Group #21424998 — subgroups

Subgroups — #21424998

Order 7552.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂²×#11745671	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273662	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837406	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273646	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14273643	C₂	1	1	yes	maximal, characteristic
6	#16837358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×#11745671	—	2	1	no	—
10	C₂×#11745671	—	2	1	no	—
11	C₂×#11745671	—	2	2	no	—
12	C₂×#11745671	C₄	1	4	yes	complemented
13	C₂×#11745664	C₂²	1	1	yes	characteristic
14	C₂×#11745664	C₂²	1	1	yes	characteristic
15	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
16	#14273662	—	2	2	no	—
17	#14273647	—	2	2	no	—
18	#14273646	—	2	2	no	—
19	#14273643	—	2	2	no	—
20	#14273638	—	2	1	no	—
21	C₄×C₅₉⋊C₈	—	2	1	no	—
22	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	#117×C₅₉	C₂²	1	1	yes	characteristic
25	#103×C₅₉	—	2	2	no	—
26	C₂³×D₅₉	—	2	1	no	—
27	C₂×Q₈×C₅₉	—	2	1	no	—
28	C₂×Q₈×C₅₉	—	2	1	no	—
29	C₂×Q₈×C₅₉	—	2	1	no	—
30	C₂×Q₈×C₅₉	—	2	1	no	—
31	C₂×Q₈×C₅₉	—	2	1	no	—
32	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₅₉	—	2	1	no	—
34	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×D₂₃₆	—	2	1	no	—
36	C₂×D₂₃₆	—	2	1	no	—
37	C₂×D₂₃₆	—	2	1	no	—
38	C₂×D₂₃₆	—	2	1	no	—
39	C₂×D₂₃₆	—	2	1	no	—
40	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
41	M₁₆×C₅₉	—	2	1	no	—
42	M₁₆×C₅₉	—	2	1	no	—
43	C₂×C₈×C₅₉	—	2	1	no	—
44	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
45	C₂²⋊C₄×C₅₉	—	2	1	no	—
46	C₄²×C₅₉	—	2	1	no	—
47	#11745671	—	2	4	no	—
48	#11745671	—	2	4	no	—
49	#11745664	—	2	1	no	—
50	#11745664	—	2	1	no	—
51	#11745664	—	2	1	no	—
52	#11745664	D₄	1	2	yes	—
53	C₂×C₅₉⋊C₈	—	2	1	no	—
54	C₂×C₅₉⋊C₈	—	2	1	no	—
55	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
56	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
57	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
58	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
59	C₂³×C₅₉	#41	1	1	yes	characteristic
60	C₂²×D₅₉	—	2	2	no	—
61	C₂²×D₅₉	—	4	1	no	—
62	C₂²×D₅₉	—	4	1	no	—
63	C₂²×D₅₉	—	8	2	no	—
64	Q₈×C₅₉	—	2	2	no	—
65	Q₈×C₅₉	—	2	2	no	—
66	Q₈×C₅₉	—	4	1	no	—
67	Q₈×C₅₉	—	4	1	no	—
68	C₂×C₄×C₅₉	—	2	2	no	—
69	C₂×C₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	—	4	1	no	—
71	C₂×C₄×C₅₉	—	4	1	no	—
72	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
78	D₂₃₆	—	2	2	no	—
79	D₂₃₆	—	2	2	no	—
80	D₂₃₆	—	4	1	no	—
81	D₂₃₆	—	4	1	no	—
82	C₈×C₅₉	—	4	1	no	—
83	C₅₉⋊C₈	—	2	2	no	—
84	C₅₉⋊C₈	—	4	1	no	—
85	C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
86	C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₅₉	—	2	2	no	—
88	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
89	C₂²×C₅₉	#121	1	1	yes	characteristic
90	C₂²×C₅₉	#124	1	1	yes	characteristic
91	C₂×D₅₉	—	4	2	no	—
92	C₂×D₅₉	—	8	2	no	—
93	C₂×D₅₉	—	8	2	no	—
94	C₂×D₅₉	—	8	2	no	—
95	C₄×C₅₉	—	4	2	no	—
96	C₄×C₅₉	—	4	2	no	—
