Group #28507176 — subgroups

Subgroups — #28507176

Order 15104.

566 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16837779	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837216	C₂	1	4	yes	maximal, complemented
3	C₄×#14273663	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₇₂	—	2	1	no	—
6	C₂²×D₄₇₂	C₂²	1	2	yes	—
7	C₂×#14273724	—	2	1	no	—
8	#16837779	—	2	2	no	—
9	#16837779	C₄	1	4	yes	—
10	C₂×#14273663	C₂²	1	2	yes	—
11	C₄×#11745664	—	2	1	no	—
12	#16837216	—	2	8	no	—
13	#16837167	—	2	1	no	—
14	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×C₄⋊C₁₆	—	2	2	no	—
17	C₂²×D₂₃₆	—	2	2	no	—
18	C₂²×D₂₃₆	—	4	2	no	—
19	C₂×D₄₇₂	—	2	4	no	—
20	C₂×D₄₇₂	—	2	4	no	—
21	C₂×D₄₇₂	—	2	4	no	—
22	C₂×D₄₇₂	—	2	2	no	—
23	C₂×D₄₇₂	—	4	4	no	—
24	C₂×D₄₇₂	C₂×C₄	1	4	yes	—
25	C₂×#11745664	—	2	2	no	—
26	#14273724	—	2	4	no	—
27	#14273724	—	4	1	no	—
28	#14273724	—	4	1	no	—
29	#14273663	—	2	2	no	—
30	#14273663	D₄	1	4	yes	complemented
31	#14273638	—	4	1	no	—
32	C₄×C₅₉⋊C₈	—	4	1	no	—
33	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
36	M₃₂×C₅₉	—	2	4	no	—
37	C₂×C₁₆×C₅₉	—	2	2	no	—
38	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₉	D₄	1	2	yes	—
40	C₂³×D₅₉	—	4	2	no	—
41	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
42	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×D₂₃₆	—	2	4	no	—
44	C₂×D₂₃₆	—	4	4	no	—
45	C₂×D₂₃₆	—	4	4	no	—
46	C₂×D₂₃₆	—	4	4	no	—
47	C₂×D₂₃₆	—	4	4	no	—
48	C₂×D₂₃₆	—	4	2	no	—
49	C₂×D₂₃₆	—	8	8	no	—
50	C₂×C₈×C₅₉	—	2	2	no	—
51	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	#41	1	2	yes	—
56	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₅₉	D₈	1	1	yes	characteristic
58	C₄²×C₅₉	D₈	1	1	yes	characteristic
59	#11745664	—	2	4	no	—
60	#11745664	—	4	2	no	—
61	C₂×C₅₉⋊C₈	—	4	2	no	—
62	D₄₇₂	—	2	8	no	—
63	D₄₇₂	—	4	8	no	—
64	D₄₇₂	—	4	2	no	—
65	D₄₇₂	—	4	2	no	—
66	C₁₆×C₅₉	—	4	2	no	—
67	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₂²×D₅₉	—	4	4	no	—
69	C₂²×D₅₉	—	8	8	no	—
70	C₂²×D₅₉	—	8	4	no	—
71	C₂×C₄×C₅₉	—	2	4	no	—
72	C₂×C₄×C₅₉	#102	1	2	yes	—
73	C₂×C₄×C₅₉	#102	1	2	yes	—
74	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#121	1	2	yes	—
76	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
78	D₂₃₆	—	4	4	no	—
79	D₂₃₆	—	4	4	no	—
80	D₂₃₆	—	8	8	no	—
81	D₂₃₆	—	8	8	no	—
82	D₂₃₆	—	8	4	no	—
83	C₈×C₅₉	—	2	2	no	—
84	C₈×C₅₉	C₄×D₄	1	2	yes	—
85	C₈×C₅₉	#124	1	1	yes	characteristic
86	C₈×C₅₉	#127	1	1	yes	characteristic
87	C₅₉⋊C₈	—	4	4	no	—
88	#13257	—	59	1	no	maximal, Sylow
89	C₂²×C₅₉	—	2	2	no	—
90	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
91	C₂²×C₅₉	#460	1	2	yes	—
92	C₂×D₅₉	—	8	8	no	—
93	C₂×D₅₉	—	8	4	no	—
94	C₂×D₅₉	—	16	8	no	—
95	C₄×C₅₉	—	2	2	no	—
96	C₄×C₅₉	#361	1	4	yes	complemented
97	C₄×C₅₉	#390	1	1	yes	characteristic
98	C₄×C₅₉	#418	1	1	yes	characteristic
99	C₄×C₅₉	#459	1	2	yes	—
100	C₂×#493	—	59	1	no	—
101	#2150	D₅₉	1	1	yes	characteristic, complemented
102	C₂×#361	—	59	4	no	—
103	C₄×#108	—	59	1	no	—
104	C₂×C₅₉	—	2	2	no	—
105	C₂×C₅₉	#1982	1	1	yes	characteristic
106	C₂×C₅₉	C₂×#361	1	2	yes	—
