Group #31916023 — subgroups

Subgroups — #31916023

Order 18176.

398 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
3	#22586432	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556128	—	2	1	no	—
6	#17986404	—	2	2	no	—
7	#17986404	C₄	1	4	yes	complemented
8	C₂×#14556062	—	2	1	no	—
9	C₂×#14556062	C₂²	1	2	yes	—
10	C₂×#14556062	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₇₁	—	2	1	no	—
12	#17985762	—	2	1	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₁	—	2	2	no	—
16	C₂²×Dic₁₄₂	—	4	1	no	—
17	#14556128	—	2	4	no	—
18	#14556128	—	4	2	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	4	1	no	—
22	C₂×C₄×Dic₇₁	—	4	1	no	—
23	#14556062	—	2	4	no	—
24	#14556062	—	2	2	no	—
25	#14556062	—	2	2	no	—
26	#14556062	C₂×C₄	1	4	yes	—
27	#14556062	Q₈	1	4	yes	complemented
28	C₇₁⋊#95	—	4	1	no	—
29	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₁	—	2	2	no	—
33	C₂×C₁₆×C₇₁	—	2	4	no	—
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₁	D₄	1	2	yes	—
36	C₂×Dic₁₄₂	—	4	4	no	—
37	C₂×Dic₁₄₂	—	8	2	no	—
38	C₂×Dic₁₄₂	—	8	2	no	—
39	C₂²×Dic₇₁	—	4	2	no	—
40	C₂²×Dic₇₁	—	4	2	no	—
41	C₄⋊Dic₇₁	—	2	4	no	—
42	C₄⋊Dic₇₁	—	2	4	no	—
43	C₄⋊Dic₇₁	—	4	4	no	—
44	C₄⋊Dic₇₁	—	4	2	no	—
45	C₄⋊Dic₇₁	—	4	2	no	—
46	C₄×Dic₇₁	—	8	2	no	—
47	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	—	2	2	no	—
51	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₇₁	#41	1	2	yes	—
56	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₇₁	SD₁₆	1	2	yes	—
58	C₁₆×C₇₁	—	4	2	no	—
59	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	#102	1	2	yes	—
62	C₂×C₄×C₇₁	#102	1	2	yes	—
63	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
64	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
65	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#137	1	2	yes	—
69	C₂×Dic₇₁	—	4	4	no	—
70	C₂×Dic₇₁	—	4	4	no	—
71	C₂×Dic₇₁	—	8	2	no	—
72	C₂×Dic₇₁	—	8	2	no	—
73	C₂×Dic₇₁	—	8	2	no	—
74	C₂×Dic₇₁	—	8	2	no	—
75	Dic₁₄₂	—	8	8	no	—
76	C₈×C₇₁	—	2	2	no	—
77	C₈×C₇₁	C₄×Q₈	1	2	yes	—
78	C₈×C₇₁	#124	1	1	yes	characteristic
79	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
81	C₂²×C₇₁	#474	1	2	yes	—
82	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
83	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
84	C₂²×C₇₁	#510	1	2	yes	—
85	C₄×C₇₁	—	2	2	no	—
86	C₄×C₇₁	#357	1	4	yes	—
87	C₄×C₇₁	#391	1	1	yes	characteristic
88	C₄×C₇₁	#419	1	1	yes	characteristic
89	C₄×C₇₁	#475	1	2	yes	—
90	Dic₇₁	—	8	4	no	—
91	Dic₇₁	—	8	4	no	—
92	#13275	—	71	1	no	maximal, Sylow
93	C₂×C₇₁	#1988	1	1	yes	characteristic
94	C₂×C₇₁	C₂×#357	1	1	yes	characteristic
95	C₂×C₇₁	#2141	1	1	yes	characteristic
96	C₂×C₇₁	#2224	1	2	yes	—
97	C₂×C₇₁	#2225	1	2	yes	—
98	C₂×#500	—	71	1	no	—
99	C₂×C₄⋊C₁₆	D₇₁	1	1	yes	characteristic
100	C₄×#107	—	71	1	no	—
101	#1380	—	71	4	no	—
