Group #32155129 — subgroups

Subgroups — #32155129

Order 18176.

528 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 165 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	Q₈×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#22590563	C₂	1	3	yes	maximal, complemented
3	#22590487	C₂	1	3	yes	maximal, complemented
4	#22590377	C₂	1	3	yes	maximal, complemented
5	Q₈×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14556072	C₂	1	3	yes	maximal, complemented
7	C₇₁×#3391	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₄₂	—	2	1	no	—
9	C₂×#14556072	C₂²	1	3	yes	—
10	#17986404	C₂²	1	3	yes	—
11	#17986295	—	2	3	no	—
12	C₄×Dic₂₈₄	C₂²	1	3	yes	—
13	#17985826	C₂²	1	3	yes	—
14	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	3	yes	—
15	#17985802	C₂²	1	1	yes	characteristic
16	#17985800	C₂²	1	6	yes	—
17	#17985800	C₂²	1	3	yes	—
18	C₄×C₇₁⋊C₁₆	C₂²	1	3	yes	—
19	#550×C₇₁	—	2	1	no	—
20	#493×C₇₁	C₂²	1	3	yes	—
21	#459×C₇₁	—	2	3	no	—
22	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×D₈×C₇₁	C₂²	1	3	yes	—
24	C₂×Dic₂₈₄	C₂³	1	1	yes	characteristic
25	#14556128	—	2	3	no	—
26	#14556127	—	4	3	no	—
27	Q₈×Dic₇₁	—	4	1	no	—
28	C₄×Dic₁₄₂	—	2	3	no	—
29	#14556072	—	2	6	no	—
30	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
31	C₂×C₇₁⋊C₁₆	C₂³	1	3	yes	—
32	#14556062	—	2	3	no	—
33	#14556062	C₂³	1	3	yes	—
34	#14556053	—	2	3	no	—
35	C₂×#41×C₇₁	—	4	1	no	—
36	C₂×D₈×C₇₁	—	2	3	no	—
37	C₂×D₈×C₇₁	C₂³	1	3	yes	—
38	#127×C₇₁	—	2	3	no	—
39	#121×C₇₁	—	4	3	no	—
40	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
41	C₄×D₄×C₇₁	—	2	3	no	—
42	#107×C₇₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₁	D₄	1	3	yes	—
44	#102×C₇₁	—	2	3	no	—
45	C₄×C₈×C₇₁	C₂³	1	3	yes	—
46	C₂×Dic₁₄₂	—	2	1	no	—
47	C₂×Dic₁₄₂	—	4	3	no	—
48	Dic₂₈₄	—	2	1	no	—
49	Dic₂₈₄	#41	1	2	yes	complemented
50	C₄⋊Dic₇₁	—	2	3	no	—
51	C₄⋊Dic₇₁	—	4	3	no	—
52	Dic₇₁⋊C₄	—	4	3	no	—
53	C₄×Dic₇₁	—	4	3	no	—
54	C₇₁⋊C₁₆	—	2	3	no	—
55	C₇₁⋊C₁₆	—	2	1	no	—
56	#41×C₇₁	—	8	4	no	—
57	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
58	C₂×D₄×C₇₁	—	2	3	no	—
59	C₂×D₄×C₇₁	—	4	3	no	—
60	C₂×D₄×C₇₁	—	4	3	no	—
61	C₂²×C₄×C₇₁	—	4	3	no	—
62	D₈×C₇₁	—	2	3	no	—
63	D₈×C₇₁	—	2	3	no	—
64	D₈×C₇₁	—	8	3	no	—
65	C₂×C₈×C₇₁	C₂×D₄	1	3	yes	—
66	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₇₁	—	2	1	no	—
68	C₄⋊C₄×C₇₁	D₈	1	3	yes	—
69	C₂²⋊C₄×C₇₁	—	4	3	no	—
70	C₄²×C₇₁	C₂×D₄	1	3	yes	—
71	C₂³×C₇₁	—	4	3	no	—
72	Q₈×C₇₁	SD₃₂	1	4	yes	complemented
73	D₄×C₇₁	—	4	3	no	—
74	D₄×C₇₁	—	4	3	no	—
75	D₄×C₇₁	—	8	6	no	—
76	D₄×C₇₁	—	16	3	no	—
77	C₂×C₄×C₇₁	—	4	1	no	—
78	C₂×C₄×C₇₁	—	8	6	no	—
79	C₂×C₄×C₇₁	C₂×D₈	1	3	yes	—
80	C₂×C₄×C₇₁	C₂×D₈	1	3	yes	—
81	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₇₁	—	4	1	no	—
83	C₂×Dic₇₁	—	4	3	no	—
84	Dic₁₄₂	—	2	2	no	—
85	Dic₁₄₂	—	4	1	no	—
86	Dic₁₄₂	—	16	3	no	—
87	C₈×C₇₁	—	2	3	no	—
88	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
89	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
90	C₂²×C₇₁	—	8	3	no	—
91	C₂²×C₇₁	—	8	3	no	—
92	C₂²×C₇₁	—	8	3	no	—
93	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₇₁	—	2	3	no	—
95	C₄×C₇₁	—	8	1	no	—
96	C₄×C₇₁	C₂×SD₃₂	1	6	yes	—
