Group #21425436 — subgroups

Subgroups — #21425436

Order 7552.

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

normal subgroups only

This group has 250 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	C₂×#14273718	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273660	C₂	1	1	yes	maximal, characteristic
3	#16837298	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×#106×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×Dic₅₉⋊C₄	—	2	1	no	—
16	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
17	#14273660	—	2	2	no	—
18	C₅₉⋊#95	—	2	1	no	—
19	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
22	#106×C₅₉	—	2	2	no	—
23	C₂²×C₄×C₅₉	—	2	1	no	—
24	C₂²×C₄×C₅₉	—	2	1	no	—
25	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₉	—	2	1	no	—
27	C₂²×Dic₅₉	—	2	1	no	—
28	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
29	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₂×C₈×C₅₉	—	2	1	no	—
32	C₂×C₈×C₅₉	Q₈	1	2	yes	—
33	C₄⋊C₄×C₅₉	—	2	1	no	—
34	C₄⋊C₄×C₅₉	—	2	1	no	—
35	C₄⋊C₄×C₅₉	—	2	1	no	—
36	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
37	C₄⋊Dic₅₉	—	2	4	no	—
38	C₄⋊Dic₅₉	—	2	1	no	—
39	C₄⋊Dic₅₉	—	2	1	no	—
40	C₄⋊Dic₅₉	D₄	1	2	yes	—
41	C₄⋊Dic₅₉	Q₈	1	2	yes	—
42	Dic₅₉⋊C₄	—	2	4	no	—
43	Dic₅₉⋊C₄	—	4	2	no	—
44	C₄×Dic₅₉	—	2	1	no	—
45	C₄×Dic₅₉	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂×C₅₉⋊C₈	—	2	1	no	—
48	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
49	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	—	2	2	no	—
51	C₂×C₄×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	—	4	1	no	—
53	C₂×C₄×C₅₉	—	4	1	no	—
54	C₂×C₄×C₅₉	—	4	1	no	—
55	C₂×C₄×C₅₉	—	4	1	no	—
56	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
62	C₂×Dic₅₉	—	2	1	no	—
63	C₂×Dic₅₉	—	2	1	no	—
64	C₂×Dic₅₉	—	2	2	no	—
65	C₂×Dic₅₉	—	2	2	no	—
66	C₂×Dic₅₉	—	2	1	no	—
67	C₂×Dic₅₉	—	2	1	no	—
68	C₂×Dic₅₉	—	2	1	no	—
69	C₂×Dic₅₉	—	2	1	no	—
70	C₂×Dic₅₉	—	4	1	no	—
71	C₂×Dic₅₉	—	4	1	no	—
72	C₂×Dic₅₉	—	4	1	no	—
73	C₂×Dic₅₉	—	4	1	no	—
74	C₈×C₅₉	—	2	2	no	—
75	C₅₉⋊C₈	—	2	2	no	—
76	C₂²×C₅₉	#120	1	1	yes	characteristic
77	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
79	C₂²×C₅₉	#135	1	1	yes	characteristic
80	C₂²×C₅₉	#135	1	1	yes	characteristic
81	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂²×C₅₉	#137	1	1	yes	characteristic
83	C₄×C₅₉	—	4	2	no	—
84	C₄×C₅₉	—	8	1	no	—
85	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
86	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
87	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
89	Dic₅₉	—	4	1	no	—
90	Dic₅₉	—	4	2	no	—
91	Dic₅₉	—	4	1	no	—
92	Dic₅₉	—	4	2	no	—
93	#2059	—	59	1	no	maximal, Sylow
94	C₂×C₅₉	#395	1	1	yes	characteristic
95	C₂×C₅₉	#449	1	1	yes	characteristic
96	C₂×C₅₉	#452	1	1	yes	characteristic
97	C₂×C₅₉	#478	1	1	yes	characteristic
98	C₂×C₅₉	#479	1	1	yes	characteristic
99	C₂×C₅₉	#499	1	1	yes	characteristic
