Group #28507233 — subgroups

Subgroups — #28507233

Order 15104.

308 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 115 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	#21433034	C₂	1	1	yes	maximal, characteristic
2	#21425862	C₂	1	1	yes	maximal, characteristic
3	#21422181	C₂	1	2	yes	maximal
4	#21422175	C₂	1	2	yes	maximal
5	C₅₉×#2110	C₂	1	1	yes	maximal, characteristic
6	#16837955	C₄	1	1	yes	characteristic
7	C₂×#14273667	C₂²	1	2	yes	—
8	#16837532	—	2	2	no	—
9	#16837530	—	2	2	no	—
10	#16837430	C₄	1	1	yes	characteristic
11	#16837429	C₄	1	1	yes	characteristic
12	#16837425	C₄	1	1	yes	characteristic
13	#16837357	C₂²	1	1	yes	characteristic
14	C₂×#14273637	C₂²	1	1	yes	characteristic
15	#16837223	—	2	2	no	—
16	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
17	C₅₉×#405	C₂²	1	1	yes	characteristic
18	C₅₉×#347	—	2	2	no	—
19	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	#14273726	—	2	1	no	—
22	#14273718	—	2	1	no	—
23	#14273704	—	2	2	no	—
24	C₄×Dic₁₁₈	—	2	2	no	—
25	#14273686	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
27	#14273667	—	2	2	no	—
28	#14273667	—	2	2	no	—
29	C₂×C₅₉⋊C₁₆	—	2	2	no	—
30	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
31	Dic₅₉⋊C₈	—	2	2	no	—
32	Dic₅₉⋊C₈	—	2	2	no	—
33	#14273637	—	2	1	no	—
34	#14273637	—	2	1	no	—
35	#14273637	—	2	1	no	—
36	C₈×Dic₅₉	—	2	1	no	—
37	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₅₉	—	2	2	no	—
41	M₃₂×C₅₉	—	2	2	no	—
42	C₂×C₁₆×C₅₉	Q₈	1	4	yes	complemented
43	#97×C₅₉	D₄	1	2	yes	—
44	C₄×C₈×C₅₉	D₄	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₉	—	2	1	no	—
47	C₂×Dic₁₁₈	—	2	2	no	—
48	M₁₆×C₅₉	—	2	2	no	—
49	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₅₉	#41	1	2	yes	—
54	C₄⋊C₄×C₅₉	—	2	1	no	—
55	C₂²⋊C₄×C₅₉	—	2	1	no	—
56	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
57	C₂²⋊Dic₅₉	M₁₆	1	2	yes	—
58	C₄⋊Dic₅₉	—	2	2	no	—
59	C₄⋊Dic₅₉	—	2	1	no	—
60	Dic₅₉⋊C₄	—	2	2	no	—
61	Dic₅₉⋊C₄	—	2	2	no	—
62	C₄×Dic₅₉	—	2	1	no	—
63	C₄×Dic₅₉	—	2	1	no	—
64	C₄×Dic₅₉	M₁₆	1	1	yes	characteristic
65	C₄×Dic₅₉	M₁₆	1	1	yes	characteristic
66	#11745664	—	2	2	no	—
67	C₂×C₅₉⋊C₈	—	2	1	no	—
68	C₂×C₅₉⋊C₈	—	2	1	no	—
69	C₂×C₅₉⋊C₈	—	2	2	no	—
70	C₁₆×C₅₉	—	2	4	no	—
71	C₅₉⋊C₁₆	—	4	2	no	—
72	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₉	—	2	2	no	—
74	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#131	1	2	yes	—
78	C₂×Dic₅₉	—	2	1	no	—
79	C₂×Dic₅₉	—	2	1	no	—
80	C₂×Dic₅₉	—	2	2	no	—
81	C₂×Dic₅₉	—	8	1	no	—
82	Dic₁₁₈	—	8	2	no	—
83	C₈×C₅₉	—	2	2	no	—
84	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
85	C₈×C₅₉	#124	1	1	yes	characteristic
86	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
87	C₅₉⋊C₈	—	4	1	no	—
88	C₅₉⋊C₈	—	4	1	no	—
89	#13269	—	59	1	no	maximal, Sylow
90	C₂²×C₅₉	—	4	1	no	—
91	C₂²×C₅₉	#407	1	1	yes	characteristic
92	C₂²×C₅₉	#446	1	2	yes	—
93	C₄×C₅₉	—	4	2	no	—
94	C₄×C₅₉	#391	1	1	yes	characteristic
95	C₄×C₅₉	#408	1	1	yes	characteristic
96	C₄×C₅₉	#446	1	2	yes	—
97	Dic₅₉	—	4	1	no	—
98	Dic₅₉	—	4	2	no	—
99	Dic₅₉	—	4	1	no	—
100	#2961	—	59	1	no	—
101	#2153	—	59	1	no	—
