Group #21424486 — subgroups

Subgroups — #21424486

Order 7552.

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

normal subgroups only

This group has 324 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	C₂×#14273725	C₂	1	1	yes	maximal, characteristic
2	#16837879	C₂	1	1	yes	maximal, characteristic
3	#16837411	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273645	C₂	1	1	yes	maximal, characteristic
5	#16837163	C₂	1	1	yes	maximal, characteristic
6	#16837161	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#419	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	C₂×#11745664	C₂²	1	1	yes	characteristic
10	#14273725	—	2	2	no	—
11	#14273725	C₄	1	4	yes	complemented
12	C₄×Dic₁₁₈	—	2	2	no	—
13	#14273692	—	2	2	no	—
14	#14273686	C₂²	1	1	yes	characteristic
15	#14273683	—	2	2	no	—
16	C₂×Dic₅₉⋊C₄	—	2	2	no	—
17	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
18	#14273651	—	2	2	no	—
19	#14273645	—	2	2	no	—
20	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	#117×C₅₉	C₂²	1	1	yes	characteristic
23	#103×C₅₉	—	2	2	no	—
24	C₂×Q₈×C₅₉	—	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	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₅₉	—	2	1	no	—
31	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅₉	—	2	1	no	—
33	C₂²×Dic₅₉	D₄	1	2	yes	—
34	C₂×Dic₁₁₈	—	2	1	no	—
35	C₂×Dic₁₁₈	—	2	1	no	—
36	C₂×Dic₁₁₈	—	2	1	no	—
37	C₂×Dic₁₁₈	—	2	1	no	—
38	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
39	C₂×Dic₁₁₈	D₄	1	2	yes	—
40	M₁₆×C₅₉	—	2	1	no	—
41	M₁₆×C₅₉	—	2	1	no	—
42	C₂×C₈×C₅₉	—	2	1	no	—
43	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
44	C₂²⋊C₄×C₅₉	—	2	1	no	—
45	C₄²×C₅₉	—	2	1	no	—
46	C₂²⋊Dic₅₉	—	2	1	no	—
47	C₄⋊Dic₅₉	D₄	1	2	yes	—
48	Dic₅₉⋊C₄	—	2	4	no	—
49	Dic₅₉⋊C₄	—	2	8	no	—
50	C₄×Dic₅₉	—	2	4	no	—
51	C₄×Dic₅₉	—	2	1	no	—
52	C₄×Dic₅₉	—	2	1	no	—
53	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
54	#11745664	—	2	1	no	—
55	#11745664	—	2	1	no	—
56	C₂×C₅₉⋊C₈	—	2	1	no	—
57	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
58	Q₈×C₅₉	—	2	2	no	—
59	Q₈×C₅₉	—	2	2	no	—
60	Q₈×C₅₉	—	4	1	no	—
61	Q₈×C₅₉	—	4	1	no	—
62	C₂×C₄×C₅₉	—	2	2	no	—
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	4	1	no	—
65	C₂×C₄×C₅₉	—	4	1	no	—
66	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
72	C₂×Dic₅₉	—	2	4	no	—
73	C₂×Dic₅₉	—	2	2	no	—
74	C₂×Dic₅₉	—	2	2	no	—
75	C₂×Dic₅₉	—	2	2	no	—
76	C₂×Dic₅₉	—	2	2	no	—
77	C₂×Dic₅₉	C₂²⋊C₄	1	4	yes	—
78	Dic₁₁₈	—	2	2	no	—
79	Dic₁₁₈	—	2	2	no	—
80	Dic₁₁₈	—	2	2	no	—
81	Dic₁₁₈	—	2	2	no	—
82	C₈×C₅₉	—	4	1	no	—
83	C₅₉⋊C₈	—	4	1	no	—
84	C₂²×C₅₉	—	2	2	no	—
85	C₂²×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₅₉	#120	1	1	yes	characteristic
87	C₂²×C₅₉	#123	1	1	yes	characteristic
88	C₄×C₅₉	—	4	2	no	—
89	C₄×C₅₉	—	4	2	no	—
90	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
91	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
92	C₄×C₅₉	#121	1	1	yes	characteristic
93	C₄×C₅₉	#121	1	1	yes	characteristic
94	Dic₅₉	—	2	4	no	—
95	Dic₅₉	—	2	4	no	—
96	Dic₅₉	—	4	2	no	—
