Group #21425245 — subgroups

Subgroups — #21425245

Order 7552.

258 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 85 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×#14273718	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273651	C₂	1	1	yes	maximal, characteristic
3	#16837309	C₂	1	1	yes	maximal, characteristic
4	#16837165	C₂	1	1	yes	maximal, characteristic
5	#16837164	C₂	1	1	yes	maximal, characteristic
6	#16837156	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	#14273718	—	2	2	no	—
10	#14273718	—	2	2	no	—
11	C₂×C₄⋊Dic₅₉	—	2	1	no	—
12	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₅₉⋊C₄	—	2	1	no	—
15	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
16	#14273651	—	2	2	no	—
17	C₅₉⋊#95	—	2	1	no	—
18	C₅₉⋊#95	—	2	1	no	—
19	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
22	#103×C₅₉	—	2	2	no	—
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×Q₈×C₅₉	—	2	1	no	—
25	C₂×Q₈×C₅₉	—	2	1	no	—
26	C₂×Q₈×C₅₉	—	2	1	no	—
27	C₂×Q₈×C₅₉	—	2	1	no	—
28	C₂×Q₈×C₅₉	Q₈	1	2	yes	—
29	C₂²×C₄×C₅₉	—	2	1	no	—
30	C₂²×C₄×C₅₉	—	2	1	no	—
31	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₉	—	2	1	no	—
33	C₂²×Dic₅₉	—	2	1	no	—
34	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
35	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₂×C₈×C₅₉	—	2	1	no	—
39	C₄⋊C₄×C₅₉	—	2	1	no	—
40	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
41	C₄⋊Dic₅₉	—	2	4	no	—
42	C₄⋊Dic₅₉	—	2	1	no	—
43	C₄⋊Dic₅₉	—	2	1	no	—
44	C₄⋊Dic₅₉	—	2	1	no	—
45	C₄⋊Dic₅₉	D₄	1	2	yes	—
46	Dic₅₉⋊C₄	—	2	4	no	—
47	C₄×Dic₅₉	—	2	1	no	—
48	C₄×Dic₅₉	—	2	1	no	—
49	C₂×C₅₉⋊C₈	—	2	1	no	—
50	C₂×C₅₉⋊C₈	—	2	1	no	—
51	C₂×C₅₉⋊C₈	—	2	1	no	—
52	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
53	Q₈×C₅₉	—	2	2	no	—
54	Q₈×C₅₉	—	2	2	no	—
55	Q₈×C₅₉	—	4	1	no	—
56	Q₈×C₅₉	—	4	1	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	4	1	no	—
60	C₂×C₄×C₅₉	—	4	1	no	—
61	C₂×C₄×C₅₉	—	4	1	no	—
62	C₂×C₄×C₅₉	—	4	1	no	—
63	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
68	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
69	C₂×Dic₅₉	—	2	1	no	—
70	C₂×Dic₅₉	—	2	1	no	—
71	C₂×Dic₅₉	—	2	2	no	—
72	C₂×Dic₅₉	—	2	1	no	—
73	C₂×Dic₅₉	—	2	1	no	—
74	C₂×Dic₅₉	—	2	1	no	—
75	C₂×Dic₅₉	—	2	1	no	—
76	C₂×Dic₅₉	—	4	1	no	—
77	C₂×Dic₅₉	—	4	1	no	—
78	C₂×Dic₅₉	—	4	1	no	—
79	C₂×Dic₅₉	—	4	1	no	—
80	C₂×Dic₅₉	—	4	1	no	—
81	C₈×C₅₉	—	4	1	no	—
82	C₅₉⋊C₈	—	4	1	no	—
83	C₂²×C₅₉	#120	1	1	yes	characteristic
84	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂²×C₅₉	#124	1	1	yes	characteristic
86	C₂²×C₅₉	#135	1	1	yes	characteristic
87	C₂²×C₅₉	#135	1	1	yes	characteristic
88	C₂²×C₅₉	#137	1	1	yes	characteristic
89	C₂²×C₅₉	#137	1	1	yes	characteristic
90	C₄×C₅₉	—	4	2	no	—
91	C₄×C₅₉	—	4	2	no	—
92	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
93	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₄×C₅₉	#124	1	1	yes	characteristic
95	C₄×C₅₉	#124	1	1	yes	characteristic
96	Dic₅₉	—	4	1	no	—
97	Dic₅₉	—	4	1	no	—
98	Dic₅₉	—	4	2	no	—
99	Dic₅₉	—	8	1	no	—
100	#2026	—	59	1	no	maximal, Sylow
101	C₂×C₅₉	#393	1	1	yes	characteristic
102	C₂×C₅₉	#452	1	1	yes	characteristic
