Group #31909452 — subgroups

Subgroups — #31909452

Order 18176.

206 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×#14556074	C₂	1	1	yes	maximal, characteristic
2	#22587237	C₂	1	1	yes	maximal, characteristic
3	#22587236	C₂	1	1	yes	maximal, characteristic
4	#22586413	C₂	1	1	yes	maximal, characteristic
5	#22586305	C₂	1	1	yes	maximal, characteristic
6	#22586256	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1332	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556074	C₄	1	1	yes	characteristic
9	#17986130	—	2	1	no	—
10	#17986126	—	2	1	no	—
11	C₄×Dic₂₈₄	C₂²	1	1	yes	characteristic
12	#17985824	C₄	1	1	yes	characteristic
13	#17985803	C₄	1	1	yes	characteristic
14	#17985802	C₄	1	1	yes	characteristic
15	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#17985752	C₂²	1	1	yes	characteristic
17	Q₈⋊(C₇₁⋊C₈)	—	2	1	no	—
18	#17985729	—	2	1	no	—
19	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	#335×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₈₄	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₄₂	—	2	1	no	—
25	#14556074	—	2	2	no	—
26	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	#14556062	C₂×C₄	1	1	yes	characteristic
29	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
30	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
31	Dic₇₁⋊C₈	—	4	1	no	—
32	#14556053	—	2	1	no	—
33	#14556039	—	4	1	no	—
34	#14556038	—	4	1	no	—
35	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
36	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
37	C₂×Q₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₇₁	—	2	1	no	—
39	C₂×C₁₆×C₇₁	—	2	1	no	—
40	#107×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₇₁	—	2	1	no	—
42	#103×C₇₁	—	2	1	no	—
43	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₄₂	—	2	1	no	—
45	Dic₂₈₄	—	2	1	no	—
46	Dic₂₈₄	—	2	1	no	—
47	C₂×C₇₁⋊C₈	—	2	1	no	—
48	C₂×C₇₁⋊C₈	—	4	1	no	—
49	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
50	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
51	C₄⋊Dic₇₁	—	2	1	no	—
52	Dic₇₁⋊C₄	—	4	1	no	—
53	C₄×Dic₇₁	—	4	1	no	—
54	C₇₁⋊C₁₆	—	2	1	no	—
55	C₇₁⋊C₁₆	M₁₆	1	2	yes	—
56	C₂×Q₈×C₇₁	—	2	1	no	—
57	Q₁₆×C₇₁	—	2	1	no	—
58	Q₁₆×C₇₁	—	2	1	no	—
59	C₂×C₈×C₇₁	—	4	1	no	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₇₁	—	2	1	no	—
63	C₄⋊C₄×C₇₁	—	4	1	no	—
64	C₄²×C₇₁	—	4	1	no	—
65	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₁₆×C₇₁	—	2	2	no	—
67	Q₈×C₇₁	—	4	1	no	—
68	Q₈×C₇₁	—	4	1	no	—
69	C₂×C₄×C₇₁	—	4	1	no	—
70	C₂×C₄×C₇₁	—	4	1	no	—
71	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
74	C₂×Dic₇₁	—	4	1	no	—
75	C₂×Dic₇₁	—	4	1	no	—
76	Dic₁₄₂	—	4	1	no	—
77	Dic₁₄₂	—	4	1	no	—
78	C₈×C₇₁	—	2	1	no	—
79	C₈×C₇₁	—	8	1	no	—
80	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
81	C₈×C₇₁	#131	1	1	yes	characteristic
82	C₇₁⋊C₈	—	2	1	no	—
83	C₇₁⋊C₈	—	2	1	no	—
84	C₇₁⋊C₈	—	4	1	no	—
85	C₇₁⋊C₈	—	8	1	no	—
86	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
87	C₄×C₇₁	—	2	1	no	—
88	C₄×C₇₁	—	8	1	no	—
89	C₄×C₇₁	—	8	1	no	—
90	C₄×C₇₁	#436	1	1	yes	characteristic
91	C₄×C₇₁	#443	1	1	yes	characteristic