100	C₂×C₅₉	#501	1	1	yes	characteristic
101	C₂×#125	—	59	1	no	—
102	C₂×#107	—	59	1	no	—
103	C₂×#106	D₅₉	1	1	yes	characteristic
104	#389	—	59	1	no	—
105	#340	—	59	1	no	—
106	#340	—	59	1	no	—
107	#336	—	59	1	no	—
108	C₅₉	#2059	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	59	1	no	—
110	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
111	#125	—	118	2	no	—
112	#125	—	118	2	no	—
113	C₂×C₄⋊C₄	—	59	1	no	—
114	C₂×C₄⋊C₄	—	59	1	no	—
115	C₂×C₄⋊C₄	—	118	1	no	—
116	C₂×C₄⋊C₄	—	118	1	no	—
117	C₂×C₄⋊C₄	—	118	1	no	—
118	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
120	C₂×C₄²	—	59	1	no	—
121	#107	—	118	2	no	—
122	#106	—	2	2	no	—
123	#95	—	118	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	59	1	no	—
127	C₂²×C₄	—	59	1	no	—
128	C₂²×C₄	—	118	1	no	—
129	C₂²×C₄	—	118	1	no	—
130	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	59	2	no	—
134	C₂×C₈	—	118	1	no	—
135	C₂×C₈	—	118	1	no	—
136	C₂×C₈	Dic₁₁₈	1	2	yes	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	2	1	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	59	2	no	—
141	C₄⋊C₄	—	59	2	no	—
142	C₄⋊C₄	—	118	4	no	—
143	C₄⋊C₄	—	118	4	no	—
144	C₄⋊C₄	—	118	1	no	—
145	C₄⋊C₄	—	118	1	no	—
146	C₄⋊C₄	—	236	2	no	—
147	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
148	C₄²	—	118	1	no	—
149	C₄²	—	118	1	no	—
150	C₂³	D₄×D₅₉	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	118	1	no	—
158	C₂×C₄	—	118	1	no	—
159	C₂×C₄	—	118	2	no	—
160	C₂×C₄	—	118	2	no	—
161	C₂×C₄	—	118	1	no	—
162	C₂×C₄	—	118	1	no	—
163	C₂×C₄	—	118	1	no	—
164	C₂×C₄	—	118	1	no	—
165	C₂×C₄	—	236	1	no	—
166	C₂×C₄	—	236	1	no	—
167	C₂×C₄	—	236	1	no	—
168	C₂×C₄	—	236	1	no	—
169	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
170	C₂×C₄	#11745685	1	1	yes	characteristic
171	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
172	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
173	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	118	2	no	—
177	C₂²	#14273695	1	1	yes	characteristic
178	C₂²	#14273705	1	1	yes	characteristic
179	C₂²	#14273726	1	1	yes	characteristic
180	C₂²	#14273748	1	1	yes	characteristic
181	C₂²	#14273748	1	1	yes	characteristic
182	C₂²	#14273754	1	1	yes	characteristic
183	C₂²	#14273756	1	1	yes	characteristic
184	C₄	—	4	2	no	—
185	C₄	—	8	1	no	—
186	C₄	—	236	2	no	—
187	C₄	—	236	1	no	—
188	C₄	—	236	2	no	—
189	C₄	—	236	1	no	—
190	C₄	#14273705	1	1	yes	characteristic
191	C₄	#14273706	1	1	yes	characteristic
192	C₄	#14273725	1	1	yes	characteristic
193	C₄	#14273726	1	1	yes	characteristic
194	C₂	#16837322	1	1	yes	characteristic
195	C₂	#16837546	1	1	yes	characteristic
196	C₂	#16837561	1	1	yes	characteristic
197	C₂	#16837701	1	1	yes	characteristic
198	C₂	#16837712	1	1	yes	characteristic
199	C₂	#16837802	1	1	yes	characteristic
200	C₂	#16837809	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273718

