Group #21435290 — subgroups

Subgroups — #21435290

Order 7552.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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	#16837959	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837808	C₂	1	2	yes	maximal, complemented
3	#16837766	C₂	1	2	yes	maximal, complemented
4	#16837723	C₂	1	4	yes	maximal, complemented
5	#16837692	C₂	1	2	yes	maximal, complemented
6	#16837682	C₂	1	2	yes	maximal, complemented
7	#16837459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#532	C₂	1	1	yes	maximal, characteristic
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273727	C₂²	1	1	yes	characteristic
11	#14273719	C₂²	1	2	yes	—
12	#14273706	—	2	1	no	—
13	#14273706	—	2	1	no	—
14	C₄×Dic₁₁₈	—	2	1	no	—
15	C₄×Dic₁₁₈	—	2	1	no	—
16	#14273686	C₂²	1	2	yes	—
17	#14273660	C₂²	1	4	yes	—
18	#14273658	C₂²	1	4	yes	—
19	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
20	#14273648	C₂²	1	4	yes	—
21	#14273648	C₂²	1	4	yes	—
22	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
23	#14273638	C₂²	1	2	yes	—
24	C₂×#41×C₅₉	—	2	1	no	—
25	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
26	#127×C₅₉	C₂²	1	1	yes	characteristic
27	#124×C₅₉	—	2	1	no	—
28	#121×C₅₉	—	2	1	no	—
29	#121×C₅₉	C₂²	1	2	yes	—
30	C₄×D₄×C₅₉	—	2	1	no	—
31	C₄×D₄×C₅₉	C₂²	1	2	yes	—
32	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
33	#41×C₅₉	—	4	4	no	—
34	C₂×Q₈×C₅₉	—	2	1	no	—
35	C₂×D₄×C₅₉	—	2	2	no	—
36	C₂×D₄×C₅₉	—	2	1	no	—
37	C₂×D₄×C₅₉	—	2	1	no	—
38	C₂×D₄×C₅₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₅₉	—	2	2	no	—
40	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₁₈	—	2	1	no	—
44	C₂×Dic₁₁₈	—	2	1	no	—
45	C₄⋊C₄×C₅₉	—	2	1	no	—
46	C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₅₉	—	2	2	no	—
48	C₂²⋊C₄×C₅₉	—	2	2	no	—
49	C₄²×C₅₉	D₄	1	1	yes	characteristic
50	C₄²×C₅₉	D₄	1	1	yes	characteristic
51	C₄²×C₅₉	C₂³	1	1	yes	characteristic
52	C₄²×C₅₉	C₂³	1	1	yes	characteristic
53	C₂²⋊Dic₅₉	—	2	2	no	—
54	C₄⋊Dic₅₉	—	2	1	no	—
55	C₄⋊Dic₅₉	—	2	1	no	—
56	C₄⋊Dic₅₉	C₂³	1	2	yes	—
57	C₄⋊Dic₅₉	C₂³	1	2	yes	—
58	Dic₅₉⋊C₄	—	2	2	no	—
59	Dic₅₉⋊C₄	—	2	2	no	—
60	C₄×Dic₅₉	—	2	2	no	—
61	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
62	C₂³×C₅₉	—	2	2	no	—
63	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
64	Q₈×C₅₉	—	4	2	no	—
65	D₄×C₅₉	—	2	4	no	—
66	D₄×C₅₉	—	4	2	no	—
67	D₄×C₅₉	—	4	2	no	—
68	D₄×C₅₉	—	4	4	no	—
69	C₂×C₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	—	4	1	no	—
73	C₂×C₄×C₅₉	—	4	4	no	—
74	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₅₉	—	2	2	no	—
81	C₂×Dic₅₉	—	2	2	no	—
82	Dic₁₁₈	—	4	2	no	—
83	Dic₁₁₈	—	4	2	no	—
84	C₅₉⋊C₈	—	2	4	no	—
85	C₂²×C₅₉	—	2	1	no	—
86	C₂²×C₅₉	—	2	1	no	—
87	C₂²×C₅₉	—	2	1	no	—
88	C₂²×C₅₉	—	4	2	no	—
89	C₂²×C₅₉	—	4	2	no	—
90	C₂²×C₅₉	—	4	2	no	—
91	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₅₉	—	2	1	no	—
