Group #28493440 — subgroups

Subgroups — #28493440

Order 15104.

350 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 119 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	#21425032	C₂	1	1	yes	maximal, characteristic
2	#21424224	C₂	1	1	yes	maximal, characteristic
3	#21424102	C₂	1	2	yes	maximal
4	#21421980	C₂	1	2	yes	maximal
5	C₅₉×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#14273652	—	2	2	no	—
9	#16837272	C₂²	1	2	yes	—
10	#16837165	—	2	2	no	—
11	#16837165	—	2	2	no	—
12	#16837156	—	2	2	no	—
13	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₁₈	D₄	1	1	yes	characteristic
16	C₂²×C₅₉⋊C₈	—	2	1	no	—
17	C₄×Dic₁₁₈	—	2	2	no	—
18	C₄×Dic₁₁₈	—	2	2	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
23	C₂×Dic₅₉⋊C₄	—	2	1	no	—
24	C₂×C₄×Dic₅₉	—	2	1	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	#14273652	—	4	4	no	—
27	C₂²×C₈×C₅₉	—	2	2	no	—
28	C₂²×C₈×C₅₉	D₄	1	2	yes	—
29	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₉	—	2	2	no	—
31	C₄⋊C₈×C₅₉	—	2	2	no	—
32	C₄×C₈×C₅₉	—	2	2	no	—
33	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
36	C₂²×Dic₅₉	—	2	1	no	—
37	C₂²×Dic₅₉	—	2	1	no	—
38	C₂²×Dic₅₉	—	4	2	no	—
39	C₂×Dic₁₁₈	—	2	2	no	—
40	C₂×Dic₁₁₈	—	2	2	no	—
41	C₂×Dic₁₁₈	—	2	1	no	—
42	C₂×Dic₁₁₈	Q₁₆	1	2	yes	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₂×C₈×C₅₉	—	2	2	no	—
46	C₂×C₈×C₅₉	—	4	2	no	—
47	C₂×C₈×C₅₉	—	4	2	no	—
48	C₂×C₈×C₅₉	—	4	2	no	—
49	C₄²×C₅₉	#41	1	2	yes	—
50	C₄²×C₅₉	#41	1	2	yes	—
51	C₄⋊Dic₅₉	—	2	2	no	—
52	C₄⋊Dic₅₉	—	4	2	no	—
53	C₄⋊Dic₅₉	—	4	2	no	—
54	C₄⋊Dic₅₉	—	4	2	no	—
55	C₄⋊Dic₅₉	—	4	2	no	—
56	C₄⋊Dic₅₉	—	8	2	no	—
57	Dic₅₉⋊C₄	—	4	2	no	—
58	C₄×Dic₅₉	—	4	1	no	—
59	C₄×Dic₅₉	—	4	1	no	—
60	C₂×C₅₉⋊C₈	—	4	1	no	—
61	C₂×C₅₉⋊C₈	—	4	2	no	—
62	C₂³×C₅₉	#121	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	#123	1	2	yes	—
65	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#126	1	2	yes	—
68	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#135	1	2	yes	—
71	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
72	C₂×C₄×C₅₉	#137	1	2	yes	—
73	C₂×Dic₅₉	—	2	2	no	—
74	C₂×Dic₅₉	—	4	1	no	—
75	C₂×Dic₅₉	—	4	1	no	—
76	C₂×Dic₅₉	—	4	1	no	—
77	C₂×Dic₅₉	—	4	2	no	—
78	C₂×Dic₅₉	—	8	2	no	—
79	C₂×Dic₅₉	—	8	2	no	—
80	C₂×Dic₅₉	—	8	2	no	—
81	Dic₁₁₈	—	2	4	no	—
82	Dic₁₁₈	—	4	2	no	—
83	C₈×C₅₉	—	4	2	no	—
84	C₈×C₅₉	—	8	2	no	—
85	C₅₉⋊C₈	—	8	1	no	—
86	#10624	—	59	1	no	maximal, Sylow
87	C₂²×C₅₉	#397	1	1	yes	characteristic
88	C₂²×C₅₉	#464	1	2	yes	—
89	C₂²×C₅₉	#467	1	1	yes	characteristic
90	C₂²×C₅₉	#469	1	1	yes	characteristic
91	C₂²×C₅₉	#471	1	2	yes	—
92	C₄×C₅₉	—	2	2	no	—
93	C₄×C₅₉	#482	1	2	yes	—
94	C₄×C₅₉	#484	1	2	yes	—
95	C₄×C₅₉	#485	1	2	yes	—
96	C₄×C₅₉	#488	1	1	yes	characteristic
97	C₄×C₅₉	#491	1	1	yes	characteristic
98	Dic₅₉	—	4	2	no	—
99	Dic₅₉	—	8	1	no	—
100	Dic₅₉	—	16	2	no	—
101	#1984	—	59	1	no	—
102	#1834	—	59	1	no	—
103	#1803	—	59	2	no	—
104	#1296	—	59	2	no	—
105	#1277	D₅₉	1	1	yes	characteristic
106	C₂×C₅₉	#1684	1	2	yes	—
107	C₂×C₅₉	#1702	1	2	yes	—
