Group #32155136 — subgroups

Subgroups — #32155136

Order 18176.

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

normal subgroups only

This group has 472 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
2	#22590567	C₂	1	3	yes	maximal
3	#22590492	C₂	1	3	yes	maximal
4	#22590387	C₂	1	3	yes	maximal
5	Q₈×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×#14556074	C₂	1	3	yes	maximal
7	C₇₁×#3395	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₄₂	—	2	1	no	—
9	C₂×#14556074	C₂²	1	3	yes	—
10	#17986404	C₂²	1	3	yes	—
11	#17986295	—	2	3	no	—
12	C₄×Dic₂₈₄	C₂²	1	3	yes	—
13	#17985824	C₂²	1	3	yes	—
14	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	3	yes	—
15	#17985803	C₂²	1	6	yes	—
16	#17985803	C₂²	1	3	yes	—
17	#17985802	C₂²	1	1	yes	characteristic
18	C₄×C₇₁⋊C₁₆	C₂²	1	3	yes	—
19	#557×C₇₁	—	2	1	no	—
20	#494×C₇₁	C₂²	1	3	yes	—
21	#462×C₇₁	—	2	3	no	—
22	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×Q₁₆×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	#14556074	—	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×Q₁₆×C₇₁	—	2	3	no	—
36	C₂×Q₁₆×C₇₁	C₂³	1	3	yes	—
37	C₄⋊Q₈×C₇₁	—	2	3	no	—
38	#125×C₇₁	—	4	3	no	—
39	C₄×Q₈×C₇₁	—	2	3	no	—
40	C₄×Q₈×C₇₁	—	4	1	no	—
41	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
42	#107×C₇₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₁	D₄	1	3	yes	—
44	#103×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	—
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	C₂×Q₈×C₇₁	—	2	3	no	—
57	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
58	Q₁₆×C₇₁	—	2	3	no	—
59	Q₁₆×C₇₁	—	2	3	no	—
60	Q₁₆×C₇₁	—	8	3	no	—
61	C₂×C₈×C₇₁	C₂×D₄	1	3	yes	—
62	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₇₁	—	2	1	no	—
64	C₄⋊C₄×C₇₁	—	4	3	no	—
65	C₄⋊C₄×C₇₁	—	4	3	no	—
66	C₄⋊C₄×C₇₁	—	4	3	no	—
67	C₄⋊C₄×C₇₁	D₈	1	3	yes	—
68	C₄²×C₇₁	—	4	3	no	—
69	C₄²×C₇₁	C₂×D₄	1	3	yes	—
70	Q₈×C₇₁	—	4	3	no	—
71	Q₈×C₇₁	—	4	3	no	—
72	Q₈×C₇₁	Q₃₂	1	4	yes	complemented
73	C₂×C₄×C₇₁	—	4	1	no	—
74	C₂×C₄×C₇₁	—	4	3	no	—
75	C₂×C₄×C₇₁	C₂×D₈	1	3	yes	—
76	C₂×C₄×C₇₁	C₂×D₈	1	3	yes	—
77	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₇₁	—	4	1	no	—
79	C₂×Dic₇₁	—	4	3	no	—
80	Dic₁₄₂	—	2	2	no	—
81	Dic₁₄₂	—	4	1	no	—
82	Dic₁₄₂	—	16	3	no	—
83	C₈×C₇₁	—	2	3	no	—
84	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
85	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
86	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
87	C₄×C₇₁	—	2	3	no	—
88	C₄×C₇₁	—	8	3	no	—
89	C₄×C₇₁	—	8	1	no	—
90	C₄×C₇₁	C₂×Q₃₂	1	6	yes	—
91	C₄×C₇₁	#548	1	1	yes	characteristic
92	C₄×C₇₁	#577	1	1	yes	characteristic
93	Dic₇₁	—	4	2	no	—
94	Dic₇₁	—	8	3	no	—
95	#33930	—	71	1	no	maximal, Sylow
96	C₂×C₇₁	#3358	1	1	yes	characteristic
97	C₂×C₇₁	C₂²×Q₃₂	1	1	yes	characteristic
98	C₂×C₇₁	#3416	1	1	yes	characteristic
99	#3395	D₇₁	1	1	yes	characteristic
100	Q₈×Q₁₆	—	71	1	no	—
101	#2241	—	71	3	no	—
102	#2227	—	71	3	no	—
103	#2209	—	71	3	no	—
104	Q₈×C₁₆	—	71	1	no	—
105	C₄×Q₃₂	—	71	3	no	—
106	C₇₁	#33930	1	1	yes	characteristic, complemented, Sylow
107	Q₈²	—	142	1	no	—
108	#557	—	2	1	no	—
109	C₂×Q₃₂	—	71	3	no	—
110	#500	—	71	3	no	—
111	#494	C₂×D₇₁	1	3	yes	—
112	#477	—	142	3	no	—
113	#462	—	2	3	no	—
114	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
115	C₄×Q₁₆	—	71	3	no	—
116	C₄×Q₁₆	C₂×D₇₁	1	3	yes	—
117	#366	—	71	3	no	—
118	C₄⋊C₁₆	—	71	3	no	—
119	#358	—	71	3	no	—
120	#358	—	71	6	no	—
121	#358	—	71	1	no	—
122	C₄×C₁₆	—	71	3	no	—
123	C₂×Q₁₆	—	2	3	no	—
124	C₂×Q₁₆	—	71	1	no	—
125	C₂×Q₁₆	C₂²×D₇₁	1	3	yes	—
126	C₄⋊Q₈	—	2	3	no	—
127	C₄⋊Q₈	—	284	3	no	—
128	C₄⋊Q₈	—	142	3	no	—
129	#125	—	4	3	no	—
130	C₄×Q₈	—	2	3	no	—
131	C₄×Q₈	—	4	1	no	—
132	C₄×Q₈	—	284	1	no	—
133	C₄×Q₈	—	142	3	no	—
134	C₄×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
135	Q₃₂	—	142	6	no	—
136	C₂×C₁₆	—	71	1	no	—
137	C₂×C₁₆	—	71	3	no	—
138	#107	—	71	3	no	—
139	#107	—	142	3	no	—
140	#107	C₂²×D₇₁	1	1	yes	characteristic
141	C₄⋊C₈	C₂²⋊D₇₁	1	3	yes	—
142	#103	—	2	3	no	—
143	#103	—	142	3	no	—
144	C₄×C₈	C₂²×D₇₁	1	3	yes	—
145	C₂×Q₈	—	2	3	no	—
146	C₂×Q₈	—	284	3	no	—
147	C₂×Q₈	—	142	1	no	—
148	C₂×Q₈	#11765850	1	1	yes	characteristic
149	Q₁₆	—	2	3	no	—
150	Q₁₆	—	2	3	no	—
151	Q₁₆	—	8	3	no	—
152	Q₁₆	—	71	2	no	—
153	Q₁₆	—	142	1	no	—
154	C₂×C₈	C₂×C₂²⋊D₇₁	1	3	yes	—
155	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	4	3	no	—
158	C₄⋊C₄	—	4	3	no	—
159	C₄⋊C₄	—	4	3	no	—
160	C₄⋊C₄	—	284	3	no	—
161	C₄⋊C₄	—	284	3	no	—
162	C₄⋊C₄	—	142	3	no	—
163	C₄⋊C₄	#11765850	1	3	yes	—
164	C₄²	—	4	3	no	—
165	C₄²	—	284	3	no	—
166	C₄²	C₂×C₂²⋊D₇₁	1	3	yes	—
167	C₁₆	—	142	3	no	—
168	C₁₆	—	142	1	no	—
169	Q₈	—	4	3	no	—
170	Q₈	—	4	3	no	—
171	Q₈	—	284	1	no	—
172	Q₈	—	1136	3	no	—
173	Q₈	—	142	2	no	—
174	Q₈	#14556074	1	4	yes	complemented
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	3	no	—
177	C₂×C₄	—	284	1	no	—
178	C₂×C₄	—	284	3	no	—
179	C₂×C₄	C₂×#11765850	1	3	yes	—
180	C₂×C₄	C₂×#11765850	1	3	yes	—
181	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
182	C₈	—	2	3	no	—
183	C₈	#41×D₇₁	1	1	yes	characteristic
184	C₈	#14556175	1	1	yes	characteristic
185	C₂²	C₂²×#11765850	1	1	yes	characteristic
186	C₄	—	2	3	no	—
187	C₄	—	8	3	no	—
188	C₄	—	8	1	no	—
189	C₄	—	284	2	no	—
190	C₄	—	568	3	no	—
191	C₄	C₂×#14556074	1	6	yes	—
192	C₄	#17986656	1	1	yes	characteristic
193	C₄	#17986810	1	1	yes	characteristic
194	C₂	#22600971	1	1	yes	characteristic
195	C₂	C₂²×#14556074	1	1	yes	characteristic
196	C₂	#22601484	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₈₄