102	C₇₁	#13275	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	142	1	no	—
104	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
105	#500	—	71	4	no	—
106	#500	—	142	2	no	—
107	C₂×#107	—	71	2	no	—
108	C₂×#107	—	71	2	no	—
109	C₂×#107	—	142	1	no	—
110	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	142	1	no	—
112	C₄⋊C₁₆	—	2	2	no	—
113	#340	—	142	1	no	—
114	C₂²×Q₈	—	284	1	no	—
115	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	C₄⋊Q₈	—	284	2	no	—
118	C₄⋊Q₈	—	142	4	no	—
119	C₂×C₄⋊C₄	—	284	1	no	—
120	C₂×C₄⋊C₄	—	142	2	no	—
121	C₂×C₄⋊C₄	—	142	2	no	—
122	C₂×C₄²	—	284	1	no	—
123	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	2	4	no	—
126	#107	—	71	4	no	—
127	#107	—	71	4	no	—
128	#107	—	142	4	no	—
129	#107	—	142	2	no	—
130	#107	—	142	2	no	—
131	C₄×C₈	C₄×D₇₁	1	2	yes	—
132	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
133	#95	—	284	1	no	—
134	C₂×Q₈	—	284	4	no	—
135	C₂×Q₈	—	568	2	no	—
136	C₂×Q₈	—	568	2	no	—
137	C₂²×C₄	—	284	2	no	—
138	C₂²×C₄	—	284	2	no	—
139	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
140	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
144	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
145	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₇₁	1	2	yes	—
147	C₂×C₈	#11765864	1	2	yes	—
148	C₄⋊C₄	—	284	2	no	—
149	C₄⋊C₄	—	284	4	no	—
150	C₄⋊C₄	—	284	2	no	—
151	C₄⋊C₄	—	142	4	no	—
152	C₄⋊C₄	—	142	4	no	—
153	C₄²	—	568	2	no	—
154	C₄²	(C₂×D₇₁)⋊C₄	1	2	yes	—
155	C₄²	#11765854	1	2	yes	—
156	C₁₆	—	4	2	no	—
157	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
158	Q₈	—	568	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	284	4	no	—
161	C₂×C₄	—	284	4	no	—
162	C₂×C₄	—	568	2	no	—
163	C₂×C₄	—	568	2	no	—
164	C₂×C₄	—	568	2	no	—
165	C₂×C₄	—	568	2	no	—
166	C₂×C₄	#14556050	1	2	yes	—
167	C₂×C₄	#14556050	1	2	yes	—
168	C₂×C₄	D₁₁₃₆	1	2	yes	—
169	C₂×C₄	#14556073	1	2	yes	—
170	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
171	C₂×C₄	#14556108	1	2	yes	—
172	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
173	C₂×C₄	#14556161	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	#14556093	1	2	yes	—
176	C₈	#14556115	1	1	yes	characteristic
177	C₈	#14556126	1	1	yes	characteristic
178	C₂²	C₂×#14556050	1	1	yes	characteristic
179	C₂²	#17986280	1	2	yes	—
180	C₂²	C₂×D₁₁₃₆	1	1	yes	characteristic
181	C₂²	C₂×#14556073	1	1	yes	characteristic
182	C₂²	#17986442	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	—	568	4	no	—
185	C₄	—	568	4	no	—
186	C₄	#17985801	1	4	yes	—
187	C₄	#17985904	1	1	yes	characteristic
188	C₄	#17986009	1	1	yes	characteristic
189	C₄	#17986285	1	2	yes	—
190	C₂	#22589291	1	1	yes	characteristic
191	C₂	C₂×#17985801	1	1	yes	characteristic
192	C₂	#22590061	1	1	yes	characteristic
193	C₂	#22590478	1	2	yes	—
194	C₂	#22590483	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17986404

