Group #28507178 — subgroups

Subgroups — #28507178

Order 15104.

566 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 143 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	#21425852	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837215	C₂	1	4	yes	maximal, complemented
3	C₄×#14273663	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₉×#493	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	2	yes	—
6	C₂×#14273663	C₂²	1	2	yes	—
7	C₄×#11745664	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
9	#16837215	—	2	8	no	—
10	#16837167	—	2	1	no	—
11	C₂²×D₈×C₅₉	—	2	1	no	—
12	C₂²×D₈×C₅₉	C₂²	1	2	yes	—
13	C₂×#127×C₅₉	—	2	1	no	—
14	C₅₉×#493	—	2	2	no	—
15	C₅₉×#493	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745664	—	2	2	no	—
18	#14273667	—	2	4	no	—
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	#14273663	—	2	2	no	—
21	#14273663	D₄	1	4	yes	complemented
22	#14273638	—	4	1	no	—
23	C₄×C₅₉⋊C₈	—	4	1	no	—
24	C₂²×D₄×C₅₉	—	2	2	no	—
25	C₂²×D₄×C₅₉	—	4	2	no	—
26	C₂×D₈×C₅₉	—	2	4	no	—
27	C₂×D₈×C₅₉	—	2	4	no	—
28	C₂×D₈×C₅₉	—	2	2	no	—
29	C₂×D₈×C₅₉	—	2	4	no	—
30	C₂×D₈×C₅₉	—	4	4	no	—
31	C₂×D₈×C₅₉	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	#127×C₅₉	—	2	4	no	—
35	#127×C₅₉	—	4	1	no	—
36	#127×C₅₉	—	4	1	no	—
37	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₉	D₄	1	2	yes	—
40	C₂⁴×C₅₉	—	4	2	no	—
41	C₂×D₄×C₅₉	—	2	4	no	—
42	C₂×D₄×C₅₉	—	4	4	no	—
43	C₂×D₄×C₅₉	—	4	4	no	—
44	C₂×D₄×C₅₉	—	4	4	no	—
45	C₂×D₄×C₅₉	—	4	2	no	—
46	C₂×D₄×C₅₉	—	4	4	no	—
47	C₂×D₄×C₅₉	—	8	8	no	—
48	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
49	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	D₈×C₅₉	—	2	8	no	—
51	D₈×C₅₉	—	4	8	no	—
52	D₈×C₅₉	—	4	2	no	—
53	D₈×C₅₉	—	4	2	no	—
54	C₂×C₈×C₅₉	—	2	2	no	—
55	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₉	#41	1	2	yes	—
60	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₅₉	D₈	1	1	yes	characteristic
62	C₄²×C₅₉	D₈	1	1	yes	characteristic
63	#11745664	—	2	4	no	—
64	#11745664	—	4	2	no	—
65	C₂×C₅₉⋊C₈	—	4	2	no	—
66	C₅₉⋊C₁₆	—	4	2	no	—
67	C₂³×C₅₉	—	4	4	no	—
68	C₂³×C₅₉	—	8	8	no	—
69	C₂³×C₅₉	—	8	4	no	—
70	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
71	D₄×C₅₉	—	4	4	no	—
72	D₄×C₅₉	—	4	4	no	—
73	D₄×C₅₉	—	8	8	no	—
74	D₄×C₅₉	—	8	8	no	—
75	D₄×C₅₉	—	8	4	no	—
76	C₂×C₄×C₅₉	—	2	4	no	—
77	C₂×C₄×C₅₉	#102	1	2	yes	—
78	C₂×C₄×C₅₉	#102	1	2	yes	—
79	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#121	1	2	yes	—
81	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₈×C₅₉	—	2	2	no	—
84	C₈×C₅₉	C₄×D₄	1	2	yes	—
85	C₈×C₅₉	#124	1	1	yes	characteristic
86	C₈×C₅₉	#127	1	1	yes	characteristic
87	C₅₉⋊C₈	—	4	4	no	—
88	#13257	—	59	1	no	maximal, Sylow
89	C₂²×C₅₉	—	2	2	no	—
90	C₂²×C₅₉	—	8	8	no	—
91	C₂²×C₅₉	—	8	4	no	—
92	C₂²×C₅₉	—	16	8	no	—
93	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
94	C₂²×C₅₉	#460	1	2	yes	—
95	C₄×C₅₉	—	2	2	no	—
96	C₄×C₅₉	#361	1	4	yes	complemented
97	C₄×C₅₉	#390	1	1	yes	characteristic
98	C₄×C₅₉	#418	1	1	yes	characteristic
99	C₄×C₅₉	#459	1	2	yes	—
100	C₂×#493	D₅₉	1	1	yes	characteristic
101	#2150	—	59	1	no	—
102	C₂×#361	—	59	4	no	—