97	#1886	—	59	1	no	maximal, Sylow
98	C₂×C₅₉	—	2	1	no	—
99	C₂×C₅₉	—	2	1	no	—
100	C₂×C₅₉	#386	1	1	yes	characteristic
101	C₂×C₅₉	#456	1	2	yes	—
102	C₂×C₄⋊Q₈	—	59	1	no	—
103	#519	—	59	1	no	—
104	#419	—	59	1	no	—
105	#419	D₅₉	1	1	yes	characteristic
106	C₂×#101	—	59	1	no	—
107	#339	—	59	1	no	—
108	#339	—	59	1	no	—
109	C₅₉	#1886	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	59	1	no	—
111	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
112	C₂×M₁₆	—	59	1	no	—
113	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
114	C₄⋊Q₈	—	59	4	no	—
115	C₄⋊Q₈	—	118	2	no	—
116	C₄×Q₈	—	118	2	no	—
117	C₄×Q₈	—	118	2	no	—
118	#117	—	59	1	no	—
119	#117	—	118	2	no	—
120	#117	C₂×D₅₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	118	2	no	—
122	C₂×C₄²	—	59	1	no	—
123	#103	—	2	2	no	—
124	#103	—	118	2	no	—
125	#101	—	118	2	no	—
126	C₂×Q₈	—	2	1	no	—
127	C₂×Q₈	—	2	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	59	2	no	—
132	C₂×Q₈	—	59	2	no	—
133	C₂×Q₈	—	118	1	no	—
134	C₂×Q₈	—	118	1	no	—
135	C₂×Q₈	—	118	1	no	—
136	C₂×Q₈	—	118	1	no	—
137	C₂×Q₈	C₄×D₅₉	1	2	yes	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	59	2	no	—
140	C₂²×C₄	—	118	1	no	—
141	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
142	M₁₆	—	2	1	no	—
143	M₁₆	—	2	1	no	—
144	M₁₆	—	118	1	no	—
145	M₁₆	—	118	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	118	1	no	—
148	C₄⋊C₄	—	59	2	no	—
149	C₄⋊C₄	—	118	8	no	—
150	C₄⋊C₄	—	118	4	no	—
151	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
152	C₂²⋊C₄	—	2	1	no	—
153	C₂²⋊C₄	—	118	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	59	2	no	—
156	C₄²	—	118	4	no	—
157	C₄²	—	118	1	no	—
158	C₄²	—	118	1	no	—
159	C₂³	D₄×D₅₉	1	1	yes	characteristic
160	Q₈	—	2	2	no	—
161	Q₈	—	2	2	no	—
162	Q₈	—	4	1	no	—
163	Q₈	—	4	1	no	—
164	Q₈	—	118	2	no	—
165	Q₈	—	118	2	no	—
166	Q₈	—	118	2	no	—
167	Q₈	—	118	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	59	4	no	—
173	C₂×C₄	—	118	4	no	—
174	C₂×C₄	—	118	2	no	—
175	C₂×C₄	—	118	2	no	—
176	C₂×C₄	—	118	2	no	—
177	C₂×C₄	—	118	2	no	—
178	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
179	C₂×C₄	#11745685	1	1	yes	characteristic
180	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
181	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
182	C₂×C₄	#11745687	1	1	yes	characteristic
183	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
184	C₈	—	4	1	no	—
185	C₈	—	236	1	no	—
186	C₂²	—	2	2	no	—
187	C₂²	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
188	C₂²	#14273695	1	1	yes	characteristic
189	C₂²	#14273709	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	118	4	no	—
193	C₄	—	118	4	no	—
194	C₄	—	236	2	no	—
195	C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
196	C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
197	C₄	#14273698	1	1	yes	characteristic
198	C₄	#14273700	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#16837278	1	1	yes	characteristic
202	C₂	#16837583	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273725

