Group #31902576 — subgroups

Subgroups — #31902576

Order 18176.

264 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22589063	C₂	1	1	yes	maximal, characteristic
2	#22586241	C₂	1	2	yes	maximal
3	#22586235	C₂	1	2	yes	maximal
4	#22586171	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1849	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×Dic₇₁⋊C₈	—	2	1	no	—
9	C₄×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
10	#17985753	—	2	1	no	—
11	C₂×#108×C₇₁	—	2	1	no	—
12	C₂×#108×C₇₁	—	2	1	no	—
13	#423×C₇₁	C₂²	1	2	yes	—
14	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
15	C₂²×C₇₁⋊C₈	—	2	1	no	—
16	C₂×C₄⋊Dic₇₁	—	2	1	no	—
17	C₂×C₄×Dic₇₁	—	2	1	no	—
18	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
19	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
20	Dic₇₁⋊C₈	—	4	2	no	—
21	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
22	C₄×C₇₁⋊C₈	—	2	2	no	—
23	C₇₁⋊#95	Q₈	1	1	yes	characteristic
24	C₇₁⋊#95	Q₈	1	1	yes	characteristic
25	C₂×M₁₆×C₇₁	—	2	2	no	—
26	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
29	#108×C₇₁	—	4	2	no	—
30	#108×C₇₁	—	4	2	no	—
31	C₄⋊C₈×C₇₁	—	2	4	no	—
32	#97×C₇₁	—	2	2	no	—
33	C₂²×Dic₇₁	—	2	1	no	—
34	C₂²×Dic₇₁	—	2	1	no	—
35	C₂×C₇₁⋊C₈	—	4	1	no	—
36	C₂×C₇₁⋊C₈	—	4	2	no	—
37	C₂×C₇₁⋊C₈	—	4	1	no	—
38	C₂×C₇₁⋊C₈	—	4	1	no	—
39	C₄⋊Dic₇₁	—	4	2	no	—
40	C₄×Dic₇₁	—	4	1	no	—
41	C₄×Dic₇₁	—	4	1	no	—
42	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
45	M₁₆×C₇₁	—	4	2	no	—
46	M₁₆×C₇₁	—	4	2	no	—
47	C₂×C₈×C₇₁	—	2	1	no	—
48	C₂×C₈×C₇₁	—	2	1	no	—
49	C₂×C₈×C₇₁	—	2	1	no	—
50	C₂×C₈×C₇₁	—	2	1	no	—
51	C₂×C₈×C₇₁	—	4	2	no	—
52	C₂×C₈×C₇₁	D₈	1	2	yes	—
53	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
54	C₄²×C₇₁	#41	1	2	yes	—
55	C₄²×C₇₁	#41	1	2	yes	—
56	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	4	1	no	—
58	C₂×C₄×C₇₁	—	4	1	no	—
59	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#125	1	2	yes	—
62	C₂×C₄×C₇₁	#126	1	2	yes	—
63	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
67	C₂×Dic₇₁	—	2	2	no	—
68	C₂×Dic₇₁	—	4	1	no	—
69	C₂×Dic₇₁	—	4	1	no	—
70	C₂×Dic₇₁	—	4	1	no	—
71	C₂×Dic₇₁	—	4	1	no	—
72	C₂×Dic₇₁	—	4	1	no	—
73	C₈×C₇₁	—	2	2	no	—
74	C₈×C₇₁	—	2	2	no	—
75	C₈×C₇₁	—	8	2	no	—
76	C₇₁⋊C₈	—	8	2	no	—
77	C₇₁⋊C₈	—	8	1	no	—
78	C₂²×C₇₁	—	2	1	no	—
79	C₂²×C₇₁	—	2	1	no	—
80	C₂²×C₇₁	#400	1	1	yes	characteristic
81	C₂²×C₇₁	#474	1	1	yes	characteristic
82	C₂²×C₇₁	#475	1	1	yes	characteristic
83	C₄×C₇₁	—	2	2	no	—
84	C₄×C₇₁	—	2	2	no	—
85	C₄×C₇₁	#480	1	1	yes	characteristic
86	C₄×C₇₁	#483	1	1	yes	characteristic
87	C₄×C₇₁	#486	1	1	yes	characteristic
88	C₄×C₇₁	#491	1	1	yes	characteristic
89	Dic₇₁	—	4	2	no	—
90	Dic₇₁	—	8	1	no	—
91	#10687	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	—	2	1	no	—
