Group #31902226 — subgroups

Subgroups — #31902226

Order 18176.

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

normal subgroups only

This group has 292 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	#22589321	C₂	1	1	yes	maximal, characteristic
2	#22586998	C₂	1	2	yes	maximal
3	#22586237	C₂	1	2	yes	maximal
4	#22586169	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1843	C₂	1	1	yes	maximal, characteristic
6	#17986556	C₂²	1	1	yes	characteristic
7	#17986052	C₂²	1	2	yes	—
8	#17986023	C₂²	1	1	yes	characteristic
9	#17985759	—	2	1	no	—
10	#17985759	—	2	1	no	—
11	#17985747	—	2	2	no	—
12	#17985744	—	2	2	no	—
13	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
14	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	#376×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#11765849	—	2	1	no	—
18	#14556128	D₄	1	2	yes	—
19	#14556120	—	2	1	no	—
20	#14556107	—	2	2	no	—
21	C₄×Dic₁₄₂	—	2	2	no	—
22	#14556087	—	2	1	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
26	C₄×C₇₁⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₇₁	—	2	2	no	—
28	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	C₄⋊C₈×C₇₁	—	2	2	no	—
33	C₄⋊C₈×C₇₁	—	4	2	no	—
34	C₄⋊C₈×C₇₁	—	4	2	no	—
35	#95×C₇₁	—	2	2	no	—
36	C₂×Dic₁₄₂	—	2	2	no	—
37	#11765849	—	4	1	no	—
38	#11765849	—	4	1	no	—
39	C₂×C₇₁⋊C₈	—	4	2	no	—
40	C₂×C₇₁⋊C₈	—	4	1	no	—
41	C₄⋊Dic₇₁	—	2	2	no	—
42	C₄⋊Dic₇₁	—	2	2	no	—
43	Dic₇₁⋊C₄	—	4	2	no	—
44	C₂²⋊Dic₇₁	—	4	1	no	—
45	C₄×Dic₇₁	—	4	1	no	—
46	C₂²×C₄×C₇₁	—	2	1	no	—
47	C₂²×C₄×C₇₁	—	2	1	no	—
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	#41	1	2	yes	—
50	C₂×C₈×C₇₁	—	4	2	no	—
51	C₂×C₈×C₇₁	—	4	2	no	—
52	C₂×C₈×C₇₁	—	4	2	no	—
53	C₄²×C₇₁	—	2	1	no	—
54	C₄²×C₇₁	—	2	1	no	—
55	C₄²×C₇₁	SD₁₆	1	2	yes	—
56	C₄²×C₇₁	Q₁₆	1	2	yes	—
57	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₁	#41	1	1	yes	characteristic
59	C₄²×C₇₁	#41	1	2	yes	—
60	C₂³×C₇₁	#123	1	1	yes	characteristic
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	—	2	2	no	—
63	C₂×C₄×C₇₁	—	4	2	no	—
64	C₂×C₄×C₇₁	—	4	2	no	—
65	C₂×C₄×C₇₁	—	4	2	no	—
66	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#123	1	2	yes	—
68	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#126	1	2	yes	—
70	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
74	C₂×Dic₇₁	—	4	2	no	—
75	Dic₁₄₂	—	8	2	no	—
76	C₈×C₇₁	—	8	2	no	—
77	C₇₁⋊C₈	—	8	2	no	—
78	C₇₁⋊C₈	—	8	1	no	—
79	C₂²×C₇₁	—	2	2	no	—
80	C₂²×C₇₁	#397	1	1	yes	characteristic
81	C₂²×C₇₁	#483	1	1	yes	characteristic
82	C₂²×C₇₁	#484	1	1	yes	characteristic
83	C₄×C₇₁	—	2	2	no	—
84	C₄×C₇₁	—	2	2	no	—
85	C₄×C₇₁	—	4	2	no	—
86	C₄×C₇₁	—	4	2	no	—
87	C₄×C₇₁	#460	1	1	yes	characteristic
88	C₄×C₇₁	#462	1	1	yes	characteristic
