Group #21425356 — subgroups

Subgroups — #21425356

Order 7552.

276 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 69 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×#11745671	C₂	1	1	yes	maximal, characteristic
2	#16837429	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837406	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273646	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14273643	C₂	1	1	yes	maximal, characteristic
6	#16837166	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₃₆	C₂²	1	1	yes	characteristic
9	C₂×#11745671	—	2	2	no	—
10	C₂×#11745671	—	2	2	no	—
11	C₂×#11745671	—	2	2	no	—
12	C₂×#11745664	C₂²	1	1	yes	characteristic
13	C₂×#11745664	C₂²	1	1	yes	characteristic
14	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
16	#14273647	—	2	2	no	—
17	#14273646	—	2	2	no	—
18	#14273643	—	2	2	no	—
19	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	#117×C₅₉	C₂²	1	1	yes	characteristic
22	#103×C₅₉	—	2	2	no	—
23	C₂³×D₅₉	—	2	1	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	D₄	1	2	yes	—
30	C₂²×C₄×C₅₉	—	2	1	no	—
31	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×D₂₃₆	—	2	1	no	—
33	C₂×D₂₃₆	—	2	1	no	—
34	C₂×D₂₃₆	—	2	1	no	—
35	C₂×D₂₃₆	—	2	1	no	—
36	C₂×D₂₃₆	—	2	1	no	—
37	C₂×D₂₃₆	D₄	1	2	yes	—
38	M₁₆×C₅₉	—	2	1	no	—
39	M₁₆×C₅₉	—	2	1	no	—
40	C₂×C₈×C₅₉	—	2	1	no	—
41	C₄⋊C₄×C₅₉	D₄	1	2	yes	—
42	C₂²⋊C₄×C₅₉	—	2	1	no	—
43	C₄²×C₅₉	—	2	1	no	—
44	#11745671	—	2	8	no	—
45	#11745664	—	2	1	no	—
46	#11745664	—	2	1	no	—
47	#11745664	—	2	1	no	—
48	#11745664	—	2	1	no	—
49	C₂×C₅₉⋊C₈	—	2	1	no	—
50	C₂×C₅₉⋊C₈	—	2	1	no	—
51	C₂×C₅₉⋊C₈	—	2	1	no	—
52	C₂×C₅₉⋊C₈	—	2	1	no	—
53	C₂×C₅₉⋊C₈	D₄	1	2	yes	—
54	C₂³×C₅₉	#41	1	1	yes	characteristic
55	C₂²×D₅₉	—	2	2	no	—
56	C₂²×D₅₉	—	4	1	no	—
57	C₂²×D₅₉	—	4	1	no	—
58	C₂²×D₅₉	—	8	2	no	—
59	Q₈×C₅₉	—	2	2	no	—
60	Q₈×C₅₉	—	2	2	no	—
61	Q₈×C₅₉	—	4	1	no	—
62	Q₈×C₅₉	—	4	1	no	—
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	2	2	no	—
65	C₂×C₄×C₅₉	—	4	1	no	—
66	C₂×C₄×C₅₉	—	4	1	no	—
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	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
73	D₂₃₆	—	2	2	no	—
74	D₂₃₆	—	2	2	no	—
75	D₂₃₆	—	4	1	no	—
76	D₂₃₆	—	4	1	no	—
77	C₈×C₅₉	—	4	1	no	—
78	C₅₉⋊C₈	—	2	2	no	—
79	C₅₉⋊C₈	—	4	1	no	—
80	C₅₉⋊C₈	—	4	1	no	—
81	C₂²×C₅₉	—	2	2	no	—
82	C₂²×C₅₉	#121	1	1	yes	characteristic
83	C₂²×C₅₉	#123	1	1	yes	characteristic
84	C₂²×C₅₉	#123	1	1	yes	characteristic
85	C₂×D₅₉	—	4	2	no	—
86	C₂×D₅₉	—	8	2	no	—
87	C₂×D₅₉	—	8	2	no	—
88	C₂×D₅₉	—	8	2	no	—
89	C₄×C₅₉	—	4	2	no	—
90	C₄×C₅₉	—	4	2	no	—
91	C₄×C₅₉	#120	1	1	yes	characteristic
92	C₄×C₅₉	#121	1	1	yes	characteristic
93	C₄×C₅₉	#121	1	1	yes	characteristic
94	C₄×C₅₉	#124	1	1	yes	characteristic
95	#2046	—	59	1	no	maximal, Sylow
96	C₂×C₅₉	—	2	1	no	—
97	C₂×C₅₉	—	2	1	no	—