103	C₂×C₅₉	#452	1	1	yes	characteristic
104	C₂×C₅₉	#479	1	1	yes	characteristic
105	C₂×C₅₉	#479	1	1	yes	characteristic
106	C₂×C₅₉	#488	1	1	yes	characteristic
107	C₂×C₅₉	#491	1	1	yes	characteristic
108	C₂×#125	—	59	1	no	—
109	C₂×#103	—	59	1	no	—
110	C₂×#103	D₅₉	1	1	yes	characteristic
111	#391	—	59	1	no	—
112	#340	—	59	1	no	—
113	#340	—	59	1	no	—
114	#336	—	59	1	no	—
115	C₅₉	#2026	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
117	C₂²×C₈	—	59	1	no	—
118	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
119	#125	—	118	2	no	—
120	#125	—	118	2	no	—
121	C₂×C₄⋊C₄	—	59	1	no	—
122	C₂×C₄⋊C₄	—	59	1	no	—
123	C₂×C₄⋊C₄	—	118	1	no	—
124	C₂×C₄⋊C₄	—	118	1	no	—
125	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
126	C₂×C₄²	—	59	1	no	—
127	#103	—	2	2	no	—
128	#103	—	118	2	no	—
129	#95	—	118	1	no	—
130	#95	—	118	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	Dic₁₁₈	1	2	yes	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	59	1	no	—
140	C₂²×C₄	—	59	1	no	—
141	C₂²×C₄	—	118	1	no	—
142	C₂²×C₄	—	118	1	no	—
143	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	118	1	no	—
148	C₂×C₈	—	118	1	no	—
149	C₂×C₈	—	118	1	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	59	2	no	—
152	C₄⋊C₄	—	118	4	no	—
153	C₄⋊C₄	—	118	4	no	—
154	C₄⋊C₄	—	118	1	no	—
155	C₄⋊C₄	—	118	1	no	—
156	C₄⋊C₄	—	118	1	no	—
157	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
158	C₄²	—	118	1	no	—
159	C₄²	—	118	1	no	—
160	C₂³	D₄×D₅₉	1	1	yes	characteristic
161	Q₈	—	2	2	no	—
162	Q₈	—	2	2	no	—
163	Q₈	—	4	1	no	—
164	Q₈	—	4	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	118	1	no	—
172	C₂×C₄	—	118	1	no	—
173	C₂×C₄	—	118	2	no	—
174	C₂×C₄	—	118	1	no	—
175	C₂×C₄	—	118	1	no	—
176	C₂×C₄	—	118	1	no	—
177	C₂×C₄	—	118	1	no	—
178	C₂×C₄	—	236	1	no	—
179	C₂×C₄	—	236	1	no	—
180	C₂×C₄	—	236	1	no	—
181	C₂×C₄	—	236	1	no	—
182	C₂×C₄	—	236	1	no	—
183	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
184	C₂×C₄	#11745685	1	1	yes	characteristic
185	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
186	C₂×C₄	#11745687	1	1	yes	characteristic
187	C₂×C₄	#11745687	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
189	C₈	—	4	1	no	—
190	C₈	—	236	1	no	—
191	C₂²	#14273695	1	1	yes	characteristic
192	C₂²	#14273704	1	1	yes	characteristic
193	C₂²	#14273713	1	1	yes	characteristic
194	C₂²	#14273748	1	1	yes	characteristic
195	C₂²	#14273750	1	1	yes	characteristic
196	C₂²	#14273756	1	1	yes	characteristic
197	C₂²	#14273759	1	1	yes	characteristic
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	472	1	no	—
201	C₄	—	236	1	no	—
202	C₄	—	236	1	no	—
203	C₄	—	236	2	no	—
204	C₄	#14273704	1	1	yes	characteristic
205	C₄	#14273706	1	1	yes	characteristic
206	C₄	#14273713	1	1	yes	characteristic
207	C₄	#14273714	1	1	yes	characteristic
208	C₂	#16837314	1	1	yes	characteristic
209	C₂	#16837561	1	1	yes	characteristic
210	C₂	#16837563	1	1	yes	characteristic
211	C₂	#16837707	1	1	yes	characteristic
212	C₂	#16837711	1	1	yes	characteristic
213	C₂	#16837758	1	1	yes	characteristic
214	C₂	#16837769	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273718

