Group #13252882 — subgroups

Subgroups — #13252882

Order 1792.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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×#2868	C₂	1	1	yes	maximal, characteristic
2	C₂×#89342	C₂	1	4	yes	maximal
3	#11727146	C₂	1	1	yes	maximal, characteristic
4	#11727145	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×C₄⋊Q₈	—	2	2	no	—
6	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₇×#384	—	2	1	no	—
8	C₇×#384	—	2	1	no	—
9	C₇×#384	C₄	1	4	yes	—
10	C₄³×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×#6377	—	2	1	no	—
14	#89342	—	2	8	no	—
15	#89319	—	2	1	no	—
16	#89319	—	2	2	no	—
17	#9703	—	7	1	no	maximal, Sylow
18	C₂²×C₇×Q₈	—	2	2	no	—
19	C₇×C₄⋊Q₈	—	2	4	no	—
20	C₇×C₄⋊Q₈	—	2	4	no	—
21	C₇×C₄⋊Q₈	—	4	8	no	—
22	C₂×C₇×C₄⋊C₄	—	2	4	no	—
23	C₂×C₇×C₄⋊C₄	—	2	4	no	—
24	C₂×C₇×C₄⋊C₄	—	2	2	no	—
25	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	D₄	1	2	yes	—
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
31	C₂²×C₇⋊C₈	—	2	2	no	—
32	C₂²×C₇⋊C₈	—	2	2	no	—
33	C₄⋊(C₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₇⋊C₈)	D₄	1	4	yes	—
36	#6377	—	4	2	no	—
37	#2868	D₇	1	1	yes	characteristic
38	#1831	—	7	1	no	—
39	#1767	—	7	1	no	—
40	C₂×#332	—	7	4	no	—
41	C₂×C₇×Q₈	—	4	2	no	—
42	C₂×C₇×Q₈	—	4	2	no	—
43	C₂×C₇×Q₈	—	4	8	no	—
44	C₂²×C₄×C₇	—	2	4	no	—
45	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₇×C₄⋊C₄	—	2	8	no	—
51	C₇×C₄⋊C₄	—	4	8	no	—
52	C₇×C₄⋊C₄	—	4	8	no	—
53	C₇×C₄⋊C₄	—	4	4	no	—
54	C₄²×C₇	—	2	4	no	—
55	C₄²×C₇	—	2	4	no	—
56	C₄²×C₇	—	2	2	no	—
57	C₄²×C₇	SD₁₆	1	2	yes	—
58	C₄²×C₇	Q₁₆	1	2	yes	—
59	C₄²×C₇	C₂×D₄	1	2	yes	—
60	C₄²×C₇	#41	1	2	yes	—
61	C₂×C₇⋊C₈	—	2	4	no	—
62	C₂×C₇⋊C₈	—	4	2	no	—
63	C₂×C₇⋊C₈	—	4	2	no	—
64	C₂×C₇⋊C₈	—	4	2	no	—
65	C₂×C₇⋊C₈	—	4	4	no	—
66	C₂×C₄⋊Q₈	—	2	2	no	—
67	C₂×C₄⋊Q₈	C₂×D₇	1	2	yes	—
68	C₂×C₄⋊C₈	—	7	2	no	—
69	C₂×C₄⋊C₈	—	7	2	no	—
70	C₂×#97	—	14	1	no	—
71	#384	—	2	1	no	—
72	#384	—	2	1	no	—
73	#384	Dic₇	1	4	yes	—
74	C₄³	C₂×D₇	1	1	yes	characteristic
75	#336	—	14	2	no	—
76	#336	—	14	1	no	—
77	#332	—	14	8	no	—
78	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₇×Q₈	—	8	4	no	—
80	C₇×Q₈	—	8	4	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
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	—	4	4	no	—
87	C₂×C₄×C₇	—	4	8	no	—
88	C₂×C₄×C₇	#101	1	4	yes	—
89	C₂×C₄×C₇	#103	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×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
97	C₂×C₄×C₇	#137	1	2	yes	—
98	C₇⋊C₈	—	4	4	no	—
99	C₇⋊C₈	—	8	2	no	—
100	C₂²×Q₈	—	2	2	no	—
101	C₂²×C₈	—	14	2	no	—
102	C₂²×C₈	—	14	2	no	—
103	C₄⋊Q₈	—	2	4	no	—
104	C₄⋊Q₈	—	2	4	no	—