C₂

yes

maximal, characteristic

#16837879

C₂

yes

maximal, characteristic

#16837411

C₂

yes

maximal, characteristic

C₂×#14273645

C₂

yes

maximal, characteristic

#16837163

C₂

yes

maximal, characteristic

#16837161

C₂

yes

maximal, characteristic

C₅₉×#419

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

#14273725

—

#14273725

C₄

yes

complemented

C₄×Dic₁₁₈

—

#14273692

—

#14273686

C₂²

yes

characteristic

#14273683

—

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273651

—

#14273645

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#117×C₅₉

C₂²

yes

characteristic

#103×C₅₉

—

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

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₄

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×Dic₁₁₈

D₄

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

D₄

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₂×C₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

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

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂²⋊C₄

yes

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#1886

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

100

C₂×C₅₉

#386

yes

characteristic

101

C₂×C₅₉

#456

yes

—

102

C₂×C₄⋊Q₈

—

103

#519

—

104

#419

—

105

#419

D₅₉

yes

characteristic

106

C₂×#101

—

107

#339

—

108

#339

—

109

C₅₉

#1886

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂²×Q₈

C₂×D₅₉

yes

characteristic

112

C₂×M₁₆

—

113

C₂×M₁₆

C₂×D₅₉

yes

characteristic

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

118

—

116

C₄×Q₈

—

118

—

117

C₄×Q₈

—

118

—

118

#117

—

119

#117

—

118

—

120

#117

C₂×D₅₉

yes

characteristic

121

C₂×C₄⋊C₄

—

118

—

122

C₂×C₄²

—

123

#103

—

124

#103

—

118

—

125

#101

—

118

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

118

—

134

C₂×Q₈

—

118

—

135

C₂×Q₈

—

118

—

136

C₂×Q₈

—

118

—

137

C₂×Q₈

C₄×D₅₉

yes

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

118

—

141

C₂²×C₄

C₂²×D₅₉

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

118

—

145

M₁₆

—

118

—

146

C₂×C₈

—

147

C₂×C₈

—

118

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

118

—

150

C₄⋊C₄

—

118

—

151

C₄⋊C₄

C₂²⋊D₅₉

yes

—

152

C₂²⋊C₄

—

153

C₂²⋊C₄

—

118

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

118

—

157

C₄²

—

118

—

158

C₄²

—

118

—

159

C₂³

D₄×D₅₉

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

118

—

165

Q₈

—

118

—

166

Q₈

—

118

—

167

Q₈

—

118

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

118

—

174

C₂×C₄

—

118

—

175

C₂×C₄

—

118

—

176

C₂×C₄

—

118

—

177

C₂×C₄

—

118

—

178

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

179

C₂×C₄

#11745685

yes

characteristic

180

C₂×C₄

D₄×D₅₉

yes

characteristic

181

C₂×C₄

D₄×D₅₉

yes

characteristic

182

C₂×C₄

#11745687

yes

characteristic

183

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

184

C₈

—

185

C₈

—

236

—

186

C₂²

—

187

C₂²

C₂²⋊C₄×D₅₉

yes

characteristic

188

C₂²

#14273695

yes

characteristic

189

C₂²

#14273709

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

118

—

193

C₄

—

118

—

194

C₄

—

236

—

195

C₄

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

196

C₄

C₄×C₂²⋊D₅₉

yes

characteristic

197

C₄

#14273698

yes

characteristic

198

C₄

#14273700

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#16837278

yes

characteristic

202

C₂

#16837583

yes

—

	subgroup	G/H
center	C₂²	#14273695
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#419	C₂
socle	C₂²×C₅₉	#120
hypercenter	#419	D₅₉
hypocenter	C₅₉	#1886
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14273695

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
11. #14273725	C₄	semidirect
109. C₅₉	#1886	semidirect

H ≅ G/N

type

11. #14273725

C₄

semidirect

109. C₅₉

#1886

semidirect

Subgroups — #21424486

Conjugacy classes of subgroups

Complements