Group #21424731 — subgroups

Subgroups — #21424731

Order 7552.

254 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 95 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×#14273659	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273652	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273652	C₂	1	1	yes	maximal, characteristic
4	#16837308	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837165	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#336	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₅₉	—	2	1	no	—
13	#14273659	—	2	2	no	—
14	#14273652	—	2	2	no	—
15	#14273652	C₄	1	4	yes	complemented
16	C₅₉⋊#95	—	2	1	no	—
17	C₅₉⋊#95	—	2	1	no	—
18	C₅₉⋊#95	—	2	1	no	—
19	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₉	—	2	1	no	—
26	C₂²×Dic₅₉	—	2	1	no	—
27	C₂²×Dic₅₉	—	2	1	no	—
28	C₂²×Dic₅₉	—	2	1	no	—
29	C₂×Dic₁₁₈	—	2	1	no	—
30	C₂×Dic₁₁₈	—	2	1	no	—
31	C₂×Dic₁₁₈	—	2	1	no	—
32	C₂×Dic₁₁₈	—	2	1	no	—
33	C₂×Dic₁₁₈	—	2	1	no	—
34	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₅₉	—	2	1	no	—
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	—	2	1	no	—
38	C₂×C₈×C₅₉	—	2	1	no	—
39	C₂×C₈×C₅₉	—	2	1	no	—
40	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
41	C₄²×C₅₉	—	2	1	no	—
42	C₄²×C₅₉	—	2	1	no	—
43	C₄⋊Dic₅₉	—	2	1	no	—
44	C₄⋊Dic₅₉	—	2	1	no	—
45	C₄⋊Dic₅₉	—	2	1	no	—
46	C₄⋊Dic₅₉	—	2	1	no	—
47	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
48	C₄⋊Dic₅₉	D₄	1	2	yes	—
49	C₄×Dic₅₉	—	4	2	no	—
50	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₉	—	2	1	no	—
52	C₂×C₄×C₅₉	—	2	1	no	—
53	C₂×C₄×C₅₉	—	2	1	no	—
54	C₂×C₄×C₅₉	—	2	1	no	—
55	C₂×C₄×C₅₉	—	2	1	no	—
56	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
57	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
63	C₂×Dic₅₉	—	2	2	no	—
64	C₂×Dic₅₉	—	2	2	no	—
65	C₂×Dic₅₉	—	2	2	no	—
66	C₂×Dic₅₉	—	4	1	no	—
67	C₂×Dic₅₉	—	4	1	no	—
68	C₂×Dic₅₉	—	4	1	no	—
69	C₂×Dic₅₉	—	4	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	C₂×Dic₅₉	—	4	1	no	—
75	Dic₁₁₈	—	2	2	no	—
76	Dic₁₁₈	—	2	2	no	—
77	Dic₁₁₈	—	4	1	no	—
78	Dic₁₁₈	—	4	1	no	—
79	C₈×C₅₉	—	2	2	no	—
80	C₈×C₅₉	—	4	1	no	—
81	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
82	C₂²×C₅₉	#121	1	1	yes	characteristic
83	C₂²×C₅₉	#123	1	1	yes	characteristic
84	C₂²×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
85	C₂²×C₅₉	#135	1	1	yes	characteristic
86	C₂²×C₅₉	#137	1	1	yes	characteristic
87	C₂²×C₅₉	#137	1	1	yes	characteristic
88	C₄×C₅₉	—	2	2	no	—
89	C₄×C₅₉	—	4	1	no	—
90	C₄×C₅₉	#117	1	1	yes	characteristic
91	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
92	C₄×C₅₉	#124	1	1	yes	characteristic
93	C₄×C₅₉	#126	1	1	yes	characteristic
94	Dic₅₉	—	4	2	no	—
95	Dic₅₉	—	4	2	no	—
96	Dic₅₉	—	4	2	no	—
97	Dic₅₉	—	8	1	no	—
98	#1927	—	59	1	no	maximal, Sylow
99	C₂×C₅₉	#388	1	1	yes	characteristic
100	C₂×C₅₉	C₄×SD₁₆	1	1	yes	characteristic
101	C₂×C₅₉	#441	1	1	yes	characteristic
102	C₂×C₅₉	#461	1	1	yes	characteristic
103	C₂×C₅₉	#464	1	1	yes	characteristic
104	C₂×C₅₉	#483	1	1	yes	characteristic