97	C₄×C₇₁	#548	1	1	yes	characteristic
98	C₄×C₇₁	#577	1	1	yes	characteristic
99	Dic₇₁	—	4	2	no	—
100	Dic₇₁	—	8	3	no	—
101	#33929	—	71	1	no	maximal, Sylow
102	C₂×C₇₁	—	16	3	no	—
103	C₂×C₇₁	#3358	1	1	yes	characteristic
104	C₂×C₇₁	C₂²×SD₃₂	1	1	yes	characteristic
105	C₂×C₇₁	#3415	1	1	yes	characteristic
106	#3391	D₇₁	1	1	yes	characteristic
107	Q₈×Q₁₆	—	71	1	no	—
108	#2240	—	71	3	no	—
109	#2226	—	71	3	no	—
110	#2207	—	71	3	no	—
111	Q₈×C₁₆	—	71	1	no	—
112	C₄×SD₃₂	—	71	3	no	—
113	C₇₁	#33929	1	1	yes	characteristic, complemented, Sylow
114	Q₈²	—	142	1	no	—
115	#550	—	2	1	no	—
116	C₂×SD₃₂	—	71	3	no	—
117	#500	—	71	3	no	—
118	#493	C₂×D₇₁	1	3	yes	—
119	#477	—	142	3	no	—
120	#459	—	2	3	no	—
121	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
122	C₄×Q₁₆	—	71	3	no	—
123	C₄×D₈	C₂×D₇₁	1	3	yes	—
124	#367	—	71	3	no	—
125	C₄⋊C₁₆	—	71	3	no	—
126	#358	—	71	1	no	—
127	#357	—	71	3	no	—
128	#357	—	71	6	no	—
129	C₄×C₁₆	—	71	3	no	—
130	C₂×#41	—	4	1	no	—
131	C₂×Q₁₆	—	71	1	no	—
132	C₂×D₈	—	2	3	no	—
133	C₂×D₈	C₂²×D₇₁	1	3	yes	—
134	C₄⋊Q₈	—	284	3	no	—
135	C₄⋊Q₈	—	142	3	no	—
136	#127	—	2	3	no	—
137	#121	—	4	3	no	—
138	C₄×Q₈	—	284	1	no	—
139	C₄×Q₈	—	142	3	no	—
140	C₄×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
141	C₄×D₄	—	2	3	no	—
142	SD₃₂	—	142	6	no	—
143	C₂×C₁₆	—	71	1	no	—
144	C₂×C₁₆	—	71	3	no	—
145	#107	—	71	3	no	—
146	#107	—	142	3	no	—
147	#107	C₂²×D₇₁	1	1	yes	characteristic
148	C₄⋊C₈	C₂²⋊D₇₁	1	3	yes	—
149	#103	—	142	3	no	—
150	#102	—	2	3	no	—
151	C₄×C₈	C₂²×D₇₁	1	3	yes	—
152	#41	—	8	4	no	—
153	C₂×Q₈	—	284	3	no	—
154	C₂×Q₈	—	142	1	no	—
155	C₂×Q₈	#11765850	1	1	yes	characteristic
156	C₂×D₄	—	2	3	no	—
157	C₂×D₄	—	4	3	no	—
158	C₂×D₄	—	4	3	no	—
159	C₂²×C₄	—	4	3	no	—
160	Q₁₆	—	71	2	no	—
161	Q₁₆	—	142	1	no	—
162	D₈	—	2	3	no	—
163	D₈	—	2	3	no	—
164	D₈	—	8	3	no	—
165	C₂×C₈	C₂×C₂²⋊D₇₁	1	3	yes	—
166	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	284	3	no	—
169	C₄⋊C₄	—	284	3	no	—
170	C₄⋊C₄	—	142	3	no	—
171	C₄⋊C₄	#11765850	1	3	yes	—
172	C₂²⋊C₄	—	4	3	no	—
173	C₄²	—	284	3	no	—
174	C₄²	C₂×C₂²⋊D₇₁	1	3	yes	—
175	C₁₆	—	142	3	no	—
176	C₁₆	—	142	1	no	—
177	C₂³	—	4	3	no	—
178	Q₈	—	284	1	no	—
179	Q₈	—	1136	3	no	—
180	Q₈	—	142	2	no	—
181	Q₈	#14556072	1	4	yes	complemented
182	D₄	—	4	3	no	—
183	D₄	—	4	3	no	—
184	D₄	—	8	6	no	—
185	D₄	—	16	3	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	8	6	no	—
188	C₂×C₄	—	284	1	no	—
189	C₂×C₄	—	284	3	no	—
190	C₂×C₄	C₂×#11765850	1	3	yes	—
191	C₂×C₄	C₂×#11765850	1	3	yes	—
192	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
193	C₈	—	2	3	no	—
194	C₈	#41×D₇₁	1	1	yes	characteristic
195	C₈	#14556175	1	1	yes	characteristic
196	C₂²	—	8	3	no	—
197	C₂²	—	8	3	no	—
198	C₂²	—	8	3	no	—
199	C₂²	C₂²×#11765850	1	1	yes	characteristic
200	C₄	—	2	3	no	—
201	C₄	—	8	1	no	—
202	C₄	—	284	2	no	—
203	C₄	—	568	3	no	—
204	C₄	C₂×#14556072	1	6	yes	—
205	C₄	#17986656	1	1	yes	characteristic
206	C₄	#17986810	1	1	yes	characteristic
207	C₂	—	16	3	no	—
208	C₂	#22600971	1	1	yes	characteristic
209	C₂	C₂²×#14556072	1	1	yes	characteristic
210	C₂	#22601479	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₈₄