97	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
99	C₄×C₅₉	#121	1	1	yes	characteristic
100	C₄×C₅₉	#127	1	1	yes	characteristic
101	#1977	—	59	1	no	maximal, Sylow
102	C₂×C₅₉	—	2	1	no	—
103	C₂×C₅₉	—	2	1	no	—
104	C₂×C₅₉	#390	1	1	yes	characteristic
105	C₂×C₅₉	#471	1	2	yes	—
106	D₅₉	—	8	2	no	—
107	C₂²×SD₁₆	—	59	1	no	—
108	C₂×#108	—	59	1	no	—
109	#419	D₅₉	1	1	yes	characteristic, complemented
110	#418	—	59	1	no	—
111	C₂×#101	—	59	1	no	—
112	C₂×#100	—	59	1	no	—
113	#405	—	59	1	no	—
114	C₅₉	#1977	1	1	yes	characteristic, complemented, Sylow
115	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
116	C₂²×D₄	—	59	1	no	—
117	C₂×SD₁₆	—	59	4	no	—
118	C₂×SD₁₆	—	118	1	no	—
119	C₂×SD₁₆	—	118	1	no	—
120	C₂×SD₁₆	—	118	2	no	—
121	C₂×M₁₆	—	59	1	no	—
122	C₂×M₁₆	—	59	1	no	—
123	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
124	C₂²×C₈	—	59	1	no	—
125	#117	C₂×D₅₉	1	1	yes	characteristic
126	#108	—	118	2	no	—
127	#103	—	2	2	no	—
128	#102	—	118	2	no	—
129	#101	—	118	2	no	—
130	#100	—	118	2	no	—
131	#97	—	118	1	no	—
132	C₄×C₈	—	118	1	no	—
133	C₂⁴	—	118	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	C₄×D₅₉	1	2	yes	—
140	C₂×D₄	—	59	2	no	—
141	C₂×D₄	—	118	1	no	—
142	C₂×D₄	—	118	1	no	—
143	C₂×D₄	—	118	1	no	—
144	C₂×D₄	—	118	1	no	—
145	C₂×D₄	—	118	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
148	SD₁₆	—	118	4	no	—
149	SD₁₆	—	118	4	no	—
150	M₁₆	—	2	1	no	—
151	M₁₆	—	2	1	no	—
152	M₁₆	—	59	2	no	—
153	M₁₆	—	118	1	no	—
154	M₁₆	—	118	1	no	—
155	M₁₆	—	118	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	59	2	no	—
158	C₂×C₈	—	59	1	no	—
159	C₂×C₈	—	59	1	no	—
160	C₂×C₈	—	59	2	no	—
161	C₂×C₈	—	118	1	no	—
162	C₂×C₈	—	118	1	no	—
163	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₂³	—	118	2	no	—
167	C₂³	—	472	2	no	—
168	C₂³	—	236	1	no	—
169	C₂³	—	236	1	no	—
170	C₂³	#11745687	1	1	yes	characteristic
171	Q₈	—	2	2	no	—
172	Q₈	—	2	2	no	—
173	Q₈	—	4	1	no	—
174	Q₈	—	4	1	no	—
175	D₄	—	118	2	no	—
176	D₄	—	118	2	no	—
177	D₄	—	236	1	no	—
178	D₄	—	236	1	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
184	C₂×C₄	#11745685	1	1	yes	characteristic
185	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
186	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
187	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
189	C₈	—	4	1	no	—
190	C₈	—	59	2	no	—
191	C₈	—	59	2	no	—
192	C₈	—	118	2	no	—
193	C₈	—	236	1	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	472	2	no	—
196	C₂²	—	472	2	no	—
197	C₂²	—	472	2	no	—
198	C₂²	—	236	2	no	—
199	C₂²	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
200	C₂²	#14273699	1	1	yes	characteristic
201	C₂²	#14273713	1	1	yes	characteristic
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
205	C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
206	C₄	#14273700	1	1	yes	characteristic
207	C₄	#14273723	1	1	yes	characteristic
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	—	472	2	no	—
211	C₂	#16837302	1	1	yes	characteristic
212	C₂	#16837667	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745671