93	C₄×C₅₉	—	2	1	no	—
94	C₄×C₅₉	—	2	1	no	—
95	C₄×C₅₉	—	2	1	no	—
96	C₄×C₅₉	—	4	2	no	—
97	C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
98	C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
99	Dic₅₉	—	4	2	no	—
100	Dic₅₉	—	4	2	no	—
101	#3235	—	59	1	no	maximal, Sylow
102	C₂×C₅₉	—	4	1	no	—
103	C₂×C₅₉	—	8	2	no	—
104	C₂×C₅₉	#536	1	1	yes	characteristic
105	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×C₅₉	#575	1	1	yes	characteristic
107	#533	—	59	1	no	—
108	#532	D₅₉	1	1	yes	characteristic
109	#501	—	59	2	no	—
110	#490	—	59	2	no	—
111	#482	—	59	4	no	—
112	#476	—	59	2	no	—
113	#474	—	59	2	no	—
114	#432	—	59	1	no	—
115	C₅₉	#3235	1	1	yes	characteristic, complemented, Sylow
116	C₂×#41	—	2	1	no	—
117	C₄⋊Q₈	—	59	1	no	—
118	C₄⋊Q₈	—	59	1	no	—
119	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
120	#127	C₂×D₅₉	1	1	yes	characteristic
121	#125	—	59	2	no	—
122	#124	—	2	1	no	—
123	C₂²⋊Q₈	—	118	1	no	—
124	C₂²⋊Q₈	—	118	1	no	—
125	#121	—	2	1	no	—
126	#121	C₂×D₅₉	1	2	yes	—
127	C₄×Q₈	—	118	1	no	—
128	C₄×Q₈	—	118	1	no	—
129	C₄×D₄	—	2	1	no	—
130	C₄×D₄	C₂×D₅₉	1	2	yes	—
131	#117	—	59	2	no	—
132	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
133	#107	—	59	4	no	—
134	#106	—	59	4	no	—
135	C₄⋊C₈	—	59	2	no	—
136	#102	—	59	4	no	—
137	#102	—	59	4	no	—
138	C₂²⋊C₈	—	59	2	no	—
139	#97	—	59	2	no	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	118	1	no	—
143	C₂×Q₈	—	118	1	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	C₂²×D₅₉	1	2	yes	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
152	C₂×C₈	—	59	4	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	59	2	no	—
155	C₄⋊C₄	—	59	2	no	—
156	C₄⋊C₄	—	118	2	no	—
157	C₄⋊C₄	—	118	2	no	—
158	C₄⋊C₄	—	118	1	no	—
159	C₄⋊C₄	—	118	1	no	—
160	C₄⋊C₄	C₂²×D₅₉	1	2	yes	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	118	2	no	—
164	C₄²	—	118	2	no	—
165	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
166	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
167	C₄²	C₂²×D₅₉	1	1	yes	characteristic
168	C₄²	C₂²×D₅₉	1	1	yes	characteristic
169	C₂³	—	2	2	no	—
170	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
171	Q₈	—	4	2	no	—
172	Q₈	—	236	2	no	—
173	Q₈	—	236	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	2	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	118	2	no	—
184	C₂×C₄	—	118	2	no	—
185	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
190	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
191	C₈	—	118	4	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	2	no	—
204	C₄	—	236	2	no	—
205	C₄	—	236	2	no	—
206	C₄	#14273753	1	2	yes	—
207	C₄	#14273774	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	—	8	2	no	—
210	C₂	#16837973	1	1	yes	characteristic
211	C₂	C₂×#14273753	1	1	yes	characteristic
212	C₂	#16838200	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837959

