Group #28493672 — subgroups

Subgroups — #28493672

Order 15104.

372 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 99 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	#21424271	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422866	C₂	1	2	yes	maximal, complemented
3	#21422521	C₂	1	2	yes	maximal
4	#21421927	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1955	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273663	—	2	2	no	—
8	#16837426	C₂²	1	2	yes	—
9	#16837152	—	2	4	no	—
10	D₄⋊(C₅₉⋊C₈)	—	2	4	no	—
11	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₈×C₅₉	—	2	2	no	—
14	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#11745664	—	2	2	no	—
16	#14273663	—	4	4	no	—
17	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
21	C₄×C₅₉⋊C₈	D₄	1	4	yes	—
22	C₂²×D₄×C₅₉	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₅₉	—	2	1	no	—
24	C₂²×C₈×C₅₉	—	2	1	no	—
25	C₂²×C₈×C₅₉	D₄	1	2	yes	—
26	C₄×D₄×C₅₉	—	2	2	no	—
27	C₄×D₄×C₅₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₉	Q₈	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	—	2	2	no	—
32	C₂²⋊C₈×C₅₉	—	4	4	no	—
33	#97×C₅₉	—	2	2	no	—
34	C₂⁴×C₅₉	—	2	1	no	—
35	C₂×D₄×C₅₉	—	2	2	no	—
36	C₂×D₄×C₅₉	—	4	2	no	—
37	C₂²×C₄×C₅₉	—	2	1	no	—
38	C₂²×C₄×C₅₉	—	4	1	no	—
39	C₂²×C₄×C₅₉	—	4	1	no	—
40	C₂²×C₄×C₅₉	—	4	2	no	—
41	C₂²×C₄×C₅₉	—	4	2	no	—
42	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
45	C₂×C₈×C₅₉	—	2	2	no	—
46	C₂×C₈×C₅₉	—	2	2	no	—
47	C₂×C₈×C₅₉	—	4	1	no	—
48	C₂×C₈×C₅₉	—	4	2	no	—
49	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
50	C₄⋊C₄×C₅₉	—	2	2	no	—
51	C₂²⋊C₄×C₅₉	—	4	2	no	—
52	C₄²×C₅₉	C₂×D₄	1	2	yes	—
53	C₄²×C₅₉	#41	1	2	yes	—
54	#11745664	—	4	4	no	—
55	C₂×C₅₉⋊C₈	—	2	4	no	—
56	C₂×C₅₉⋊C₈	—	4	2	no	—
57	C₂³×C₅₉	—	2	2	no	—
58	C₂³×C₅₉	—	4	1	no	—
59	C₂³×C₅₉	—	4	2	no	—
60	C₂³×C₅₉	—	4	1	no	—
61	C₂³×C₅₉	—	4	2	no	—
62	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
63	D₄×C₅₉	—	2	4	no	—
64	D₄×C₅₉	—	4	2	no	—
65	C₂×C₄×C₅₉	—	4	1	no	—
66	C₂×C₄×C₅₉	—	4	1	no	—
67	C₂×C₄×C₅₉	—	4	1	no	—
68	C₂×C₄×C₅₉	—	4	2	no	—
69	C₂×C₄×C₅₉	—	8	2	no	—
70	C₂×C₄×C₅₉	—	8	1	no	—
71	C₂×C₄×C₅₉	—	8	1	no	—
72	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#123	1	2	yes	—
75	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
78	C₈×C₅₉	—	2	4	no	—
79	C₈×C₅₉	—	8	1	no	—
80	C₅₉⋊C₈	—	4	4	no	—
81	C₅₉⋊C₈	—	8	2	no	—
82	#10661	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	—	2	1	no	—
84	C₂²×C₅₉	—	2	1	no	—
85	C₂²×C₅₉	—	4	4	no	—
86	C₂²×C₅₉	—	4	2	no	—
87	C₂²×C₅₉	—	8	2	no	—
88	C₂²×C₅₉	—	8	1	no	—
89	C₂²×C₅₉	—	8	1	no	—
90	C₂²×C₅₉	#399	1	1	yes	characteristic
91	C₂²×C₅₉	#476	1	2	yes	—
92	C₄×C₅₉	—	2	2	no	—
93	C₄×C₅₉	—	2	2	no	—
94	C₄×C₅₉	—	8	1	no	—
95	C₄×C₅₉	#481	1	2	yes	—
96	C₄×C₅₉	#492	1	1	yes	characteristic
