×C₈

—

#6384

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#1984

—

#1834

D₇

yes

—

—

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

characteristic

C₂²×Dic₇

—

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

Q₁₆

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

#41

yes

—

C₄²×C₇

#41

yes

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₄×Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₂×C₄×C₈

C₂×D₇

yes

characteristic

#384

C₂×D₇

yes

—

#340

—

#340

—

#336

—

C₂³×C₇

#121

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#123

yes

—

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

—

C₂×C₄×C₇

C₂×Q₁₆

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

—

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇

#137

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇⋊C₈

—

100

C₇⋊C₈

—

101

C₂²×Q₈

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

C₂²⋊D₇

yes

—

105

C₄×Q₈

—

106

C₄×Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄²

—

113

C₂×C₄²

C₂²×D₇

yes

characteristic

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

#103

—

118

C₄×C₈

—

119

C₂²×C₇

#397

yes

characteristic

120

C₂²×C₇

#464

yes

—

121

C₂²×C₇

#467

yes

characteristic

122

C₂²×C₇

#469

yes

characteristic

123

C₂²×C₇

#471

yes

—

124

C₄×C₇

—

125

C₄×C₇

—

126

C₄×C₇

#482

yes

—

127

C₄×C₇

#484

yes

—

128

C₄×C₇

#485

yes

—

129

C₄×C₇

#488

yes

characteristic

130

C₄×C₇

#491

yes

characteristic

131

Dic₇

—

132

Dic₇

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

D₄×D₇

yes

characteristic

141

C₂²×C₄

#1159

yes

characteristic

142

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄²

—

159

C₄²

—

160

C₄²

#1157

yes

—

161

C₄²

#1161

yes

—

162

C₂×C₇

#1684

yes

—

163

C₂×C₇

#1702

yes

—

164

C₂×C₇

#2074

yes

characteristic

165

C₂×C₇

#2076

yes

characteristic

166

C₂×C₇

#2079

yes

characteristic

167

C₂³

#6501

yes

characteristic

168

Q₈

—

169

Q₈

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#6457

yes

—

180

C₂×C₄

#6461

yes

—

181

C₂×C₄

#6474

yes

—

182

C₂×C₄

D₇×Q₁₆

yes

characteristic

183

C₂×C₄

#6482

yes

characteristic

184

C₂×C₄

#6492

yes

characteristic

185

C₂×C₄

#6510

yes

characteristic

186

C₂×C₄

#6513

yes

—

187

C₂×C₄

#6514

yes

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₇

#10624

yes

characteristic, complemented, Sylow

192

C₂²

#89765

yes

characteristic

193

C₂²

#89826

yes

—

194

C₂²

#89931

yes

characteristic

195

C₂²

#89949

yes

—

196

C₂²

#89960

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

#89656

yes

—

202

C₄

#89657

yes

—

203

C₄

#89768

yes

—

204

C₄

#89840

yes

characteristic

205

C₄

#89841

yes

characteristic

206

C₂

#11727314

yes

characteristic

207

C₂

#11728347

yes

characteristic

208

C₂

#11728391

yes

characteristic

209

C₂

#11728965

yes

—

210

C₂

#11728995

yes

—

	subgroup	G/H
center	C₂³	#6501
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1834	C₂
socle, prime radical	C₂³×C₇	#121
hypercenter	#1834	D₇
hypocenter	C₇	#10624
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#6501

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 ◁ #1834

N	H ≅ G/N	type
191. C₇	#10624	semidirect

H ≅ G/N

type

191. C₇

#10624

semidirect

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

subgroup	index
C₇×#1834	2
#11727542	2
#11727342	2
#11725460	2
#11725440	2
#10624	7

subgroup

index

Subgroups — #13258546

Conjugacy classes of subgroups

Complements

Maximal subgroups