94	C₂×C₇₁	#2099	1	1	yes	characteristic
95	C₂×C₇₁	#2102	1	1	yes	characteristic
96	C₂×C₇₁	#2103	1	1	yes	characteristic
97	#1940	—	71	1	no	—
98	#1849	D₇₁	1	1	yes	characteristic
99	#1303	—	71	2	no	—
100	#1301	—	71	2	no	—
101	#1275	—	71	1	no	—
102	C₇₁	#10687	1	1	yes	characteristic, complemented, Sylow
103	#431	—	71	2	no	—
104	C₂×#108	—	2	1	no	—
105	C₂×#108	—	2	1	no	—
106	#423	C₂×D₇₁	1	2	yes	—
107	C₂×C₄⋊C₈	—	71	1	no	—
108	C₂×C₄⋊C₈	—	142	1	no	—
109	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	71	1	no	—
111	#336	—	142	1	no	—
112	C₂×M₁₆	—	2	2	no	—
113	C₂²×C₈	—	142	1	no	—
114	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	142	1	no	—
117	C₂×C₄²	—	142	1	no	—
118	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
119	#108	—	4	2	no	—
120	#108	—	4	2	no	—
121	C₄⋊C₈	—	2	4	no	—
122	C₄⋊C₈	—	284	2	no	—
123	C₄⋊C₈	—	142	2	no	—
124	C₄⋊C₈	—	142	2	no	—
125	C₂²⋊C₈	—	142	2	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	142	2	no	—
128	#95	—	71	1	no	—
129	#95	—	71	1	no	—
130	C₂²×C₄	—	142	1	no	—
131	C₂²×C₄	—	142	1	no	—
132	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
134	C₂²×C₄	#11765866	1	1	yes	characteristic
135	M₁₆	—	4	2	no	—
136	M₁₆	—	4	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	284	1	no	—
143	C₂×C₈	—	284	2	no	—
144	C₂×C₈	—	284	1	no	—
145	C₂×C₈	—	284	1	no	—
146	C₂×C₈	#11765850	1	2	yes	—
147	C₂×C₈	#11765853	1	2	yes	—
148	C₄⋊C₄	—	284	2	no	—
149	C₄²	—	284	1	no	—
150	C₄²	—	284	1	no	—
151	C₄²	#11765864	1	2	yes	—
152	C₄²	#11765865	1	2	yes	—
153	C₂³	#14556104	1	1	yes	characteristic
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	284	1	no	—
157	C₂×C₄	—	284	1	no	—
158	C₂×C₄	—	284	1	no	—
159	C₂×C₄	—	284	1	no	—
160	C₂×C₄	—	284	1	no	—
161	C₂×C₄	—	142	2	no	—
162	C₂×C₄	#14556111	1	1	yes	characteristic
163	C₂×C₄	#14556115	1	1	yes	characteristic
164	C₂×C₄	#14556118	1	2	yes	—
165	C₂×C₄	#14556123	1	2	yes	—
166	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
167	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
168	C₂×C₄	#14556159	1	1	yes	characteristic
169	C₂×C₄	#14556159	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	8	2	no	—
173	C₈	—	568	2	no	—
174	C₈	—	568	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	#17985937	1	1	yes	characteristic
178	C₂²	#17986276	1	1	yes	characteristic
179	C₂²	#17986281	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	284	2	no	—
183	C₄	—	568	1	no	—
184	C₄	#17986312	1	1	yes	characteristic
185	C₄	#17986329	1	1	yes	characteristic
186	C₄	#17986345	1	1	yes	characteristic
187	C₄	#17986367	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#22589897	1	1	yes	characteristic
191	C₂	#22589914	1	1	yes	characteristic
192	C₂	#22589919	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22589063

