Group #28482358 — subgroups

Subgroups — #28482358

Order 15104.

362 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 147 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	#21424247	C₂	1	2	yes	maximal
2	#21423841	C₂	1	1	yes	maximal, characteristic
3	#21421931	C₂	1	1	yes	maximal, characteristic
4	#21421930	C₂	1	2	yes	maximal
5	C₅₉×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×Dic₅₉⋊C₈	—	2	2	no	—
8	C₂×Dic₅₉⋊C₈	—	2	2	no	—
9	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
11	#16837156	C₄	1	2	yes	—
12	#16837156	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×C₅₉⋊C₈	—	2	2	no	—
16	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₉	Q₈	1	2	yes	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	Dic₅₉⋊C₈	—	4	4	no	—
25	Dic₅₉⋊C₈	—	4	4	no	—
26	C₄×C₅₉⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₅₉	—	2	2	no	—
28	C₂²×C₈×C₅₉	C₂×C₄	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	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
36	C₂²×Dic₅₉	C₄⋊C₄	1	2	yes	—
37	C₂²×Dic₅₉	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₅₉	—	2	2	no	—
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	—	2	2	no	—
41	C₂×C₈×C₅₉	—	4	2	no	—
42	C₂×C₈×C₅₉	—	4	2	no	—
43	C₂×C₈×C₅₉	—	4	2	no	—
44	C₄²×C₅₉	#41	1	2	yes	—
45	C₄²×C₅₉	#41	1	2	yes	—
46	C₄×Dic₅₉	—	2	2	no	—
47	C₄×Dic₅₉	—	2	2	no	—
48	C₄×Dic₅₉	—	4	2	no	—
49	C₄×Dic₅₉	D₈	1	2	yes	—
50	C₄×Dic₅₉	SD₁₆	1	4	yes	—
51	C₄×Dic₅₉	Q₁₆	1	2	yes	—
52	C₂×C₅₉⋊C₈	—	2	1	no	—
53	C₂×C₅₉⋊C₈	—	2	1	no	—
54	C₂×C₅₉⋊C₈	—	2	2	no	—
55	C₂×C₅₉⋊C₈	—	2	2	no	—
56	C₂×C₅₉⋊C₈	—	4	2	no	—
57	C₂×C₅₉⋊C₈	—	4	2	no	—
58	C₂×C₅₉⋊C₈	—	4	2	no	—
59	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#124	1	2	yes	—
64	C₂×C₄×C₅₉	#125	1	2	yes	—
65	C₂×C₄×C₅₉	#131	1	2	yes	—
66	C₂×C₄×C₅₉	#131	1	2	yes	—
67	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
69	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₅₉	—	2	2	no	—
71	C₂×Dic₅₉	—	2	2	no	—
72	C₂×Dic₅₉	—	2	2	no	—
73	C₂×Dic₅₉	—	2	2	no	—
74	C₂×Dic₅₉	#106	1	4	yes	—
75	C₂×Dic₅₉	#107	1	4	yes	—
76	C₈×C₅₉	—	4	2	no	—
77	C₈×C₅₉	—	8	2	no	—
78	C₅₉⋊C₈	—	4	1	no	—
79	C₅₉⋊C₈	—	4	1	no	—
80	C₅₉⋊C₈	—	8	2	no	—
81	#8787	—	59	1	no	maximal, Sylow
82	C₂²×C₅₉	#382	1	1	yes	characteristic
83	C₂²×C₅₉	#424	1	2	yes	—
84	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
85	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
86	C₂²×C₅₉	#430	1	2	yes	—
87	C₄×C₅₉	—	2	2	no	—
88	C₄×C₅₉	#433	1	2	yes	—
89	C₄×C₅₉	#433	1	1	yes	characteristic
90	C₄×C₅₉	#433	1	1	yes	characteristic
91	C₄×C₅₉	#486	1	2	yes	—
92	C₄×C₅₉	#487	1	2	yes	—
93	Dic₅₉	—	2	4	no	—
94	Dic₅₉	—	2	4	no	—
95	#1842	—	59	2	no	—
96	#1744	—	59	1	no	—
97	#1277	—	59	2	no	—
98	#1277	—	59	1	no	—
99	#1277	D₅₉	1	1	yes	characteristic
100	C₂×C₅₉	#1570	1	2	yes	—