103	C₄×#108	—	59	1	no	—
104	C₂×C₅₉	—	2	2	no	—
105	C₂×C₅₉	—	16	4	no	—
106	C₂×C₅₉	#1982	1	1	yes	characteristic
107	C₂×C₅₉	C₂×#361	1	2	yes	—
108	C₂²×D₈	—	2	1	no	—
109	C₂²×D₈	C₂×D₅₉	1	2	yes	—
110	C₂×#127	—	2	1	no	—
111	C₂×M₃₂	—	59	2	no	—
112	#493	—	2	2	no	—
113	#493	Dic₅₉	1	4	yes	—
114	C₂×#108	—	59	2	no	—
115	C₄×M₁₆	—	118	1	no	—
116	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
117	C₄⋊C₁₆	—	118	2	no	—
118	#361	—	118	8	no	—
119	#341	—	118	1	no	—
120	C₅₉	#13257	1	1	yes	characteristic, complemented, Sylow
121	C₂²×D₄	—	2	2	no	—
122	C₂²×D₄	—	4	2	no	—
123	C₂×D₈	—	2	4	no	—
124	C₂×D₈	—	2	4	no	—
125	C₂×D₈	—	2	4	no	—
126	C₂×D₈	—	2	2	no	—
127	C₂×D₈	—	4	4	no	—
128	C₂×D₈	C₂×Dic₅₉	1	4	yes	—
129	C₂×M₁₆	—	118	2	no	—
130	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
132	#127	—	2	4	no	—
133	#127	—	4	1	no	—
134	#127	—	4	1	no	—
135	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
136	M₃₂	—	118	4	no	—
137	C₂×C₁₆	—	118	2	no	—
138	#108	—	59	4	no	—
139	#108	—	118	2	no	—
140	#97	—	236	1	no	—
141	C₄×C₈	—	236	1	no	—
142	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
143	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
144	C₂⁴	—	4	2	no	—
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	—	4	4	no	—
147	C₂×D₄	—	4	4	no	—
148	C₂×D₄	—	4	4	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	8	8	no	—
152	C₂²×C₄	#11745670	1	2	yes	—
153	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
154	D₈	—	2	8	no	—
155	D₈	—	4	8	no	—
156	D₈	—	4	2	no	—
157	D₈	—	4	2	no	—
158	M₁₆	—	118	4	no	—
159	M₁₆	—	236	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	236	2	no	—
162	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
163	C₂×C₈	D₄×D₅₉	1	2	yes	—
164	C₂×C₈	#11745687	1	2	yes	—
165	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
167	C₄²	#11745669	1	1	yes	characteristic
168	C₄²	#11745669	1	1	yes	characteristic
169	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
170	C₁₆	—	236	2	no	—
171	C₂³	—	4	4	no	—
172	C₂³	—	8	8	no	—
173	C₂³	—	8	4	no	—
174	C₂³	C₂×#11745670	1	1	yes	characteristic
175	D₄	—	4	4	no	—
176	D₄	—	4	4	no	—
177	D₄	—	8	8	no	—
178	D₄	—	8	8	no	—
179	D₄	—	8	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	#14273648	1	2	yes	—
182	C₂×C₄	#14273648	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
184	C₂×C₄	#14273698	1	2	yes	—
185	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
186	C₂×C₄	#14273753	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	236	4	no	—
189	C₈	D₄×Dic₅₉	1	2	yes	—
190	C₈	#14273714	1	1	yes	characteristic
191	C₈	#14273723	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	8	4	no	—
194	C₂²	—	8	8	no	—
195	C₂²	—	16	8	no	—
196	C₂²	C₂×#14273648	1	1	yes	characteristic
197	C₂²	#16837599	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	#16837215	1	4	yes	complemented
200	C₄	#16837305	1	1	yes	characteristic
201	C₄	#16837407	1	1	yes	characteristic
202	C₄	#16837594	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	16	4	no	—
205	C₂	#21425024	1	1	yes	characteristic
206	C₂	C₂×#16837215	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425852

