Group #31913803 — subgroups

Subgroups — #31913803

Order 18176.

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

normal subgroups only

This group has 298 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	C₂×#17985819	C₂	1	2	yes	maximal
2	#22588028	C₂	1	1	yes	maximal, characteristic
3	#22586424	C₂	1	1	yes	maximal, characteristic
4	#22586419	C₂	1	2	yes	maximal
5	C₇₁×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₂×#14556038	C₄	1	1	yes	characteristic
8	C₂×#14556038	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
12	#17985845	C₄	1	1	yes	characteristic
13	#17985819	—	2	4	no	—
14	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
15	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
20	#14556068	—	2	2	no	—
21	#14556068	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	#14556038	—	2	2	no	—
24	#14556038	—	2	2	no	—
25	C₄×C₇₁⋊C₈	—	2	2	no	—
26	C₈×Dic₇₁	C₈	1	4	yes	—
27	C₈×Dic₇₁	D₄	1	2	yes	—
28	C₈×Dic₇₁	Q₈	1	2	yes	—
29	C₇₁⋊#95	—	2	2	no	—
30	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₇₁	—	2	2	no	—
34	M₃₂×C₇₁	—	2	2	no	—
35	C₂×C₁₆×C₇₁	—	2	2	no	—
36	#97×C₇₁	—	2	2	no	—
37	C₂²×Dic₇₁	—	2	1	no	—
38	C₂²×Dic₇₁	—	2	1	no	—
39	C₂×C₇₁⋊C₈	—	2	1	no	—
40	C₂×C₇₁⋊C₈	—	2	1	no	—
41	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
42	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
43	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
44	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
45	C₄×Dic₇₁	—	2	1	no	—
46	C₄×Dic₇₁	—	2	1	no	—
47	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
48	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
49	C₇₁⋊C₁₆	—	4	2	no	—
50	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
51	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	—	2	1	no	—
53	C₂×C₈×C₇₁	—	2	1	no	—
54	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₁	#41	1	2	yes	—
58	C₂×C₈×C₇₁	#41	1	2	yes	—
59	C₄²×C₇₁	—	2	1	no	—
60	C₄²×C₇₁	—	2	1	no	—
61	C₁₆×C₇₁	—	4	2	no	—
62	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₁	—	2	2	no	—
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	#117	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
71	C₂×Dic₇₁	—	2	2	no	—
72	C₂×Dic₇₁	—	2	2	no	—
73	C₂×Dic₇₁	—	4	2	no	—
74	C₂×Dic₇₁	—	4	2	no	—
75	C₈×C₇₁	—	2	2	no	—
76	C₈×C₇₁	#124	1	2	yes	—
77	C₈×C₇₁	#125	1	2	yes	—
78	C₇₁⋊C₈	—	2	2	no	—
79	C₇₁⋊C₈	C₄⋊C₈	1	4	yes	—
80	C₂²×C₇₁	—	2	2	no	—
81	C₂²×C₇₁	#423	1	1	yes	characteristic
82	C₂²×C₇₁	#431	1	1	yes	characteristic
83	C₂²×C₇₁	#432	1	1	yes	characteristic
84	C₄×C₇₁	—	4	2	no	—
85	C₄×C₇₁	#382	1	1	yes	characteristic
86	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₄×C₇₁	#431	1	1	yes	characteristic
88	C₄×C₇₁	#432	1	1	yes	characteristic
89	Dic₇₁	—	4	2	no	—
90	Dic₇₁	—	4	2	no	—
91	#12881	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	—	2	1	no	—
