Group #28497897 — subgroups

Subgroups — #28497897

Order 15104.

314 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 89 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	#21432736	C₂	1	1	yes	maximal, characteristic
2	#21425892	C₂	1	2	yes	maximal, complemented
3	#21422037	C₂	1	2	yes	maximal, complemented
4	#21422022	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1315	C₂	1	1	yes	maximal, characteristic
6	#41×Dic₅₉	C₄	1	1	yes	characteristic
7	#16837460	—	2	2	no	—
8	#16837452	—	2	2	no	—
9	#16837430	C₄	1	1	yes	characteristic
10	#16837428	C₂²	1	1	yes	characteristic
11	#16837369	C₄	1	1	yes	characteristic
12	M₁₆×Dic₅₉	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
14	Dic₅₉⋊C₁₆	—	2	2	no	—
15	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
16	C₁₆×Dic₅₉	—	2	2	no	—
17	C₅₉×#408	C₂²	1	1	yes	characteristic
18	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
19	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	Q₈×Dic₅₉	C₈	1	2	yes	—
22	D₄×Dic₅₉	—	2	2	no	—
23	D₄×Dic₅₉	C₈	1	2	yes	—
24	#14273686	—	2	1	no	—
25	#14273686	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅₉	—	2	1	no	—
27	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅₉⋊C₁₆	—	2	2	no	—
29	Dic₅₉⋊C₈	—	2	2	no	—
30	Dic₅₉⋊C₈	—	2	2	no	—
31	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
32	#14273637	—	2	1	no	—
33	C₈×Dic₅₉	—	2	1	no	—
34	C₈×Dic₅₉	D₄	1	2	yes	—
35	C₈×Dic₅₉	D₄	1	2	yes	—
36	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	C₂×C₁₆×C₅₉	—	2	2	no	—
40	C₂²⋊C₈×C₅₉	—	2	2	no	—
41	#41×C₅₉	—	4	1	no	—
42	#41×C₅₉	—	4	1	no	—
43	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
44	C₂×D₄×C₅₉	—	2	1	no	—
45	C₂×D₄×C₅₉	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	—	2	1	no	—
47	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₅₉	—	2	2	no	—
49	C₂²×Dic₅₉	M₁₆	1	2	yes	—
50	M₁₆×C₅₉	—	4	1	no	—
51	C₂×C₈×C₅₉	—	2	1	no	—
52	C₂×C₈×C₅₉	—	4	1	no	—
53	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
54	C₂²⋊Dic₅₉	—	2	1	no	—
55	C₂²⋊Dic₅₉	—	2	2	no	—
56	C₄⋊Dic₅₉	—	2	2	no	—
57	C₄⋊Dic₅₉	C₂×C₈	1	2	yes	—
58	C₄×Dic₅₉	—	2	1	no	—
59	C₄×Dic₅₉	—	2	2	no	—
60	C₄×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₄×Dic₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
63	#11745664	—	4	1	no	—
64	C₂×C₅₉⋊C₈	—	2	1	no	—
65	C₂×C₅₉⋊C₈	—	4	1	no	—
66	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
67	C₁₆×C₅₉	—	4	2	no	—
68	C₅₉⋊C₁₆	—	4	2	no	—
69	C₂³×C₅₉	—	2	1	no	—
70	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₅₉	—	4	1	no	—
72	D₄×C₅₉	—	4	1	no	—
73	D₄×C₅₉	—	4	2	no	—
74	C₂×C₄×C₅₉	—	2	1	no	—
75	C₂×C₄×C₅₉	—	2	1	no	—
76	C₂×C₄×C₅₉	—	2	1	no	—
77	C₂×C₄×C₅₉	—	4	1	no	—
78	C₂×C₄×C₅₉	—	4	1	no	—
79	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
81	C₂×Dic₅₉	—	2	4	no	—
82	C₂×Dic₅₉	—	2	2	no	—
83	C₂×Dic₅₉	—	4	4	no	—
84	C₂×Dic₅₉	C₂²⋊C₈	1	2	yes	—
85	C₂×Dic₅₉	C₂²⋊C₈	1	2	yes	—
86	C₈×C₅₉	—	2	2	no	—
87	C₈×C₅₉	—	4	1	no	—
88	C₅₉⋊C₈	—	2	2	no	—
89	C₅₉⋊C₈	—	4	1	no	—
90	#11462	—	59	1	no	maximal, Sylow
91	C₂²×C₅₉	—	2	2	no	—
92	C₂²×C₅₉	—	2	1	no	—
93	C₂²×C₅₉	—	4	2	no	—
94	C₂²×C₅₉	—	4	1	no	—
