Group #31902572 — subgroups

Subgroups — #31902572

Order 18176.

410 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22589302	C₂	1	2	yes	maximal
2	#22588497	C₂	1	1	yes	maximal, characteristic
3	#22588467	C₂	1	2	yes	maximal
4	C₇₁⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556060	—	2	2	no	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
8	#17985869	C₂²	1	2	yes	—
9	#17985761	—	2	4	no	—
10	#17985761	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×#103×C₇₁	—	2	2	no	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#384×C₇₁	C₂²	1	1	yes	characteristic
15	C₂²×C₇₁⋊C₈	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₇₁	—	2	2	no	—
17	C₂×C₄⋊Dic₇₁	—	2	2	no	—
18	C₂×C₄⋊Dic₇₁	—	2	2	no	—
19	#14556060	—	4	4	no	—
20	C₄×C₇₁⋊C₈	—	2	2	no	—
21	C₄×C₇₁⋊C₈	—	2	2	no	—
22	C₂²×Q₈×C₇₁	—	2	1	no	—
23	C₂²×C₈×C₇₁	D₄	1	2	yes	—
24	C₄⋊Q₈×C₇₁	—	2	2	no	—
25	C₄⋊Q₈×C₇₁	—	2	1	no	—
26	C₄⋊Q₈×C₇₁	Q₈	1	2	yes	—
27	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	#103×C₇₁	—	4	4	no	—
32	C₄×C₈×C₇₁	—	2	2	no	—
33	C₂²×Dic₇₁	—	4	2	no	—
34	C₂×C₇₁⋊C₈	—	2	4	no	—
35	C₂×C₇₁⋊C₈	—	2	4	no	—
36	C₂×C₇₁⋊C₈	—	2	4	no	—
37	C₄⋊Dic₇₁	—	4	4	no	—
38	C₄⋊Dic₇₁	—	4	4	no	—
39	C₄⋊Dic₇₁	—	4	4	no	—
40	C₂×Q₈×C₇₁	—	2	2	no	—
41	C₂×Q₈×C₇₁	—	4	1	no	—
42	C₂×Q₈×C₇₁	—	4	2	no	—
43	C₂×Q₈×C₇₁	—	4	2	no	—
44	C₂²×C₄×C₇₁	—	4	1	no	—
45	C₂²×C₄×C₇₁	—	4	1	no	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	—	2	2	no	—
49	C₂×C₈×C₇₁	—	2	2	no	—
50	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
51	C₄⋊C₄×C₇₁	—	2	4	no	—
52	C₄⋊C₄×C₇₁	—	4	2	no	—
53	C₄⋊C₄×C₇₁	—	4	2	no	—
54	C₄⋊C₄×C₇₁	—	4	4	no	—
55	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₇₁	#41	1	2	yes	—
57	C₄²×C₇₁	#41	1	1	yes	characteristic
58	C₂³×C₇₁	#127	1	1	yes	characteristic
59	Q₈×C₇₁	—	4	4	no	—
60	Q₈×C₇₁	—	8	2	no	—
61	C₂×C₄×C₇₁	—	4	2	no	—
62	C₂×C₄×C₇₁	—	8	1	no	—
63	C₂×C₄×C₇₁	—	8	1	no	—
64	C₂×C₄×C₇₁	—	8	1	no	—
65	C₂×C₄×C₇₁	—	8	2	no	—
66	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#123	1	2	yes	—
70	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₇₁	#135	1	4	yes	—
72	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₇₁	#137	1	4	yes	—
74	C₂×Dic₇₁	—	8	2	no	—
75	C₂×Dic₇₁	—	8	2	no	—
76	C₂×Dic₇₁	—	8	2	no	—
77	C₈×C₇₁	—	2	4	no	—
78	C₇₁⋊C₈	—	4	4	no	—
79	C₂²×C₇₁	#399	1	1	yes	characteristic
80	C₂²×C₇₁	#492	1	1	yes	characteristic
81	C₂²×C₇₁	#495	1	2	yes	—
82	C₂²×C₇₁	#496	1	1	yes	characteristic
83	C₂²×C₇₁	#497	1	2	yes	—
84	C₄×C₇₁	—	8	2	no	—
85	C₄×C₇₁	—	16	1	no	—
86	C₄×C₇₁	#476	1	2	yes	—
87	C₄×C₇₁	#479	1	4	yes	—
88	C₄×C₇₁	#481	1	2	yes	—
89	C₄×C₇₁	#485	1	4	yes	—
90	Dic₇₁	—	16	2	no	—