107	D₅₉	—	16	4	no	—
108	C₂²×D₈	—	59	2	no	—
109	C₂²×D₈	—	118	1	no	—
110	C₂×#127	—	118	1	no	—
111	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
112	#493	—	59	4	no	—
113	#493	—	118	2	no	—
114	C₂×#108	—	59	2	no	—
115	C₄×M₁₆	—	118	1	no	—
116	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
117	C₄⋊C₁₆	—	2	2	no	—
118	#361	—	118	8	no	—
119	#341	—	118	1	no	—
120	C₅₉	#13257	1	1	yes	characteristic, complemented, Sylow
121	C₂²×D₄	—	118	2	no	—
122	C₂²×D₄	—	236	2	no	—
123	C₂×D₈	—	59	4	no	—
124	C₂×D₈	—	118	4	no	—
125	C₂×D₈	—	118	4	no	—
126	C₂×D₈	—	118	4	no	—
127	C₂×D₈	—	118	2	no	—
128	C₂×D₈	—	236	4	no	—
129	C₂×M₁₆	—	118	2	no	—
130	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
132	#127	—	118	4	no	—
133	#127	—	236	1	no	—
134	#127	—	236	1	no	—
135	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
136	M₃₂	—	2	4	no	—
137	C₂×C₁₆	—	2	2	no	—
138	#108	—	59	4	no	—
139	#108	—	118	2	no	—
140	#97	—	236	1	no	—
141	C₄×C₈	—	236	1	no	—
142	C₄×C₈	C₄×D₅₉	1	2	yes	—
143	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
144	C₂⁴	—	236	2	no	—
145	C₂×D₄	—	118	4	no	—
146	C₂×D₄	—	472	8	no	—
147	C₂×D₄	—	236	4	no	—
148	C₂×D₄	—	236	4	no	—
149	C₂×D₄	—	236	4	no	—
150	C₂×D₄	—	236	2	no	—
151	C₂×D₄	—	236	4	no	—
152	C₂²×C₄	#11745660	1	2	yes	—
153	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
154	D₈	—	118	8	no	—
155	D₈	—	236	8	no	—
156	D₈	—	236	2	no	—
157	D₈	—	236	2	no	—
158	M₁₆	—	118	4	no	—
159	M₁₆	—	236	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	236	2	no	—
162	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
164	C₂×C₈	D₄×D₅₉	1	2	yes	—
165	C₂×C₈	#11745689	1	2	yes	—
166	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
167	C₄²	D₄₇₂	1	1	yes	characteristic
168	C₄²	D₄₇₂	1	1	yes	characteristic
169	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
170	C₁₆	—	4	2	no	—
171	C₂³	—	472	8	no	—
172	C₂³	—	472	4	no	—
173	C₂³	—	236	4	no	—
174	C₂³	C₂×#11745660	1	1	yes	characteristic
175	D₄	—	472	8	no	—
176	D₄	—	472	8	no	—
177	D₄	—	472	4	no	—
178	D₄	—	236	4	no	—
179	D₄	—	236	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	#14273649	1	2	yes	—
182	C₂×C₄	#14273649	1	2	yes	—
183	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
184	C₂×C₄	#14273702	1	2	yes	—
185	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
186	C₂×C₄	#14273755	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	236	4	no	—
189	C₈	#14273689	1	2	yes	—
190	C₈	#14273716	1	1	yes	characteristic
191	C₈	#14273723	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	944	8	no	—
194	C₂²	—	472	4	no	—
195	C₂²	—	472	8	no	—
196	C₂²	C₂×#14273649	1	1	yes	characteristic
197	C₂²	#16837603	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	#16837216	1	4	yes	complemented
200	C₄	#16837303	1	1	yes	characteristic
201	C₄	#16837408	1	1	yes	characteristic
202	C₄	#16837598	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	944	4	no	—
205	C₂	#21425022	1	1	yes	characteristic
206	C₂	C₂×#16837216	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837779

