Group #21435557 — subgroups

Subgroups — #21435557

Order 7552.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16838023	C₂	1	2	yes	maximal, complemented
2	#16837698	C₂	1	2	yes	maximal, complemented
3	#16837683	C₂	1	2	yes	maximal, complemented
4	#16837648	C₂	1	2	yes	maximal, complemented
5	#16837643	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837638	C₂	1	1	yes	maximal, characteristic
7	#16837625	C₂	1	2	yes	maximal, complemented
8	#16837618	C₂	1	2	yes	maximal, complemented
9	C₅₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
11	C₂×#11745660	C₂²	1	2	yes	—
12	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
13	#14273727	C₂²	1	2	yes	—
14	#14273722	—	2	2	no	—
15	#14273715	C₂²	1	2	yes	—
16	#14273713	—	2	2	no	—
17	#14273709	—	2	2	no	—
18	#14273706	C₂²	1	2	yes	—
19	#14273705	—	2	2	no	—
20	#14273701	C₂²	1	2	yes	—
21	C₄×Dic₁₁₈	C₂²	1	2	yes	—
22	C₄×D₂₃₆	C₂²	1	2	yes	—
23	#14273661	C₂²	1	1	yes	characteristic
24	#14273661	C₂²	1	1	yes	characteristic
25	#14273661	C₂²	1	2	yes	—
26	#14273652	C₂²	1	2	yes	—
27	#14273652	C₂²	1	2	yes	—
28	#14273649	C₂²	1	2	yes	—
29	#14273649	C₂²	1	2	yes	—
30	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
32	#117×C₅₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
34	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
36	C₂×C₂²⋊D₅₉	—	2	2	no	—
37	C₂×D₂₃₆	C₂³	1	2	yes	—
38	C₂×C₄×D₅₉	—	2	2	no	—
39	C₂×Dic₁₁₈	—	2	2	no	—
40	C₂×Dic₁₁₈	C₂³	1	2	yes	—
41	M₁₆×C₅₉	—	4	1	no	—
42	C₂×C₈×C₅₉	—	4	1	no	—
43	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂³	1	2	yes	—
46	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₅₉	D₄	1	2	yes	—
49	C₄²×C₅₉	C₂³	1	2	yes	—
50	C₂²⋊Dic₅₉	—	2	2	no	—
51	(C₂×D₅₉)⋊C₄	—	2	2	no	—
52	(C₂×D₅₉)⋊C₄	—	2	2	no	—
53	(C₂×D₅₉)⋊C₄	—	2	2	no	—
54	(C₂×D₅₉)⋊C₄	—	2	2	no	—
55	C₄⋊Dic₅₉	—	2	2	no	—
56	C₄⋊Dic₅₉	C₂³	1	2	yes	—
57	C₄⋊Dic₅₉	C₂³	1	2	yes	—
58	Dic₅₉⋊C₄	—	2	2	no	—
59	Dic₅₉⋊C₄	—	2	2	no	—
60	C₄×Dic₅₉	—	2	2	no	—
61	Dic₂₃₆	—	4	1	no	—
62	D₄₇₂	—	4	1	no	—
63	#11745660	—	4	2	no	—
64	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₅₉	—	2	2	no	—
66	C₂×C₄×C₅₉	—	2	1	no	—
67	C₂×C₄×C₅₉	—	2	1	no	—
68	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₅₉	—	8	2	no	—
73	C₂×Dic₅₉	—	2	2	no	—
74	C₂×Dic₅₉	—	2	2	no	—
75	C₂×Dic₅₉	—	2	2	no	—
76	D₂₃₆	—	2	2	no	—
77	D₂₃₆	—	2	2	no	—
78	C₄×D₅₉	—	8	2	no	—
79	Dic₁₁₈	—	2	2	no	—
80	Dic₁₁₈	—	2	2	no	—
81	Dic₁₁₈	—	8	2	no	—
82	C₈×C₅₉	—	2	2	no	—
83	C₈×C₅₉	—	2	1	no	—
84	C₈×C₅₉	—	2	1	no	—
85	C₂²×C₅₉	—	2	1	no	—
86	C₂²×C₅₉	—	2	1	no	—
87	C₂²×C₅₉	—	2	1	no	—
88	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₅₉	—	4	2	no	—
90	C₂×D₅₉	—	4	2	no	—
91	C₂×D₅₉	—	4	2	no	—
92	C₄×C₅₉	—	2	1	no	—
93	C₄×C₅₉	—	2	2	no	—
94	C₄×C₅₉	—	2	2	no	—
95	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
96	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
97	Dic₅₉	—	4	2	no	—
98	Dic₅₉	—	4	2	no	—
99	Dic₅₉	—	4	2	no	—
100	#3262	—	59	1	no	maximal, Sylow
