Group #21424909 — subgroups

Subgroups — #21424909

Order 7552.

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

normal subgroups only

This group has 344 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×#14273716	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273651	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14273649	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273647	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273637	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837271	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#103×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₃₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
10	#14273716	—	2	2	no	—
11	#14273716	C₄	1	4	yes	complemented
12	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₅₉⋊C₄	—	2	2	no	—
14	C₂×(C₂×D₅₉)⋊C₄	—	2	2	no	—
15	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
16	#14273651	—	2	2	no	—
17	#14273649	—	2	2	no	—
18	#14273647	—	2	2	no	—
19	#14273637	—	2	2	no	—
20	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
23	#103×C₅₉	—	2	2	no	—
24	C₂³×D₅₉	—	2	1	no	—
25	C₂×Q₈×C₅₉	—	2	1	no	—
26	C₂×Q₈×C₅₉	—	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	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₅₉	—	2	1	no	—
32	C₂²×C₄×C₅₉	—	2	1	no	—
33	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₅₉	—	2	1	no	—
35	C₂²×Dic₅₉	D₄	1	2	yes	—
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	—	2	1	no	—
41	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₅₉	—	2	1	no	—
43	C₂×C₈×C₅₉	—	2	1	no	—
44	C₂×C₈×C₅₉	D₄	1	2	yes	—
45	C₄⋊C₄×C₅₉	—	2	1	no	—
46	C₄⋊C₄×C₅₉	—	2	1	no	—
47	(C₂×D₅₉)⋊C₄	—	2	8	no	—
48	C₄⋊Dic₅₉	—	2	1	no	—
49	C₄⋊Dic₅₉	—	2	1	no	—
50	Dic₅₉⋊C₄	—	4	4	no	—
51	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
52	C₄×Dic₅₉	D₄	1	2	yes	—
53	C₂×C₅₉⋊C₈	—	2	1	no	—
54	C₂×C₅₉⋊C₈	—	2	1	no	—
55	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
56	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×D₅₉	—	2	2	no	—
58	C₂²×D₅₉	—	4	1	no	—
59	C₂²×D₅₉	—	4	1	no	—
60	C₂²×D₅₉	—	4	2	no	—
61	C₂²×D₅₉	—	4	2	no	—
62	Q₈×C₅₉	—	2	2	no	—
63	Q₈×C₅₉	—	2	2	no	—
64	Q₈×C₅₉	—	4	1	no	—
65	Q₈×C₅₉	—	4	1	no	—
66	C₂×C₄×C₅₉	—	2	2	no	—
67	C₂×C₄×C₅₉	—	4	1	no	—
68	C₂×C₄×C₅₉	—	4	1	no	—
69	C₂×C₄×C₅₉	—	4	1	no	—
70	C₂×C₄×C₅₉	—	4	2	no	—
71	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×D₄	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	#41	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
77	C₂×Dic₅₉	—	2	2	no	—
78	C₂×Dic₅₉	—	2	2	no	—
79	C₂×Dic₅₉	—	4	1	no	—
80	C₂×Dic₅₉	—	4	2	no	—
81	C₂×Dic₅₉	C₂²⋊C₄	1	4	yes	—
82	D₂₃₆	—	2	2	no	—
83	D₂₃₆	—	2	2	no	—
84	D₂₃₆	—	4	1	no	—
85	D₂₃₆	—	4	1	no	—
86	C₈×C₅₉	—	2	2	no	—
87	C₅₉⋊C₈	—	2	2	no	—
88	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₅₉	#124	1	1	yes	characteristic
90	C₂²×C₅₉	#127	1	1	yes	characteristic
91	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
92	C₂²×C₅₉	#137	1	2	yes	—
93	C₂×D₅₉	—	4	2	no	—
94	C₂×D₅₉	—	4	4	no	—
95	C₂×D₅₉	—	8	2	no	—
96	C₂×D₅₉	—	8	1	no	—
97	C₂×D₅₉	—	8	1	no	—
98	C₄×C₅₉	—	4	2	no	—
99	C₄×C₅₉	—	8	1	no	—
100	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
101	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
102	C₄×C₅₉	#121	1	1	yes	characteristic
