Group #28488359 — subgroups

Subgroups — #28488359

Order 15104.

434 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16837807	C₂	1	1	yes	maximal, characteristic
2	#21424939	C₂	1	2	yes	maximal
3	#21422769	C₂	1	2	yes	maximal
4	#21422385	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273727	C₂²	1	2	yes	—
7	#16837807	—	2	1	no	—
8	#16837807	—	2	1	no	—
9	#16837807	—	2	2	no	—
10	#16837455	C₂²	1	1	yes	characteristic
11	C₂×#14273661	C₄	1	4	yes	—
12	C₂×#14273645	—	2	4	no	—
13	C₄×#11745664	C₂²	1	1	yes	characteristic
14	#16837149	—	2	4	no	—
15	#16837124	—	2	1	no	—
16	#16837123	—	2	1	no	—
17	C₅₉×#423	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#334	—	2	2	no	—
20	C₂²×Dic₁₁₈	—	2	2	no	—
21	C₂×#11745664	D₄	1	2	yes	—
22	#14273727	—	2	2	no	—
23	#14273727	—	2	2	no	—
24	#14273727	—	2	2	no	—
25	#14273727	—	2	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
28	#14273661	—	2	4	no	—
29	#14273661	—	2	4	no	—
30	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
31	#14273645	—	4	8	no	—
32	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
33	#14273638	—	2	1	no	—
34	C₄×C₅₉⋊C₈	—	2	1	no	—
35	C₄×C₅₉⋊C₈	—	2	1	no	—
36	C₂×M₁₆×C₅₉	—	2	2	no	—
37	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₉	—	2	2	no	—
40	C₄⋊C₈×C₅₉	—	2	2	no	—
41	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₉	D₄	1	2	yes	—
44	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
47	C₂²×Dic₅₉	—	2	4	no	—
48	C₂×Dic₁₁₈	—	4	2	no	—
49	C₂×Dic₁₁₈	—	4	2	no	—
50	C₂×Dic₁₁₈	—	4	4	no	—
51	C₂×Dic₁₁₈	—	4	4	no	—
52	M₁₆×C₅₉	—	4	4	no	—
53	C₂×C₈×C₅₉	—	2	4	no	—
54	C₂×C₈×C₅₉	—	4	2	no	—
55	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₉	#41	1	1	yes	characteristic
61	C₄⋊Dic₅₉	—	2	4	no	—
62	C₄⋊Dic₅₉	—	2	4	no	—
63	C₄⋊Dic₅₉	—	4	2	no	—
64	C₄⋊Dic₅₉	—	4	2	no	—
65	#11745664	—	2	4	no	—
66	C₂×C₅₉⋊C₈	—	2	2	no	—
67	C₂×C₅₉⋊C₈	—	2	2	no	—
68	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
77	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
78	C₂×Dic₅₉	—	4	4	no	—
79	C₂×Dic₅₉	—	4	8	no	—
80	Dic₁₁₈	—	8	4	no	—
81	Dic₁₁₈	—	8	2	no	—
82	Dic₁₁₈	—	8	2	no	—
83	C₈×C₅₉	—	2	2	no	—
84	C₈×C₅₉	—	8	2	no	—
85	C₈×C₅₉	#101	1	4	yes	complemented
86	C₅₉⋊C₈	—	4	2	no	—
87	C₅₉⋊C₈	—	4	2	no	—
88	#9808	—	59	1	no	maximal, Sylow
89	C₂²×C₅₉	—	2	1	no	—
90	C₂²×C₅₉	—	2	1	no	—
91	C₂²×C₅₉	#390	1	1	yes	characteristic
92	C₂²×C₅₉	#442	1	2	yes	—
93	C₄×C₅₉	—	2	1	no	—
94	C₄×C₅₉	—	2	1	no	—
95	C₄×C₅₉	C₂×#101	1	2	yes	—
96	C₄×C₅₉	#469	1	2	yes	—
97	C₄×C₅₉	#473	1	2	yes	—
98	C₄×C₅₉	#496	1	2	yes	—
99	Dic₅₉	—	8	4	no	—
100	C₂×#500	—	59	1	no	—
101	#1966	—	59	2	no	—
102	#1566	D₅₉	1	1	yes	characteristic
103	#1546	—	59	2	no	—
104	#1451	—	59	1	no	—
105	C₂×C₅₉	—	2	2	no	—
106	C₂×C₅₉	#1966	1	1	yes	characteristic
107	C₂×C₅₉	#1972	1	1	yes	characteristic
108	C₂×C₅₉	#1979	1	1	yes	characteristic
109	C₂×C₄⋊Q₈	—	59	2	no	—
110	#500	—	118	1	no	—
111	#500	—	118	1	no	—
112	#500	—	118	2	no	—
113	#431	—	59	1	no	—