C₂

yes

maximal, characteristic, complemented

#22590563

C₂

yes

maximal, complemented

#22590487

C₂

yes

maximal, complemented

#22590377

C₂

yes

maximal, complemented

Q₈×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄×#14556072

C₂

yes

maximal, complemented

C₇₁×#3391

C₂

yes

maximal, characteristic

Q₈×Dic₁₄₂

—

C₂×#14556072

C₂²

yes

—

#17986404

C₂²

yes

—

#17986295

—

C₄×Dic₂₈₄

C₂²

yes

—

#17985826

C₂²

yes

—

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

—

#17985802

C₂²

yes

characteristic

#17985800

C₂²

yes

—

#17985800

C₂²

yes

—

C₄×C₇₁⋊C₁₆

C₂²

yes

—

#550×C₇₁

—

#493×C₇₁

C₂²

yes

—

#459×C₇₁

—

C₈×Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₈×C₇₁

C₂²

yes

—

C₂×Dic₂₈₄

C₂³

yes

characteristic

#14556128

—

#14556127

—

Q₈×Dic₇₁

—

C₄×Dic₁₄₂

—

#14556072

—

C₂×C₇₁⋊C₁₆

C₂³

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂³

yes

—

#14556062

—

#14556062

C₂³

yes

—

#14556053

—

C₂×#41×C₇₁

—

C₂×D₈×C₇₁

—

C₂×D₈×C₇₁

C₂³

yes

—

#127×C₇₁

—

#121×C₇₁

—

C₄×Q₈×C₇₁

D₄

yes

characteristic

C₄×D₄×C₇₁

—

#107×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

D₄

yes

—

#102×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

#41

yes

complemented

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

#41×C₇₁

—

C₂×Q₈×C₇₁

D₈

yes

characteristic

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂²×C₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₈