C₂

yes

maximal, characteristic, complemented

C₂×#16837215

C₂

yes

maximal, complemented

C₄×#14273663

C₂

yes

maximal, characteristic, complemented

C₂×C₅₉×#493

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₂×#14273663

C₂²

yes

—

C₄×#11745664

—

C₄⋊(C₅₉⋊C₁₆)

—

#16837215

—

#16837167

—

C₂²×D₈×C₅₉

—

C₂²×D₈×C₅₉

C₂²

yes

—

C₂×#127×C₅₉

—

C₅₉×#493

—

C₅₉×#493

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

#14273663

—

#14273663

D₄

yes

complemented

#14273638

—

C₄×C₅₉⋊C₈

—

C₂²×D₄×C₅₉

—

C₂²×D₄×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#127×C₅₉

—

#127×C₅₉

—

#127×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂⁴×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

SD₁₆

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

D₈×C₅₉

—

D₈×C₅₉

—

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×SD₁₆

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₄×D₄

yes

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

#13257

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#102

yes

characteristic

C₂²×C₅₉

#460

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

100

C₂×#493

D₅₉

yes

characteristic

101

#2150

—

102

C₂×#361

—

103

C₄×#108

—

104

C₂×C₅₉

—

105

C₂×C₅₉

—

106

C₂×C₅₉

#1982

yes

characteristic

107

C₂×C₅₉

C₂×#361

yes

—

108

C₂²×D₈

—

109

C₂²×D₈

C₂×D₅₉

yes

—

110

C₂×#127

—

111

C₂×M₃₂

—

112

#493

—

113

#493

Dic₅₉

yes

—

114

C₂×#108

—

115

C₄×M₁₆

—

118

—

116

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

117

C₄⋊C₁₆

—

118

—

118

#361

—

118

—

119

#341

—

118

—

120

C₅₉

#13257

yes

characteristic, complemented, Sylow

121

C₂²×D₄

—

122

C₂²×D₄

—

123

C₂×D₈

—

124

C₂×D₈

—

125

C₂×D₈

—

126

C₂×D₈

—

127

C₂×D₈

—

128

C₂×D₈

C₂×Dic₅₉

yes

—

129

C₂×M₁₆

—

118

—

130

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

131

C₂²×C₈

C₂²×D₅₉

yes

characteristic

132

#127

—

133

#127

—

134

#127

—

135

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

136

M₃₂

—

118

—

137

C₂×C₁₆

—

118

—

138

#108

—

139

#108

—

118

—

140

#97

—

236

—

141

C₄×C₈

—

236

—

142

C₄×C₈

C₂×Dic₅₉

yes

—

143

C₄×C₈

C₂²⋊D₅₉

yes

—

144

C₂⁴

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂²×C₄

#11745670

yes

—

153

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

154

D₈

—

155

D₈

—

156

D₈

—

157

D₈

—

158

M₁₆

—

118

—

159

M₁₆

—

236

—

160

C₂×C₈

—

161

C₂×C₈

—

236

—

162

C₂×C₈

C₂²⋊Dic₅₉

yes

—

163

C₂×C₈

D₄×D₅₉

yes

—

164

C₂×C₈

#11745687

yes

—

165

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

166

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

167

C₄²

#11745669

yes

characteristic

168

C₄²

#11745669

yes

characteristic

169

C₄²

C₂²⋊Dic₅₉

yes

—

170

C₁₆

—

236

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

C₂×#11745670

yes

characteristic

175

D₄

—

176

D₄

—

177

D₄

—

178

D₄

—

179

D₄

—

180

C₂×C₄

—

181

C₂×C₄

#14273648

yes

—

182

C₂×C₄

#14273648

yes

—

183

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

184

C₂×C₄

#14273698

yes

—

185

C₂×C₄

C₂×#11745669

yes

characteristic

186

C₂×C₄

#14273753

yes

—

187

C₈

—

188

C₈

—

236

—

189

C₈

D₄×Dic₅₉

yes

—

190

C₈

#14273714

yes

characteristic

191

C₈

#14273723

yes

characteristic

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

C₂×#14273648

yes

characteristic

197

C₂²

#16837599

yes

—

198

C₄

—

199

C₄

#16837215

yes

complemented

200

C₄

#16837305

yes

characteristic

201

C₄

#16837407

yes

characteristic

202

C₄

#16837594

yes

—

203

C₂

—

204

C₂

—

205

C₂

#21425024

yes

characteristic

206

C₂

C₂×#16837215

yes

—

	subgroup	G/H
center	C₂²	C₂×#14273648
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting, prime radical	C₂×C₅₉×#493	C₂
socle	C₂²×C₅₉	C₂×#102
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#493	D₅₉
hypocenter	C₅₉	#13257
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

subgroup

G/H

center

C₂²

C₂×#14273648

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting, prime radical

C₂×C₅₉×#493

C₂

socle

C₂²×C₅₉

C₂×#102

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

N	H ≅ G/N	type
1. #21425852	C₂	semidirect
2. C₂×#16837215	C₂	semidirect
3. C₄×#14273663	C₂	semidirect
21. #14273663	D₄	semidirect
96. C₄×C₅₉	#361	semidirect
120. C₅₉	#13257	semidirect
199. C₄	#16837215	semidirect

H ≅ G/N

type

1. #21425852

C₂

semidirect

2. C₂×#16837215

C₂

semidirect

3. C₄×#14273663

C₂

semidirect

21. #14273663

D₄

semidirect

96. C₄×C₅₉

#361

semidirect

120. C₅₉

#13257

semidirect

199. C₄

#16837215

semidirect

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

subgroup	index
#21425852	2
C₂×#16837215	2
C₄×#14273663	2
C₂×C₅₉×#493	2
#13257	59

subgroup

index

C₂×C₅₉×#493

Subgroups — #28507178

Conjugacy classes of subgroups

Complements

Maximal subgroups