C₂

yes

maximal, characteristic

C₂×#16837216

C₂

yes

maximal, complemented

C₄×#14273663

C₂

yes

maximal, characteristic, complemented

C₅₉×#2150

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₇₂

—

C₂²×D₄₇₂

C₂²

yes

—

C₂×#14273724

—

#16837779

—

#16837779

C₄

yes

—

C₂×#14273663

C₂²

yes

—

C₄×#11745664

—

#16837216

—

#16837167

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₂²×D₂₃₆

—

C₂²×D₂₃₆

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

—

C₂×D₄₇₂

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

complemented

#14273638

—

C₄×C₅₉⋊C₈

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂³×D₅₉

—

C₂²×C₄×C₅₉

SD₁₆

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

D₄₇₂

—

D₄₇₂

—

D₄₇₂

—

D₄₇₂

—

C₁₆×C₅₉

—

C₂³×C₅₉

C₂×SD₁₆

yes

characteristic

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₈×C₅₉

C₄×D₄

yes

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

#13257

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#102

yes

characteristic

C₂²×C₅₉

#460

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

100

C₂×#493

—

101

#2150

D₅₉

yes

characteristic, complemented

102

C₂×#361

—

103

C₄×#108

—

104

C₂×C₅₉

—

105

C₂×C₅₉

#1982

yes

characteristic

106

C₂×C₅₉

C₂×#361

yes

—

107

D₅₉

—

108

C₂²×D₈

—

109

C₂²×D₈

—

118

—

110

C₂×#127

—

118

—

111

C₂×M₃₂

C₂×D₅₉

yes

—

112

#493

—

113

#493

—

118

—

114

C₂×#108

—

115

C₄×M₁₆

—

118

—

116

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

117

C₄⋊C₁₆

—

118

#361

—

118

—

119

#341

—

118

—

120

C₅₉

#13257

yes

characteristic, complemented, Sylow

121

C₂²×D₄

—

118

—

122

C₂²×D₄

—

236

—

123

C₂×D₈

—

124

C₂×D₈

—

118

—

125

C₂×D₈

—

118

—

126

C₂×D₈

—

118

—

127

C₂×D₈

—

118

—

128

C₂×D₈

—

236

—

129

C₂×M₁₆

—

118

—

130

C₂²×C₈

D₂₃₆

yes

characteristic

131

C₂²×C₈

C₂²×D₅₉

yes

characteristic

132

#127

—

118

—

133

#127

—

236

—

134

#127

—

236

—

135

C₂×C₄²

D₂₃₆

yes

characteristic

136

M₃₂

—

137

C₂×C₁₆

—

138

#108

—

139

#108

—

118

—

140

#97

—

236

—

141

C₄×C₈

—

236

—

142

C₄×C₈

C₄×D₅₉

yes

—

143

C₄×C₈

C₂²⋊D₅₉

yes

—

144

C₂⁴

—

236

—

145

C₂×D₄

—

118

—

146

C₂×D₄

—

472

—

147

C₂×D₄

—

236

—

148

C₂×D₄

—

236

—

149

C₂×D₄

—

236

—

150

C₂×D₄

—

236

—

151

C₂×D₄

—

236

—

152

C₂²×C₄

#11745660

yes

—

153

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

154

D₈

—

118

—

155

D₈

—

236

—

156

D₈

—

236

—

157

D₈

—

236

—

158

M₁₆

—

118

—

159

M₁₆

—

236

—

160

C₂×C₈

—

161

C₂×C₈

—

236

—

162

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

163

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

164

C₂×C₈

D₄×D₅₉

yes

—

165

C₂×C₈

#11745689

yes

—

166

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

167

C₄²

D₄₇₂

yes

characteristic

168

C₄²

D₄₇₂

yes

characteristic

169

C₄²

(C₂×D₅₉)⋊C₄

yes

—

170

C₁₆

—

171

C₂³

—

472

—

172

C₂³

—

472

—

173

C₂³

—

236

—

174

C₂³

C₂×#11745660

yes

characteristic

175

D₄

—

472

—

176

D₄

—

472

—

177

D₄

—

472

—

178

D₄

—

236

—

179

D₄

—

236

—

180

C₂×C₄

—

181

C₂×C₄

#14273649

yes

—

182

C₂×C₄

#14273649

yes

—

183

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

184

C₂×C₄

#14273702

yes

—

185

C₂×C₄

C₂×D₄₇₂

yes

characteristic

186

C₂×C₄

#14273755

yes

—

187

C₈

—

188

C₈

—

236

—

189

C₈

#14273689

yes

—

190

C₈

#14273716

yes

characteristic

191

C₈

#14273723

yes

characteristic

192

C₂²

—

193

C₂²

—

944

—

194

C₂²

—

472

—

195

C₂²

—

472

—

196

C₂²

C₂×#14273649

yes

characteristic

197

C₂²

#16837603

yes

—

198

C₄

—

199

C₄

#16837216

yes

complemented

200

C₄

#16837303

yes

characteristic

201

C₄

#16837408

yes

characteristic

202

C₄

#16837598

yes

—

203

C₂

—

204

C₂

—

944

—

205

C₂

#21425022

yes

characteristic

206

C₂

C₂×#16837216

yes

—

	subgroup	G/H
center	C₂²	C₂×#14273649
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂×#16837779	C₂
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#13257
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂²

C₂×#14273649

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#2150

C₂

C₂²×C₅₉

C₂

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
2. C₂×#16837216	C₂	semidirect
3. C₄×#14273663	C₂	semidirect
4. C₅₉×#2150	C₂	semidirect
30. #14273663	D₄	semidirect
96. C₄×C₅₉	#361	semidirect
101. #2150	D₅₉	semidirect
120. C₅₉	#13257	semidirect
199. C₄	#16837216	semidirect

H ≅ G/N

type

2. C₂×#16837216

C₂

semidirect

3. C₄×#14273663

C₂

semidirect

4. C₅₉×#2150

C₂

semidirect

30. #14273663

D₄

semidirect

96. C₄×C₅₉

#361

semidirect

101. #2150

D₅₉

semidirect

120. C₅₉

#13257

semidirect

199. C₄

#16837216

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16837779	2
C₂×#16837216	2
C₄×#14273663	2
C₅₉×#2150	2
#13257	59

subgroup

index

C₅₉×#2150

Subgroups — #28507176

Conjugacy classes of subgroups

Complements

Maximal subgroups