Group #21425010 — subgroups

Subgroups — #21425010

Order 7552.

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

normal subgroups only

This group has 276 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	C₂²×#11745672	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273662	C₂	1	1	yes	maximal, characteristic
3	#16837410	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273646	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273645	C₂	1	1	yes	maximal, characteristic
6	#16837358	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×#11745672	—	2	1	no	—
10	C₂×#11745672	—	2	1	no	—
11	C₂×#11745672	—	2	2	no	—
12	C₂×#11745672	C₄	1	4	yes	complemented
13	C₂×#11745664	C₂²	1	1	yes	characteristic
14	C₂×#11745664	C₂²	1	1	yes	characteristic
15	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
16	#14273662	—	2	2	no	—
17	#14273650	—	2	2	no	—
18	#14273646	—	2	2	no	—
19	#14273645	—	2	2	no	—
20	#14273638	—	2	1	no	—
21	C₄×C₅₉⋊C₈	—	2	1	no	—
22	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	#117×C₅₉	C₂²	1	1	yes	characteristic
25	#103×C₅₉	—	2	2	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	—	2	1	no	—
31	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅₉	—	2	1	no	—
33	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₅₉	—	2	1	no	—
35	C₂×Dic₁₁₈	—	2	1	no	—
36	C₂×Dic₁₁₈	—	2	1	no	—
37	C₂×Dic₁₁₈	—	2	1	no	—
38	C₂×Dic₁₁₈	—	2	1	no	—
39	C₂×Dic₁₁₈	—	2	1	no	—
40	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
41	M₁₆×C₅₉	—	2	1	no	—
42	M₁₆×C₅₉	—	2	1	no	—
43	C₂×C₈×C₅₉	—	2	1	no	—
44	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
45	C₂²⋊C₄×C₅₉	—	2	1	no	—
46	C₄²×C₅₉	—	2	1	no	—
47	#11745672	—	2	4	no	—
48	#11745672	—	2	4	no	—
49	#11745664	—	2	1	no	—
50	#11745664	—	2	1	no	—
51	#11745664	—	2	1	no	—
52	#11745664	D₄	1	2	yes	—
53	C₂×C₅₉⋊C₈	—	2	1	no	—
54	C₂×C₅₉⋊C₈	—	2	1	no	—
55	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
56	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
57	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
58	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
59	C₂³×C₅₉	#41	1	1	yes	characteristic
60	Q₈×C₅₉	—	2	2	no	—
61	Q₈×C₅₉	—	2	2	no	—
62	Q₈×C₅₉	—	4	1	no	—
63	Q₈×C₅₉	—	4	1	no	—
64	C₂×C₄×C₅₉	—	2	2	no	—
65	C₂×C₄×C₅₉	—	2	2	no	—
66	C₂×C₄×C₅₉	—	4	1	no	—
67	C₂×C₄×C₅₉	—	4	1	no	—
68	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
74	C₂×Dic₅₉	—	2	2	no	—
75	C₂×Dic₅₉	—	4	1	no	—
76	C₂×Dic₅₉	—	4	1	no	—
77	Dic₁₁₈	—	2	2	no	—
78	Dic₁₁₈	—	2	2	no	—
79	Dic₁₁₈	—	4	1	no	—
80	Dic₁₁₈	—	4	1	no	—
81	C₈×C₅₉	—	4	1	no	—
82	C₅₉⋊C₈	—	2	2	no	—
83	C₅₉⋊C₈	—	4	1	no	—
84	C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
85	C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
86	C₂²×C₅₉	—	2	2	no	—
87	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₅₉	#121	1	1	yes	characteristic
89	C₂²×C₅₉	#124	1	1	yes	characteristic
90	C₄×C₅₉	—	4	2	no	—
91	C₄×C₅₉	—	4	2	no	—
92	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
94	C₄×C₅₉	#121	1	1	yes	characteristic
95	C₄×C₅₉	#127	1	1	yes	characteristic
96	Dic₅₉	—	4	2	no	—
97	#1979	—	59	1	no	maximal, Sylow
98	C₂×C₅₉	—	2	1	no	—
99	C₂×C₅₉	—	2	1	no	—
100	C₂×C₅₉	#390	1	1	yes	characteristic
101	C₂×C₅₉	#473	1	2	yes	—
102	C₂²×Q₁₆	—	59	1	no	—
103	C₂×#108	—	59	1	no	—
104	#419	—	59	1	no	—
105	#419	D₅₉	1	1	yes	characteristic
106	C₂×#101	—	59	1	no	—
107	C₂×#101	—	59	1	no	—
108	#405	—	59	1	no	—
109	C₅₉	#1979	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	59	1	no	—
111	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
112	C₂×Q₁₆	—	59	4	no	—
113	C₂×Q₁₆	—	118	1	no	—
114	C₂×Q₁₆	—	118	1	no	—
115	C₂×Q₁₆	—	118	2	no	—
116	C₂×M₁₆	—	59	1	no	—
117	C₂×M₁₆	—	59	1	no	—
118	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
119	C₂²×C₈	—	59	1	no	—
120	#117	C₂×D₅₉	1	1	yes	characteristic
121	#108	—	118	2	no	—
122	#103	—	2	2	no	—
123	#103	—	118	2	no	—
124	#101	—	118	2	no	—
125	#101	—	118	2	no	—
126	#97	—	118	1	no	—
127	C₄×C₈	—	118	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	59	2	no	—
134	C₂×Q₈	—	118	1	no	—
135	C₂×Q₈	—	118	1	no	—
136	C₂×Q₈	—	118	1	no	—
137	C₂×Q₈	—	118	1	no	—
138	C₂×Q₈	—	118	1	no	—
139	C₂×Q₈	C₄×D₅₉	1	2	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	118	1	no	—
142	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
143	Q₁₆	—	118	4	no	—
144	Q₁₆	—	118	4	no	—
145	M₁₆	—	2	1	no	—
146	M₁₆	—	2	1	no	—
147	M₁₆	—	59	2	no	—
148	M₁₆	—	118	1	no	—
149	M₁₆	—	118	1	no	—
150	M₁₆	—	118	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	59	2	no	—
153	C₂×C₈	—	59	1	no	—
154	C₂×C₈	—	59	1	no	—
155	C₂×C₈	—	59	2	no	—
156	C₂×C₈	—	118	1	no	—
157	C₂×C₈	—	118	1	no	—
158	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
159	C₂²⋊C₄	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₂³	#11745687	1	1	yes	characteristic
162	Q₈	—	2	2	no	—
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	4	1	no	—
166	Q₈	—	118	2	no	—
167	Q₈	—	118	2	no	—
168	Q₈	—	236	1	no	—
169	Q₈	—	236	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	118	2	no	—
175	C₂×C₄	—	236	1	no	—
176	C₂×C₄	—	236	1	no	—
177	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
178	C₂×C₄	#11745685	1	1	yes	characteristic
179	C₂×C₄	D₄×D₅₉	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	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
183	C₈	—	4	1	no	—
184	C₈	—	59	2	no	—
185	C₈	—	59	2	no	—
186	C₈	—	118	2	no	—
187	C₈	—	236	1	no	—
188	C₂²	—	2	2	no	—
189	C₂²	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
190	C₂²	#14273699	1	1	yes	characteristic
191	C₂²	#14273713	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	236	2	no	—
195	C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
196	C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
197	C₄	#14273700	1	1	yes	characteristic
198	C₄	#14273723	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#16837302	1	1	yes	characteristic
202	C₂	#16837675	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745672