97	C₄×C₅₉	#496	1	1	yes	characteristic
98	#1955	D₅₉	1	1	yes	characteristic
99	#1849	—	59	1	no	—
100	#1570	—	59	2	no	—
101	#1486	—	59	2	no	—
102	#1275	—	59	1	no	—
103	C₂×C₅₉	—	2	2	no	—
104	C₂×C₅₉	—	8	2	no	—
105	C₂×C₅₉	#2091	1	1	yes	characteristic
106	C₂×C₅₉	#2094	1	1	yes	characteristic
107	C₂×C₅₉	#2098	1	1	yes	characteristic
108	C₂×C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
109	#431	—	59	2	no	—
110	C₂×#108	—	118	2	no	—
111	#423	—	59	2	no	—
112	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
113	C₂×C₂²⋊C₈	—	2	2	no	—
114	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
115	#334	—	118	4	no	—
116	D₄⋊C₈	—	118	4	no	—
117	C₅₉	#10661	1	1	yes	characteristic, complemented, Sylow
118	C₂²×D₄	Dic₁₁₈	1	1	yes	characteristic
119	C₂³×C₄	—	2	1	no	—
120	C₂×M₁₆	—	118	2	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
123	C₄×D₄	—	2	2	no	—
124	C₄×D₄	—	2	2	no	—
125	C₂×C₄⋊C₄	Dic₁₁₈	1	1	yes	characteristic
126	C₂×C₂²⋊C₄	—	2	1	no	—
127	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
128	#108	—	236	4	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	118	2	no	—
131	C₄⋊C₈	—	118	2	no	—
132	C₄⋊C₈	—	118	2	no	—
133	C₂²⋊C₈	—	4	4	no	—
134	C₂²⋊C₈	—	118	2	no	—
135	#97	—	2	2	no	—
136	C₄×C₈	—	59	4	no	—
137	C₂⁴	—	2	1	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	4	1	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
146	C₂²×C₄	#11745685	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
148	M₁₆	—	236	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	118	4	no	—
154	C₂×C₈	—	236	2	no	—
155	C₂×C₈	#11745670	1	4	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	D₄×D₅₉	1	2	yes	—
159	C₄²	#11745687	1	2	yes	—
160	C₂³	—	2	2	no	—
161	C₂³	—	4	1	no	—
162	C₂³	—	4	2	no	—
163	C₂³	—	4	2	no	—
164	C₂³	—	4	1	no	—
165	C₂³	#14273706	1	1	yes	characteristic
166	D₄	—	2	4	no	—
167	D₄	—	4	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	#14273704	1	2	yes	—
176	C₂×C₄	#14273708	1	1	yes	characteristic
177	C₂×C₄	#14273709	1	2	yes	—
178	C₂×C₄	#14273723	1	1	yes	characteristic
179	C₂×C₄	C₂×#11745670	1	2	yes	—
180	C₂×C₄	#14273752	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	8	1	no	—
183	C₈	—	472	2	no	—
184	C₈	—	236	4	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	4	4	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	1	no	—
191	C₂²	—	8	1	no	—
192	C₂²	#16837338	1	1	yes	characteristic
193	C₂²	#16837692	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	8	1	no	—
197	C₄	#16837718	1	2	yes	—
198	C₄	#16837774	1	1	yes	characteristic
199	C₄	#16837792	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	—	8	2	no	—
202	C₂	#21425612	1	1	yes	characteristic
203	C₂	#21425629	1	1	yes	characteristic
204	C₂	#21425649	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424271