108	C₂×C₅₉	#2074	1	1	yes	characteristic
109	C₂×C₅₉	#2076	1	1	yes	characteristic
110	C₂×C₅₉	#2079	1	1	yes	characteristic
111	C₂×C₄×Q₈	—	59	1	no	—
112	C₂×C₄⋊C₈	—	59	1	no	—
113	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
114	C₂×#103	—	118	2	no	—
115	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
116	#384	—	59	2	no	—
117	#340	—	118	2	no	—
118	#340	—	118	2	no	—
119	#336	—	118	2	no	—
120	C₅₉	#10624	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	59	1	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	118	1	no	—
124	C₂²×C₈	D₂₃₆	1	2	yes	—
125	C₄×Q₈	—	118	2	no	—
126	C₄×Q₈	—	118	2	no	—
127	C₂×C₄⋊C₄	—	59	1	no	—
128	C₂×C₄⋊C₄	—	118	2	no	—
129	C₂×C₄⋊C₄	—	118	1	no	—
130	C₂×C₄⋊C₄	—	118	2	no	—
131	C₂×C₄⋊C₄	—	118	2	no	—
132	C₂×C₄²	—	118	1	no	—
133	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	118	2	no	—
137	#103	—	236	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	59	2	no	—
140	C₂×Q₈	—	118	1	no	—
141	C₂×Q₈	—	118	2	no	—
142	C₂×Q₈	—	118	2	no	—
143	C₂²×C₄	—	118	1	no	—
144	C₂²×C₄	—	118	1	no	—
145	C₂²×C₄	—	236	2	no	—
146	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
147	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
148	C₂²×C₄	#11745689	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	236	1	no	—
156	C₂×C₈	—	236	2	no	—
157	C₄⋊C₄	—	118	2	no	—
158	C₄⋊C₄	—	472	2	no	—
159	C₄⋊C₄	—	236	2	no	—
160	C₄⋊C₄	—	236	2	no	—
161	C₄⋊C₄	—	236	2	no	—
162	C₄⋊C₄	—	236	2	no	—
163	C₄⋊C₄	—	236	2	no	—
164	C₄²	—	236	1	no	—
165	C₄²	—	236	1	no	—
166	C₄²	#11745685	1	2	yes	—
167	C₄²	#11745687	1	2	yes	—
168	C₂³	#14273702	1	1	yes	characteristic
169	Q₈	—	118	4	no	—
170	Q₈	—	236	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	118	2	no	—
173	C₂×C₄	—	472	2	no	—
174	C₂×C₄	—	472	2	no	—
175	C₂×C₄	—	472	2	no	—
176	C₂×C₄	—	236	2	no	—
177	C₂×C₄	—	236	1	no	—
178	C₂×C₄	—	236	1	no	—
179	C₂×C₄	—	236	1	no	—
180	C₂×C₄	#14273709	1	2	yes	—
181	C₂×C₄	#14273711	1	1	yes	characteristic
182	C₂×C₄	#14273715	1	1	yes	characteristic
183	C₂×C₄	#14273722	1	2	yes	—
184	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
185	C₂×C₄	#14273746	1	1	yes	characteristic
186	C₂×C₄	#14273749	1	2	yes	—
187	C₂×C₄	#14273754	1	2	yes	—
188	C₂×C₄	#14273760	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	2	no	—
191	C₈	—	472	1	no	—
192	C₂²	#16837329	1	1	yes	characteristic
193	C₂²	#16837625	1	2	yes	—
194	C₂²	#16837644	1	1	yes	characteristic
195	C₂²	#16837656	1	1	yes	characteristic
196	C₂²	#16837666	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	944	2	no	—
199	C₄	—	472	1	no	—
200	C₄	—	236	2	no	—
201	C₄	#16837724	1	2	yes	—
202	C₄	#16837736	1	2	yes	—
203	C₄	#16837744	1	2	yes	—
204	C₄	#16837757	1	1	yes	characteristic
205	C₄	#16837771	1	1	yes	characteristic
206	C₂	#21423555	1	2	yes	—
207	C₂	#21423665	1	2	yes	—
208	C₂	#21425516	1	1	yes	characteristic
209	C₂	#21425526	1	1	yes	characteristic
210	C₂	#21425543	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425032