101	C₂×C₅₉	—	4	1	no	—
102	C₂×C₅₉	#537	1	1	yes	characteristic
103	C₂×C₅₉	#576	1	1	yes	characteristic
104	C₂×C₅₉	#578	1	1	yes	characteristic
105	D₅₉	—	8	2	no	—
106	#542	—	59	2	no	—
107	#477	—	59	2	no	—
108	#474	—	59	2	no	—
109	#468	—	59	2	no	—
110	#467	—	59	1	no	—
111	#466	—	59	1	no	—
112	#464	—	59	2	no	—
113	#463	—	59	2	no	—
114	#424	D₅₉	1	1	yes	characteristic, complemented
115	C₅₉	#3262	1	1	yes	characteristic, complemented, Sylow
116	C₂×Q₁₆	—	59	1	no	—
117	C₂×SD₁₆	—	59	2	no	—
118	C₂×D₈	—	59	1	no	—
119	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
121	C₄⋊Q₈	—	59	2	no	—
122	#126	—	118	2	no	—
123	#124	—	59	2	no	—
124	#124	—	118	2	no	—
125	#123	—	118	2	no	—
126	C₂²⋊Q₈	—	59	2	no	—
127	C₂²⋊Q₈	—	118	2	no	—
128	#121	—	59	2	no	—
129	C₄×Q₈	—	59	2	no	—
130	C₄×D₄	—	59	2	no	—
131	#117	C₂×D₅₉	1	1	yes	characteristic
132	#107	—	59	1	no	—
133	#107	—	59	1	no	—
134	#107	—	59	2	no	—
135	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
136	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
137	#103	—	59	2	no	—
138	#103	—	59	2	no	—
139	#102	—	59	2	no	—
140	#102	—	59	2	no	—
141	C₂×Q₈	—	59	2	no	—
142	C₂×Q₈	—	118	2	no	—
143	C₂×D₄	—	59	2	no	—
144	C₂×D₄	—	118	2	no	—
145	C₂²×C₄	—	118	2	no	—
146	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
147	Q₁₆	—	236	1	no	—
148	SD₁₆	—	236	2	no	—
149	D₈	—	236	1	no	—
150	M₁₆	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
153	C₂×C₈	C₂²×D₅₉	1	2	yes	—
154	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
155	C₄⋊C₄	—	59	2	no	—
156	C₄⋊C₄	—	59	2	no	—
157	C₄⋊C₄	—	118	2	no	—
158	C₄⋊C₄	—	118	2	no	—
159	C₄⋊C₄	—	118	2	no	—
160	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
161	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
162	C₂²⋊C₄	—	118	2	no	—
163	C₂²⋊C₄	—	118	2	no	—
164	C₂²⋊C₄	—	118	2	no	—
165	C₂²⋊C₄	—	118	2	no	—
166	C₂²⋊C₄	—	118	2	no	—
167	C₂²⋊C₄	D₂₃₆	1	2	yes	—
168	C₄²	—	118	2	no	—
169	C₄²	C₂²×D₅₉	1	2	yes	—
170	C₂³	—	118	2	no	—
171	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
172	Q₈	—	118	2	no	—
173	Q₈	—	118	2	no	—
174	Q₈	—	472	2	no	—
175	D₄	—	118	2	no	—
176	D₄	—	118	2	no	—
177	D₄	—	472	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	118	2	no	—
181	C₂×C₄	—	118	2	no	—
182	C₂×C₄	—	118	2	no	—
183	C₂×C₄	—	472	2	no	—
184	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
185	C₂×C₄	C₂×D₂₃₆	1	2	yes	—
186	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
187	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	236	2	no	—
195	C₂²	—	236	2	no	—
196	C₂²	—	236	2	no	—
197	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	236	2	no	—
202	C₄	—	236	2	no	—
203	C₄	—	236	2	no	—
204	C₄	#14273772	1	1	yes	characteristic
205	C₄	#14273775	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	—	472	2	no	—
208	C₂	#16837983	1	1	yes	characteristic
209	C₂	#16838207	1	1	yes	characteristic
210	C₂	#16838219	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838023