92	C₄×C₇₁	#510	1	1	yes	characteristic
93	C₄×C₇₁	#510	1	1	yes	characteristic
94	Dic₇₁	—	8	1	no	—
95	Dic₇₁	—	8	1	no	—
96	#12085	—	71	1	no	maximal, Sylow
97	C₂×C₇₁	#1589	1	1	yes	characteristic
98	C₂×C₇₁	#2176	1	1	yes	characteristic
99	C₂×C₇₁	#2179	1	1	yes	characteristic
100	C₄×Q₃₂	—	71	1	no	—
101	#1591	—	71	1	no	—
102	#1591	—	71	1	no	—
103	#1372	—	71	1	no	—
104	#1332	—	71	1	no	—
105	#1332	D₇₁	1	1	yes	characteristic
106	#1311	—	71	1	no	—
107	C₇₁	#12085	1	1	yes	characteristic, complemented, Sylow
108	C₂×Q₃₂	—	71	1	no	—
109	#446	—	142	1	no	—
110	#446	—	142	1	no	—
111	C₄×Q₁₆	—	71	1	no	—
112	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
113	#366	—	71	1	no	—
114	#358	—	71	1	no	—
115	#358	—	71	1	no	—
116	C₄×C₁₆	—	71	1	no	—
117	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
118	#335	—	71	1	no	—
119	#335	C₂×D₇₁	1	1	yes	characteristic
120	Q₈⋊C₈	—	142	1	no	—
121	Q₈⋊C₈	—	142	1	no	—
122	C₈²	—	71	1	no	—
123	C₂×Q₁₆	—	71	1	no	—
124	C₂×Q₁₆	C₄×D₇₁	1	1	yes	characteristic
125	C₄×Q₈	—	2	1	no	—
126	C₄×Q₈	—	142	1	no	—
127	Q₃₂	—	142	2	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	71	1	no	—
130	C₂×C₁₆	—	71	1	no	—
131	#107	—	71	1	no	—
132	#107	C₄×D₇₁	1	1	yes	characteristic
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	—	284	1	no	—
135	C₄⋊C₈	—	284	1	no	—
136	C₄⋊C₈	—	142	1	no	—
137	#103	—	2	1	no	—
138	#103	—	142	1	no	—
139	#97	—	284	1	no	—
140	#97	—	284	1	no	—
141	C₄×C₈	—	71	1	no	—
142	C₄×C₈	—	71	1	no	—
143	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
144	C₂×Q₈	—	2	1	no	—
145	C₂×Q₈	—	142	1	no	—
146	Q₁₆	—	2	1	no	—
147	Q₁₆	—	2	1	no	—
148	Q₁₆	—	142	1	no	—
149	Q₁₆	—	142	1	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	284	1	no	—
152	C₂×C₈	—	71	1	no	—
153	C₂×C₈	—	71	1	no	—
154	C₂×C₈	—	142	1	no	—
155	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
156	C₂×C₈	#11765864	1	1	yes	characteristic
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	4	1	no	—
159	C₄⋊C₄	—	284	1	no	—
160	C₄⋊C₄	—	142	1	no	—
161	C₄²	—	4	1	no	—
162	C₄²	—	284	1	no	—
163	C₄²	D₄×D₇₁	1	1	yes	characteristic
164	C₁₆	—	2	2	no	—
165	C₁₆	—	71	2	no	—
166	C₁₆	—	142	1	no	—
167	Q₈	—	4	1	no	—
168	Q₈	—	4	1	no	—
169	Q₈	—	284	1	no	—
170	Q₈	—	284	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	284	1	no	—
174	C₂×C₄	—	284	1	no	—
175	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
177	C₂×C₄	#14556149	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	8	1	no	—
180	C₈	—	284	1	no	—
181	C₈	—	568	1	no	—
182	C₈	—	142	1	no	—
183	C₈	—	142	1	no	—
184	C₈	M₁₆×D₇₁	1	1	yes	characteristic
185	C₈	#14556134	1	1	yes	characteristic
186	C₂²	#17986079	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	8	1	no	—
189	C₄	—	8	1	no	—
190	C₄	—	568	1	no	—
191	C₄	—	568	1	no	—
192	C₄	#17986074	1	1	yes	characteristic
193	C₄	#17986113	1	1	yes	characteristic
194	C₄	#17986438	1	1	yes	characteristic
195	C₄	#17986441	1	1	yes	characteristic
196	C₂	#22587222	1	1	yes	characteristic
197	C₂	#22590208	1	1	yes	characteristic
198	C₂	#22590223	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556074