103	C₄×C₅₉	#121	1	1	yes	characteristic
104	Dic₅₉	—	2	4	no	—
105	Dic₅₉	—	8	1	no	—
106	#1959	—	59	1	no	maximal, Sylow
107	C₂×C₅₉	#390	1	1	yes	characteristic
108	C₂×C₅₉	#417	1	2	yes	—
109	C₂×C₅₉	#489	1	2	yes	—
110	C₂×C₅₉	#496	1	1	yes	characteristic
111	C₂×C₅₉	#497	1	1	yes	characteristic
112	D₅₉	—	8	2	no	—
113	C₂×#124	—	59	1	no	—
114	C₂×#103	—	59	1	no	—
115	C₂×#103	D₅₉	1	1	yes	characteristic, complemented
116	C₂×#102	—	59	1	no	—
117	C₂×#102	—	59	1	no	—
118	C₂×#97	—	59	1	no	—
119	#384	—	59	1	no	—
120	C₅₉	#1959	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
122	C₂²×D₄	—	59	1	no	—
123	C₂²×C₈	—	59	1	no	—
124	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
125	#124	—	59	4	no	—
126	#124	—	118	2	no	—
127	C₂×C₄⋊C₄	—	59	1	no	—
128	C₂×C₄⋊C₄	—	118	2	no	—
129	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	—	118	2	no	—
131	C₂×C₄²	—	59	1	no	—
132	#103	—	2	2	no	—
133	#103	—	118	2	no	—
134	#102	—	118	2	no	—
135	#102	—	118	2	no	—
136	#97	—	118	2	no	—
137	C₂⁴	—	118	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×Q₈	C₄×D₅₉	1	2	yes	—
144	C₂×D₄	—	59	2	no	—
145	C₂×D₄	—	118	1	no	—
146	C₂×D₄	—	118	1	no	—
147	C₂×D₄	—	118	1	no	—
148	C₂×D₄	—	118	1	no	—
149	C₂×D₄	—	118	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	59	2	no	—
153	C₂²×C₄	—	118	1	no	—
154	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	59	2	no	—
158	C₂×C₈	—	118	1	no	—
159	C₂×C₈	—	118	1	no	—
160	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
161	C₄⋊C₄	—	2	1	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	118	1	no	—
164	C₄⋊C₄	—	118	1	no	—
165	C₄⋊C₄	—	236	4	no	—
166	C₂²⋊C₄	—	118	8	no	—
167	C₄²	—	59	2	no	—
168	C₄²	—	59	2	no	—
169	C₂³	—	118	2	no	—
170	C₂³	—	236	1	no	—
171	C₂³	—	236	1	no	—
172	C₂³	—	236	2	no	—
173	C₂³	—	236	2	no	—
174	C₂³	D₄×D₅₉	1	1	yes	characteristic
175	Q₈	—	2	2	no	—
176	Q₈	—	2	2	no	—
177	Q₈	—	4	1	no	—
178	Q₈	—	4	1	no	—
179	D₄	—	118	2	no	—
180	D₄	—	118	2	no	—
181	D₄	—	236	1	no	—
182	D₄	—	236	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	59	4	no	—
189	C₂×C₄	—	118	2	no	—
190	C₂×C₄	—	118	2	no	—
191	C₂×C₄	—	236	1	no	—
192	C₂×C₄	—	236	2	no	—
193	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
194	C₂×C₄	#11745685	1	1	yes	characteristic
195	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
196	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
197	C₂×C₄	#11745687	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	118	2	no	—
201	C₂²	—	472	2	no	—
202	C₂²	—	472	1	no	—
203	C₂²	—	472	1	no	—
204	C₂²	—	236	4	no	—
205	C₂²	—	236	2	no	—
206	C₂²	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
207	C₂²	#14273713	1	1	yes	characteristic
208	C₂²	#14273723	1	1	yes	characteristic
209	C₂²	#14273754	1	2	yes	—
210	C₂²	#14273759	1	2	yes	—
211	C₄	—	4	2	no	—
212	C₄	—	8	1	no	—
213	C₄	—	118	4	no	—
214	C₄	—	472	1	no	—
215	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
216	C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
217	C₄	#14273699	1	1	yes	characteristic
218	C₄	#14273700	1	1	yes	characteristic
219	C₂	—	472	2	no	—
220	C₂	#16837302	1	1	yes	characteristic
221	C₂	#16837402	1	2	yes	—
222	C₂	#16837763	1	2	yes	—
223	C₂	#16837788	1	1	yes	characteristic
224	C₂	#16837795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273716