105	C₂×C₅₉	#485	1	1	yes	characteristic
106	C₂×#106	—	59	1	no	—
107	C₂×#103	—	59	1	no	—
108	C₂×#103	—	59	1	no	—
109	#391	—	59	1	no	—
110	C₄×C₄⋊C₄	—	59	1	no	—
111	#340	—	59	1	no	—
112	#336	D₅₉	1	1	yes	characteristic
113	C₅₉	#1927	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	59	1	no	—
115	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
116	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	59	1	no	—
118	C₂×C₄⋊C₄	—	59	1	no	—
119	C₂×C₄⋊C₄	—	59	1	no	—
120	C₂×C₄²	—	118	1	no	—
121	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
122	#106	—	118	2	no	—
123	#103	—	59	4	no	—
124	#103	—	118	2	no	—
125	#95	—	118	1	no	—
126	#95	—	118	1	no	—
127	#95	—	118	1	no	—
128	C₂×Q₈	—	59	2	no	—
129	C₂×Q₈	—	118	1	no	—
130	C₂×Q₈	—	118	1	no	—
131	C₂×Q₈	—	118	1	no	—
132	C₂×Q₈	—	118	1	no	—
133	C₂×Q₈	—	118	1	no	—
134	C₂²×C₄	—	118	1	no	—
135	C₂²×C₄	—	118	1	no	—
136	C₂²×C₄	—	118	1	no	—
137	C₂²×C₄	—	118	1	no	—
138	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
139	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	C₄×D₅₉	1	2	yes	—
147	C₄⋊C₄	—	59	2	no	—
148	C₄⋊C₄	—	59	2	no	—
149	C₄⋊C₄	—	118	1	no	—
150	C₄⋊C₄	—	118	1	no	—
151	C₄⋊C₄	—	118	1	no	—
152	C₄⋊C₄	—	118	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	236	2	no	—
156	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
157	Q₈	—	118	2	no	—
158	Q₈	—	118	2	no	—
159	Q₈	—	236	1	no	—
160	Q₈	—	236	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	118	2	no	—
167	C₂×C₄	—	118	2	no	—
168	C₂×C₄	—	118	2	no	—
169	C₂×C₄	—	236	1	no	—
170	C₂×C₄	—	236	1	no	—
171	C₂×C₄	—	236	1	no	—
172	C₂×C₄	—	236	1	no	—
173	C₂×C₄	—	236	1	no	—
174	C₂×C₄	—	236	1	no	—
175	C₂×C₄	—	236	1	no	—
176	C₂×C₄	—	236	1	no	—
177	C₂×C₄	—	236	1	no	—
178	C₂×C₄	#11745660	1	2	yes	—
179	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
180	C₂×C₄	#11745685	1	1	yes	characteristic
181	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
182	C₂×C₄	#11745687	1	1	yes	characteristic
183	C₂×C₄	#11745687	1	1	yes	characteristic
184	C₂×C₄	#11745689	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	4	1	no	—
187	C₂²	C₄×D₂₃₆	1	1	yes	characteristic
188	C₂²	#14273702	1	1	yes	characteristic
189	C₂²	#14273711	1	1	yes	characteristic
190	C₂²	C₂×#11745660	1	1	yes	characteristic
191	C₂²	#14273749	1	1	yes	characteristic
192	C₂²	#14273760	1	1	yes	characteristic
193	C₂²	#14273760	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	4	1	no	—
196	C₄	—	472	1	no	—
197	C₄	—	236	2	no	—
198	C₄	—	236	2	no	—
199	C₄	—	236	2	no	—
200	C₄	#14273685	1	1	yes	characteristic
201	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
202	C₄	#14273713	1	1	yes	characteristic
203	C₄	#14273722	1	1	yes	characteristic
204	C₂	#16837294	1	1	yes	characteristic
205	C₂	C₄×#11745660	1	1	yes	characteristic
206	C₂	#16837510	1	1	yes	characteristic
207	C₂	#16837608	1	1	yes	characteristic
208	C₂	#16837625	1	1	yes	characteristic
209	C₂	#16837733	1	1	yes	characteristic
210	C₂	#16837744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273659