101	C₂×C₅₉	#1571	1	2	yes	—
102	C₂×C₅₉	#1832	1	1	yes	characteristic
103	C₂×C₅₉	#1834	1	1	yes	characteristic
104	C₂×C₅₉	#1838	1	1	yes	characteristic
105	C₂×C₄⋊C₈	—	59	2	no	—
106	C₂×C₄⋊C₈	—	118	2	no	—
107	C₂×C₄⋊C₈	—	118	2	no	—
108	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
109	C₂×C₄×C₈	—	59	1	no	—
110	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
111	C₄³	—	59	1	no	—
112	#336	—	59	2	no	—
113	#336	—	59	2	no	—
114	C₅₉	#8787	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	—	59	1	no	—
117	C₂²×C₈	—	59	1	no	—
118	C₂²×C₈	—	118	2	no	—
119	C₂²×C₈	C₄×D₅₉	1	2	yes	—
120	C₂×C₄²	—	59	1	no	—
121	C₂×C₄²	—	59	1	no	—
122	C₂×C₄²	—	59	2	no	—
123	C₂×C₄²	—	59	2	no	—
124	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	118	2	no	—
128	C₄⋊C₈	—	118	2	no	—
129	C₄⋊C₈	—	236	4	no	—
130	C₄⋊C₈	—	236	4	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	118	2	no	—
133	C₂²×C₄	—	59	2	no	—
134	C₂²×C₄	—	59	2	no	—
135	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
136	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
137	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	118	1	no	—
145	C₂×C₈	—	118	1	no	—
146	C₂×C₈	—	118	2	no	—
147	C₂×C₈	—	118	2	no	—
148	C₂×C₈	—	236	2	no	—
149	C₂×C₈	—	236	2	no	—
150	C₂×C₈	—	236	2	no	—
151	C₄²	—	59	2	no	—
152	C₄²	—	59	4	no	—
153	C₄²	—	59	2	no	—
154	C₄²	—	118	2	no	—
155	C₄²	—	118	2	no	—
156	C₄²	—	236	2	no	—
157	C₄²	#11745685	1	2	yes	—
158	C₄²	#11745687	1	2	yes	—
159	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	59	4	no	—
162	C₂×C₄	—	59	4	no	—
163	C₂×C₄	—	118	2	no	—
164	C₂×C₄	—	118	2	no	—
165	C₂×C₄	—	118	2	no	—
166	C₂×C₄	—	118	2	no	—
167	C₂×C₄	#14273683	1	1	yes	characteristic
168	C₂×C₄	#14273684	1	1	yes	characteristic
169	C₂×C₄	#14273713	1	2	yes	—
170	C₂×C₄	#14273717	1	2	yes	—
171	C₂×C₄	#14273733	1	2	yes	—
172	C₂×C₄	#14273735	1	2	yes	—
173	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
174	C₂×C₄	SD₁₆×D₅₉	1	2	yes	—
175	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
176	C₈	—	4	2	no	—
177	C₈	—	8	2	no	—
178	C₈	—	472	2	no	—
179	C₈	—	236	1	no	—
180	C₈	—	236	1	no	—
181	C₂²	#16837266	1	1	yes	characteristic
182	C₂²	#16837427	1	2	yes	—
183	C₂²	#106×D₅₉	1	1	yes	characteristic
184	C₂²	#107×D₅₉	1	1	yes	characteristic
185	C₂²	#16837449	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	118	4	no	—
188	C₄	—	118	4	no	—
189	C₄	#16837460	1	2	yes	—
190	C₄	#16837461	1	1	yes	characteristic
191	C₄	#16837464	1	1	yes	characteristic
192	C₄	#16837747	1	2	yes	—
193	C₄	#16837751	1	2	yes	—
194	C₂	#21422865	1	2	yes	—
195	C₂	#21422869	1	2	yes	—
196	C₂	#21424214	1	1	yes	characteristic
197	C₂	#21424222	1	1	yes	characteristic
198	C₂	#21424235	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424247