C₂

yes

maximal, characteristic

C₂×#14273662

C₂

yes

maximal, characteristic

#16837410

C₂

yes

maximal, characteristic

C₂×#14273646

C₂

yes

maximal, characteristic

C₂×#14273645

C₂

yes

maximal, characteristic

#16837358

C₂

yes

maximal, characteristic

C₅₉×#419

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₈

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#11745664

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

—

—

—

—

C₄×C₅₉⋊C₈

—

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

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

C₂×C₄

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

—

—

—

—

—

D₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

#41

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

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₅₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₄×D₄

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1979

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

100

C₂×C₅₉

#390

yes

characteristic

101

C₂×C₅₉

#473

yes

—

102

C₂²×Q₁₆

—

103

C₂×#108

—

104

#419

—

105

#419

D₅₉

yes

characteristic

106

C₂×#101

—

107

C₂×#101

—

108

#405

—

109

C₅₉

#1979

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂²×Q₈

C₂×D₅₉

yes

characteristic

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

118

—

114

C₂×Q₁₆

—

118

—

115

C₂×Q₁₆

—

118

—

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

C₂×D₅₉

yes

characteristic

119

C₂²×C₈

—

120

#117

C₂×D₅₉

yes

characteristic

121

#108

—

118

—

122

#103

—

123

#103

—

118

—

124

#101

—

118

—

125

#101

—

118

—

126

#97

—

118

—

127

C₄×C₈

—

118

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

118

—

135

C₂×Q₈

—

118

—

136

C₂×Q₈

—

118

—

137

C₂×Q₈

—

118

—

138

C₂×Q₈

—

118

—

139

C₂×Q₈

C₄×D₅₉

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

118

—

142

C₂²×C₄

C₂²×D₅₉

yes

characteristic

143

Q₁₆

—

118

—

144

Q₁₆

—

118

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

118

—

149

M₁₆

—

118

—

150

M₁₆

—

118

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

118

—

157

C₂×C₈

—

118

—

158

C₄⋊C₄

C₂²⋊D₅₉

yes

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₂³

#11745687

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

118

—

167

Q₈

—

118

—

168

Q₈

—

236

—

169

Q₈

—

236

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

118

—

175

C₂×C₄

—

236

—

176

C₂×C₄

—

236

—

177

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

178

C₂×C₄

#11745685

yes

characteristic

179

C₂×C₄

D₄×D₅₉

yes

characteristic

180

C₂×C₄

D₄×D₅₉

yes

characteristic

181

C₂×C₄

D₄×D₅₉

yes

characteristic

182

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

118

—

187

C₈

—

236

—

188

C₂²

—

189

C₂²

(C₂²⋊D₅₉)⋊C₄

yes

characteristic

190

C₂²

#14273699

yes

characteristic

191

C₂²

#14273713

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

236

—

195

C₄

C₂²⋊C₄×D₅₉

yes

characteristic

196

C₄

C₄×C₂²⋊D₅₉

yes

characteristic

197

C₄

#14273700

yes

characteristic

198

C₄

#14273723

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#16837302

yes

characteristic

202

C₂

#16837675

yes

—

	subgroup	G/H
center	C₂²	#14273699
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#419	C₂
socle	C₂²×C₅₉	#121
hypercenter	#419	D₅₉
hypocenter	C₅₉	#1979
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14273699

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
12. C₂×#11745672	C₄	semidirect
109. C₅₉	#1979	semidirect

H ≅ G/N

type

12. C₂×#11745672

C₄

semidirect

109. C₅₉

#1979

semidirect

Subgroups — #21425010

Conjugacy classes of subgroups

Complements