94	C₂×C₇₁	#1831	1	1	yes	characteristic
95	C₂×C₇₁	#2166	1	2	yes	—
96	C₂×#364	—	71	2	no	—
97	#1726	—	71	1	no	—
98	#1377	—	71	1	no	—
99	#1377	D₇₁	1	1	yes	characteristic
100	#1375	—	71	2	no	—
101	C₇₁	#12881	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	—	71	2	no	—
103	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
104	C₂×#97	—	71	1	no	—
105	C₂×#97	—	71	1	no	—
106	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
107	C₂×C₄×C₈	—	71	1	no	—
108	C₂×C₄×C₈	—	71	1	no	—
109	C₂×C₄×C₈	—	71	1	no	—
110	#376	—	71	1	no	—
111	#364	—	142	4	no	—
112	C₂²×C₈	—	71	1	no	—
113	C₂²×C₈	—	71	1	no	—
114	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
116	C₂×C₄²	—	71	1	no	—
117	C₂×C₄²	—	71	1	no	—
118	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
119	M₃₂	—	2	2	no	—
120	M₃₂	—	2	2	no	—
121	M₃₂	—	142	2	no	—
122	M₃₂	—	142	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	142	2	no	—
125	#97	—	2	2	no	—
126	#97	—	142	2	no	—
127	#97	—	142	2	no	—
128	C₄×C₈	—	71	4	no	—
129	C₄×C₈	—	71	2	no	—
130	C₄×C₈	—	71	2	no	—
131	C₄×C₈	—	142	2	no	—
132	#95	—	142	2	no	—
133	C₂²×C₄	—	142	1	no	—
134	C₂²×C₄	—	142	1	no	—
135	C₂²×C₄	#11765848	1	2	yes	—
136	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	71	2	no	—
140	C₂×C₈	—	71	2	no	—
141	C₂×C₈	—	71	2	no	—
142	C₂×C₈	—	71	2	no	—
143	C₂×C₈	—	142	1	no	—
144	C₂×C₈	—	142	1	no	—
145	C₂×C₈	C₈×D₇₁	1	2	yes	—
146	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
147	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
148	C₂×C₈	#11765864	1	2	yes	—
149	C₂×C₈	#11765865	1	2	yes	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	71	2	no	—
153	C₄²	—	71	2	no	—
154	C₄²	—	142	1	no	—
155	C₄²	—	142	1	no	—
156	C₁₆	—	4	2	no	—
157	C₁₆	—	284	2	no	—
158	C₂³	C₂×#11765848	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	284	2	no	—
162	C₂×C₄	—	284	2	no	—
163	C₂×C₄	—	142	2	no	—
164	C₂×C₄	—	142	2	no	—
165	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
166	C₂×C₄	#14556084	1	1	yes	characteristic
167	C₂×C₄	#14556085	1	1	yes	characteristic
168	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
169	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
170	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	71	4	no	—
173	C₈	—	142	2	no	—
174	C₈	#14556114	1	2	yes	—
175	C₈	#14556118	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	#17986021	1	1	yes	characteristic
178	C₂²	#17986050	1	1	yes	characteristic
179	C₂²	#17986053	1	1	yes	characteristic
180	C₄	—	4	2	no	—
181	C₄	—	284	2	no	—
182	C₄	—	284	2	no	—
183	C₄	#17985863	1	1	yes	characteristic
184	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
185	C₄	#17986049	1	1	yes	characteristic
186	C₄	#17986054	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	#22588455	1	1	yes	characteristic
190	C₂	#22590153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985819