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

—

C₂³×C₇₁

—

Q₈×C₇₁

SD₃₂

yes

complemented

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×C₄×C₇₁

C₂²×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×#41

yes

characteristic

C₈×C₇₁

2₋¹⁺⁴

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₃₂

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

100

Dic₇₁

—

101

#33929

—

maximal, Sylow

102

C₂×C₇₁

—

103

C₂×C₇₁

#3358

yes

characteristic

104

C₂×C₇₁

C₂²×SD₃₂

yes

characteristic

105

C₂×C₇₁

#3415

yes

characteristic

106

#3391

D₇₁

yes

characteristic

107

Q₈×Q₁₆

—

108

#2240

—

109

#2226

—

110

#2207

—

111

Q₈×C₁₆

—

112

C₄×SD₃₂

—

113

C₇₁

#33929

yes

characteristic, complemented, Sylow

114

Q₈²

—

142

—

115

#550

—

116

C₂×SD₃₂

—

117

#500

—

118

#493

C₂×D₇₁

yes

—

119

#477

—

142

—

120

#459

—

121

C₈×Q₈

C₂×D₇₁

yes

characteristic

122

C₄×Q₁₆

—

123

C₄×D₈

C₂×D₇₁

yes

—

124

#367

—

125

C₄⋊C₁₆

—

126

#358

—

127

#357

—

128

#357

—

129

C₄×C₁₆

—

130

C₂×#41

—

131

C₂×Q₁₆

—

132

C₂×D₈

—

133

C₂×D₈

C₂²×D₇₁

yes

—

134

C₄⋊Q₈

—

284

—

135

C₄⋊Q₈

—

142

—

136

#127

—

137

#121

—

138

C₄×Q₈

—

284

—

139

C₄×Q₈

—

142

—

140

C₄×Q₈

C₂²⋊D₇₁

yes

characteristic

141

C₄×D₄

—

142

SD₃₂

—

142

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

145

#107

—

146

#107

—

142

—

147

#107

C₂²×D₇₁

yes

characteristic

148

C₄⋊C₈

C₂²⋊D₇₁

yes

—

149

#103

—

142

—

150

#102

—

151

C₄×C₈

C₂²×D₇₁

yes

—

152

#41

—

153

C₂×Q₈

—

284

—

154

C₂×Q₈

—

142

—

155

C₂×Q₈

#11765850

yes

characteristic

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂²×C₄

—

160

Q₁₆

—

161

Q₁₆

—

142

—

162

D₈

—

163

D₈

—

164

D₈

—

165

C₂×C₈

C₂×C₂²⋊D₇₁

yes

—

166

C₂×C₈

C₂³×D₇₁

yes

characteristic

167

C₄⋊C₄

—

168

C₄⋊C₄

—

284

—

169

C₄⋊C₄

—

284

—

170

C₄⋊C₄

—

142

—

171

C₄⋊C₄

#11765850

yes

—

172

C₂²⋊C₄

—

173

C₄²

—

284

—

174

C₄²

C₂×C₂²⋊D₇₁

yes

—

175

C₁₆

—

142

—

176

C₁₆

—

142

—

177

C₂³

—

178

Q₈

—

284

—

179

Q₈

—

1136

—

180

Q₈

—

142

—

181

Q₈

#14556072

yes

complemented

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

284

—

189

C₂×C₄

—

284

—

190

C₂×C₄

C₂×#11765850

yes

—

191

C₂×C₄

C₂×#11765850

yes

—

192

C₂×C₄

C₂²×C₂²⋊D₇₁

yes

characteristic

193

C₈

—

194

C₈

#41×D₇₁

yes

characteristic

195

C₈

#14556175

yes

characteristic

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

C₂²×#11765850

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

284

—

203

C₄

—

568

—

204

C₄

C₂×#14556072

yes

—

205

C₄

#17986656

yes

characteristic

206

C₄

#17986810

yes

characteristic

207

C₂

—

208

C₂

#22600971

yes

characteristic

209

C₂

C₂²×#14556072

yes

characteristic

210

C₂

#22601479

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#11765850
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting, prime radical	C₇₁×#3391	C₂
socle	C₂²×C₇₁	C₂²×D₈
hypercenter	#3391	D₇₁
hypocenter	C₇₁	#33929
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

subgroup

G/H

center

C₂²

C₂²×#11765850

commutator, cosocle

C₂×C₈×C₇₁

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₁

Fitting, prime radical

C₇₁×#3391

C₂

socle

C₂²×C₇₁

C₂²×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

N	H ≅ G/N	type
1. Q₈×Dic₂₈₄	C₂	semidirect
2. #22590563	C₂	semidirect
3. #22590487	C₂	semidirect
4. #22590377	C₂	semidirect
5. Q₈×C₇₁⋊C₁₆	C₂	semidirect
6. C₄×#14556072	C₂	semidirect
49. Dic₂₈₄	#41	semidirect
72. Q₈×C₇₁	SD₃₂	semidirect
113. C₇₁	#33929	semidirect
181. Q₈	#14556072	semidirect

H ≅ G/N

type

1. Q₈×Dic₂₈₄

C₂

semidirect

2. #22590563

C₂

semidirect

3. #22590487

C₂

semidirect

4. #22590377

C₂

semidirect

5. Q₈×C₇₁⋊C₁₆

C₂

semidirect

6. C₄×#14556072

C₂

semidirect

49. Dic₂₈₄

#41

semidirect

72. Q₈×C₇₁

SD₃₂

semidirect

113. C₇₁

#33929

semidirect

181. Q₈

#14556072

semidirect

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

subgroup	index
Q₈×Dic₂₈₄	2
#22590563	2
#22590487	2
#22590377	2
Q₈×C₇₁⋊C₁₆	2
C₄×#14556072	2
C₇₁×#3391	2
#33929	71

subgroup

index

Q₈×Dic₂₈₄

Q₈×C₇₁⋊C₁₆

C₇₁×#3391

Subgroups — #32155129

Conjugacy classes of subgroups

Complements

Maximal subgroups