Group #31891099 — subgroups

Subgroups — #31891099

Order 18176.

310 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 123 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	#22588028	C₂	1	1	yes	maximal, characteristic
2	#22587086	C₂	1	2	yes	maximal
3	#22586185	C₂	1	1	yes	maximal, characteristic
4	#22586168	C₂	1	2	yes	maximal
5	C₇₁×#1281	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×#14556038	C₄	1	1	yes	characteristic
8	C₂×#14556038	C₄	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
12	#17985845	C₂²	1	1	yes	characteristic
13	#17985751	—	2	2	no	—
14	#17985749	—	2	2	no	—
15	#431×C₇₁	C₂²	1	2	yes	—
16	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
21	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
22	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
23	#14556038	—	2	2	no	—
24	#14556038	C₈	1	4	yes	complemented
25	C₄×C₇₁⋊C₈	—	2	2	no	—
26	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
27	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
28	C₈×Dic₇₁	—	2	2	no	—
29	C₈×Dic₇₁	—	2	2	no	—
30	C₇₁⋊#95	—	2	2	no	—
31	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₁	—	2	2	no	—
35	C₂²⋊C₈×C₇₁	—	2	2	no	—
36	#97×C₇₁	—	2	2	no	—
37	C₄×C₈×C₇₁	—	2	2	no	—
38	C₂²×Dic₇₁	—	2	1	no	—
39	C₂²×Dic₇₁	—	2	1	no	—
40	C₂×C₇₁⋊C₈	—	2	1	no	—
41	C₂×C₇₁⋊C₈	—	2	1	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
46	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
47	C₄×Dic₇₁	—	2	1	no	—
48	C₄×Dic₇₁	—	2	1	no	—
49	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
50	C₄×Dic₇₁	M₁₆	1	2	yes	—
51	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	#41	1	2	yes	—
53	C₂×C₈×C₇₁	—	2	1	no	—
54	C₂×C₈×C₇₁	—	2	1	no	—
55	C₂×C₈×C₇₁	—	2	1	no	—
56	C₂×C₈×C₇₁	—	2	1	no	—
57	C₂×C₈×C₇₁	—	4	2	no	—
58	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
60	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₇₁	#41	1	2	yes	—
63	C₂³×C₇₁	#117	1	1	yes	characteristic
64	C₂×C₄×C₇₁	—	4	2	no	—
65	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#124	1	2	yes	—
68	C₂×C₄×C₇₁	#125	1	2	yes	—
69	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
73	C₂×Dic₇₁	—	2	2	no	—
74	C₂×Dic₇₁	—	2	2	no	—
75	C₂×Dic₇₁	—	4	2	no	—
76	C₂×Dic₇₁	—	4	2	no	—
77	C₈×C₇₁	—	2	2	no	—
78	C₈×C₇₁	—	2	2	no	—
79	C₈×C₇₁	—	8	2	no	—
80	C₇₁⋊C₈	—	2	2	no	—
81	C₇₁⋊C₈	—	8	2	no	—
82	C₇₁⋊C₈	C₄⋊C₈	1	4	yes	—
83	C₂²×C₇₁	—	2	2	no	—
84	C₂²×C₇₁	#382	1	1	yes	characteristic
85	C₂²×C₇₁	#431	1	1	yes	characteristic
86	C₂²×C₇₁	#432	1	1	yes	characteristic
87	C₄×C₇₁	—	2	2	no	—
88	C₄×C₇₁	—	2	2	no	—
89	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
90	C₄×C₇₁	#423	1	1	yes	characteristic
91	C₄×C₇₁	#431	1	1	yes	characteristic
92	C₄×C₇₁	#432	1	1	yes	characteristic
93	Dic₇₁	—	4	2	no	—
94	Dic₇₁	—	4	2	no	—
95	#8781	—	71	1	no	maximal, Sylow
96	C₂×C₇₁	—	2	1	no	—
97	C₂×C₇₁	—	2	1	no	—
98	C₂×C₇₁	#1831	1	1	yes	characteristic
99	C₂×C₇₁	#1833	1	2	yes	—
