Group #31897136 — subgroups

Subgroups — #31897136

Order 18176.

402 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₂×#17986404	C₂	1	1	yes	maximal, characteristic
2	#22589329	C₂	1	2	yes	maximal
3	#22587096	C₂	1	1	yes	maximal, characteristic
4	#22587024	C₂	1	2	yes	maximal
5	C₇₁×#1451	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	C₂²	1	2	yes	—
7	#17986404	—	2	2	no	—
8	#17986404	—	2	1	no	—
9	#17986404	—	2	1	no	—
10	C₂×#14556062	C₄	1	4	yes	—
11	#17986023	C₂²	1	2	yes	—
12	#17985767	—	2	4	no	—
13	#17985750	—	2	2	no	—
14	#17985746	—	2	4	no	—
15	#431×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#322×C₇₁	—	2	1	no	—
19	#322×C₇₁	—	2	1	no	—
20	C₂²×Dic₁₄₂	—	2	2	no	—
21	C₂×#11765849	—	2	2	no	—
22	#14556128	—	2	2	no	—
23	#14556128	—	2	2	no	—
24	#14556128	—	2	2	no	—
25	#14556128	—	2	2	no	—
26	C₂×C₄⋊Dic₇₁	—	2	2	no	—
27	C₂×C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
28	#14556062	—	2	4	no	—
29	#14556062	—	2	4	no	—
30	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
31	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
32	C₂×M₁₆×C₇₁	D₄	1	2	yes	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
36	C₂²⋊C₈×C₇₁	—	2	1	no	—
37	#97×C₇₁	—	2	1	no	—
38	C₄×C₈×C₇₁	—	2	1	no	—
39	C₄×C₈×C₇₁	—	2	1	no	—
40	C₄×C₈×C₇₁	D₄	1	2	yes	—
41	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
43	C₂×Dic₁₄₂	—	4	2	no	—
44	C₂×Dic₁₄₂	—	4	2	no	—
45	C₂×Dic₁₄₂	—	8	4	no	—
46	C₂²×Dic₇₁	—	2	4	no	—
47	#11765849	—	4	4	no	—
48	C₂×C₇₁⋊C₈	—	4	2	no	—
49	C₄⋊Dic₇₁	—	2	4	no	—
50	C₄⋊Dic₇₁	—	2	4	no	—
51	C₄⋊Dic₇₁	—	4	2	no	—
52	C₄⋊Dic₇₁	—	4	2	no	—
53	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
56	M₁₆×C₇₁	—	2	4	no	—
57	C₂×C₈×C₇₁	—	2	4	no	—
58	C₂×C₈×C₇₁	—	2	2	no	—
59	C₂×C₈×C₇₁	—	2	2	no	—
60	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₁	#41	1	1	yes	characteristic
66	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	—	2	2	no	—
69	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
73	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#137	1	4	yes	—
76	C₂×Dic₇₁	—	4	4	no	—
77	C₂×Dic₇₁	—	8	4	no	—
78	Dic₁₄₂	—	8	4	no	—
79	Dic₁₄₂	—	8	4	no	—
80	C₈×C₇₁	—	2	2	no	—
81	C₈×C₇₁	—	4	2	no	—
82	C₈×C₇₁	—	4	2	no	—
83	C₈×C₇₁	#101	1	4	yes	—
84	C₇₁⋊C₈	—	8	2	no	—
85	C₂²×C₇₁	—	2	1	no	—
86	C₂²×C₇₁	—	2	1	no	—
87	C₂²×C₇₁	#390	1	1	yes	characteristic
88	C₂²×C₇₁	#441	1	2	yes	—
89	C₄×C₇₁	—	2	1	no	—
90	C₄×C₇₁	—	2	1	no	—
91	C₄×C₇₁	C₂×#101	1	2	yes	—
92	C₄×C₇₁	#470	1	2	yes	—
93	C₄×C₇₁	#472	1	2	yes	—
94	C₄×C₇₁	#497	1	2	yes	—
95	Dic₇₁	—	8	4	no	—
96	#9810	—	71	1	no	maximal, Sylow
97	C₂×C₇₁	—	2	2	no	—
98	C₂×C₇₁	#1966	1	1	yes	characteristic
99	C₂×C₇₁	#1971	1	1	yes	characteristic
100	C₂×C₇₁	#1978	1	1	yes	characteristic
101	C₂×#500	—	71	1	no	—
102	#1997	—	71	2	no	—
103	#1566	—	71	1	no	—
104	#1549	—	71	2	no	—
105	#1451	D₇₁	1	1	yes	characteristic