C₂

yes

maximal, characteristic, complemented

#21422866

C₂

yes

maximal, complemented

#21422521

C₂

yes

maximal

#21421927

C₂

yes

maximal, characteristic, complemented

C₅₉×#1955

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

C₂×#14273663

—

#16837426

C₂²

yes

—

#16837152

—

D₄⋊(C₅₉⋊C₈)

—

C₂×C₄×D₄×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₅₉

—

C₂×#97×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₂²×D₄×C₅₉

Q₈

yes

characteristic

C₂³×C₄×C₅₉

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₄×D₄×C₅₉

—

C₄×D₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

Q₈

yes

characteristic

C₂×C₂²⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₂⁴×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

#41

yes

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂²⋊Q₈

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

#123

yes

characteristic

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#10661

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#399

yes

characteristic

C₂²×C₅₉

#476

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#481

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

characteristic

#1849

—

100

#1570

—

101

#1486

—

102

#1275

—

103

C₂×C₅₉

—

104

C₂×C₅₉

—

105

C₂×C₅₉

#2091

yes

characteristic

106

C₂×C₅₉

#2094

yes

characteristic

107

C₂×C₅₉

#2098

yes

characteristic

108

C₂×C₄×D₄

C₂×D₅₉

yes

characteristic

109

#431

—

110

C₂×#108

—

118

—

111

#423

—

112

C₂×C₄⋊C₈

C₂×D₅₉

yes

characteristic

113

C₂×C₂²⋊C₈

—

114

C₂×#97

C₂×D₅₉

yes

characteristic

115

#334

—

118

—

116

D₄⋊C₈

—

118

—

117

C₅₉

#10661

yes

characteristic, complemented, Sylow

118

C₂²×D₄

Dic₁₁₈

yes

characteristic

119

C₂³×C₄

—

120

C₂×M₁₆

—

118

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂²⋊D₅₉

yes

—

123

C₄×D₄

—

124

C₄×D₄

—

125

C₂×C₄⋊C₄

Dic₁₁₈

yes

characteristic

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

C₂²×D₅₉

yes

characteristic

128

#108

—

236

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

118

—

131

C₄⋊C₈

—

118

—

132

C₄⋊C₈

—

118

—

133

C₂²⋊C₈

—

134

C₂²⋊C₈

—

118

—

135

#97

—

136

C₄×C₈

—

137

C₂⁴

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂×Dic₁₁₈

yes

characteristic

146

C₂²×C₄

#11745685

yes

characteristic

147

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

148

M₁₆

—

236

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

118

—

154

C₂×C₈

—

236

—

155

C₂×C₈

#11745670

yes

—

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

D₄×D₅₉

yes

—

159

C₄²

#11745687

yes

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

#14273706

yes

characteristic

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#14273704

yes

—

176

C₂×C₄

#14273708

yes

characteristic

177

C₂×C₄

#14273709

yes

—

178

C₂×C₄

#14273723

yes

characteristic

179

C₂×C₄

C₂×#11745670

yes

—

180

C₂×C₄

#14273752

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

472

—

184

C₈

—

236

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

#16837338

yes

characteristic

193

C₂²

#16837692

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#16837718

yes

—

198

C₄

#16837774

yes

characteristic

199

C₄

#16837792

yes

characteristic

200

C₂

—

201

C₂

—

202

C₂

#21425612

yes

characteristic

203

C₂

#21425629

yes

characteristic

204

C₂

#21425649

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14273723
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1955	C₂
socle	C₂²×C₅₉	#399
prime radical	C₂²×D₄×C₅₉	Q₈
hypercenter	#1955	D₅₉
hypocenter	C₅₉	#10661
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1955

subgroup

G/H

center

C₂×C₄

#14273723

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1955

C₂

socle

C₂²×C₅₉

#399

prime radical

C₂²×D₄×C₅₉

Q₈

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1955

N	H ≅ G/N	type
1. #21424271	C₂	semidirect
2. #21422866	C₂	semidirect
4. #21421927	C₂	semidirect
117. C₅₉	#10661	semidirect

H ≅ G/N

type

1. #21424271

C₂

semidirect

2. #21422866

C₂

semidirect

4. #21421927

C₂

semidirect

117. C₅₉

#10661

semidirect

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

subgroup	index
#21424271	2
#21422866	2
#21422521	2
#21421927	2
C₅₉×#1955	2
#10661	59

subgroup

index

C₅₉×#1955

Subgroups — #28493672

Conjugacy classes of subgroups

Complements

Maximal subgroups