C₂

yes

maximal, characteristic

C₂×#14273651

C₂

yes

maximal, characteristic, complemented

C₂×#14273649

C₂

yes

maximal, characteristic

C₂×#14273647

C₂

yes

maximal, characteristic

C₂×#14273637

C₂

yes

maximal, characteristic, complemented

#16837271

C₂

yes

maximal, characteristic, complemented

C₂×#103×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273716

—

#14273716

C₄

yes

complemented

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×Dic₅₉⋊C₄

—

C₂×(C₂×D₅₉)⋊C₄

—

C₂×C₄×Dic₅₉

C₂²

yes

—

—

—

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×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

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

D₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂²×D₅₉

—

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

—

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

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²⋊C₄

yes

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#137

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

100

C₄×C₅₉

C₄×D₄

yes

characteristic

101

C₄×C₅₉

C₄×D₄

yes

characteristic

102

C₄×C₅₉

#121

yes

characteristic

103

C₄×C₅₉

#121

yes

characteristic

104

Dic₅₉

—

105

Dic₅₉

—

106

#1959

—

maximal, Sylow

107

C₂×C₅₉

#390

yes

characteristic

108

C₂×C₅₉

#417

yes

—

109

C₂×C₅₉

#489

yes

—

110

C₂×C₅₉

#496

yes

characteristic

111

C₂×C₅₉

#497

yes

characteristic

112

D₅₉

—

113

C₂×#124

—

114

C₂×#103

—

115

C₂×#103

D₅₉

yes

characteristic, complemented

116

C₂×#102

—

117

C₂×#102

—

118

C₂×#97

—

119

#384

—

120

C₅₉

#1959

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

C₂×D₅₉

yes

characteristic

122

C₂²×D₄

—

123

C₂²×C₈

—

124

C₂²×C₈

C₂×D₅₉

yes

characteristic

125

#124

—

126

#124

—

118

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

118

—

129

C₂×C₄⋊C₄

C₂×D₅₉

yes

characteristic

130

C₂×C₂²⋊C₄

—

118

—

131

C₂×C₄²

—

132

#103

—

133

#103

—

118

—

134

#102

—

118

—

135

#102

—

118

—

136

#97

—

118

—

137

C₂⁴

—

118

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

C₄×D₅₉

yes

—

144

C₂×D₄

—

145

C₂×D₄

—

118

—

146

C₂×D₄

—

118

—

147

C₂×D₄

—

118

—

148

C₂×D₄

—

118

—

149

C₂×D₄

—

118

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

118

—

154

C₂²×C₄

C₂²×D₅₉

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

118

—

159

C₂×C₈

—

118

—

160

C₂×C₈

C₂²⋊D₅₉

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

118

—

164

C₄⋊C₄

—

118

—

165

C₄⋊C₄

—

236

—

166

C₂²⋊C₄

—

118

—

167

C₄²

—

168

C₄²

—

169

C₂³

—

118

—

170

C₂³

—

236

—

171

C₂³

—

236

—

172

C₂³

—

236

—

173

C₂³

—

236

—

174

C₂³

D₄×D₅₉

yes

characteristic

175

Q₈

—

176

Q₈

—

177

Q₈

—

178

Q₈

—

179

D₄

—

118

—

180

D₄

—

118

—

181

D₄

—

236

—

182

D₄

—

236

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

118

—

190

C₂×C₄

—

118

—

191

C₂×C₄

—

236

—

192

C₂×C₄

—

236

—

193

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

194

C₂×C₄

#11745685

yes

characteristic

195

C₂×C₄

D₄×D₅₉

yes

characteristic

196

C₂×C₄

D₄×D₅₉

yes

characteristic

197

C₂×C₄

#11745687

yes

characteristic

198

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

199

C₈

—

200

C₈

—

118

—

201

C₂²

—

472

—

202

C₂²

—

472

—

203

C₂²

—

472

—

204

C₂²

—

236

—

205

C₂²

—

236

—

206

C₂²

C₂²⋊C₄×D₅₉

yes

characteristic

207

C₂²

#14273713

yes

characteristic

208

C₂²

#14273723

yes

characteristic

209

C₂²

#14273754

yes

—

210

C₂²

#14273759

yes

—

211

C₄

—

212

C₄

—

213

C₄

—

118

—

214

C₄

—

472

—

215

C₄

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

216

C₄

C₄×C₂²⋊D₅₉

yes

characteristic

217

C₄

#14273699

yes

characteristic

218

C₄

#14273700

yes

characteristic

219

C₂

—

472

—

220

C₂

#16837302

yes

characteristic

221

C₂

#16837402

yes

—

222

C₂

#16837763

yes

—

223

C₂

#16837788

yes

characteristic

224

C₂

#16837795

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#103×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×#103	D₅₉
hypocenter	C₅₉	#1959
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×#103×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
2. C₂×#14273651	C₂	semidirect
5. C₂×#14273637	C₂	semidirect
6. #16837271	C₂	semidirect
7. C₂×#103×C₅₉	C₂	semidirect
11. #14273716	C₄	semidirect
115. C₂×#103	D₅₉	semidirect
120. C₅₉	#1959	semidirect

H ≅ G/N

type

2. C₂×#14273651

C₂

semidirect

5. C₂×#14273637

C₂

semidirect

6. #16837271

C₂

semidirect

7. C₂×#103×C₅₉

C₂

semidirect

11. #14273716

C₄

semidirect

115. C₂×#103

D₅₉

semidirect

120. C₅₉

#1959

semidirect

Subgroups — #21424909

Conjugacy classes of subgroups

Complements