106	C₇₁	#9810	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊Q₈	—	71	2	no	—
108	#500	—	142	2	no	—
109	#500	—	142	1	no	—
110	#500	—	142	1	no	—
111	#431	C₂×D₇₁	1	1	yes	characteristic
112	C₂×#107	—	71	4	no	—
113	#423	—	71	2	no	—
114	C₄×M₁₆	C₂×D₇₁	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
116	#343	—	142	4	no	—
117	#334	—	142	2	no	—
118	#332	—	142	4	no	—
119	#322	—	2	1	no	—
120	#322	—	2	1	no	—
121	C₂²×Q₈	—	142	2	no	—
122	C₂×M₁₆	—	142	2	no	—
123	C₂×M₁₆	D₂₈₄	1	2	yes	—
124	C₂²×C₈	C₄×D₇₁	1	2	yes	—
125	C₄⋊Q₈	—	142	2	no	—
126	C₄⋊Q₈	—	142	2	no	—
127	C₄⋊Q₈	—	142	2	no	—
128	C₄⋊Q₈	—	142	2	no	—
129	C₂×C₄⋊C₄	—	71	4	no	—
130	C₂×C₄⋊C₄	—	142	2	no	—
131	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
132	#107	—	142	4	no	—
133	#107	—	142	4	no	—
134	C₄⋊C₈	—	142	2	no	—
135	C₄⋊C₈	—	142	2	no	—
136	C₄⋊C₈	D₂₈₄	1	2	yes	—
137	C₂²⋊C₈	—	2	1	no	—
138	#97	—	2	1	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	—	2	1	no	—
141	C₄×C₈	D₂₈₄	1	2	yes	—
142	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
144	C₂×Q₈	—	284	2	no	—
145	C₂×Q₈	—	284	2	no	—
146	C₂×Q₈	—	568	4	no	—
147	C₂²×C₄	—	142	4	no	—
148	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
150	C₂²×C₄	#11765865	1	1	yes	characteristic
151	M₁₆	—	2	4	no	—
152	M₁₆	—	284	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	284	2	no	—
157	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
158	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
159	C₄⋊C₄	—	284	2	no	—
160	C₄⋊C₄	—	284	2	no	—
161	C₄⋊C₄	—	142	4	no	—
162	C₄⋊C₄	—	142	4	no	—
163	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
164	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
165	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
166	C₄²	#11765865	1	1	yes	characteristic
167	C₂³	C₄×D₂₈₄	1	1	yes	characteristic
168	Q₈	—	568	4	no	—
169	Q₈	—	568	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	284	4	no	—
173	C₂×C₄	—	568	4	no	—
174	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
176	C₂×C₄	#14556102	1	1	yes	characteristic
177	C₂×C₄	#14556102	1	1	yes	characteristic
178	C₂×C₄	#14556116	1	1	yes	characteristic
179	C₂×C₄	#14556125	1	1	yes	characteristic
180	C₂×C₄	#14556161	1	4	yes	—
181	C₈	—	2	2	no	—
182	C₈	—	4	2	no	—
183	C₈	—	4	2	no	—
184	C₈	—	568	2	no	—
185	C₈	#14556046	1	4	yes	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#17985901	1	1	yes	characteristic
189	C₂²	#17986107	1	2	yes	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	568	4	no	—
193	C₄	C₂×#14556046	1	2	yes	—
194	C₄	#17986257	1	2	yes	—
195	C₄	#17986269	1	2	yes	—
196	C₄	#17986393	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#22589184	1	1	yes	characteristic
199	C₂	#22589211	1	1	yes	characteristic
200	C₂	#22589250	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986404