102	#2110	D₅₉	1	1	yes	characteristic
103	#1378	—	59	2	no	—
104	#1375	—	59	2	no	—
105	C₂×C₅₉	—	2	1	no	—
106	C₂×C₅₉	—	2	2	no	—
107	C₂×C₅₉	#1985	1	1	yes	characteristic
108	#533	—	59	1	no	—
109	C₂×M₃₂	—	59	2	no	—
110	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
111	#446	—	118	2	no	—
112	#446	—	118	2	no	—
113	#424	—	59	1	no	—
114	#424	—	59	1	no	—
115	#423	—	59	1	no	—
116	#405	—	59	1	no	—
117	#405	C₂×D₅₉	1	1	yes	characteristic
118	C₂×#97	—	59	1	no	—
119	#364	—	118	2	no	—
120	#347	—	2	2	no	—
121	C₅₉	#13269	1	1	yes	characteristic, complemented, Sylow
122	C₂×M₁₆	—	59	1	no	—
123	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
124	C₂²×C₈	—	59	1	no	—
125	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
126	C₄⋊Q₈	—	118	1	no	—
127	#125	—	118	1	no	—
128	C₂²⋊Q₈	—	118	2	no	—
129	C₄×Q₈	—	118	2	no	—
130	#117	—	59	1	no	—
131	#117	C₄×D₅₉	1	1	yes	characteristic
132	C₂×C₄²	—	59	1	no	—
133	M₃₂	—	2	2	no	—
134	M₃₂	—	2	2	no	—
135	M₃₂	—	118	2	no	—
136	M₃₂	—	118	2	no	—
137	C₂×C₁₆	—	118	2	no	—
138	C₂×C₁₆	Dic₁₁₈	1	4	yes	complemented
139	C₄⋊C₈	—	118	2	no	—
140	C₄⋊C₈	—	118	2	no	—
141	C₄⋊C₈	—	118	2	no	—
142	#97	—	118	1	no	—
143	#97	—	118	1	no	—
144	#97	—	118	1	no	—
145	#97	C₂²⋊D₅₉	1	2	yes	—
146	C₄×C₈	—	118	1	no	—
147	C₄×C₈	D₂₃₆	1	2	yes	—
148	C₂×Q₈	—	118	2	no	—
149	C₂²×C₄	—	118	1	no	—
150	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
151	M₁₆	—	2	2	no	—
152	M₁₆	—	118	2	no	—
153	C₂×C₈	—	118	1	no	—
154	C₂×C₈	—	118	1	no	—
155	C₂×C₈	—	118	2	no	—
156	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
157	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
158	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
159	C₂×C₈	#11745685	1	2	yes	—
160	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	118	2	no	—
163	C₄⋊C₄	—	118	2	no	—
164	C₄⋊C₄	—	118	2	no	—
165	C₄⋊C₄	—	118	1	no	—
166	C₂²⋊C₄	—	2	1	no	—
167	C₂²⋊C₄	—	59	2	no	—
168	C₄²	—	59	1	no	—
169	C₄²	—	59	1	no	—
170	C₄²	—	118	1	no	—
171	C₄²	—	118	1	no	—
172	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	236	2	no	—
175	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
176	Q₈	—	472	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	118	1	no	—
179	C₂×C₄	—	118	1	no	—
180	C₂×C₄	—	118	2	no	—
181	C₂×C₄	—	472	1	no	—
182	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
183	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
184	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
185	C₂×C₄	#14273735	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	236	1	no	—
188	C₈	—	236	1	no	—
189	C₈	#14273706	1	2	yes	—
190	C₈	#14273715	1	1	yes	characteristic
191	C₈	#14273727	1	1	yes	characteristic
192	C₂²	—	4	1	no	—
193	C₂²	#16837365	1	1	yes	characteristic
194	C₂²	#16837530	1	2	yes	—
195	C₄	—	4	2	no	—
196	C₄	—	236	1	no	—
197	C₄	—	236	1	no	—
198	C₄	—	236	2	no	—
199	C₄	#16837308	1	1	yes	characteristic
200	C₄	#16837370	1	1	yes	characteristic
201	C₄	#16837530	1	2	yes	—
202	C₂	—	2	1	no	—
203	C₂	—	2	2	no	—
204	C₂	#21425036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21433034