114	C₂×#107	—	59	4	no	—
115	#423	C₂×D₅₉	1	2	yes	—
116	C₂×#101	—	118	4	no	—
117	C₄×M₁₆	—	59	1	no	—
118	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
119	#334	—	2	2	no	—
120	#332	—	118	4	no	—
121	#322	—	118	1	no	—
122	#322	—	118	1	no	—
123	C₅₉	#9808	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	118	2	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂×M₁₆	—	59	2	no	—
127	C₂²×C₈	C₄×D₅₉	1	2	yes	—
128	C₄⋊Q₈	—	118	2	no	—
129	C₄⋊Q₈	—	118	2	no	—
130	C₄⋊Q₈	—	118	2	no	—
131	C₄⋊Q₈	—	118	2	no	—
132	C₂×C₄⋊C₄	—	59	4	no	—
133	C₂×C₄⋊C₄	—	118	2	no	—
134	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
135	#107	—	118	4	no	—
136	#107	—	118	4	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	59	2	no	—
140	#101	—	236	8	no	—
141	C₂²⋊C₈	—	118	1	no	—
142	#97	—	118	1	no	—
143	C₄×C₈	—	118	1	no	—
144	C₄×C₈	—	118	1	no	—
145	C₄×C₈	D₂₃₆	1	1	yes	characteristic
146	C₄×C₈	D₂₃₆	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
148	C₂×Q₈	—	236	2	no	—
149	C₂×Q₈	—	236	2	no	—
150	C₂×Q₈	—	236	4	no	—
151	C₂×Q₈	—	236	4	no	—
152	C₂²×C₄	—	118	4	no	—
153	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
154	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
155	C₂²×C₄	#11745689	1	1	yes	characteristic
156	M₁₆	—	4	4	no	—
157	M₁₆	—	118	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	118	2	no	—
161	C₂×C₈	—	118	2	no	—
162	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
163	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
164	C₄⋊C₄	—	118	4	no	—
165	C₄⋊C₄	—	118	4	no	—
166	C₄⋊C₄	—	236	2	no	—
167	C₄⋊C₄	—	236	2	no	—
168	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
169	C₄²	D₄×D₅₉	1	1	yes	characteristic
170	C₄²	#11745689	1	1	yes	characteristic
171	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
172	C₂³	#14273689	1	1	yes	characteristic
173	Q₈	—	472	4	no	—
174	Q₈	—	472	2	no	—
175	Q₈	—	472	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	236	8	no	—
179	C₂×C₄	—	236	4	no	—
180	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
181	C₂×C₄	#14273689	1	1	yes	characteristic
182	C₂×C₄	#14273702	1	1	yes	characteristic
183	C₂×C₄	#14273702	1	1	yes	characteristic
184	C₂×C₄	#14273716	1	1	yes	characteristic
185	C₂×C₄	#14273723	1	1	yes	characteristic
186	C₂×C₄	#14273754	1	4	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	8	2	no	—
189	C₈	—	236	2	no	—
190	C₈	—	236	2	no	—
191	C₈	#14273645	1	4	yes	complemented
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	#16837303	1	1	yes	characteristic
195	C₂²	#16837513	1	2	yes	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	472	4	no	—
199	C₄	C₂×#14273645	1	2	yes	—
200	C₄	#16837656	1	2	yes	—
201	C₄	#16837678	1	2	yes	—
202	C₄	#16837788	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#21424939	1	1	yes	characteristic
205	C₂	#21424970	1	1	yes	characteristic
206	C₂	#21425009	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837807