91	#10686	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	#1708	1	1	yes	characteristic
93	C₂×C₇₁	#1716	1	1	yes	characteristic
94	C₂×C₇₁	#1719	1	2	yes	—
95	C₂×C₇₁	#2094	1	1	yes	characteristic
96	C₂×C₇₁	#2097	1	2	yes	—
97	#1990	—	71	2	no	—
98	#1956	D₇₁	1	1	yes	characteristic
99	#1844	—	71	1	no	—
100	#1834	—	71	2	no	—
101	#1273	—	71	1	no	—
102	C₇₁	#10686	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
104	C₂×#106	—	142	2	no	—
105	C₂×#103	—	2	2	no	—
106	C₂×C₄×C₈	—	71	2	no	—
107	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
108	#384	—	71	2	no	—
109	#384	C₂×D₇₁	1	1	yes	characteristic
110	#340	—	142	4	no	—
111	#340	—	142	4	no	—
112	C₂²×Q₈	—	2	1	no	—
113	C₂²×C₈	—	71	4	no	—
114	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	Dic₁₄₂	1	2	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	—	142	2	no	—
122	C₂×C₄⋊C₄	—	142	2	no	—
123	C₂×C₄⋊C₄	—	142	2	no	—
124	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
125	#106	—	284	4	no	—
126	#103	—	4	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	142	2	no	—
129	C₄×C₈	—	142	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	284	2	no	—
137	C₂²×C₄	D₄×D₇₁	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	142	4	no	—
142	C₂×C₈	—	142	4	no	—
143	C₂×C₈	—	142	4	no	—
144	C₂×C₈	#11765853	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	284	4	no	—
150	C₄⋊C₄	—	284	4	no	—
151	C₄⋊C₄	—	284	4	no	—
152	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
153	C₄²	#11765864	1	2	yes	—
154	C₄²	#11765865	1	1	yes	characteristic
155	C₂³	#14556124	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	568	2	no	—
164	C₂×C₄	—	568	2	no	—
165	C₂×C₄	—	568	2	no	—
166	C₂×C₄	#14556105	1	2	yes	—
167	C₂×C₄	#14556107	1	1	yes	characteristic
168	C₂×C₄	#14556109	1	1	yes	characteristic
169	C₂×C₄	#14556110	1	2	yes	—
170	C₂×C₄	C₂×#11765853	1	2	yes	—
171	C₂×C₄	#14556149	1	4	yes	—
172	C₂×C₄	#14556153	1	2	yes	—
173	C₂×C₄	#14556160	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	284	4	no	—
176	C₂²	#17985935	1	1	yes	characteristic
177	C₂²	#17986371	1	1	yes	characteristic
178	C₂²	#17986380	1	2	yes	—
179	C₂²	#17986389	1	1	yes	characteristic
180	C₂²	#17986392	1	2	yes	—
181	C₄	—	8	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	1136	2	no	—
184	C₄	#17986289	1	2	yes	—
185	C₄	#17986304	1	4	yes	—
186	C₄	#17986315	1	2	yes	—
187	C₄	#17986338	1	4	yes	—
188	C₂	#22587939	1	1	yes	characteristic
189	C₂	#22587971	1	1	yes	characteristic
190	C₂	#22587995	1	2	yes	—
191	C₂	#22589873	1	1	yes	characteristic
192	C₂	#22589890	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22589302