95	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
96	C₄×C₅₉	—	2	1	no	—
97	C₄×C₅₉	—	4	1	no	—
98	C₄×C₅₉	#410	1	1	yes	characteristic
99	C₄×C₅₉	#413	1	1	yes	characteristic
100	Dic₅₉	—	2	2	no	—
101	Dic₅₉	—	4	2	no	—
102	Dic₅₉	#350	1	4	yes	—
103	#2922	—	59	1	no	—
104	#2162	—	59	2	no	—
105	#1323	—	59	2	no	—
106	#1315	—	59	1	no	—
107	#1315	D₅₉	1	1	yes	characteristic
108	C₂×C₅₉	—	4	1	no	—
109	C₂×C₅₉	—	8	1	no	—
110	C₂×C₅₉	#1458	1	1	yes	characteristic
111	C₂×C₅₉	C₂×#350	1	2	yes	—
112	C₄×#41	—	59	1	no	—
113	#433	—	118	2	no	—
114	#431	—	118	2	no	—
115	#424	—	59	1	no	—
116	#424	—	59	1	no	—
117	#408	—	59	1	no	—
118	#408	C₂×D₅₉	1	1	yes	characteristic
119	C₄×M₁₆	—	59	1	no	—
120	C₂×C₄×C₈	—	59	1	no	—
121	C₄⋊C₁₆	—	118	2	no	—
122	C₂²⋊C₁₆	—	59	2	no	—
123	C₂²⋊C₁₆	C₂×D₅₉	1	2	yes	—
124	C₄×C₁₆	—	118	2	no	—
125	C₅₉	#11462	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	C₄×D₅₉	1	1	yes	characteristic
127	C₂×M₁₆	—	59	1	no	—
128	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
129	C₂²×C₈	—	59	1	no	—
130	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
131	C₄×Q₈	—	59	2	no	—
132	C₄×D₄	—	59	2	no	—
133	C₄×D₄	—	118	2	no	—
134	#117	—	59	1	no	—
135	#117	—	118	1	no	—
136	C₂×C₄²	—	59	1	no	—
137	C₂×C₄²	—	118	1	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	118	2	no	—
140	C₄⋊C₈	—	118	2	no	—
141	C₄⋊C₈	—	118	2	no	—
142	C₂²⋊C₈	—	2	2	no	—
143	C₂²⋊C₈	—	118	2	no	—
144	#97	—	118	1	no	—
145	C₄×C₈	—	59	2	no	—
146	C₄×C₈	—	59	2	no	—
147	C₄×C₈	—	118	1	no	—
148	#41	—	4	1	no	—
149	#41	—	4	1	no	—
150	C₂×Q₈	C₈×D₅₉	1	1	yes	characteristic
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	C₈×D₅₉	1	1	yes	characteristic
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	59	2	no	—
155	C₂²×C₄	—	118	2	no	—
156	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
157	M₁₆	—	4	1	no	—
158	M₁₆	—	236	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	59	2	no	—
162	C₂×C₈	—	118	1	no	—
163	C₂×C₈	—	236	1	no	—
164	C₂×C₈	D₄×D₅₉	1	2	yes	—
165	C₄⋊C₄	—	59	2	no	—
166	C₄⋊C₄	—	118	2	no	—
167	C₂²⋊C₄	—	118	2	no	—
168	C₂²⋊C₄	—	118	1	no	—
169	C₄²	—	59	1	no	—
170	C₄²	—	59	1	no	—
171	C₄²	—	59	2	no	—
172	C₄²	—	118	2	no	—
173	C₄²	—	118	1	no	—
174	C₁₆	—	4	2	no	—
175	C₁₆	—	236	2	no	—
176	C₂³	—	2	1	no	—
177	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
178	Q₈	—	4	1	no	—
179	D₄	—	4	1	no	—
180	D₄	—	4	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	59	2	no	—
187	C₂×C₄	—	59	2	no	—
188	C₂×C₄	—	118	4	no	—
189	C₂×C₄	—	118	2	no	—
190	C₂×C₄	—	236	4	no	—
191	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
192	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	4	1	no	—
195	C₈	—	118	2	no	—
196	C₈	—	236	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	C₂²⋊C₈×D₅₉	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	4	1	no	—
204	C₄	—	59	4	no	—
205	C₄	—	118	2	no	—
206	C₄	—	236	2	no	—
207	C₄	#16837374	1	1	yes	characteristic
208	C₄	#16837383	1	1	yes	characteristic
209	C₂	—	4	1	no	—
210	C₂	—	8	1	no	—
211	C₂	#21422407	1	1	yes	characteristic
212	C₂	#350×D₅₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21432736