C₂

yes

maximal, characteristic

C₂×#14273662

C₂

yes

maximal, characteristic, complemented

#16837406

C₂

yes

maximal, characteristic

C₂×#14273646

C₂

yes

maximal, characteristic, complemented

C₂×#14273643

C₂

yes

maximal, characteristic

#16837358

C₂

yes

maximal, characteristic, complemented

C₅₉×#419

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#11745664

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

—

—

—

—

C₄×C₅₉⋊C₈

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

C₂³×D₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₄

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

—

—

—

—

—

D₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

#41

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×D₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

#121

yes

characteristic

100

C₄×C₅₉

yes

101

—

102

C₂×C₅₉

—

103

C₂×C₅₉

—

104

C₂×C₅₉

#390

yes

characteristic

105

C₂×C₅₉

#471

yes

—

106

D₅₉

—

107

C₂²×SD₁₆

—

108

C₂×#108

—

109

#419

D₅₉

yes

characteristic, complemented

110

#418

—

111

C₂×#101

—

112

C₂×#100

—

113

#405

—

114

C₅₉

#1977

yes

characteristic, complemented, Sylow

115

C₂²×Q₈

C₂×D₅₉

yes

characteristic

116

C₂²×D₄

—

117

C₂×SD₁₆

—

118

C₂×SD₁₆

—

118

—

119

C₂×SD₁₆

—

118

—

120

C₂×SD₁₆

—

118

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂×M₁₆

C₂×D₅₉

yes

characteristic

124

C₂²×C₈

—

125

#117

C₂×D₅₉

yes

characteristic

126

#108

—

118

—

127

#103

—

128

#102

—

118

—

129

#101

—

118

—

130

#100

—

118

—

131

#97

—

118

—

132

C₄×C₈

—

118

—

133

C₂⁴

—

118

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

C₄×D₅₉

yes

—

140

C₂×D₄

—

141

C₂×D₄

—

118

—

142

C₂×D₄

—

118

—

143

C₂×D₄

—

118

—

144

C₂×D₄

—

118

—

145

C₂×D₄

—

118

—

146

C₂²×C₄

—

147

C₂²×C₄

C₂²×D₅₉

yes

characteristic

148

SD₁₆

—

118

—

149

SD₁₆

—

118

—

150

M₁₆

—

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

118

—

154

M₁₆

—

118

—

155

M₁₆

—

118

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

118

—

162

C₂×C₈

—

118

—

163

C₄⋊C₄

C₂²⋊D₅₉

yes

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₂³

—

118

—

167

C₂³

—

472

—

168

C₂³

—

236

—

169

C₂³

—

236

—

170

C₂³

#11745687

yes

characteristic

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

Q₈

—

175

D₄

—

118

—

176

D₄

—

118

—

177

D₄

—

236

—

178

D₄

—

236

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

184

C₂×C₄

#11745685

yes

characteristic

185

C₂×C₄

D₄×D₅₉

yes

characteristic

186

C₂×C₄

D₄×D₅₉

yes

characteristic

187

C₂×C₄

D₄×D₅₉

yes

characteristic

188

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

189

C₈

—

190

C₈

—

191

C₈

—

192

C₈

—

118

—

193

C₈

—

236

—

194

C₂²

—

195

C₂²

—

472

—

196

C₂²

—

472

—

197

C₂²

—

472

—

198

C₂²

—

236

—

199

C₂²

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

200

C₂²

#14273699

yes

characteristic

201

C₂²

#14273713

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

C₂²⋊C₄×D₅₉

yes

characteristic

205

C₄

C₄×C₂²⋊D₅₉

yes

characteristic

206

C₄

#14273700

yes

characteristic

207

C₄

#14273723

yes

characteristic

208

C₂

—

209

C₂

—

210

C₂

—

472

—

211

C₂

#16837302

yes

characteristic

212

C₂

#16837667

yes

—

	subgroup	G/H
center	C₂²	#14273699
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#419	C₂
socle	C₂²×C₅₉	#121
hypercenter	#419	D₅₉
hypocenter	C₅₉	#1977
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14273699

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#419

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
2. C₂×#14273662	C₂	semidirect
4. C₂×#14273646	C₂	semidirect
6. #16837358	C₂	semidirect
7. C₅₉×#419	C₂	semidirect
12. C₂×#11745671	C₄	semidirect
109. #419	D₅₉	semidirect
114. C₅₉	#1977	semidirect

H ≅ G/N

type

2. C₂×#14273662

C₂

semidirect

4. C₂×#14273646

C₂

semidirect

6. #16837358

C₂

semidirect

7. C₅₉×#419

C₂

semidirect

12. C₂×#11745671

C₄

semidirect

109. #419

D₅₉

semidirect

114. C₅₉

#1977

semidirect

Subgroups — #21424998

Conjugacy classes of subgroups

Complements