98	C₂×C₅₉	#394	1	1	yes	characteristic
99	C₂×C₅₉	#471	1	2	yes	—
100	D₅₉	—	8	2	no	—
101	C₂²×SD₁₆	—	59	1	no	—
102	#424	—	59	1	no	—
103	#419	D₅₉	1	1	yes	characteristic, complemented
104	#418	—	59	1	no	—
105	C₂×#101	—	59	1	no	—
106	C₂×#100	—	59	1	no	—
107	#341	—	59	1	no	—
108	C₅₉	#2046	1	1	yes	characteristic, complemented, Sylow
109	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
110	C₂²×D₄	—	59	1	no	—
111	C₂×SD₁₆	—	118	2	no	—
112	C₂×SD₁₆	—	118	2	no	—
113	C₂×SD₁₆	—	118	2	no	—
114	C₂×M₁₆	—	59	1	no	—
115	C₂×M₁₆	—	59	1	no	—
116	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
117	C₂²×C₈	—	59	1	no	—
118	#117	C₂×D₅₉	1	1	yes	characteristic
119	C₄⋊C₈	—	118	2	no	—
120	#103	—	2	2	no	—
121	#102	—	118	2	no	—
122	#101	—	118	2	no	—
123	#100	—	118	2	no	—
124	C₂⁴	—	118	1	no	—
125	C₂×Q₈	—	2	1	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	D₂₃₆	1	2	yes	—
131	C₂×D₄	—	59	2	no	—
132	C₂×D₄	—	118	1	no	—
133	C₂×D₄	—	118	1	no	—
134	C₂×D₄	—	118	1	no	—
135	C₂×D₄	—	118	1	no	—
136	C₂×D₄	—	118	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
139	SD₁₆	—	118	8	no	—
140	M₁₆	—	2	1	no	—
141	M₁₆	—	2	1	no	—
142	M₁₆	—	118	1	no	—
143	M₁₆	—	118	1	no	—
144	M₁₆	—	118	1	no	—
145	M₁₆	—	118	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	59	2	no	—
148	C₂×C₈	—	118	1	no	—
149	C₂×C₈	—	118	1	no	—
150	C₂×C₈	—	118	1	no	—
151	C₂×C₈	—	118	1	no	—
152	C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
153	C₂²⋊C₄	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₂³	—	118	2	no	—
156	C₂³	—	472	2	no	—
157	C₂³	—	236	1	no	—
158	C₂³	—	236	1	no	—
159	C₂³	#11745687	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	D₄	—	118	2	no	—
165	D₄	—	118	2	no	—
166	D₄	—	236	1	no	—
167	D₄	—	236	1	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	C₂×D₂₃₆	1	1	yes	characteristic
173	C₂×C₄	#11745685	1	1	yes	characteristic
174	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
175	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
176	C₂×C₄	#11745687	1	1	yes	characteristic
177	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
178	C₈	—	4	1	no	—
179	C₈	—	118	2	no	—
180	C₈	—	236	1	no	—
181	C₈	—	236	1	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	472	2	no	—
184	C₂²	—	472	2	no	—
185	C₂²	—	472	2	no	—
186	C₂²	—	236	2	no	—
187	C₂²	#14273699	1	1	yes	characteristic
188	C₂²	#14273709	1	1	yes	characteristic
189	C₂²	#14273711	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	4	2	no	—
192	C₄	#14273697	1	1	yes	characteristic
193	C₄	#14273698	1	1	yes	characteristic
194	C₄	#14273701	1	1	yes	characteristic
195	C₄	#14273713	1	1	yes	characteristic
196	C₂	—	2	1	no	—
197	C₂	—	2	1	no	—
198	C₂	—	472	2	no	—
199	C₂	#16837318	1	1	yes	characteristic
200	C₂	#16837667	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11745671