105	C₄⋊Q₈	—	4	8	no	—
106	C₂×C₄⋊C₄	—	2	4	no	—
107	C₂×C₄⋊C₄	—	2	4	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
110	C₂×C₄²	C₂×Dic₇	1	2	yes	—
111	C₂×C₄²	C₂²⋊D₇	1	2	yes	—
112	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
113	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
114	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
115	C₄⋊C₈	—	7	4	no	—
116	C₄⋊C₈	—	14	4	no	—
117	C₄⋊C₈	—	14	2	no	—
118	#97	—	28	2	no	—
119	C₂²×C₇	#390	1	1	yes	characteristic
120	C₂²×C₇	C₂×#101	1	2	yes	—
121	C₂²×C₇	C₂×#103	1	1	yes	characteristic
122	C₂²×C₇	#418	1	2	yes	—
123	C₂²×C₇	#419	1	1	yes	characteristic
124	C₄×C₇	—	2	4	no	—
125	C₄×C₇	—	2	4	no	—
126	C₄×C₇	—	2	2	no	—
127	C₄×C₇	—	8	4	no	—
128	C₄×C₇	#461	1	2	yes	—
129	C₄×C₇	#462	1	2	yes	—
130	C₄×C₇	#490	1	2	yes	—
131	C₄×C₇	#496	1	2	yes	—
132	C₂×Q₈	—	4	8	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
137	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
141	C₂×C₈	—	14	4	no	—
142	C₂×C₈	—	28	2	no	—
143	C₂×C₈	—	28	2	no	—
144	C₂×C₈	—	28	4	no	—
145	C₂×C₈	—	28	2	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	4	8	no	—
148	C₄⋊C₄	—	4	8	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	2	2	no	—
153	C₄²	#1143	1	2	yes	—
154	C₄²	#1144	1	2	yes	—
155	C₄²	D₄×D₇	1	2	yes	—
156	C₄²	#1159	1	2	yes	—
157	C₂×C₇	#1549	1	2	yes	—
158	C₂×C₇	#1556	1	2	yes	—
159	C₂×C₇	#1961	1	1	yes	characteristic
160	C₂×C₇	#1966	1	1	yes	characteristic
161	C₂×C₇	#1981	1	1	yes	characteristic
162	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
163	Q₈	—	8	4	no	—
164	Q₈	—	8	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	8	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	#6409	1	4	yes	—
173	C₂×C₄	#6410	1	4	yes	—
174	C₂×C₄	#6495	1	4	yes	—
175	C₂×C₄	D₄×Dic₇	1	2	yes	—
176	C₂×C₄	#6499	1	1	yes	characteristic
177	C₂×C₄	#6501	1	2	yes	—
178	C₂×C₄	#6503	1	1	yes	characteristic
179	C₂×C₄	C₂×#1143	1	1	yes	characteristic
180	C₂×C₄	#6505	1	2	yes	—
181	C₂×C₄	C₂×#1144	1	1	yes	characteristic
182	C₈	—	28	4	no	—
183	C₈	—	56	2	no	—
184	C₇	#9703	1	1	yes	characteristic, complemented, Sylow
185	C₂²	#89987	1	2	yes	—
186	C₂²	#89993	1	1	yes	characteristic
187	C₂²	C₂×#6409	1	1	yes	characteristic
188	C₂²	#89997	1	1	yes	characteristic
189	C₂²	C₂×#6410	1	2	yes	—
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	8	4	no	—
194	C₄	#89843	1	2	yes	—
195	C₄	#89846	1	2	yes	—
196	C₄	#89855	1	2	yes	—
197	C₄	#89862	1	2	yes	—
198	C₂	#11729519	1	1	yes	characteristic
199	C₂	#11729529	1	2	yes	—
200	C₂	#11729530	1	2	yes	—
201	C₂	#11729532	1	1	yes	characteristic
202	C₂	#11729543	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2868