C₂

yes

maximal, characteristic

#22589329

C₂

yes

maximal

#22587096

C₂

yes

maximal, characteristic

#22587024

C₂

yes

maximal

C₇₁×#1451

C₂

yes

maximal, characteristic

C₂×#14556128

C₂²

yes

—

—

—

—

C₄

yes

—

#17986023

C₂²

yes

—

#17985767

—

#17985750

—

#17985746

—

#431×C₇₁

C₂²

yes

characteristic

C₄×M₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#322×C₇₁

—

#322×C₇₁

—

C₂²×Dic₁₄₂

—

—

—

—

—

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

C₂×C₄

yes

—

#14556062

—

#14556062

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂×M₁₆×C₇₁

D₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

D₄

yes

—

C₂²⋊C₈×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

#41

yes

characteristic

C₂³×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#101

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

#441

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#101

yes

—

C₄×C₇₁

#470

yes

—

C₄×C₇₁

#472

yes

—

C₄×C₇₁

#497

yes

—

Dic₇₁

—

#9810

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

100

C₂×C₇₁

#1978

yes

characteristic

101

C₂×#500

—

102

#1997

—

103

#1566

—

104

#1549

—

105

#1451

D₇₁

yes

characteristic

106

C₇₁

#9810

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊Q₈

—

108

#500

—

142

—

109

#500

—

142

—

110

#500

—

142

—

111

#431

C₂×D₇₁

yes

characteristic

112

C₂×#107

—

113

#423

—

114

C₄×M₁₆

C₂×D₇₁

yes

characteristic

115

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

116

#343

—

142

—

117

#334

—

142

—

118

#332

—

142

—

119

#322

—

120

#322

—

121

C₂²×Q₈

—

142

—

122

C₂×M₁₆

—

142

—

123

C₂×M₁₆

D₂₈₄

yes

—

124

C₂²×C₈

C₄×D₇₁

yes

—

125

C₄⋊Q₈

—

142

—

126

C₄⋊Q₈

—

142

—

127

C₄⋊Q₈

—

142

—

128

C₄⋊Q₈

—

142

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

142

—

131

C₂×C₄²

C₂²×D₇₁

yes

characteristic

132

#107

—

142

—

133

#107

—

142

—

134

C₄⋊C₈

—

142

—

135

C₄⋊C₈

—

142

—

136

C₄⋊C₈

D₂₈₄

yes

—

137

C₂²⋊C₈

—

138

#97

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

D₂₈₄

yes

—

142

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

143

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

144

C₂×Q₈

—

284

—

145

C₂×Q₈

—

284

—

146

C₂×Q₈

—

568

—

147

C₂²×C₄

—

142

—

148

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

149

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

150

C₂²×C₄

#11765865

yes

characteristic

151

M₁₆

—

152

M₁₆

—

284

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

284

—

157

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

158

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

159

C₄⋊C₄

—

284

—

160

C₄⋊C₄

—

284

—

161

C₄⋊C₄

—

142

—

162

C₄⋊C₄

—

142

—

163

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

164

C₄²

C₂×D₂₈₄

yes

characteristic

165

C₄²

C₂×D₂₈₄

yes

characteristic

166

C₄²

#11765865

yes

characteristic

167

C₂³

C₄×D₂₈₄

yes

characteristic

168

Q₈

—

568

—

169

Q₈

—

568

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

284

—

173

C₂×C₄

—

568

—

174

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

175

C₂×C₄

C₄×D₂₈₄

yes

characteristic

176

C₂×C₄

#14556102

yes

characteristic

177

C₂×C₄

#14556102

yes

characteristic

178

C₂×C₄

#14556116

yes

characteristic

179

C₂×C₄

#14556125

yes

characteristic

180

C₂×C₄

#14556161

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

568

—

185

C₈

#14556046

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

#17985901

yes

characteristic

189

C₂²

#17986107

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

568

—

193

C₄

C₂×#14556046

yes

—

194

C₄

#17986257

yes

—

195

C₄

#17986269

yes

—

196

C₄

#17986393

yes

—

197

C₂

—

198

C₂

#22589184

yes

characteristic

199

C₂

#22589211

yes

characteristic

200

C₂

#22589250

yes

characteristic

	subgroup	G/H
center	C₂²	#17985901
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1451	C₂
socle	C₂²×C₇₁	#390
prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1451	D₇₁
hypocenter	C₇₁	#9810
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451

subgroup

G/H

center

C₂²

#17985901

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1451

C₂

socle

C₂²×C₇₁

#390

prime radical

C₂³×C₇₁

C₄×D₄

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451

N	H ≅ G/N	type
106. C₇₁	#9810	semidirect

H ≅ G/N

type

106. C₇₁

#9810

semidirect

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

subgroup	index
C₂×#17986404	2
#22589329	2
#22587096	2
#22587024	2
C₇₁×#1451	2
#9810	71

subgroup

index

C₇₁×#1451

Subgroups — #31897136

Conjugacy classes of subgroups

Complements

Maximal subgroups