C₂

yes

maximal, characteristic

#16837429

C₂

yes

maximal, characteristic, complemented

#16837406

C₂

yes

maximal, characteristic

C₂×#14273646

C₂

yes

maximal, characteristic, complemented

C₂×#14273643

C₂

yes

maximal, characteristic

#16837166

C₂

yes

maximal, characteristic, complemented

C₅₉×#419

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₃₆

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#11745664

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

#14273647

—

#14273646

—

#14273643

—

C₂²×Q₈×C₅₉

C₂²

yes

characteristic

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#117×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

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

D₄

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

—

—

—

—

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂³×C₅₉

#41

yes

characteristic

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×D₄

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

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#394

yes

characteristic

C₂×C₅₉

#471

yes

—

100

D₅₉

—

101

C₂²×SD₁₆

—

102

#424

—

103

#419

D₅₉

yes

characteristic, complemented

104

#418

—

105

C₂×#101

—

106

C₂×#100

—

107

#341

—

108

C₅₉

#2046

yes

characteristic, complemented, Sylow

109

C₂²×Q₈

C₂×D₅₉

yes

characteristic

110

C₂²×D₄

—

111

C₂×SD₁₆

—

118

—

112

C₂×SD₁₆

—

118

—

113

C₂×SD₁₆

—

118

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

C₂×D₅₉

yes

characteristic

117

C₂²×C₈

—

118

#117

C₂×D₅₉

yes

characteristic

119

C₄⋊C₈

—

118

—

120

#103

—

121

#102

—

118

—

122

#101

—

118

—

123

#100

—

118

—

124

C₂⁴

—

118

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

D₂₃₆

yes

—

131

C₂×D₄

—

132

C₂×D₄

—

118

—

133

C₂×D₄

—

118

—

134

C₂×D₄

—

118

—

135

C₂×D₄

—

118

—

136

C₂×D₄

—

118

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂²×D₅₉

yes

characteristic

139

SD₁₆

—

118

—

140

M₁₆

—

141

M₁₆

—

142

M₁₆

—

118

—

143

M₁₆

—

118

—

144

M₁₆

—

118

—

145

M₁₆

—

118

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

118

—

149

C₂×C₈

—

118

—

150

C₂×C₈

—

118

—

151

C₂×C₈

—

118

—

152

C₄⋊C₄

C₂²⋊D₅₉

yes

—

153

C₂²⋊C₄

—

154

C₄²

—

155

C₂³

—

118

—

156

C₂³

—

472

—

157

C₂³

—

236

—

158

C₂³

—

236

—

159

C₂³

#11745687

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

D₄

—

118

—

165

D₄

—

118

—

166

D₄

—

236

—

167

D₄

—

236

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

C₂×D₂₃₆

yes

characteristic

173

C₂×C₄

#11745685

yes

characteristic

174

C₂×C₄

D₄×D₅₉

yes

characteristic

175

C₂×C₄

D₄×D₅₉

yes

characteristic

176

C₂×C₄

#11745687

yes

characteristic

177

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

178

C₈

—

179

C₈

—

118

—

180

C₈

—

236

—

181

C₈

—

236

—

182

C₂²

—

183

C₂²

—

472

—

184

C₂²

—

472

—

185

C₂²

—

472

—

186

C₂²

—

236

—

187

C₂²

#14273699

yes

characteristic

188

C₂²

#14273709

yes

characteristic

189

C₂²

#14273711

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

#14273697

yes

characteristic

193

C₄

#14273698

yes

characteristic

194

C₄

#14273701

yes

characteristic

195

C₄

#14273713

yes

characteristic

196

C₂

—

197

C₂

—

198

C₂

—

472

—

199

C₂

#16837318

yes

characteristic

200

C₂

#16837667

yes

—

	subgroup	G/H
center	C₂²	#14273699
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#419	C₂
socle	C₂²×C₅₉	#121
hypercenter	#419	D₅₉
hypocenter	C₅₉	#2046
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14273699

commutator

C₂²×C₄×C₅₉

C₂³

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#419

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
2. #16837429	C₂	semidirect
4. C₂×#14273646	C₂	semidirect
6. #16837166	C₂	semidirect
7. C₅₉×#419	C₂	semidirect
103. #419	D₅₉	semidirect
108. C₅₉	#2046	semidirect

H ≅ G/N

type

2. #16837429

C₂

semidirect

4. C₂×#14273646

C₂

semidirect

6. #16837166

C₂

semidirect

7. C₅₉×#419

C₂

semidirect

103. #419

D₅₉

semidirect

108. C₅₉

#2046

semidirect

Subgroups — #21425356

Conjugacy classes of subgroups

Complements