C₂

yes

maximal, characteristic

#22587237

C₂

yes

maximal, characteristic

#22587236

C₂

yes

maximal, characteristic

#22586413

C₂

yes

maximal, characteristic

#22586305

C₂

yes

maximal, characteristic

#22586256

C₂

yes

maximal, characteristic

C₇₁×#1332

C₂

yes

maximal, characteristic

C₂×#14556074

C₄

yes

characteristic

#17986130

—

#17986126

—

C₄×Dic₂₈₄

C₂²

yes

characteristic

#17985824

C₄

yes

characteristic

#17985803

C₄

yes

characteristic

#17985802

C₄

yes

characteristic

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985752

C₂²

yes

characteristic

Q₈⋊(C₇₁⋊C₈)

—

#17985729

—

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×Q₁₆×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#335×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

C₂×C₄

yes

characteristic

C₄×Dic₁₄₂

—

#14556074

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

#14556062

C₂×C₄

yes

characteristic

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

#14556053

—

#14556039

—

#14556038

—

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₇₁

C₂×C₄

yes

characteristic

C₄×Q₈×C₇₁

—

C₂×C₁₆×C₇₁

—

#107×C₇₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₁

—

#103×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

M₁₆

yes

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₄×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

100

C₄×Q₃₂

—

101

#1591

—

102

#1591

—

103

#1372

—

104

#1332

—

105

#1332

D₇₁

yes

characteristic

106

#1311

—

107

C₇₁

#12085

yes

characteristic, complemented, Sylow

108

C₂×Q₃₂

—

109

#446

—

142

—

110

#446

—

142

—

111

C₄×Q₁₆

—

112

C₄×Q₁₆

C₂×D₇₁

yes

characteristic

113

#366

—

114

#358

—

115

#358

—

116

C₄×C₁₆

—

117

C₄×C₁₆

C₂×D₇₁

yes

characteristic

118

#335

—

119

#335

C₂×D₇₁

yes

characteristic

120

Q₈⋊C₈

—

142

—

121

Q₈⋊C₈

—

142

—

122

C₈²

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

C₄×D₇₁

yes

characteristic

125

C₄×Q₈

—

126

C₄×Q₈

—

142

—

127

Q₃₂

—

142

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

#107

—

132

#107

C₄×D₇₁

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

284

—

135

C₄⋊C₈

—

284

—

136

C₄⋊C₈

—

142

—

137

#103

—

138

#103

—

142

—

139

#97

—

284

—

140

#97

—

284

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

C₂²×D₇₁

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

—

142

—

146

Q₁₆

—

147

Q₁₆

—

148

Q₁₆

—

142

—

149

Q₁₆

—

142

—

150

C₂×C₈

—

151

C₂×C₈

—

284

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

142

—

155

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

156

C₂×C₈

#11765864

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

284

—

160

C₄⋊C₄

—

142

—

161

C₄²

—

162

C₄²

—

284

—

163

C₄²

D₄×D₇₁

yes

characteristic

164

C₁₆

—

165

C₁₆

—

166

C₁₆

—

142

—

167

Q₈

—

168

Q₈

—

169

Q₈

—

284

—

170

Q₈

—

284

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

284

—

174

C₂×C₄

—

284

—

175

C₂×C₄

(C₂²⋊D₇₁)⋊C₄

yes

characteristic

176

C₂×C₄

D₈×D₇₁

yes

characteristic

177

C₂×C₄

#14556149

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

284

—

181

C₈

—

568

—

182

C₈

—

142

—

183

C₈

—

142

—

184

C₈

M₁₆×D₇₁

yes

characteristic

185

C₈

#14556134

yes

characteristic

186

C₂²

#17986079

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

568

—

191

C₄

—

568

—

192

C₄

#17986074

yes

characteristic

193

C₄

#17986113

yes

characteristic

194

C₄

#17986438

yes

characteristic

195

C₄

#17986441

yes

characteristic

196

C₂

#22587222

yes

characteristic

197

C₂

#22590208

yes

characteristic

198

C₂

#22590223

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1332	C₂
socle, prime radical	C₂²×C₇₁	C₄×D₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1332	D₇₁
hypocenter	C₇₁	#12085
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1332

subgroup

G/H

center

C₂×C₄

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1332

C₂

socle, prime radical

C₂²×C₇₁

C₄×D₈

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1332

N	H ≅ G/N	type
107. C₇₁	#12085	semidirect

H ≅ G/N

type

107. C₇₁

#12085

semidirect

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

subgroup	index
C₄×#14556074	2
#22587237	2
#22587236	2
#22586413	2
#22586305	2
#22586256	2
C₇₁×#1332	2
#12085	71

subgroup

index

C₇₁×#1332

Subgroups — #31909452

Conjugacy classes of subgroups

Complements

Maximal subgroups