C₂

yes

maximal, characteristic

#21425862

C₂

yes

maximal, characteristic

#21422181

C₂

yes

maximal

#21422175

C₂

yes

maximal

C₅₉×#2110

C₂

yes

maximal, characteristic

#16837955

C₄

yes

characteristic

C₂×#14273667

C₂²

yes

—

#16837532

—

#16837530

—

#16837430

C₄

yes

characteristic

#16837429

C₄

yes

characteristic

#16837425

C₄

yes

characteristic

#16837357

C₂²

yes

characteristic

C₂×#14273637

C₂²

yes

characteristic

#16837223

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₅₉×#405

C₂²

yes

characteristic

C₅₉×#347

—

C₂×#11745664

C₂×C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

—

—

C₄×Dic₁₁₈

—

#14273686

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₉

C₂×C₄

yes

characteristic

#14273667

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₄⋊(C₅₉⋊C₈)

—

Dic₅₉⋊C₈

—

Dic₅₉⋊C₈

—

#14273637

—

#14273637

—

#14273637

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

C₂×C₄

yes

characteristic

M₃₂×C₅₉

—

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

Q₈

yes

complemented

#97×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₂²⋊Dic₅₉

M₁₆

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

M₁₆

yes

characteristic

C₄×Dic₅₉

M₁₆

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×Q₈

yes

—

C₂×C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₉

#131

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊Q₈

yes

—

C₈×C₅₉

#124

yes

characteristic

C₈×C₅₉

C₄⋊Q₈

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#13269

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#407

yes

characteristic

C₂²×C₅₉

#446

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

100

#2961

—

101

#2153

—

102

#2110

D₅₉

yes

characteristic

103

#1378

—

104

#1375

—

105

C₂×C₅₉

—

106

C₂×C₅₉

—

107

C₂×C₅₉

#1985

yes

characteristic

108

#533

—

109

C₂×M₃₂

—

110

C₂×M₃₂

C₂×D₅₉

yes

—

111

#446

—

118

—

112

#446

—

118

—

113

#424

—

114

#424

—

115

#423

—

116

#405

—

117

#405

C₂×D₅₉

yes

characteristic

118

C₂×#97

—

119

#364

—

118

—

120

#347

—

121

C₅₉

#13269

yes

characteristic, complemented, Sylow

122

C₂×M₁₆

—

123

C₂×M₁₆

C₄×D₅₉

yes

characteristic

124

C₂²×C₈

—

125

C₂²×C₈

C₂²×D₅₉

yes

characteristic

126

C₄⋊Q₈

—

118

—

127

#125

—

118

—

128

C₂²⋊Q₈

—

118

—

129

C₄×Q₈

—

118

—

130

#117

—

131

#117

C₄×D₅₉

yes

characteristic

132

C₂×C₄²

—

133

M₃₂

—

134

M₃₂

—

135

M₃₂

—

118

—

136

M₃₂

—

118

—

137

C₂×C₁₆

—

118

—

138

C₂×C₁₆

Dic₁₁₈

yes

complemented

139

C₄⋊C₈

—