100	#1726	—	71	1	no	—
101	#1564	—	71	2	no	—
102	#1281	—	71	1	no	—
103	#1281	D₇₁	1	1	yes	characteristic
104	#1274	—	71	2	no	—
105	C₇₁	#8781	1	1	yes	characteristic, complemented, Sylow
106	#431	—	71	2	no	—
107	#431	C₂×D₇₁	1	2	yes	—
108	C₂×#97	—	71	1	no	—
109	C₂×#97	—	71	1	no	—
110	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
111	C₂×C₄×C₈	—	71	1	no	—
112	C₂×C₄×C₈	—	71	1	no	—
113	C₂×C₄×C₈	—	71	1	no	—
114	#376	—	71	1	no	—
115	#335	—	142	2	no	—
116	#334	—	142	2	no	—
117	C₂²×C₈	—	71	1	no	—
118	C₂²×C₈	—	71	1	no	—
119	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
120	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
121	C₂×C₄²	—	71	1	no	—
122	C₂×C₄²	—	71	1	no	—
123	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	142	2	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	142	2	no	—
128	#97	—	2	2	no	—
129	#97	—	71	4	no	—
130	#97	—	142	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	71	2	no	—
133	C₄×C₈	—	71	2	no	—
134	C₄×C₈	—	142	2	no	—
135	C₄×C₈	—	142	2	no	—
136	C₄×C₈	—	142	2	no	—
137	#95	—	142	2	no	—
138	C₂²×C₄	—	142	1	no	—
139	C₂²×C₄	—	142	1	no	—
140	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
141	C₂²×C₄	#11765865	1	2	yes	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	284	2	no	—
148	C₂×C₈	—	71	2	no	—
149	C₂×C₈	—	71	2	no	—
150	C₂×C₈	—	71	2	no	—
151	C₂×C₈	—	71	2	no	—
152	C₂×C₈	—	142	1	no	—
153	C₂×C₈	—	142	1	no	—
154	C₂×C₈	C₈×D₇₁	1	2	yes	—
155	C₂×C₈	#11765848	1	2	yes	—
156	C₄²	—	71	2	no	—
157	C₄²	—	71	2	no	—
158	C₄²	—	142	1	no	—
159	C₄²	—	142	1	no	—
160	C₄²	D₄×D₇₁	1	1	yes	characteristic
161	C₄²	Q₈×D₇₁	1	1	yes	characteristic
162	C₄²	#11765864	1	2	yes	—
163	C₂³	#14556085	1	1	yes	characteristic
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	284	2	no	—
166	C₂×C₄	—	284	2	no	—
167	C₂×C₄	—	142	2	no	—
168	C₂×C₄	—	142	2	no	—
169	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
170	C₂×C₄	#14556084	1	1	yes	characteristic
171	C₂×C₄	#14556114	1	2	yes	—
172	C₂×C₄	#14556118	1	2	yes	—
173	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
174	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
175	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
176	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	—	2	2	no	—
179	C₈	—	8	2	no	—
180	C₈	—	568	2	no	—
181	C₈	—	71	4	no	—
182	C₈	—	142	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	#17985863	1	1	yes	characteristic
185	C₂²	#17986050	1	1	yes	characteristic
186	C₂²	#17986054	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	284	2	no	—
190	C₄	—	284	2	no	—
191	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
192	C₄	#17986021	1	1	yes	characteristic
193	C₄	#17986049	1	1	yes	characteristic
194	C₄	#17986053	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#22588455	1	1	yes	characteristic
198	C₂	#22588461	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588028

