Group #28488374 — subgroups

Subgroups — #28488374

Order 15104.

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

normal subgroups only

This group has 402 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	#21425086	C₂	1	2	yes	maximal
2	#21422852	C₂	1	1	yes	maximal, characteristic
3	#21422781	C₂	1	2	yes	maximal
4	#21422385	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×#500	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	1	yes	characteristic
7	#16837426	C₂²	1	2	yes	—
8	C₄×#11745664	C₂²	1	1	yes	characteristic
9	#16837171	—	2	4	no	—
10	#16837153	—	2	2	no	—
11	#16837148	—	2	4	no	—
12	#16837124	—	2	1	no	—
13	#16837123	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#500	—	2	1	no	—
16	C₅₉×#500	—	2	1	no	—
17	C₅₉×#500	—	2	2	no	—
18	C₂×#107×C₅₉	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×#11745664	—	2	2	no	—
21	C₂×#11745664	D₄	1	2	yes	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
25	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
26	#14273638	—	2	1	no	—
27	C₄×C₅₉⋊C₈	—	2	1	no	—
28	C₄×C₅₉⋊C₈	—	2	1	no	—
29	C₂²×Q₈×C₅₉	—	2	2	no	—
30	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₅₉	—	2	2	no	—
32	C₄⋊Q₈×C₅₉	—	2	2	no	—
33	C₄⋊Q₈×C₅₉	—	2	2	no	—
34	C₄⋊Q₈×C₅₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	#107×C₅₉	—	2	4	no	—
39	#107×C₅₉	—	2	4	no	—
40	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₉	D₄	1	2	yes	—
43	C₂×Q₈×C₅₉	—	4	2	no	—
44	C₂×Q₈×C₅₉	—	4	2	no	—
45	C₂×Q₈×C₅₉	—	8	4	no	—
46	C₂²×C₄×C₅₉	—	2	4	no	—
47	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
50	C₂×C₈×C₅₉	—	2	4	no	—
51	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
53	C₄⋊C₄×C₅₉	—	2	4	no	—
54	C₄⋊C₄×C₅₉	—	2	4	no	—
55	C₄⋊C₄×C₅₉	—	4	2	no	—
56	C₄⋊C₄×C₅₉	—	4	2	no	—
57	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₉	#41	1	1	yes	characteristic
61	#11745664	—	2	4	no	—
62	#11745664	—	4	4	no	—
63	C₂×C₅₉⋊C₈	—	2	2	no	—
64	C₂×C₅₉⋊C₈	—	2	2	no	—
65	C₂×C₅₉⋊C₈	—	4	2	no	—
66	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
67	Q₈×C₅₉	—	8	4	no	—
68	Q₈×C₅₉	—	8	4	no	—
69	C₂×C₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	4	4	no	—
72	C₂×C₄×C₅₉	—	8	4	no	—
73	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
78	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
79	C₂×C₄×C₅₉	#137	1	4	yes	—
80	C₈×C₅₉	—	2	2	no	—
81	C₈×C₅₉	#101	1	4	yes	—
82	C₅₉⋊C₈	—	4	2	no	—
83	C₅₉⋊C₈	—	4	2	no	—
84	C₅₉⋊C₈	—	8	2	no	—
85	#9810	—	59	1	no	maximal, Sylow
86	C₂²×C₅₉	—	2	1	no	—
87	C₂²×C₅₉	—	2	1	no	—
88	C₂²×C₅₉	#390	1	1	yes	characteristic
89	C₂²×C₅₉	#441	1	2	yes	—
90	C₄×C₅₉	—	2	1	no	—
91	C₄×C₅₉	—	2	1	no	—
92	C₄×C₅₉	—	8	4	no	—
93	C₄×C₅₉	C₂×#101	1	2	yes	—
94	C₄×C₅₉	#470	1	2	yes	—
95	C₄×C₅₉	#472	1	2	yes	—
96	C₄×C₅₉	#497	1	2	yes	—
97	C₂×#500	D₅₉	1	1	yes	characteristic
98	#1997	—	59	2	no	—
99	#1566	—	59	1	no	—
100	#1549	—	59	2	no	—
101	#1451	—	59	1	no	—
102	C₂×C₅₉	—	2	2	no	—
103	C₂×C₅₉	#1966	1	1	yes	characteristic
104	C₂×C₅₉	#1971	1	1	yes	characteristic
105	C₂×C₅₉	#1978	1	1	yes	characteristic
106	C₂×C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
107	#500	—	2	1	no	—
108	#500	—	2	1	no	—
109	#500	—	2	2	no	—
110	#431	—	59	1	no	—
111	C₂×#107	Dic₅₉	1	4	yes	—
112	#423	—	59	2	no	—
113	C₄×M₁₆	—	59	1	no	—
114	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
115	#343	—	118	4	no	—
116	#334	—	118	2	no	—
117	#332	—	118	4	no	—
118	#322	—	118	1	no	—
119	#322	—	118	1	no	—
120	C₅₉	#9810	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	2	2	no	—
122	C₂×M₁₆	—	59	2	no	—
123	C₂×M₁₆	—	118	2	no	—
124	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
131	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
132	#107	—	2	4	no	—
133	#107	—	2	4	no	—
134	C₄⋊C₈	—	59	2	no	—
135	C₄⋊C₈	—	118	2	no	—
136	C₄⋊C₈	—	118	2	no	—
137	C₂²⋊C₈	—	118	1	no	—
138	#97	—	118	1	no	—
139	C₄×C₈	—	118	1	no	—
140	C₄×C₈	—	118	1	no	—
141	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	8	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
149	C₂²×C₄	#11745687	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
151	M₁₆	—	118	4	no	—
152	M₁₆	—	236	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	118	2	no	—
155	C₂×C₈	—	118	2	no	—
156	C₂×C₈	—	236	2	no	—
157	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
158	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄²	D₄×D₅₉	1	1	yes	characteristic
164	C₄²	#11745687	1	1	yes	characteristic
165	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
166	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
167	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
168	Q₈	—	8	4	no	—
169	Q₈	—	8	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	4	no	—
174	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
176	C₂×C₄	#14273698	1	1	yes	characteristic
177	C₂×C₄	#14273698	1	1	yes	characteristic
178	C₂×C₄	#14273714	1	1	yes	characteristic
179	C₂×C₄	#14273723	1	1	yes	characteristic
180	C₂×C₄	#14273759	1	4	yes	—
181	C₈	—	2	2	no	—
182	C₈	—	472	2	no	—
183	C₈	—	236	2	no	—
184	C₈	—	236	2	no	—
185	C₈	#14273646	1	4	yes	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#16837305	1	1	yes	characteristic
189	C₂²	#16837509	1	2	yes	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	8	4	no	—
193	C₄	C₂×#14273646	1	2	yes	—
194	C₄	#16837657	1	2	yes	—
195	C₄	#16837669	1	2	yes	—
196	C₄	#16837795	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#21424941	1	1	yes	characteristic
199	C₂	#21424968	1	1	yes	characteristic
200	C₂	#21425007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425086