C₂

yes

maximal, characteristic

C₄×#14556062

C₂

yes

maximal, characteristic

#22586432

C₂

yes

maximal

C₂×C₄⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556128

—

#17986404

—

#17986404

C₄

yes

complemented

C₂×#14556062

—

C₂×#14556062

C₂²

yes

—

C₂×#14556062

C₂²

yes

—

C₄×C₄⋊Dic₇₁

—

#17985762

—

C₂²×C₁₆×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₁

—

C₂²×Dic₁₄₂

—

#14556128

—

#14556128

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄×Dic₇₁

—

—

—

—

C₂×C₄

yes

—

#14556062

Q₈

yes

complemented

C₇₁⋊#95

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

D₈

yes

characteristic

C₂²×C₄×C₇₁

D₈

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

SD₁₆

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

D₁₆

yes

—

C₂×C₄×C₇₁

SD₃₂

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

#137

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₄×Q₈

yes

—

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₂²×C₇₁

C₂×SD₃₂

yes

characteristic

C₂²×C₇₁

#510

yes

—

C₄×C₇₁

—

C₄×C₇₁

#357

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#2225

yes

—

C₂×#500

—

C₂×C₄⋊C₁₆

D₇₁

yes

characteristic

100

C₄×#107

—

101

#1380

—

102

C₇₁

#13275

yes

characteristic, complemented, Sylow

103

C₂×C₄⋊Q₈

—

142

—

104

C₂²×C₁₆

C₂×D₇₁

yes

—

105

#500

—

106

#500

—

142

—

107

C₂×#107

—

108

C₂×#107

—

109

C₂×#107

—

142

—

110

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

111

C₄×C₄⋊C₄

—

142

—

112

C₄⋊C₁₆

—

113

#340

—

142

—

114

C₂²×Q₈

—

284

—

115

C₂²×C₈

D₂₈₄

yes

characteristic

116

C₂²×C₈

C₂²×D₇₁

yes

characteristic

117

C₄⋊Q₈

—

284

—

118

C₄⋊Q₈

—

142

—

119

C₂×C₄⋊C₄

—

284

—

120

C₂×C₄⋊C₄

—

142

—

121

C₂×C₄⋊C₄

—

142

—

122

C₂×C₄²

—

284

—

123

C₂×C₄²

D₂₈₄

yes

characteristic

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

#107

—

127

#107

—

128

#107

—

142

—

129

#107

—

142

—

130

#107

—

142

—

131

C₄×C₈

C₄×D₇₁

yes

—

132

C₄×C₈

C₂²⋊D₇₁

yes

—

133

#95

—

284

—

134

C₂×Q₈

—

284

—

135

C₂×Q₈

—

568

—

136

C₂×Q₈

—

568

—

137

C₂²×C₄

—

284

—

138

C₂²×C₄

—

284

—

139

C₂²×C₄

D₅₆₈

yes

characteristic

140

C₂²×C₄

D₅₆₈

yes

characteristic

141

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

—

144

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

145

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

146

C₂×C₈

Q₈×D₇₁

yes

—

147

C₂×C₈

#11765864

yes

—

148

C₄⋊C₄

—

284

—

149

C₄⋊C₄

—

284

—

150

C₄⋊C₄

—

284

—

151

C₄⋊C₄

—

142

—

152

C₄⋊C₄

—

142

—

153

C₄²

—

568

—

154

C₄²

(C₂×D₇₁)⋊C₄

yes

—

155

C₄²

#11765854

yes

—

156

C₁₆

—

157

C₂³

C₂×D₅₆₈

yes

characteristic

158

Q₈

—

568

—

159

C₂×C₄

—

160

C₂×C₄

—

284

—

161

C₂×C₄

—

284

—

162

C₂×C₄

—

568

—

163

C₂×C₄

—

568

—

164

C₂×C₄

—

568

—

165

C₂×C₄

—

568

—

166

C₂×C₄

#14556050

yes

—

167

C₂×C₄

#14556050

yes

—

168

C₂×C₄

D₁₁₃₆

yes

—

169

C₂×C₄

#14556073

yes

—

170

C₂×C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

171

C₂×C₄

#14556108

yes

—

172

C₂×C₄

C₂×#11765854

yes

characteristic

173

C₂×C₄

#14556161

yes

—

174

C₈

—

175

C₈

#14556093

yes

—

176

C₈

#14556115

yes

characteristic

177

C₈

#14556126

yes

characteristic

178

C₂²

C₂×#14556050

yes

characteristic

179

C₂²

#17986280

yes

—

180

C₂²

C₂×D₁₁₃₆

yes

characteristic

181

C₂²

C₂×#14556073

yes

characteristic

182

C₂²

#17986442

yes

—

183

C₄

—

184

C₄

—

568

—

185

C₄

—

568

—

186

C₄

#17985801

yes

—

187

C₄

#17985904

yes

characteristic

188

C₄

#17986009

yes

characteristic

189

C₄

#17986285

yes

—

190

C₂

#22589291

yes

characteristic

191

C₂

C₂×#17985801

yes

characteristic

192

C₂

#22590061

yes

characteristic

193

C₂

#22590478

yes

—

194

C₂

#22590483

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄⋊C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#13275
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂³

C₂×D₅₆₈

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×C₄⋊C₁₆×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂×D₈

cosocle

C₂²×C₈×C₇₁

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₇₁

hypocenter

C₇₁

#13275

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
7. #17986404	C₄	semidirect
27. #14556062	Q₈	semidirect
102. C₇₁	#13275	semidirect

H ≅ G/N

type

7. #17986404

C₄

semidirect

27. #14556062

Q₈

semidirect

102. C₇₁

#13275

semidirect

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

subgroup	index
C₂×#17986404	2
C₄×#14556062	2
#22586432	2
C₂×C₄⋊C₁₆×C₇₁	2
#13275	71

subgroup

index

C₂×C₄⋊C₁₆×C₇₁

Subgroups — #31916023

Conjugacy classes of subgroups

Complements

Maximal subgroups