C₂

yes

maximal, complemented

#16837698

C₂

yes

maximal, complemented

#16837683

C₂

yes

maximal, complemented

#16837648

C₂

yes

maximal, complemented

#16837643

C₂

yes

maximal, characteristic, complemented

#16837638

C₂

yes

maximal, characteristic

#16837625

C₂

yes

maximal, complemented

#16837618

C₂

yes

maximal, complemented

C₅₉×#424

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745660

C₂²

yes

—

C₂×D₄₇₂

C₂²

yes

characteristic

#14273727

C₂²

yes

—

#14273722

—

#14273715

C₂²

yes

—

#14273713

—

#14273709

—

#14273706

C₂²

yes

—

#14273705

—

#14273701

C₂²

yes

—

C₄×Dic₁₁₈

C₂²

yes

—

C₄×D₂₃₆

C₂²

yes

—

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

—

#14273652

C₂²

yes

—

#14273652

C₂²

yes

—

#14273649

C₂²

yes

—

#14273649

C₂²

yes

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄⋊C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₂²⋊D₅₉

—

C₂×D₂₃₆

C₂³

yes

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂³

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₂²⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

—

C₂²⋊Dic₅₉

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

D₄₇₂

—

#11745660

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

H(5,𝔽₂)

yes

characteristic

C₄×C₅₉

2₋¹⁺⁴

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

100

#3262

—

maximal, Sylow

101

C₂×C₅₉

—

102

C₂×C₅₉

#537

yes

characteristic

103

C₂×C₅₉

#576

yes

characteristic

104

C₂×C₅₉

#578

yes

characteristic

105

D₅₉

—

106

#542

—

107

#477

—

108

#474

—

109

#468

—

110

#467

—

111

#466

—

112

#464

—

113

#463

—

114

#424

D₅₉

yes

characteristic, complemented

115

C₅₉

#3262

yes

characteristic, complemented, Sylow

116

C₂×Q₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₂×M₁₆

C₂×D₅₉

yes

characteristic

120

C₂²×C₈

C₂×D₅₉

yes

characteristic

121

C₄⋊Q₈

—

122

#126

—

118

—

123

#124

—

124

#124

—

118

—

125

#123

—

118

—

126

C₂²⋊Q₈

—

127

C₂²⋊Q₈

—

118

—

128

#121

—

129

C₄×Q₈

—

130

C₄×D₄

—

131

#117

C₂×D₅₉

yes

characteristic

132

#107

—

133

#107

—

134

#107

—

135

C₄⋊C₈

C₂×D₅₉

yes

—

136

C₄⋊C₈

C₂×D₅₉

yes

—

137

#103

—

138

#103

—

139

#102

—

140

#102

—

141

C₂×Q₈

—

142

C₂×Q₈

—

118

—

143

C₂×D₄

—

144

C₂×D₄

—

118

—

145

C₂²×C₄

—

118

—

146

C₂²×C₄

C₂²×D₅₉

yes

characteristic

147

Q₁₆

—

236

—

148

SD₁₆

—

236

—

149

D₈

—

236

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

C₂²×D₅₉

yes

characteristic

153

C₂×C₈

C₂²×D₅₉

yes

—

154

C₂×C₈

C₂²×D₅₉

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

118

—

158

C₄⋊C₄

—

118

—

159

C₄⋊C₄

—

118

—

160

C₄⋊C₄

D₂₃₆

yes

characteristic

161

C₄⋊C₄

D₂₃₆

yes

characteristic

162

C₂²⋊C₄

—

118

—

163

C₂²⋊C₄

—

118

—

164

C₂²⋊C₄

—

118

—

165

C₂²⋊C₄

—

118

—

166

C₂²⋊C₄

—

118

—

167

C₂²⋊C₄

D₂₃₆

yes

—

168

C₄²

—

118

—

169

C₄²

C₂²×D₅₉

yes

—

170

C₂³

—

118

—

171

C₂³

C₂×D₂₃₆

yes

characteristic

172

Q₈

—

118

—

173

Q₈

—

118

—

174

Q₈

—

472

—

175

D₄

—

118

—

176

D₄

—

118

—

177

D₄

—

472

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

118

—

181

C₂×C₄

—

118

—

182

C₂×C₄

—

118

—

183

C₂×C₄

—

472

—

184

C₂×C₄

C₂×D₂₃₆

yes

—

185

C₂×C₄

C₂×D₂₃₆

yes

—

186

C₂×C₄

C₂×D₂₃₆

yes

characteristic

187

C₂×C₄

C₂³×D₅₉

yes

characteristic

188

C₈

—

189

C₈

—

190

C₈

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

236

—

195

C₂²

—

236

—

196

C₂²

—

236

—

197

C₂²

C₂²×D₂₃₆

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

236

—

202

C₄

—

236

—

203

C₄

—

236

—

204

C₄

#14273772

yes

characteristic

205

C₄

#14273775

yes

characteristic

206

C₂

—

207

C₂

—

472

—

208

C₂

#16837983

yes

characteristic

209

C₂

#16838207

yes

characteristic

210

C₂

#16838219

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#424	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#424	D₅₉
hypocenter	C₅₉	#3262
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#424

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #16838023	C₂	semidirect
2. #16837698	C₂	semidirect
3. #16837683	C₂	semidirect
4. #16837648	C₂	semidirect
5. #16837643	C₂	semidirect
7. #16837625	C₂	semidirect
8. #16837618	C₂	semidirect
9. C₅₉×#424	C₂	semidirect
114. #424	D₅₉	semidirect
115. C₅₉	#3262	semidirect

H ≅ G/N

type

1. #16838023

C₂

semidirect

2. #16837698

C₂

semidirect

3. #16837683

C₂

semidirect

4. #16837648

C₂

semidirect

5. #16837643

C₂

semidirect

7. #16837625

C₂

semidirect

8. #16837618

C₂

semidirect

9. C₅₉×#424

C₂

semidirect

114. #424

D₅₉

semidirect

115. C₅₉

#3262

semidirect

Subgroups — #21435557

Conjugacy classes of subgroups

Complements