C₂

yes

maximal, characteristic

#22590567

C₂

yes

maximal

#22590492

C₂

yes

maximal

#22590387

C₂

yes

maximal

Q₈×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic

C₄×#14556074

C₂

yes

maximal

C₇₁×#3395

C₂

yes

maximal, characteristic

Q₈×Dic₁₄₂

—

C₂×#14556074

C₂²

yes

—

#17986404

C₂²

yes

—

#17986295

—

C₄×Dic₂₈₄

C₂²

yes

—

#17985824

C₂²

yes

—

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

—

#17985803

C₂²

yes

—

#17985803

C₂²

yes

—

#17985802

C₂²

yes

characteristic

C₄×C₇₁⋊C₁₆

C₂²

yes

—

#557×C₇₁

—

#494×C₇₁

C₂²

yes

—

#462×C₇₁

—

C₈×Q₈×C₇₁

C₂²

yes

characteristic

C₄×Q₁₆×C₇₁

C₂²

yes

—

C₂×Dic₂₈₄

C₂³

yes

characteristic

#14556128

—

#14556127

—

Q₈×Dic₇₁

—

C₄×Dic₁₄₂

—

#14556074

—

C₂×C₇₁⋊C₁₆

C₂³

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂³

yes

—

#14556062

—

#14556062

C₂³

yes

—

#14556053

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

C₂³

yes

—

C₄⋊Q₈×C₇₁

—

#125×C₇₁

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

D₄

yes

characteristic

#107×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

D₄

yes

—

#103×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

#41

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₈

yes

characteristic

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₈

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

Q₃₂

yes

complemented

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×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×Q₃₂