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal, characteristic

C₂×#14273652

C₂

yes

maximal, characteristic

#16837308

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₉

C₂

yes

maximal, characteristic

#16837165

C₂

yes

maximal, characteristic

C₅₉×#336

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄×Dic₅₉

—

#14273659

—

#14273652

—

#14273652

C₄

yes

complemented

C₅₉⋊#95

—

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄⋊Dic₅₉

D₄

yes

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

C₂×C₄×C₅₉

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

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

C₄×D₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

C₂×SD₁₆

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

100

C₂×C₅₉

C₄×SD₁₆

yes

characteristic

101

C₂×C₅₉

#441

yes

characteristic

102

C₂×C₅₉

#461

yes

characteristic

103

C₂×C₅₉

#464

yes

characteristic

104

C₂×C₅₉

#483

yes

characteristic

105

C₂×C₅₉

#485

yes

characteristic

106

C₂×#106

—

107

C₂×#103

—

108

C₂×#103

—

109

#391

—

110

C₄×C₄⋊C₄

—

111

#340

—

112

#336

D₅₉

yes

characteristic

113

C₅₉

#1927

yes

characteristic, complemented, Sylow

114

C₂²×Q₈

—

115

C₂²×C₈

C₂×D₅₉

yes

characteristic

116

C₂²×C₈

C₂×D₅₉

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

—

118

—

121

C₂×C₄²

C₂×D₅₉

yes

characteristic

122

#106

—

118

—

123

#103

—

124

#103

—

118

—

125

#95

—

118

—

126

#95

—

118

—

127

#95

—

118

—

128

C₂×Q₈

—

129

C₂×Q₈

—

118

—

130

C₂×Q₈

—

118

—

131

C₂×Q₈

—

118

—

132

C₂×Q₈

—

118

—

133

C₂×Q₈

—

118

—

134

C₂²×C₄

—

118

—

135

C₂²×C₄

—

118

—

136

C₂²×C₄

—

118

—

137

C₂²×C₄

—

118

—

138

C₂²×C₄

D₂₃₆

yes

characteristic

139

C₂²×C₄

D₂₃₆

yes

characteristic

140

C₂²×C₄

C₂²×D₅₉

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

C₄×D₅₉

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

118

—

150

C₄⋊C₄

—

118

—

151

C₄⋊C₄

—

118

—

152

C₄⋊C₄

—

118

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

236

—

156

C₂³

C₂×D₂₃₆

yes

characteristic

157

Q₈

—

118

—

158

Q₈

—

118

—

159

Q₈

—

236

—

160

Q₈

—

236

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

118

—

167

C₂×C₄

—

118

—

168

C₂×C₄

—

118

—

169

C₂×C₄

—

236

—

170

C₂×C₄

—

236

—

171

C₂×C₄

—

236

—

172

C₂×C₄

—

236

—

173

C₂×C₄

—

236

—

174

C₂×C₄

—

236

—

175

C₂×C₄

—

236

—

176

C₂×C₄

—

236

—

177

C₂×C₄

—

236

—

178

C₂×C₄

#11745660

yes

—

179

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

180

C₂×C₄

#11745685

yes

characteristic

181

C₂×C₄

D₄×D₅₉

yes

characteristic

182

C₂×C₄

#11745687

yes

characteristic

183

C₂×C₄

#11745687

yes

characteristic

184

C₂×C₄

#11745689

yes

characteristic

185

C₈

—

186

C₈

—

187

C₂²

C₄×D₂₃₆

yes

characteristic

188

C₂²

#14273702

yes

characteristic

189

C₂²

#14273711

yes

characteristic

190

C₂²

C₂×#11745660

yes

characteristic

191

C₂²

#14273749

yes

characteristic

192

C₂²

#14273760

yes

characteristic

193

C₂²

#14273760

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

472

—

197

C₄

—

236

—

198

C₄

—

236

—

199

C₄

—

236

—

200

C₄

#14273685

yes

characteristic

201

C₄

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

202

C₄

#14273713

yes

characteristic

203

C₄

#14273722

yes

characteristic

204

C₂

#16837294

yes

characteristic

205

C₂

C₄×#11745660

yes

characteristic

206

C₂

#16837510

yes

characteristic

207

C₂

#16837608

yes

characteristic

208

C₂

#16837625

yes

characteristic

209

C₂

#16837733

yes

characteristic

210

C₂

#16837744

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#336	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#336	D₅₉
hypocenter	C₅₉	#1927
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#336

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
15. #14273652	C₄	semidirect
113. C₅₉	#1927	semidirect

H ≅ G/N

type

15. #14273652

C₄

semidirect

113. C₅₉

#1927

semidirect

Subgroups — #21424731

Conjugacy classes of subgroups

Complements