118

—

140

C₄⋊C₈

—

118

—

141

C₄⋊C₈

—

118

—

142

#97

—

118

—

143

#97

—

118

—

144

#97

—

118

—

145

#97

C₂²⋊D₅₉

yes

—

146

C₄×C₈

—

118

—

147

C₄×C₈

D₂₃₆

yes

—

148

C₂×Q₈

—

118

—

149

C₂²×C₄

—

118

—

150

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

151

M₁₆

—

152

M₁₆

—

118

—

153

C₂×C₈

—

118

—

154

C₂×C₈

—

118

—

155

C₂×C₈

—

118

—

156

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

157

C₂×C₈

C₂×Dic₁₁₈

yes

—

158

C₂×C₈

C₂×D₂₃₆

yes

characteristic

159

C₂×C₈

#11745685

yes

—

160

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

118

—

163

C₄⋊C₄

—

118

—

164

C₄⋊C₄

—

118

—

165

C₄⋊C₄

—

118

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

118

—

171

C₄²

—

118

—

172

C₄²

(C₂×D₅₉)⋊C₄

yes

—

173

C₁₆

—

174

C₁₆

—

236

—

175

C₂³

M₁₆×D₅₉

yes

characteristic

176

Q₈

—

472

—

177

C₂×C₄

—

178

C₂×C₄

—

118

—

179

C₂×C₄

—

118

—

180

C₂×C₄

—

118

—

181

C₂×C₄

—

472

—

182

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

183

C₂×C₄

C₄×Dic₁₁₈

yes

—

184

C₂×C₄

M₁₆×D₅₉

yes

characteristic

185

C₂×C₄

#14273735

yes

—

186

C₈

—

187

C₈

—

236

—

188

C₈

—

236

—

189

C₈

#14273706

yes

—

190

C₈

#14273715

yes

characteristic

191

C₈

#14273727

yes

characteristic

192

C₂²

—

193

C₂²

#16837365

yes

characteristic

194

C₂²

#16837530

yes

—

195

C₄

—

196

C₄

—

236

—

197

C₄

—

236

—

198

C₄

—

236

—

199

C₄

#16837308

yes

characteristic

200

C₄

#16837370

yes

characteristic

201

C₄

#16837530

yes

—

202

C₂

—

203

C₂

—

204

C₂

#21425036

yes

characteristic

	subgroup	G/H
center	C₄	#16837308
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2110	C₂
socle	C₂×C₅₉	#1985
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2110	D₅₉
hypocenter	C₅₉	#13269
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #2110

subgroup

G/H

center

C₄

#16837308

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#2110

C₂

socle

C₂×C₅₉

#1985

prime radical

C₂³×C₅₉

C₂×M₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #2110

N	H ≅ G/N	type
42. C₂×C₁₆×C₅₉	Q₈	semidirect
121. C₅₉	#13269	semidirect
138. C₂×C₁₆	Dic₁₁₈	semidirect

H ≅ G/N

type

42. C₂×C₁₆×C₅₉

Q₈

semidirect

121. C₅₉

#13269

semidirect

138. C₂×C₁₆

Dic₁₁₈

semidirect

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

subgroup	index
#21433034	2
#21425862	2
#21422181	2
#21422175	2
C₅₉×#2110	2
#13269	59

subgroup

index

C₅₉×#2110

Subgroups — #28507233

Conjugacy classes of subgroups

Complements

Maximal subgroups