Group #21424836 — subgroups

Subgroups — #21424836

Order 7552.

240 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 99 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×#14273660	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273658	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273638	C₂	1	1	yes	maximal, characteristic
4	#16837300	C₂	1	1	yes	maximal, characteristic
5	#16837164	C₂	1	1	yes	maximal, characteristic
6	#16837164	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#389	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₈	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×Dic₅₉⋊C₄	—	2	1	no	—
13	#14273660	—	2	2	no	—
14	#14273658	C₄	1	4	yes	complemented
15	#14273638	—	2	2	no	—
16	C₅₉⋊#95	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
18	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	#95×C₅₉	—	2	1	no	—
22	C₂²×C₄×C₅₉	—	2	1	no	—
23	C₂²×C₄×C₅₉	—	2	1	no	—
24	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₉	—	2	1	no	—
28	C₂²×Dic₅₉	—	2	1	no	—
29	C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₄⋊C₄×C₅₉	—	4	2	no	—
33	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
34	C₄²×C₅₉	—	2	1	no	—
35	C₄²×C₅₉	—	2	1	no	—
36	C₄⋊Dic₅₉	—	2	1	no	—
37	C₄⋊Dic₅₉	—	2	1	no	—
38	C₄⋊Dic₅₉	—	2	1	no	—
39	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
40	Dic₅₉⋊C₄	—	4	2	no	—
41	C₂×C₅₉⋊C₈	—	2	1	no	—
42	C₂×C₅₉⋊C₈	—	2	1	no	—
43	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
44	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
45	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
46	C₂×C₅₉⋊C₈	Q₈	1	2	yes	—
47	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₉	—	2	2	no	—
49	C₂×C₄×C₅₉	—	2	1	no	—
50	C₂×C₄×C₅₉	—	2	1	no	—
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	—	4	1	no	—
56	C₂×C₄×C₅₉	—	4	1	no	—
57	C₂×C₄×C₅₉	—	4	1	no	—
58	C₂×C₄×C₅₉	—	4	1	no	—
59	C₂×C₄×C₅₉	—	4	1	no	—
60	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
66	C₂×Dic₅₉	—	2	2	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	C₅₉⋊C₈	—	2	2	no	—
73	C₅₉⋊C₈	C₄⋊C₄	1	4	yes	complemented
74	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
75	C₂²×C₅₉	#121	1	1	yes	characteristic
76	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂²×C₅₉	#137	1	1	yes	characteristic
80	C₂²×C₅₉	#137	1	1	yes	characteristic
81	C₄×C₅₉	—	4	2	no	—
82	C₄×C₅₉	—	4	1	no	—
83	C₄×C₅₉	—	4	1	no	—
84	C₄×C₅₉	—	8	1	no	—
85	C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
87	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
88	C₄×C₅₉	#125	1	1	yes	characteristic
89	Dic₅₉	—	4	2	no	—
90	Dic₅₉	—	8	1	no	—
91	#1944	—	59	1	no	maximal, Sylow
92	C₂×C₅₉	#389	1	1	yes	characteristic
93	C₂×C₅₉	#440	1	1	yes	characteristic
94	C₂×C₅₉	#440	1	1	yes	characteristic
95	C₂×C₅₉	#469	1	1	yes	characteristic
96	C₂×C₅₉	#470	1	1	yes	characteristic
97	C₂×C₅₉	#501	1	1	yes	characteristic
98	C₂×C₅₉	#501	1	1	yes	characteristic
99	C₂×#107	—	59	1	no	—
100	C₂×#106	—	59	1	no	—
101	C₂×#97	—	59	1	no	—
102	#389	—	59	1	no	—
103	#389	D₅₉	1	1	yes	characteristic
104	#340	—	59	1	no	—
105	#340	—	59	1	no	—
106	C₅₉	#1944	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	—	59	1	no	—
108	C₂²×C₈	—	59	1	no	—
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₄⋊C₄	—	59	1	no	—
111	C₂×C₄⋊C₄	—	59	1	no	—
112	C₂×C₄⋊C₄	—	118	1	no	—
113	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
116	#107	—	118	2	no	—
117	#106	—	59	4	no	—
118	#97	—	118	2	no	—
119	#95	—	2	1	no	—
120	#95	—	118	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	118	1	no	—
124	C₂²×C₄	—	118	1	no	—
125	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
128	C₂×C₈	—	59	2	no	—
129	C₂×C₈	—	59	2	no	—
130	C₂×C₈	—	59	2	no	—
131	C₂×C₈	—	59	2	no	—
132	C₂×C₈	—	118	1	no	—
133	C₂×C₈	—	118	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	59	2	no	—
139	C₄⋊C₄	—	118	1	no	—
140	C₄⋊C₄	—	118	1	no	—
141	C₄⋊C₄	—	118	1	no	—
142	C₄⋊C₄	—	236	2	no	—
143	C₄⋊C₄	C₄×D₅₉	1	2	yes	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	1	no	—
146	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	118	2	no	—
160	C₂×C₄	—	236	1	no	—
161	C₂×C₄	—	236	1	no	—
162	C₂×C₄	—	236	1	no	—
163	C₂×C₄	—	236	1	no	—
164	C₂×C₄	—	236	1	no	—
165	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
166	C₂×C₄	#11745685	1	1	yes	characteristic
167	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
168	C₂×C₄	#11745687	1	1	yes	characteristic
169	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
170	C₂×C₄	Q₈×D₅₉	1	1	yes	characteristic
171	C₈	—	59	4	no	—
172	C₈	—	118	2	no	—
173	C₂²	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
174	C₂²	#14273699	1	1	yes	characteristic
175	C₂²	#14273725	1	1	yes	characteristic
176	C₂²	#14273752	1	1	yes	characteristic
177	C₂²	#14273753	1	1	yes	characteristic
178	C₂²	#14273757	1	1	yes	characteristic
179	C₂²	#14273758	1	1	yes	characteristic
180	C₄	—	4	1	no	—
181	C₄	—	4	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	—	472	1	no	—
185	C₄	—	236	2	no	—
186	C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
187	C₄	#14273692	1	1	yes	characteristic
188	C₄	#14273705	1	1	yes	characteristic
189	C₄	#14273717	1	1	yes	characteristic
190	C₂	#16837297	1	1	yes	characteristic
191	C₂	#16837500	1	1	yes	characteristic
192	C₂	#16837502	1	1	yes	characteristic
193	C₂	#16837653	1	1	yes	characteristic
194	C₂	#16837658	1	1	yes	characteristic
195	C₂	#16837808	1	1	yes	characteristic
196	C₂	#16837812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273660