C₂

yes

maximal, characteristic

#21424939

C₂

yes

maximal

#21422769

C₂

yes

maximal

#21422385

C₂

yes

maximal, characteristic

C₅₉×#1566

C₂

yes

maximal, characteristic

C₂×#14273727

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#14273661

C₄

yes

—

C₂×#14273645

—

C₄×#11745664

C₂²

yes

—

—

—

C₅₉×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#334

—

C₂²×Dic₁₁₈

—

C₂×#11745664

D₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂×C₄

yes

—

#14273661

—

#14273661

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

#14273645

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

#41

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#101

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9808

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#442

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#101

yes

—

C₄×C₅₉

#469

yes

—

C₄×C₅₉

#473

yes

—

C₄×C₅₉

#496

yes

—

Dic₅₉

—

100

C₂×#500

—

101

#1966

—

102

#1566

D₅₉

yes

characteristic

103

#1546

—

104

#1451

—

105

C₂×C₅₉

—

106

C₂×C₅₉

#1966

yes

characteristic

107

C₂×C₅₉

#1972

yes

characteristic

108

C₂×C₅₉

#1979

yes

characteristic

109

C₂×C₄⋊Q₈

—

110

#500

—

118

—

111

#500

—

118

—

112

#500

—

118

—

113

#431

—

114

C₂×#107

—

115

#423

C₂×D₅₉

yes

—

116

C₂×#101

—

118

—

117

C₄×M₁₆

—

118

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

119

#334

—

120

#332

—

118

—

121

#322

—

118

—

122

#322

—

118

—

123

C₅₉

#9808

yes

characteristic, complemented, Sylow

124

C₂²×Q₈

—

118

—

125

C₂×M₁₆

—

126

C₂×M₁₆

—

127

C₂²×C₈

C₄×D₅₉

yes

—

128

C₄⋊Q₈

—

118

—

129

C₄⋊Q₈

—

118

—

130

C₄⋊Q₈

—

118

—

131

C₄⋊Q₈

—

118

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

118

—

134

C₂×C₄²

C₂²×D₅₉

yes

characteristic

135

#107

—

118

—

136

#107

—

118

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

#101

—

236

—

141

C₂²⋊C₈

—

118

—

142

#97

—

118

—

143

C₄×C₈

—

118

—

144

C₄×C₈

—

118

—

145

C₄×C₈

D₂₃₆

yes

characteristic

146

C₄×C₈

D₂₃₆

yes

characteristic

147

C₄×C₈

C₂²⋊D₅₉

yes

—

148

C₂×Q₈

—

236

—

149

C₂×Q₈

—

236

—

150

C₂×Q₈

—

236

—

151

C₂×Q₈

—

236

—

152

C₂²×C₄

—

118

—

153

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

154

C₂²×C₄

D₄×D₅₉

yes

characteristic

155

C₂²×C₄

#11745689

yes

characteristic

156

M₁₆

—

157

M₁₆

—

118

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

118

—

161

C₂×C₈

—

118

—

162

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

163

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

164

C₄⋊C₄

—

118

—

165

C₄⋊C₄

—

118

—

166

C₄⋊C₄

—

236

—

167

C₄⋊C₄

—

236

—

168

C₄²

C₂×D₂₃₆

yes

characteristic

169

C₄²

D₄×D₅₉

yes

characteristic

170

C₄²

#11745689

yes

characteristic

171

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

172

C₂³

#14273689

yes

characteristic

173

Q₈

—

472

—

174

Q₈

—

472

—

175

Q₈

—

472

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

236

—

179

C₂×C₄

—

236

—

180

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

181

C₂×C₄

#14273689

yes

characteristic

182

C₂×C₄

#14273702

yes

characteristic

183

C₂×C₄

#14273702

yes

characteristic

184

C₂×C₄

#14273716

yes

characteristic

185

C₂×C₄

#14273723

yes

characteristic

186

C₂×C₄

#14273754

yes

—

187

C₈

—

188

C₈

—

189

C₈

—

236

—

190

C₈

—

236

—

191

C₈

#14273645

yes

complemented

192

C₂²

—

193

C₂²

—

194

C₂²

#16837303

yes

characteristic

195

C₂²

#16837513

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

472

—

199

C₄

C₂×#14273645

yes

—

200

C₄

#16837656

yes

—

201

C₄

#16837678

yes

—

202

C₄

#16837788

yes

—

203

C₂

—

204

C₂

#21424939

yes

characteristic

205

C₂

#21424970

yes

characteristic

206

C₂

#21425009

yes

characteristic

	subgroup	G/H
center	C₂²	#16837303
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1566	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1566	D₅₉
hypocenter	C₅₉	#9808
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂²

#16837303

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1566

C₂

socle

C₂²×C₅₉

#390

prime radical

C₂³×C₅₉

C₄×D₄

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
85. C₈×C₅₉	#101	semidirect
123. C₅₉	#9808	semidirect
191. C₈	#14273645	semidirect

H ≅ G/N

type

85. C₈×C₅₉

#101

semidirect

123. C₅₉

#9808

semidirect

191. C₈

#14273645

semidirect

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

subgroup	index
C₂×#16837807	2
#21424939	2
#21422769	2
#21422385	2
C₅₉×#1566	2
#9808	59

subgroup

index

C₅₉×#1566

Subgroups — #28488359

Conjugacy classes of subgroups

Complements

Maximal subgroups