C₂

yes

maximal

#22588028

C₂

yes

maximal, characteristic

#22586424

C₂

yes

maximal, characteristic

#22586419

C₂

yes

maximal

C₇₁×#1377

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

—

C₂×#14556038

C₄

yes

characteristic

C₂×#14556038

C₂²

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

#17985845

C₄

yes

characteristic

#17985819

—

C₂×M₃₂×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

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

#14556038

—

#14556038

—

C₄×C₇₁⋊C₈

—

C₈×Dic₇₁

C₈

yes

—

C₈×Dic₇₁

D₄

yes

—

C₈×Dic₇₁

Q₈

yes

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

#97×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₂×C₇₁⋊C₈

M₁₆

yes

—

C₂×C₇₁⋊C₈

M₁₆

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₄⋊C₄

yes

—

C₄×Dic₇₁

C₂×C₈

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

M₁₆

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂²×C₈

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

#124

yes

—

C₈×C₇₁

#125

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄⋊C₈

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

#382

yes

characteristic

C₄×C₇₁

C₂×C₄⋊C₈

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#12881

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1831

yes

characteristic

C₂×C₇₁

#2166

yes

—

—

—

—

D₇₁

yes

characteristic

100

#1375

—

101

C₇₁

#12881

yes

characteristic, complemented, Sylow

102

C₂×M₃₂

—

103

C₂×M₃₂

C₂×D₇₁

yes

—

104

C₂×#97

—

105

C₂×#97

—

106

C₂×#97

C₂×D₇₁

yes

characteristic

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

#376

—

111

#364

—

142

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₄×D₇₁

yes

characteristic

115

C₂²×C₈

C₂²×D₇₁

yes

characteristic

116

C₂×C₄²

—

117

C₂×C₄²

—

118

C₂×C₄²

C₄×D₇₁

yes

characteristic

119

M₃₂

—

120

M₃₂

—

121

M₃₂

—

142

—

122

M₃₂

—

142

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

142

—

125

#97

—

126

#97

—

142

—

127

#97

—

142

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

142

—

132

#95

—

142

—

133

C₂²×C₄

—

142

—

134

C₂²×C₄

—

142

—

135

C₂²×C₄

#11765848

yes

—

136

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

142

—

144

C₂×C₈

—

142

—

145

C₂×C₈

C₈×D₇₁

yes

—

146

C₂×C₈

D₄×D₇₁

yes

characteristic

147

C₂×C₈

Q₈×D₇₁

yes

characteristic

148

C₂×C₈

#11765864

yes

—

149

C₂×C₈

#11765865

yes

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

142

—

155

C₄²

—

142

—

156

C₁₆

—

157

C₁₆

—

284

—

158

C₂³

C₂×#11765848

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

284

—

162

C₂×C₄

—

284

—

163

C₂×C₄

—

142

—

164

C₂×C₄

—

142

—

165

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

166

C₂×C₄

#14556084

yes

characteristic

167

C₂×C₄

#14556085

yes

characteristic

168

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

169

C₂×C₄

M₁₆×D₇₁

yes

characteristic

170

C₂×C₄

M₁₆×D₇₁

yes

characteristic

171

C₈

—

172

C₈

—

173

C₈

—

142

—

174

C₈

#14556114

yes

—

175

C₈

#14556118

yes

—

176

C₂²

—

177

C₂²

#17986021

yes

characteristic

178

C₂²

#17986050

yes

characteristic

179

C₂²

#17986053

yes

characteristic

180

C₄

—

181

C₄

—

284

—

182

C₄

—

284

—

183

C₄

#17985863

yes

characteristic

184

C₄

C₄⋊C₈×D₇₁

yes

characteristic

185

C₄

#17986049

yes

characteristic

186

C₄

#17986054

yes

characteristic

187

C₂

—

188

C₂

—

189

C₂

#22588455

yes

characteristic

190

C₂

#22590153

yes

—

	subgroup	G/H
center	C₂×C₄	#14556084
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1377	C₂
socle	C₂²×C₇₁	#432
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1377	D₇₁
hypocenter	C₇₁	#12881
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

subgroup

G/H

center

C₂×C₄

#14556084

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1377

C₂

socle

C₂²×C₇₁

#432

prime radical

C₂³×C₇₁

C₂×M₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1377

N	H ≅ G/N	type
101. C₇₁	#12881	semidirect

H ≅ G/N

type

101. C₇₁

#12881

semidirect

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

subgroup	index
C₂×#17985819	2
#22588028	2
#22586424	2
#22586419	2
C₇₁×#1377	2
#12881	71

subgroup

index

C₇₁×#1377

Subgroups — #31913803

Conjugacy classes of subgroups

Complements

Maximal subgroups