C₂

yes

maximal, characteristic

C₂×#89342

C₂

yes

maximal

#11727146

C₂

yes

maximal, characteristic

#11727145

C₂

yes

maximal, characteristic

C₂×C₇×C₄⋊Q₈

—

C₂×C₇×C₄⋊Q₈

C₂²

yes

—

C₇×#384

—

C₇×#384

—

C₇×#384

C₄

yes

—

C₄³×C₇

C₂²

yes

characteristic

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₇⋊C₈)

C₂²

yes

—

—

—

—

—

—

C₂²×C₇×Q₈

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₇×C₄⋊Q₈

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

D₄

yes

characteristic

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂²×C₇⋊C₈

—

C₂²×C₇⋊C₈

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

D₄

yes

—

#6377

—

#2868

D₇

yes

—

—

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

—

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

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

SD₁₆

yes

—

C₄²×C₇

Q₁₆

yes

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

#41

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₂×D₇

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#384

—

#384

—

#384

Dic₇

yes

—

C₄³

C₂×D₇

yes

—

—

—

C₂³×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#101

yes

—

C₂×C₄×C₇

#103

yes

—

C₂×C₄×C₇

C₄×D₄

yes

—

C₂×C₄×C₇

#121

yes

—

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇

#137

yes

—

C₇⋊C₈

—

C₇⋊C₈

—

100

C₂²×Q₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₄⋊Q₈

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×Dic₇

yes

—

110

C₂×C₄²

C₂×Dic₇

yes

—

111

C₂×C₄²

C₂²⋊D₇

yes

—

112

C₂×C₄²

C₂²⋊D₇

yes

characteristic

113

C₂×C₄²

C₂²⋊D₇

yes

characteristic

114

C₂×C₄²

C₂²×D₇

yes

characteristic

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

#97

—

119

C₂²×C₇

#390

yes

characteristic

120

C₂²×C₇

C₂×#101

yes

—

121

C₂²×C₇

C₂×#103

yes

characteristic

122

C₂²×C₇

#418

yes

—

123

C₂²×C₇

#419

yes

characteristic

124

C₄×C₇

—

125

C₄×C₇

—

126

C₄×C₇

—

127

C₄×C₇

—

128

C₄×C₇

#461

yes

—

129

C₄×C₇

#462

yes

—

130

C₄×C₇

#490

yes

—

131

C₄×C₇

#496

yes

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂²⋊Dic₇

yes

—

137

C₂²×C₄

C₂²⋊Dic₇

yes

—

138

C₂²×C₄

C₂²×Dic₇

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₇

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

#1143

yes

—

154

C₄²

#1144

yes

—

155

C₄²

D₄×D₇

yes

—

156

C₄²

#1159

yes

—

157

C₂×C₇

#1549

yes

—

158

C₂×C₇

#1556

yes

—

159

C₂×C₇

#1961

yes

characteristic

160

C₂×C₇

#1966

yes

characteristic

161

C₂×C₇

#1981

yes

characteristic

162

C₂³

C₂×C₂²⋊Dic₇

yes

characteristic

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

#6409

yes

—

173

C₂×C₄

#6410

yes

—

174

C₂×C₄

#6495

yes

—

175

C₂×C₄

D₄×Dic₇

yes

—

176

C₂×C₄

#6499

yes

characteristic

177

C₂×C₄

#6501

yes

—

178

C₂×C₄

#6503

yes

characteristic

179

C₂×C₄

C₂×#1143

yes

characteristic

180

C₂×C₄

#6505

yes

—

181

C₂×C₄

C₂×#1144

yes

characteristic

182

C₈

—

183

C₈

—

184

C₇

#9703

yes

characteristic, complemented, Sylow

185

C₂²

#89987

yes

—

186

C₂²

#89993

yes

characteristic

187

C₂²

C₂×#6409

yes

characteristic

188

C₂²

#89997

yes

characteristic

189

C₂²

C₂×#6410

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

#89843

yes

—

195

C₄

#89846

yes

—

196

C₄

#89855

yes

—

197

C₄

#89862

yes

—

198

C₂

#11729519

yes

characteristic

199

C₂

#11729529

yes

—

200

C₂

#11729530

yes

—

201

C₂

#11729532

yes

characteristic

202

C₂

#11729543

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2868	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2868	D₇
hypocenter	C₇	#9703
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₇

commutator

C₂²×C₄×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

Fitting

C₇×#2868

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 ◁ #2868

N	H ≅ G/N	type
184. C₇	#9703	semidirect

H ≅ G/N

type

184. C₇

#9703

semidirect

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

subgroup	index
C₇×#2868	2
C₂×#89342	2
#11727146	2
#11727145	2
#9703	7

subgroup

index

Subgroups — #13252882

Conjugacy classes of subgroups

Complements

Maximal subgroups