C₂

yes

maximal

#22588497

C₂

yes

maximal, characteristic

#22588467

C₂

yes

maximal

C₇₁⋊#1273

C₂

yes

maximal, characteristic

C₇₁×#1956

C₂

yes

maximal, characteristic

C₂×#14556060

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

#17985869

C₂²

yes

—

#17985761

—

#17985761

—

C₂×C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₂×#103×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#384×C₇₁

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556060

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

Q₈

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#103×C₇₁

—

C₄×C₈×C₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₄²×C₇₁

yes

C₂³×C₇₁

yes

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

#135

yes

—

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#496

yes

characteristic

C₂²×C₇₁

#497

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#476

yes

—

C₄×C₇₁

#479

yes

—

C₄×C₇₁

#481

yes

—

C₄×C₇₁

#485

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2094

yes

characteristic

C₂×C₇₁

#2097

yes

—

#1990

—

#1956

D₇₁

yes

characteristic

#1844

—

100

#1834

—

101

#1273

—

102

C₇₁

#10686

yes

characteristic, complemented, Sylow

103

C₂×C₄⋊Q₈

C₂×D₇₁

yes

characteristic

104

C₂×#106

—

142

—

105

C₂×#103

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

108

#384

—

109

#384

C₂×D₇₁

yes

characteristic

110

#340

—

142

—

111

#340

—

142

—

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²⋊D₇₁

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

Dic₁₄₂

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

142

—

122

C₂×C₄⋊C₄

—

142

—

123

C₂×C₄⋊C₄

—

142

—

124

C₂×C₄²

C₂²×D₇₁

yes

characteristic

125

#106

—

284

—

126

#103

—

127

C₄×C₈

—

128

C₄×C₈

—

142

—

129

C₄×C₈

—

142

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

284

—

137

C₂²×C₄

D₄×D₇₁

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

—

142

C₂×C₈

—

142

—

143

C₂×C₈

—

142

—

144

C₂×C₈

#11765853

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

284

—

150

C₄⋊C₄

—

284

—

151

C₄⋊C₄

—

284

—

152

C₄²

C₂×Dic₁₄₂

yes

characteristic

153

C₄²

#11765864

yes

—

154

C₄²

#11765865

yes

characteristic

155

C₂³

#14556124

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

568

—

164

C₂×C₄

—

568

—

165

C₂×C₄

—

568

—

166

C₂×C₄

#14556105

yes

—

167

C₂×C₄

#14556107

yes

characteristic

168

C₂×C₄

#14556109

yes

characteristic

169

C₂×C₄

#14556110

yes

—

170

C₂×C₄

C₂×#11765853

yes

—

171

C₂×C₄

#14556149

yes

—

172

C₂×C₄

#14556153

yes

—

173

C₂×C₄

#14556160

yes

—

174

C₈

—

175

C₈

—

284

—

176

C₂²

#17985935

yes

characteristic

177

C₂²

#17986371

yes

characteristic

178

C₂²

#17986380

yes

—

179

C₂²

#17986389

yes

characteristic

180

C₂²

#17986392

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

1136

—

184

C₄

#17986289

yes

—

185

C₄

#17986304

yes

—

186

C₄

#17986315

yes

—

187

C₄

#17986338

yes

—

188

C₂

#22587939

yes

characteristic

189

C₂

#22587971

yes

characteristic

190

C₂

#22587995

yes

—

191

C₂

#22589873

yes

characteristic

192

C₂

#22589890

yes

—

	subgroup	G/H
center	C₂³	#14556124
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1956	C₂
socle, prime radical	C₂³×C₇₁	#127
hypercenter	#1956	D₇₁
hypocenter	C₇₁	#10686
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#14556124

commutator, cosocle

C₂×C₄²×C₇₁

C₂³

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1956

C₂

C₂³×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
102. C₇₁	#10686	semidirect

H ≅ G/N

type

102. C₇₁

#10686

semidirect

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

subgroup	index
#22589302	2
#22588497	2
#22588467	2
C₇₁⋊#1273	2
C₇₁×#1956	2
#10686	71

subgroup

index

#22589302

#22588497

#22588467

C₇₁⋊#1273

C₇₁×#1956

#10686

Subgroups — #31902572

Conjugacy classes of subgroups

Complements

Maximal subgroups