C₂

yes

maximal, characteristic

C₂×#14273658

C₂

yes

maximal, characteristic

C₂×#14273638

C₂

yes

maximal, characteristic

#16837300

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal, characteristic

#16837164

C₂

yes

maximal, characteristic

C₅₉×#389

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

#14273660

—

#14273658

C₄

yes

complemented

#14273638

—

C₅₉⋊#95

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂×C₅₉⋊C₈

Q₈

yes

—

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×C₄×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×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂²×C₅₉

#121

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

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

C₄×Q₈

yes

characteristic

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#125

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

100

C₂×#106

—

101

C₂×#97

—

102

#389

—

103

#389

D₅₉

yes

characteristic

104

#340

—

105

#340

—

106

C₅₉

#1944

yes

characteristic, complemented, Sylow

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

118

—

113

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

114

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

115

C₂×C₄²

C₂×D₅₉

yes

characteristic

116

#107

—

118

—

117

#106

—

118

#97

—

118

—

119

#95

—

120

#95

—

118

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

118

—

124

C₂²×C₄

—

118

—

125

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

126

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

127

C₂²×C₄

C₂²×D₅₉

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

118

—

133

C₂×C₈

—

118

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

118

—

140

C₄⋊C₄

—

118

—

141

C₄⋊C₄

—

118

—

142

C₄⋊C₄

—

236

—

143

C₄⋊C₄

C₄×D₅₉

yes

—

144

C₄²

—

145

C₄²

—

146

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

118

—

160

C₂×C₄

—

236

—

161

C₂×C₄

—

236

—

162

C₂×C₄

—

236

—

163

C₂×C₄

—

236

—

164

C₂×C₄

—

236

—

165

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

166

C₂×C₄

#11745685

yes

characteristic

167

C₂×C₄

D₄×D₅₉

yes

characteristic

168

C₂×C₄

#11745687

yes

characteristic

169

C₂×C₄

Q₈×D₅₉

yes

characteristic

170

C₂×C₄

Q₈×D₅₉

yes

characteristic

171

C₈

—

172

C₈

—

118

—

173

C₂²

C₄×C₂²⋊D₅₉

yes

characteristic

174

C₂²

#14273699

yes

characteristic

175

C₂²

#14273725

yes

characteristic

176

C₂²

#14273752

yes

characteristic

177

C₂²

#14273753

yes

characteristic

178

C₂²

#14273757

yes

characteristic

179

C₂²

#14273758

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

472

—

185

C₄

—

236

—

186

C₄

C₄⋊C₄×D₅₉

yes

characteristic

187

C₄

#14273692

yes

characteristic

188

C₄

#14273705

yes

characteristic

189

C₄

#14273717

yes

characteristic

190

C₂

#16837297

yes

characteristic

191

C₂

#16837500

yes

characteristic

192

C₂

#16837502

yes

characteristic

193

C₂

#16837653

yes

characteristic

194

C₂

#16837658

yes

characteristic

195

C₂

#16837808

yes

characteristic

196

C₂

#16837812

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#389	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#389	D₅₉
hypocenter	C₅₉	#1944
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#389

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
14. #14273658	C₄	semidirect
73. C₅₉⋊C₈	C₄⋊C₄	semidirect
106. C₅₉	#1944	semidirect

H ≅ G/N

type

14. #14273658

C₄

semidirect

73. C₅₉⋊C₈

C₄⋊C₄

semidirect

106. C₅₉

#1944

semidirect

Subgroups — #21424836

Conjugacy classes of subgroups

Complements