C₂

yes

maximal, characteristic, complemented

#16837808

C₂

yes

maximal, complemented

#16837766

C₂

yes

maximal, complemented

#16837723

C₂

yes

maximal, complemented

#16837692

C₂

yes

maximal, complemented

#16837682

C₂

yes

maximal, complemented

#16837459

C₂

yes

maximal, characteristic, complemented

C₅₉×#532

C₂

yes

maximal, characteristic

#14273727

C₂²

yes

characteristic

#14273727

C₂²

yes

characteristic

#14273719

C₂²

yes

—

#14273706

—

#14273706

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

#14273686

C₂²

yes

—

#14273660

C₂²

yes

—

#14273658

C₂²

yes

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273648

C₂²

yes

—

#14273648

C₂²

yes

—

C₂²⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273638

C₂²

yes

—

C₂×#41×C₅₉

—

C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

#127×C₅₉

C₂²

yes

characteristic

#124×C₅₉

—

#121×C₅₉

—

#121×C₅₉

C₂²

yes

—

C₄×D₄×C₅₉

—

C₄×D₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#41×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂³

yes

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂³

yes

—

C₄⋊Dic₅₉

C₂³

yes

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

C₂³

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×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

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₄×C₅₉

2₋¹⁺⁴

yes

—

Dic₅₉

—

100

Dic₅₉

—

101

#3235

—

maximal, Sylow

102

C₂×C₅₉

—

103

C₂×C₅₉

—

104

C₂×C₅₉

#536

yes

characteristic

105

C₂×C₅₉

C₂×AGL(1,ℤ/8)

yes

characteristic

106

C₂×C₅₉

#575

yes

characteristic

107

#533

—

108

#532

D₅₉

yes

characteristic

109

#501

—

110

#490

—

111

#482

—

112

#476

—

113

#474

—

114

#432

—

115

C₅₉

#3235

yes

characteristic, complemented, Sylow

116

C₂×#41

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

C₂×D₅₉

yes

characteristic

120

#127

C₂×D₅₉

yes

characteristic

121

#125

—

122

#124

—

123

C₂²⋊Q₈

—

118

—

124

C₂²⋊Q₈

—

118

—

125

#121

—

126

#121

C₂×D₅₉

yes

—

127

C₄×Q₈

—

118

—

128

C₄×Q₈

—

118

—

129

C₄×D₄

—

130

C₄×D₄

C₂×D₅₉

yes

—

131

#117

—

132

C₂×C₄²

C₂×D₅₉

yes

characteristic

133

#107

—

134

#106

—

135

C₄⋊C₈

—

136

#102

—

137

#102

—

138

C₂²⋊C₈

—

139

#97

—

140

#41

—

141

C₂×Q₈

—

142

C₂×Q₈

—

118

—

143

C₂×Q₈

—

118

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

C₂²×D₅₉

yes

—

148

C₂²×C₄

—

149

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

150

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

151

C₂²×C₄

C₂²×D₅₉

yes

characteristic

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

118

—

157

C₄⋊C₄

—

118

—

158

C₄⋊C₄

—

118

—

159

C₄⋊C₄

—

118

—

160

C₄⋊C₄

C₂²×D₅₉

yes

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

118

—

164

C₄²

—

118

—

165

C₄²

C₂²⋊D₅₉

yes

characteristic

166

C₄²

C₂²⋊D₅₉

yes

characteristic

167

C₄²

C₂²×D₅₉

yes

characteristic

168

C₄²

C₂²×D₅₉

yes

characteristic

169

C₂³

—

170

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

171

Q₈

—

172

Q₈

—

236

—

173

Q₈

—

236

—

174

D₄

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

118

—

184

C₂×C₄

—

118

—

185

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

186

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

187

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

188

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

189

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

190

C₂×C₄

C₂³×D₅₉

yes

characteristic

191

C₈

—

118

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

C₂²×C₂²⋊D₅₉

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

236

—

205

C₄

—

236

—

206

C₄

#14273753

yes

—

207

C₄

#14273774

yes

—

208

C₂

—

209

C₂

—

210

C₂

#16837973

yes

characteristic

211

C₂

C₂×#14273753

yes

characteristic

212

C₂

#16838200

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#532	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#532	D₅₉
hypocenter	C₅₉	#3235
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₅₉×#532

C₂

socle

C₂²×C₅₉

C₂²×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #16837959	C₂	semidirect
2. #16837808	C₂	semidirect
3. #16837766	C₂	semidirect
4. #16837723	C₂	semidirect
5. #16837692	C₂	semidirect
6. #16837682	C₂	semidirect
7. #16837459	C₂	semidirect
115. C₅₉	#3235	semidirect

H ≅ G/N

type

1. #16837959

C₂

semidirect

2. #16837808

C₂

semidirect

3. #16837766

C₂

semidirect

4. #16837723

C₂

semidirect

5. #16837692

C₂

semidirect

6. #16837682

C₂

semidirect

7. #16837459

C₂

semidirect

115. C₅₉

#3235

semidirect

Subgroups — #21435290

Conjugacy classes of subgroups

Complements