C₂

yes

maximal

#21423841

C₂

yes

maximal, characteristic

#21421931

C₂

yes

maximal, characteristic

#21421930

C₂

yes

maximal

C₅₉×#1277

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂×Dic₅₉⋊C₈

—

C₂×Dic₅₉⋊C₈

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

characteristic

C₄²×Dic₅₉

C₂²

yes

characteristic

#16837156

C₄

yes

—

#16837156

C₄

yes

—

C₂×C₄⋊C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₉

C₂×C₄

yes

—

C₂×C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₄×Dic₅₉

Q₈

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

Dic₅₉⋊C₈

—

Dic₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂²×Dic₅₉

C₄⋊C₄

yes

—

C₂²×Dic₅₉

C₄⋊C₄

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

D₈

yes

—

C₄×Dic₅₉

SD₁₆

yes

—

C₄×Dic₅₉

Q₁₆

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

#125

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

#131

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

#106

yes

—

C₂×Dic₅₉

#107

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

#433

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₄×C₅₉

#487

yes

—

Dic₅₉

—

Dic₅₉

—

—

—

—

—

D₅₉

yes

characteristic

100

C₂×C₅₉

#1570

yes

—

101

C₂×C₅₉

#1571

yes

—

102

C₂×C₅₉

#1832

yes

characteristic

103

C₂×C₅₉

#1834

yes

characteristic

104

C₂×C₅₉

#1838

yes

characteristic

105

C₂×C₄⋊C₈

—

106

C₂×C₄⋊C₈

—

118

—

107

C₂×C₄⋊C₈

—

118

—

108

C₂×C₄⋊C₈

C₂×D₅₉

yes

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

111

C₄³

—

112

#336

—

113

#336

—

114

C₅₉

#8787

yes

characteristic, complemented, Sylow

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

—

118

—

119

C₂²×C₈

C₄×D₅₉

yes

—

120

C₂×C₄²

—

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

—

124

C₂×C₄²

C₂²×D₅₉

yes

characteristic

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

118

—

128

C₄⋊C₈

—

118

—

129

C₄⋊C₈

—

236

—

130

C₄⋊C₈

—

236

—

131

C₄×C₈

—

132

C₄×C₈

—

118

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

136

C₂²×C₄

D₄×D₅₉

yes

characteristic

137

C₂²×C₄

Q₈×D₅₉

yes

characteristic

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

118

—

145

C₂×C₈

—

118

—

146

C₂×C₈

—

118

—

147

C₂×C₈

—

118

—

148

C₂×C₈

—

236

—

149

C₂×C₈

—

236

—

150

C₂×C₈

—

236

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

118

—

155

C₄²

—

118

—

156

C₄²

—

236

—

157

C₄²

#11745685

yes

—

158

C₄²

#11745687

yes

—

159

C₂³

C₄⋊C₄×D₅₉

yes

characteristic

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

118

—

164

C₂×C₄

—

118

—

165

C₂×C₄

—

118

—

166

C₂×C₄

—

118

—

167

C₂×C₄

#14273683

yes

characteristic

168

C₂×C₄

#14273684

yes

characteristic

169

C₂×C₄

#14273713

yes

—

170

C₂×C₄

#14273717

yes

—

171

C₂×C₄

#14273733

yes

—

172

C₂×C₄

#14273735

yes

—

173

C₂×C₄

D₈×D₅₉

yes

characteristic

174

C₂×C₄

SD₁₆×D₅₉

yes

—

175

C₂×C₄

Q₁₆×D₅₉

yes

characteristic

176

C₈

—

177

C₈

—

178

C₈

—

472

—

179

C₈

—

236

—

180

C₈

—

236

—

181

C₂²

#16837266

yes

characteristic

182

C₂²

#16837427

yes

—

183

C₂²

#106×D₅₉

yes

characteristic

184

C₂²

#107×D₅₉

yes

characteristic

185

C₂²

#16837449

yes

—

186

C₄

—

187

C₄

—

118

—

188

C₄

—

118

—

189

C₄

#16837460

yes

—

190

C₄

#16837461

yes

characteristic

191

C₄

#16837464

yes

characteristic

192

C₄

#16837747

yes

—

193

C₄

#16837751

yes

—

194

C₂

#21422865

yes

—

195

C₂

#21422869

yes

—

196

C₂

#21424214

yes

characteristic

197

C₂

#21424222

yes

characteristic

198

C₂

#21424235

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1277	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1277	D₅₉
hypocenter	C₅₉	#8787
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

subgroup

G/H

center

C₂²×C₄

D₄×D₅₉

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1277

C₂

socle, prime radical

C₂³×C₅₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

N	H ≅ G/N	type
114. C₅₉	#8787	semidirect

H ≅ G/N

type

114. C₅₉

#8787

semidirect

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

subgroup	index
#21424247	2
#21423841	2
#21421931	2
#21421930	2
C₅₉×#1277	2
#8787	59

subgroup

index

C₅₉×#1277

Subgroups — #28482358

Conjugacy classes of subgroups

Complements

Maximal subgroups