C₂

yes

maximal, characteristic

#22586241

C₂

yes

maximal

#22586235

C₂

yes

maximal

#22586171

C₂

yes

maximal, characteristic

C₇₁×#1849

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₂×Dic₇₁⋊C₈

—

C₄×C₄⋊Dic₇₁

C₂²

yes

characteristic

#17985753

—

C₂×#108×C₇₁

—

C₂×#108×C₇₁

—

#423×C₇₁

C₂²

yes

—

C₂×#97×C₇₁

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄×Dic₇₁

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₇₁⋊#95

Q₈

yes

characteristic

C₇₁⋊#95

Q₈

yes

characteristic

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#108×C₇₁

—

#108×C₇₁

—

C₄⋊C₈×C₇₁

—

#97×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₄²×C₇₁

#41

yes

—

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#126

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#10687

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

#1303

—

100

#1301

—

101

#1275

—

102

C₇₁

#10687

yes

characteristic, complemented, Sylow

103

#431

—

104

C₂×#108

—

105

C₂×#108

—

106

#423

C₂×D₇₁

yes

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

—

142

—

109

C₂×#97

C₂×D₇₁

yes

characteristic

110

C₄×C₄⋊C₄

—

111

#336

—

142

—

112

C₂×M₁₆

—

113

C₂²×C₈

—

142

—

114

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

115

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

116

C₂×C₄⋊C₄

—

142

—

117

C₂×C₄²

—

142

—

118

C₂×C₄²

C₂²×D₇₁

yes

characteristic

119

#108

—

120

#108

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

284

—

123

C₄⋊C₈

—

142

—

124

C₄⋊C₈

—

142

—

125

C₂²⋊C₈

—

142

—

126

#97

—

127

C₄×C₈

—

142

—

128

#95

—

129

#95

—

130

C₂²×C₄

—

142

—

131

C₂²×C₄

—

142

—

132

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

133

C₂²×C₄

Q₈×D₇₁

yes

characteristic

134

C₂²×C₄

#11765866

yes

characteristic

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

284

—

143

C₂×C₈

—

284

—

144

C₂×C₈

—

284

—

145

C₂×C₈

—

284

—

146

C₂×C₈

#11765850

yes

—

147

C₂×C₈

#11765853

yes

—

148

C₄⋊C₄

—

284

—

149

C₄²

—

284

—

150

C₄²

—

284

—

151

C₄²

#11765864

yes

—

152

C₄²

#11765865

yes

—

153

C₂³

#14556104

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

284

—

157

C₂×C₄

—

284

—

158

C₂×C₄

—

284

—

159

C₂×C₄

—

284

—

160

C₂×C₄

—

284

—

161

C₂×C₄

—

142

—

162

C₂×C₄

#14556111

yes

characteristic

163

C₂×C₄

#14556115

yes

characteristic

164

C₂×C₄

#14556118

yes

—

165

C₂×C₄

#14556123

yes

—

166

C₂×C₄

C₂×#11765850

yes

characteristic

167

C₂×C₄

C₂×#11765853

yes

characteristic

168

C₂×C₄

#14556159

yes

characteristic

169

C₂×C₄

#14556159

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

173

C₈

—

568

—

174

C₈

—

568

—

175

C₂²

—

176

C₂²

—

177

C₂²

#17985937

yes

characteristic

178

C₂²

#17986276

yes

characteristic

179

C₂²

#17986281

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

284

—

183

C₄

—

568

—

184

C₄

#17986312

yes

characteristic

185

C₄

#17986329

yes

characteristic

186

C₄

#17986345

yes

characteristic

187

C₄

#17986367

yes

characteristic

188

C₂

—

189

C₂

—

190

C₂

#22589897

yes

characteristic

191

C₂

#22589914

yes

characteristic

192

C₂

#22589919

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556115
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1849	C₂
socle	C₂²×C₇₁	#400
prime radical	C₂³×C₇₁	C₂²⋊Q₈
hypercenter	#1849	D₇₁
hypocenter	C₇₁	#10687
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂×C₄

#14556115

commutator, cosocle

C₂×C₄²×C₇₁

C₂³

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1849

C₂

socle

C₂²×C₇₁

#400

prime radical

C₂³×C₇₁

C₂²⋊Q₈

D₇₁

C₇₁

G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
102. C₇₁	#10687	semidirect

H ≅ G/N

type

102. C₇₁

#10687

semidirect

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

subgroup	index
#22589063	2
#22586241	2
#22586235	2
#22586171	2
C₇₁×#1849	2
#10687	71

subgroup

index

C₇₁×#1849

Subgroups — #31902576

Conjugacy classes of subgroups

Complements

Maximal subgroups