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

C₂²×Q₃₂

yes

C₂×C₇₁

yes

D₇₁

yes

characteristic

100

Q₈×Q₁₆

—

101

#2241

—

102

#2227

—

103

#2209

—

104

Q₈×C₁₆

—

105

C₄×Q₃₂

—

106

C₇₁

#33930

yes

characteristic, complemented, Sylow

107

Q₈²

—

142

—

108

#557

—

109

C₂×Q₃₂

—

110

#500

—

111

#494

C₂×D₇₁

yes

—

112

#477

—

142

—

113

#462

—

114

C₈×Q₈

C₂×D₇₁

yes

characteristic

115

C₄×Q₁₆

—

116

C₄×Q₁₆

C₂×D₇₁

yes

—

117

#366

—

118

C₄⋊C₁₆

—

119

#358

—

120

#358

—

121

#358

—

122

C₄×C₁₆

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

125

C₂×Q₁₆

C₂²×D₇₁

yes

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

284

—

128

C₄⋊Q₈

—

142

—

129

#125

—

130

C₄×Q₈

—

131

C₄×Q₈

—

132

C₄×Q₈

—

284

—

133

C₄×Q₈

—

142

—

134

C₄×Q₈

C₂²⋊D₇₁

yes

characteristic

135

Q₃₂

—

142

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

#107

—

139

#107

—

142

—

140

#107

C₂²×D₇₁

yes

characteristic

141

C₄⋊C₈

C₂²⋊D₇₁

yes

—

142

#103

—

143

#103

—

142

—

144

C₄×C₈

C₂²×D₇₁

yes

—

145

C₂×Q₈

—

146

C₂×Q₈

—

284

—

147

C₂×Q₈

—

142

—

148

C₂×Q₈

#11765850

yes

characteristic

149

Q₁₆

—

150

Q₁₆

—

151

Q₁₆

—

152

Q₁₆

—

153

Q₁₆

—

142

—

154

C₂×C₈

C₂×C₂²⋊D₇₁

yes

—

155

C₂×C₈

C₂³×D₇₁

yes

characteristic

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

284

—

161

C₄⋊C₄

—

284

—

162

C₄⋊C₄

—

142

—

163

C₄⋊C₄

#11765850

yes

—

164

C₄²

—

165

C₄²

—

284

—

166

C₄²

C₂×C₂²⋊D₇₁

yes

—

167

C₁₆

—

142

—

168

C₁₆

—

142

—

169

Q₈

—

170

Q₈

—

171

Q₈

—

284

—

172

Q₈

—

1136

—

173

Q₈

—

142

—

174

Q₈

#14556074

yes

complemented

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

284

—

178

C₂×C₄

—

284

—

179

C₂×C₄

C₂×#11765850

yes

—

180

C₂×C₄

C₂×#11765850

yes

—

181

C₂×C₄

C₂²×C₂²⋊D₇₁

yes

characteristic

182

C₈

—

183

C₈

#41×D₇₁

yes

characteristic

184

C₈

#14556175

yes

characteristic

185

C₂²

C₂²×#11765850

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

284

—

190

C₄

—

568

—

191

C₄

C₂×#14556074

yes

—

192

C₄

#17986656

yes

characteristic

193

C₄

#17986810

yes

characteristic

194

C₂

#22600971

yes

characteristic

195

C₂

C₂²×#14556074

yes

characteristic

196

C₂

#22601484

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#11765850
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#3395	C₂
socle, prime radical	C₂²×C₇₁	C₂²×D₈
hypercenter	#3395	D₇₁
hypocenter	C₇₁	#33930
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

subgroup

G/H

center

C₂²

C₂²×#11765850

commutator, cosocle

C₂×C₈×C₇₁

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₁

Fitting

C₇₁×#3395

C₂

socle, prime radical

C₂²×C₇₁

C₂²×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

N	H ≅ G/N	type
72. Q₈×C₇₁	Q₃₂	semidirect
106. C₇₁	#33930	semidirect
174. Q₈	#14556074	semidirect

H ≅ G/N

type

72. Q₈×C₇₁

Q₃₂

semidirect

106. C₇₁

#33930

semidirect

174. Q₈

#14556074

semidirect

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

subgroup	index
Q₈×Dic₂₈₄	2
#22590567	2
#22590492	2
#22590387	2
Q₈×C₇₁⋊C₁₆	2
C₄×#14556074	2
C₇₁×#3395	2
#33930	71

subgroup

index

Q₈×Dic₂₈₄

Q₈×C₇₁⋊C₁₆

C₇₁×#3395

Subgroups — #32155136

Conjugacy classes of subgroups

Complements

Maximal subgroups