C₂

yes

maximal, characteristic

C₂×#14273651

C₂

yes

maximal, characteristic

#16837309

C₂

yes

maximal, characteristic

#16837165

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal, characteristic

#16837156

C₂

yes

maximal, characteristic

C₂×#103×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273718

—

#14273718

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273651

—

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

characteristic

C₂²×Dic₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

#120

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

100

#2026

—

maximal, Sylow

101

C₂×C₅₉

#393

yes

characteristic

102

C₂×C₅₉

#452

yes

characteristic

103

C₂×C₅₉

#452

yes

characteristic

104

C₂×C₅₉

#479

yes

characteristic

105

C₂×C₅₉

#479

yes

characteristic

106

C₂×C₅₉

#488

yes

characteristic

107

C₂×C₅₉

#491

yes

characteristic

108

C₂×#125

—

109

C₂×#103

—

110

C₂×#103

D₅₉

yes

characteristic

111

#391

—

112

#340

—

113

#340

—

114

#336

—

115

C₅₉

#2026

yes

characteristic, complemented, Sylow

116

C₂²×Q₈

C₂×D₅₉

yes

characteristic

117

C₂²×C₈

—

118

C₂²×C₈

C₂×D₅₉

yes

characteristic

119

#125

—

118

—

120

#125

—

118

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

118

—

124

C₂×C₄⋊C₄

—

118

—

125

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

126

C₂×C₄²

—

127

#103

—

128

#103

—

118

—

129

#95

—

118

—

130

#95

—

118

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

Dic₁₁₈

yes

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

118

—

142

C₂²×C₄

—

118

—

143

C₂²×C₄

C₂²×D₅₉

yes

characteristic

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

118

—

148

C₂×C₈

—

118

—

149

C₂×C₈

—

118

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

118

—

153

C₄⋊C₄

—

118

—

154

C₄⋊C₄

—

118

—

155

C₄⋊C₄

—

118

—

156

C₄⋊C₄

—

118

—

157

C₄⋊C₄

C₂²⋊D₅₉

yes

—

158

C₄²

—

118

—

159

C₄²

—

118

—

160

C₂³

D₄×D₅₉

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

118

—

172

C₂×C₄

—

118

—

173

C₂×C₄

—

118

—

174

C₂×C₄

—

118

—

175

C₂×C₄

—

118

—

176

C₂×C₄

—

118

—

177

C₂×C₄

—

118

—

178

C₂×C₄

—

236

—

179

C₂×C₄

—

236

—

180

C₂×C₄

—

236

—

181

C₂×C₄

—

236

—

182

C₂×C₄

—

236

—

183

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

184

C₂×C₄

#11745685

yes

characteristic

185

C₂×C₄

D₄×D₅₉

yes

characteristic

186

C₂×C₄

#11745687

yes

characteristic

187

C₂×C₄

#11745687

yes

characteristic

188

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

189

C₈

—

190

C₈

—

236

—

191

C₂²

#14273695

yes

characteristic

192

C₂²

#14273704

yes

characteristic

193

C₂²

#14273713

yes

characteristic

194

C₂²

#14273748

yes

characteristic

195

C₂²

#14273750

yes

characteristic

196

C₂²

#14273756

yes

characteristic

197

C₂²

#14273759

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

472

—

201

C₄

—

236

—

202

C₄

—

236

—

203

C₄

—

236

—

204

C₄

#14273704

yes

characteristic

205

C₄

#14273706

yes

characteristic

206

C₄

#14273713

yes

characteristic

207

C₄

#14273714

yes

characteristic

208

C₂

#16837314

yes

characteristic

209

C₂

#16837561

yes

characteristic

210

C₂

#16837563

yes

characteristic

211

C₂

#16837707

yes

characteristic

212

C₂

#16837711

yes

characteristic

213

C₂

#16837758

yes

characteristic

214

C₂

#16837769

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#103×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#103	D₅₉
hypocenter	C₅₉	#2026
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#103×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
115. C₅₉	#2026	semidirect

H ≅ G/N

type

115. C₅₉

#2026

semidirect

Subgroups — #21425245

Conjugacy classes of subgroups

Complements