89	C₄×C₇₁	#487	1	1	yes	characteristic
90	C₄×C₇₁	#489	1	1	yes	characteristic
91	Dic₇₁	—	8	2	no	—
92	#10628	—	71	1	no	maximal, Sylow
93	C₂×C₇₁	—	2	1	no	—
94	C₂×C₇₁	—	2	1	no	—
95	C₂×C₇₁	#2072	1	1	yes	characteristic
96	C₂×C₇₁	#2080	1	2	yes	—
97	#1995	—	71	1	no	—
98	#1843	D₇₁	1	1	yes	characteristic
99	#1542	—	71	2	no	—
100	#1301	—	71	2	no	—
101	#1274	—	71	1	no	—
102	C₇₁	#10628	1	1	yes	characteristic, complemented, Sylow
103	#533	—	71	1	no	—
104	#431	—	71	2	no	—
105	#423	—	71	1	no	—
106	C₂×C₄⋊C₈	—	2	1	no	—
107	C₂×C₄⋊C₈	—	2	1	no	—
108	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
109	#376	C₂×D₇₁	1	1	yes	characteristic
110	#339	—	142	1	no	—
111	#339	—	142	1	no	—
112	#333	—	142	2	no	—
113	#332	—	142	2	no	—
114	C₂×M₁₆	—	142	1	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₄⋊Q₈	—	71	2	no	—
117	#125	—	142	1	no	—
118	C₂²⋊Q₈	—	142	2	no	—
119	C₄×Q₈	—	142	2	no	—
120	#117	—	142	1	no	—
121	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	—	142	2	no	—
129	C₄⋊C₈	—	142	2	no	—
130	C₂²⋊C₈	—	142	2	no	—
131	C₄×C₈	—	142	2	no	—
132	#95	—	2	2	no	—
133	C₂×Q₈	—	142	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
137	C₂²×C₄	#11765865	1	2	yes	—
138	M₁₆	—	284	1	no	—
139	M₁₆	—	284	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	284	2	no	—
144	C₂×C₈	—	284	1	no	—
145	C₄⋊C₄	—	284	2	no	—
146	C₄⋊C₄	—	142	2	no	—
147	C₄⋊C₄	—	142	2	no	—
148	C₂²⋊C₄	—	284	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	284	1	no	—
152	C₄²	#11765853	1	2	yes	—
153	C₄²	#11765855	1	2	yes	—
154	C₄²	D₄×D₇₁	1	1	yes	characteristic
155	C₄²	#11765864	1	1	yes	characteristic
156	C₄²	#11765866	1	2	yes	—
157	C₂³	#14556111	1	1	yes	characteristic
158	Q₈	—	568	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	284	2	no	—
165	C₂×C₄	#14556099	1	1	yes	characteristic
166	C₂×C₄	#14556113	1	2	yes	—
167	C₂×C₄	#14556117	1	1	yes	characteristic
168	C₂×C₄	#14556123	1	2	yes	—
169	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
170	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
171	C₂×C₄	#14556154	1	1	yes	characteristic
172	C₂×C₄	#14556159	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	—	568	2	no	—
175	C₈	—	568	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#17985925	1	1	yes	characteristic
178	C₂²	#17986329	1	1	yes	characteristic
179	C₂²	#17986334	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	568	2	no	—
185	C₄	#17986196	1	1	yes	characteristic
186	C₄	#17986206	1	1	yes	characteristic
187	C₄	#17986351	1	1	yes	characteristic
188	C₄	#17986361	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#22589749	1	1	yes	characteristic
192	C₂	#22589791	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22589321