C₂

yes

maximal

#21422852

C₂

yes

maximal, characteristic

#21422781

C₂

yes

maximal

#21422385

C₂

yes

maximal, characteristic

C₂×C₅₉×#500

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

characteristic

#16837426

C₂²

yes

—

C₄×#11745664

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₅₉

C₂²

yes

—

C₅₉×#500

—

C₅₉×#500

—

C₅₉×#500

—

C₂×#107×C₅₉

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

C₂×#11745664

D₄

yes

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂²×Q₈×C₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#107×C₅₉

—

#107×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

C₄²×C₅₉

yes

—

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄×D₄

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₈×C₅₉

—

C₈×C₅₉

#101

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9810

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#441

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#101

yes

—

C₄×C₅₉

#470

yes

—

C₄×C₅₉

#472

yes

—

C₄×C₅₉

#497

yes

—

C₂×#500

D₅₉

yes

characteristic

#1997

—

#1566

—

100

#1549

—

101

#1451

—

102

C₂×C₅₉

—

103

C₂×C₅₉

#1966

yes

characteristic

104

C₂×C₅₉

#1971

yes

characteristic

105

C₂×C₅₉

#1978

yes

characteristic

106

C₂×C₄⋊Q₈

C₂×D₅₉

yes

—

107

#500

—

108

#500

—

109

#500

—

110

#431

—

111

C₂×#107

Dic₅₉

yes

—

112

#423

—

113

C₄×M₁₆

—

114

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

115

#343

—

118

—

116

#334

—

118

—

117

#332

—

118

—

118

#322

—

118

—

119

#322

—

118

—

120

C₅₉

#9810

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

118

—

124

C₂²×C₈

C₂×Dic₅₉

yes

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

C₂×Dic₅₉

yes

—

131

C₂×C₄²

C₂²×D₅₉

yes

characteristic

132

#107

—

133

#107

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

118

—

136

C₄⋊C₈

—

118

—

137

C₂²⋊C₈

—

118

—

138

#97

—

118

—

139

C₄×C₈

—

118

—

140

C₄×C₈

—

118

—

141

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

142

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

143

C₄×C₈

C₂²⋊D₅₉

yes

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

D₄×D₅₉

yes

characteristic

149

C₂²×C₄

#11745687

yes

characteristic

150

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

151

M₁₆

—

118

—

152

M₁₆

—

236

—

153

C₂×C₈

—

154

C₂×C₈

—

118

—

155

C₂×C₈

—

118

—

156

C₂×C₈

—

236

—

157

C₂×C₈

C₂²⋊Dic₅₉

yes

—

158

C₂×C₈

C₂²⋊Dic₅₉

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

D₄×D₅₉

yes

characteristic

164

C₄²

#11745687

yes

characteristic

165

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

166

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

167

C₂³

D₄×Dic₅₉

yes

characteristic

168

Q₈

—

169

Q₈

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

175

C₂×C₄

D₄×Dic₅₉

yes

characteristic

176

C₂×C₄

#14273698

yes

characteristic

177

C₂×C₄

#14273698

yes

characteristic

178

C₂×C₄

#14273714

yes

characteristic

179

C₂×C₄

#14273723

yes

characteristic

180

C₂×C₄

#14273759

yes

—

181

C₈

—

182

C₈

—

472

—

183

C₈

—

236

—

184

C₈

—

236

—

185

C₈

#14273646

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

#16837305

yes

characteristic

189

C₂²

#16837509

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

C₂×#14273646

yes

—

194

C₄

#16837657

yes

—

195

C₄

#16837669

yes

—

196

C₄

#16837795

yes

—

197

C₂

—

198

C₂

#21424941

yes

characteristic

199

C₂

#21424968

yes

characteristic

200

C₂

#21425007

yes

characteristic

	subgroup	G/H
center	C₂²	#16837305
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#500	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#500	D₅₉
hypocenter	C₅₉	#9810
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#16837305

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₅₉×#500

C₂

socle

C₂²×C₅₉

#390

prime radical

C₂³×C₅₉

C₄×D₄

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
120. C₅₉	#9810	semidirect

H ≅ G/N

type

120. C₅₉

#9810

semidirect

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

subgroup	index
#21425086	2
#21422852	2
#21422781	2
#21422385	2
C₂×C₅₉×#500	2
#9810	59

subgroup

index

C₂×C₅₉×#500

Subgroups — #28488374

Conjugacy classes of subgroups

Complements

Maximal subgroups