C₂

yes

maximal, characteristic

#21425892

C₂

yes

maximal, complemented

#21422037

C₂

yes

maximal, complemented

#21422022

C₂

yes

maximal, characteristic

C₅₉×#1315

C₂

yes

maximal, characteristic

#41×Dic₅₉

C₄

yes

—

—

C₄

yes

characteristic

#16837428

C₂²

yes

characteristic

#16837369

C₄

yes

characteristic

M₁₆×Dic₅₉

C₄

yes

characteristic

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

Dic₅₉⋊C₁₆

—

C₂²⋊(C₅₉⋊C₁₆)

C₂²

yes

—

C₁₆×Dic₅₉

—

C₅₉×#408

C₂²

yes

characteristic

C₅₉×C₂²⋊C₁₆

C₂²

yes

—

C₂×#11745664

C₂×C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

Q₈×Dic₅₉

C₈

yes

—

D₄×Dic₅₉

—

D₄×Dic₅₉

C₈

yes

—

#14273686

—

#14273686

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

Dic₅₉⋊C₈

—

Dic₅₉⋊C₈

—

C₂²⋊(C₅₉⋊C₈)

—

#14273637

—

C₈×Dic₅₉

—

C₈×Dic₅₉

D₄

yes

—

C₈×Dic₅₉

D₄

yes

—

C₂×#41×C₅₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂²⋊C₈×C₅₉

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₈

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

M₁₆

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂²⋊Dic₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₈

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₉

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₉

C₂×C₈

yes

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²⋊C₈

yes

—

C₂×Dic₅₉

C₂²⋊C₈

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#11462

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

100

Dic₅₉

—

101

Dic₅₉

—

102

Dic₅₉

#350

yes

—

103

#2922

—

104

#2162

—

105

#1323

—

106

#1315

—

107

#1315

D₅₉

yes

characteristic

108

C₂×C₅₉

—

109

C₂×C₅₉

—

110

C₂×C₅₉

#1458

yes

characteristic

111

C₂×C₅₉

C₂×#350

yes

—

112

C₄×#41

—

113

#433

—

118

—

114

#431

—

118

—

115

#424

—

116

#424

—

117

#408

—

118

#408

C₂×D₅₉

yes

characteristic

119

C₄×M₁₆

—

120

C₂×C₄×C₈

—

121

C₄⋊C₁₆

—

118

—

122

C₂²⋊C₁₆

—

123

C₂²⋊C₁₆

C₂×D₅₉

yes

—

124

C₄×C₁₆

—

118

—

125

C₅₉

#11462

yes

characteristic, complemented, Sylow

126

C₂×#41

C₄×D₅₉

yes

characteristic

127

C₂×M₁₆

—

128

C₂×M₁₆

C₄×D₅₉

yes

characteristic

129

C₂²×C₈

—

130

C₂²×C₈

C₂²×D₅₉

yes

characteristic

131

C₄×Q₈

—

132

C₄×D₄

—

133

C₄×D₄

—

118

—

134

#117

—

135

#117

—

118

—

136

C₂×C₄²

—

137

C₂×C₄²

—

118

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

118

—

140

C₄⋊C₈

—

118

—

141

C₄⋊C₈

—

118

—

142

C₂²⋊C₈

—

143

C₂²⋊C₈

—

118

—

144

#97

—

118

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

118

—

148

#41

—

149

#41

—

150

C₂×Q₈

C₈×D₅₉

yes

characteristic

151

C₂×D₄

—

152

C₂×D₄

C₈×D₅₉

yes

characteristic

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

118

—

156

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

157

M₁₆

—

158

M₁₆

—

236

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

118

—

163

C₂×C₈

—

236

—

164

C₂×C₈

D₄×D₅₉

yes

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

118

—

167

C₂²⋊C₄

—

118

—

168

C₂²⋊C₄

—

118

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₄²

—

118

—

173

C₄²

—

118

—

174

C₁₆

—

175

C₁₆

—

236

—

176

C₂³

—

177

C₂³

M₁₆×D₅₉

yes

characteristic

178

Q₈

—

179

D₄

—

180

D₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

118

—

189

C₂×C₄

—

118

—

190

C₂×C₄

—

236

—

191

C₂×C₄

C₂²⋊C₄×D₅₉

yes

characteristic

192

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

193

C₈

—

194

C₈

—

195

C₈

—

118

—

196

C₈

—

236

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

C₂²⋊C₈×D₅₉

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

118

—

206

C₄

—

236

—

207

C₄

#16837374

yes

characteristic

208

C₄

#16837383

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

#21422407

yes

characteristic

212

C₂

#350×D₅₉

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1315	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₅₉	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1315	D₅₉
hypocenter	C₅₉	#11462
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1315

subgroup

G/H

center

C₂×C₄

C₂²⋊C₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1315

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₈

prime radical

C₂×#41×C₅₉

C₂×C₄

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1315

N	H ≅ G/N	type
2. #21425892	C₂	semidirect
3. #21422037	C₂	semidirect
125. C₅₉	#11462	semidirect

H ≅ G/N

type

2. #21425892

C₂

semidirect

3. #21422037

C₂

semidirect

125. C₅₉

#11462

semidirect

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

subgroup	index
#21432736	2
#21425892	2
#21422037	2
#21422022	2
C₅₉×#1315	2
#11462	59

subgroup

index

C₅₉×#1315

Subgroups — #28497897

Conjugacy classes of subgroups

Complements

Maximal subgroups