C₂

yes

maximal, characteristic

C₂×#14273660

C₂

yes

maximal, characteristic

#16837298

C₂

yes

maximal, characteristic

#16837165

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal, characteristic

#16837156

C₂

yes

maximal, characteristic

C₂×#106×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×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273660

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

#106×C₅₉

—

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

Q₈

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

—

C₄⋊Dic₅₉

Q₈

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

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

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

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

characteristic

C₂²×C₅₉

C₄⋊Q₈

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

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₄×C₅₉

C₄⋊Q₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

100

C₂×C₅₉

#501

yes

characteristic

101

C₂×#125

—

102

C₂×#107

—

103

C₂×#106

D₅₉

yes

characteristic

104

#389

—

105

#340

—

106

#340

—

107

#336

—

108

C₅₉

#2059

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

110

C₂²×C₈

C₂×D₅₉

yes

characteristic

111

#125

—

118

—

112

#125

—

118

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

118

—

116

C₂×C₄⋊C₄

—

118

—

117

C₂×C₄⋊C₄

—

118

—

118

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

119

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

120

C₂×C₄²

—

121

#107

—

118

—

122

#106

—

123

#95

—

118

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

118

—

129

C₂²×C₄

—

118

—

130

C₂²×C₄

C₂²×D₅₉

yes

characteristic

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

118

—

135

C₂×C₈

—

118

—

136

C₂×C₈

Dic₁₁₈

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

118

—

143

C₄⋊C₄

—

118

—

144

C₄⋊C₄

—

118

—

145

C₄⋊C₄

—

118

—

146

C₄⋊C₄

—

236

—

147

C₄⋊C₄

C₂²⋊D₅₉

yes

—

148

C₄²

—

118

—

149

C₄²

—

118

—

150

C₂³

D₄×D₅₉

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

118

—

158

C₂×C₄

—

118

—

159

C₂×C₄

—

118

—

160

C₂×C₄

—

118

—

161

C₂×C₄

—

118

—

162

C₂×C₄

—

118

—

163

C₂×C₄

—

118

—

164

C₂×C₄

—

118

—

165

C₂×C₄

—

236

—

166

C₂×C₄

—

236

—

167

C₂×C₄

—

236

—

168

C₂×C₄

—

236

—

169

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

170

C₂×C₄

#11745685

yes

characteristic

171

C₂×C₄

D₄×D₅₉

yes

characteristic

172

C₂×C₄

Q₈×D₅₉

yes

characteristic

173

C₂×C₄

Q₈×D₅₉

yes

characteristic

174

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

175

C₈

—

176

C₈

—

118

—

177

C₂²

#14273695

yes

characteristic

178

C₂²

#14273705

yes

characteristic

179

C₂²

#14273726

yes

characteristic

180

C₂²

#14273748

yes

characteristic

181

C₂²

#14273748

yes

characteristic

182

C₂²

#14273754

yes

characteristic

183

C₂²

#14273756

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

—

236

—

187

C₄

—

236

—

188

C₄

—

236

—

189

C₄

—

236

—

190

C₄

#14273705

yes

characteristic

191

C₄

#14273706

yes

characteristic

192

C₄

#14273725

yes

characteristic

193

C₄

#14273726

yes

characteristic

194

C₂

#16837322

yes

characteristic

195

C₂

#16837546

yes

characteristic

196

C₂

#16837561

yes

characteristic

197

C₂

#16837701

yes

characteristic

198

C₂

#16837712

yes

characteristic

199

C₂

#16837802

yes

characteristic

200

C₂

#16837809

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#106×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#106	D₅₉
hypocenter	C₅₉	#2059
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#106×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
108. C₅₉	#2059	semidirect

H ≅ G/N

type

108. C₅₉

#2059

semidirect

Subgroups — #21425436

Conjugacy classes of subgroups

Complements