C₂

yes

maximal, characteristic

#21424224

C₂

yes

maximal, characteristic

#21424102

C₂

yes

maximal

#21421980

C₂

yes

maximal

C₅₉×#1277

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₁₈

C₂²

yes

characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

characteristic

C₂×#14273652

—

#16837272

C₂²

yes

—

#16837165

—

#16837165

—

#16837156

—

C₂×C₄⋊C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×Dic₁₁₈

D₄

yes

characteristic

C₂²×C₅₉⋊C₈

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

D₄

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

—

C₄⋊(C₅₉⋊C₈)

—

#14273652

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

Q₁₆

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

#41

yes

—

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

#482

yes

—

C₄×C₅₉

#484

yes

—

C₄×C₅₉

#485

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

100

Dic₅₉

—

101

#1984

—

102

#1834

—

103

#1803

—

104

#1296

—

105

#1277

D₅₉

yes

characteristic

106

C₂×C₅₉

#1684

yes

—

107

C₂×C₅₉

#1702

yes

—

108

C₂×C₅₉

#2074

yes

characteristic

109

C₂×C₅₉

#2076

yes

characteristic

110

C₂×C₅₉

#2079

yes

characteristic

111

C₂×C₄×Q₈

—

112

C₂×C₄⋊C₈

—

113

C₂×C₄⋊C₈

C₂×D₅₉

yes

—

114

C₂×#103

—

118

—

115

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

116

#384

—

117

#340

—

118

—

118

#340

—

118

—

119

#336

—

118

—

120

C₅₉

#10624

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂²×C₈

—

123

C₂²×C₈

—

118

—

124

C₂²×C₈

D₂₃₆

yes

—

125

C₄×Q₈

—

118

—

126

C₄×Q₈

—

118

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

118

—

129

C₂×C₄⋊C₄

—

118

—

130

C₂×C₄⋊C₄

—

118

—

131

C₂×C₄⋊C₄

—

118

—

132

C₂×C₄²

—

118

—

133

C₂×C₄²

C₂²×D₅₉

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

118

—

137

#103

—

236

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

118

—

141

C₂×Q₈

—

118

—

142

C₂×Q₈

—

118

—

143

C₂²×C₄

—

118

—

144

C₂²×C₄

—

118

—

145

C₂²×C₄

—

236

—

146

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

147

C₂²×C₄

D₄×D₅₉

yes

characteristic

148

C₂²×C₄

#11745689

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

236

—

156

C₂×C₈

—

236

—

157

C₄⋊C₄

—

118

—

158

C₄⋊C₄

—

472

—

159

C₄⋊C₄

—

236

—

160

C₄⋊C₄

—

236

—

161

C₄⋊C₄

—

236

—

162

C₄⋊C₄

—

236

—

163

C₄⋊C₄

—

236

—

164

C₄²

—

236

—

165

C₄²

—

236

—

166

C₄²

#11745685

yes

—

167

C₄²

#11745687

yes

—

168

C₂³

#14273702

yes

characteristic

169

Q₈

—

118

—

170

Q₈

—

236

—

171

C₂×C₄

—

172

C₂×C₄

—

118

—

173

C₂×C₄

—

472

—

174

C₂×C₄

—

472

—

175

C₂×C₄

—

472

—

176

C₂×C₄

—

236

—

177

C₂×C₄

—

236

—

178

C₂×C₄

—

236

—

179

C₂×C₄

—

236

—

180

C₂×C₄

#14273709

yes

—

181

C₂×C₄

#14273711

yes

characteristic

182

C₂×C₄

#14273715

yes

characteristic

183

C₂×C₄

#14273722

yes

—

184

C₂×C₄

Q₁₆×D₅₉

yes

characteristic

185

C₂×C₄

#14273746

yes

characteristic

186

C₂×C₄

#14273749

yes

—

187

C₂×C₄

#14273754

yes

—

188

C₂×C₄

#14273760

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

472

—

192

C₂²

#16837329

yes

characteristic

193

C₂²

#16837625

yes

—

194

C₂²

#16837644

yes

characteristic

195

C₂²

#16837656

yes

characteristic

196

C₂²

#16837666

yes

—

197

C₄

—

198

C₄

—

944

—

199

C₄

—

472

—

200

C₄

—

236

—

201

C₄

#16837724

yes

—

202

C₄

#16837736

yes

—

203

C₄

#16837744

yes

—

204

C₄

#16837757

yes

characteristic

205

C₄

#16837771

yes

characteristic

206

C₂

#21423555

yes

—

207

C₂

#21423665

yes

—

208

C₂

#21425516

yes

characteristic

209

C₂

#21425526

yes

characteristic

210

C₂

#21425543

yes

characteristic

	subgroup	G/H
center	C₂³	#14273702
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1277	C₂
socle, prime radical	C₂³×C₅₉	#121
hypercenter	#1277	D₅₉
hypocenter	C₅₉	#10624
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂³

#14273702

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1277

C₂

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
120. C₅₉	#10624	semidirect

H ≅ G/N

type

120. C₅₉

#10624

semidirect

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

subgroup	index
#21425032	2
#21424224	2
#21424102	2
#21421980	2
C₅₉×#1277	2
#10624	59

subgroup

index

C₅₉×#1277

Subgroups — #28493440

Conjugacy classes of subgroups

Complements

Maximal subgroups