C₂

yes

maximal, characteristic

#22587086

C₂

yes

maximal

#22586185

C₂

yes

maximal, characteristic

#22586168

C₂

yes

maximal

C₇₁×#1281

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×#14556038

C₄

yes

characteristic

C₂×#14556038

C₄

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

characteristic

#17985845

C₂²

yes

characteristic

#17985751

—

#17985749

—

#431×C₇₁

C₂²

yes

—

C₂×#97×C₇₁

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

#14556038

—

#14556038

C₈

yes

complemented

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₄×C₇₁⋊C₈

Q₈

yes

—

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₂×C₇₁⋊C₈

M₁₆

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₂×C₈

yes

—

C₄×Dic₇₁

M₁₆

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#125

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄⋊C₈

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₈

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#8781

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1831

yes

characteristic

C₂×C₇₁

#1833

yes

—

100

#1726

—

101

#1564

—

102

#1281

—

103

#1281

D₇₁

yes

characteristic

104

#1274

—

105

C₇₁

#8781

yes

characteristic, complemented, Sylow

106

#431

—

107

#431

C₂×D₇₁

yes

—

108

C₂×#97

—

109

C₂×#97

—

110

C₂×#97

C₂×D₇₁

yes

characteristic

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

#376

—

115

#335

—

142

—

116

#334

—

142

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₄×D₇₁

yes

characteristic

120

C₂²×C₈

C₄×D₇₁

yes

characteristic

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

C₂²×D₇₁

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

142

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

142

—

128

#97

—

129

#97

—

130

#97

—

142

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

142

—

135

C₄×C₈

—

142

—

136

C₄×C₈

—

142

—

137

#95

—

142

—

138

C₂²×C₄

—

142

—

139

C₂²×C₄

—

142

—

140

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

141

C₂²×C₄

#11765865

yes

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

284

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

142

—

153

C₂×C₈

—

142

—

154

C₂×C₈

C₈×D₇₁

yes

—

155

C₂×C₈

#11765848

yes

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

142

—

159

C₄²

—

142

—

160

C₄²

D₄×D₇₁

yes

characteristic

161

C₄²

Q₈×D₇₁

yes

characteristic

162

C₄²

#11765864

yes

—

163

C₂³

#14556085

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

284

—

166

C₂×C₄

—

284

—

167

C₂×C₄

—

142

—

168

C₂×C₄

—

142

—

169

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

170

C₂×C₄

#14556084

yes

characteristic

171

C₂×C₄

#14556114

yes

—

172

C₂×C₄

#14556118

yes

—

173

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

174

C₂×C₄

M₁₆×D₇₁

yes

characteristic

175

C₂×C₄

M₁₆×D₇₁

yes

characteristic

176

C₂×C₄

C₂×#11765848

yes

characteristic

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

568

—

181

C₈

—

182

C₈

—

142

—

183

C₂²

—

184

C₂²

#17985863

yes

characteristic

185

C₂²

#17986050

yes

characteristic

186

C₂²

#17986054

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

284

—

190

C₄

—

284

—

191

C₄

C₄⋊C₈×D₇₁

yes

characteristic

192

C₄

#17986021

yes

characteristic

193

C₄

#17986049

yes

characteristic

194

C₄

#17986053

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#22588455

yes

characteristic

198

C₂

#22588461

yes

—

	subgroup	G/H
center	C₂×C₄	#14556084
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1281	C₂
socle	C₂²×C₇₁	#382
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1281	D₇₁
hypocenter	C₇₁	#8781
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂×C₄

#14556084

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1281

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
24. #14556038	C₈	semidirect
105. C₇₁	#8781	semidirect

H ≅ G/N

type

24. #14556038

C₈

semidirect

105. C₇₁

#8781

semidirect

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

subgroup	index
#22588028	2
#22587086	2
#22586185	2
#22586168	2
C₇₁×#1281	2
#8781	71

subgroup

index

C₇₁×#1281

Subgroups — #31891099

Conjugacy classes of subgroups

Complements

Maximal subgroups