C₂

yes

maximal, characteristic

#22586998

C₂

yes

maximal

#22586237

C₂

yes

maximal

#22586169

C₂

yes

maximal, characteristic

C₇₁×#1843

C₂

yes

maximal, characteristic

#17986556

C₂²

yes

characteristic

#17986052

C₂²

yes

—

#17986023

C₂²

yes

—

—

—

—

C₂×C₄⋊C₈×C₇₁

—

C₂×C₄⋊C₈×C₇₁

—

C₂×C₄⋊C₈×C₇₁

C₂²

yes

—

#376×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

#14556128

D₄

yes

—

#14556120

—

#14556107

—

C₄×Dic₁₄₂

—

#14556087

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

#95×C₇₁

—

C₂×Dic₁₄₂

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

SD₁₆

yes

—

C₄²×C₇₁

Q₁₆

yes

—

C₄²×C₇₁

C₂×D₄

yes

C₄²×C₇₁

yes

C₄²×C₇₁

yes

—

C₂³×C₇₁

#123

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

#123

yes

—

C₂×C₄×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

#126

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₁₆

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

#10628

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#2072

yes

characteristic

C₂×C₇₁

#2080

yes

—

#1995

—

#1843

D₇₁

yes

characteristic

#1542

—

100

#1301

—

101

#1274

—

102

C₇₁

#10628

yes

characteristic, complemented, Sylow

103

#533

—

104

#431

—

105

#423

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

C₂×D₇₁

yes

—

109

#376

C₂×D₇₁

yes

characteristic

110

#339

—

142

—

111

#339

—

142

—

112

#333

—

142

—

113

#332

—

142

—

114

C₂×M₁₆

—

142

—

115

C₂²×C₈

—

116

C₄⋊Q₈

—

117

#125

—

142

—

118

C₂²⋊Q₈

—

142

—

119

C₄×Q₈

—

142

—

120

#117

—

142

—

121

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

122

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

123

C₂×C₄²

C₂²×D₇₁

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

142

—

129

C₄⋊C₈

—

142

—

130

C₂²⋊C₈

—

142

—

131

C₄×C₈

—

142

—

132

#95

—

133

C₂×Q₈

—

142

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

137

C₂²×C₄

#11765865

yes

—

138

M₁₆

—

284

—

139

M₁₆

—

284

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

284

—

144

C₂×C₈

—

284

—

145

C₄⋊C₄

—

284

—

146

C₄⋊C₄

—

142

—

147

C₄⋊C₄

—

142

—

148

C₂²⋊C₄

—

284

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

284

—

152

C₄²

#11765853

yes

—

153

C₄²

#11765855

yes

—

154

C₄²

D₄×D₇₁

yes

characteristic

155

C₄²

#11765864

yes

characteristic

156

C₄²

#11765866

yes

—

157

C₂³

#14556111

yes

characteristic

158

Q₈

—

568

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

284

—

165

C₂×C₄

#14556099

yes

characteristic

166

C₂×C₄

#14556113

yes

—

167

C₂×C₄

#14556117

yes

characteristic

168

C₂×C₄

#14556123

yes

—

169

C₂×C₄

C₂×#11765853

yes

characteristic

170

C₂×C₄

C₂×#11765855

yes

characteristic

171

C₂×C₄

#14556154

yes

characteristic

172

C₂×C₄

#14556159

yes

characteristic

173

C₈

—

174

C₈

—

568

—

175

C₈

—

568

—

176

C₂²

—

177

C₂²

#17985925

yes

characteristic

178

C₂²

#17986329

yes

characteristic

179

C₂²

#17986334

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

568

—

185

C₄

#17986196

yes

characteristic

186

C₄

#17986206

yes

characteristic

187

C₄

#17986351

yes

characteristic

188

C₄

#17986361

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#22589749

yes

characteristic

192

C₂

#22589791

yes

—

	subgroup	G/H
center	C₂×C₄	#14556117
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1843	C₂
socle	C₂²×C₇₁	#397
prime radical	C₂³×C₇₁	#123
hypercenter	#1843	D₇₁
hypocenter	C₇₁	#10628
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#14556117

commutator, cosocle

C₂×C₄²×C₇₁

C₂³

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1843

C₂

C₂²×C₇₁

C₂³×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
102. C₇₁	#10628	semidirect

H ≅ G/N

type

102. C₇₁

#10628

semidirect

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

subgroup	index
#22589321	2
#22586998	2
#22586237	2
#22586169	2
C₇₁×#1843	2
#10628	71

subgroup

index

C₇₁×#1843

Subgroups — #31902226

Conjugacy classes of subgroups

Complements

Maximal subgroups