Group #28486294 — subgroups

Subgroups — #28486294

Order 15104.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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	#21433198	C₂	1	1	yes	maximal, characteristic
2	#21422834	C₂	1	1	yes	maximal, characteristic
3	#21422781	C₂	1	2	yes	maximal
4	#21421986	C₂	1	2	yes	maximal
5	C₅₉×#3139	C₂	1	1	yes	maximal, characteristic
6	#16837533	—	2	2	no	—
7	Q₈×C₅₉⋊C₈	—	2	2	no	—
8	#16837463	—	2	2	no	—
9	#16837455	C₂²	1	2	yes	—
10	#16837426	C₂²	1	2	yes	—
11	#16837148	—	2	2	no	—
12	#16837148	—	2	2	no	—
13	#16837148	—	2	2	no	—
14	C₅₉×#533	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#491	C₄	1	4	yes	—
17	C₅₉×#487	—	2	2	no	—
18	C₅₉×#419	—	2	2	no	—
19	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×#11745664	—	2	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
25	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
26	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
27	#14273638	—	2	2	no	—
28	C₄×C₅₉⋊C₈	—	2	2	no	—
29	C₄×C₅₉⋊C₈	—	2	2	no	—
30	C₂²×Q₈×C₅₉	—	2	1	no	—
31	C₂×M₁₆×C₅₉	—	2	1	no	—
32	C₄⋊Q₈×C₅₉	—	2	2	no	—
33	C₄⋊Q₈×C₅₉	—	2	1	no	—
34	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
35	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
36	#125×C₅₉	C₂×C₄	1	2	yes	—
37	C₂²⋊Q₈×C₅₉	—	2	2	no	—
38	C₄×Q₈×C₅₉	—	2	2	no	—
39	#117×C₅₉	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
41	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	#103×C₅₉	—	2	8	no	—
43	#97×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₉	—	2	1	no	—
45	C₂×Q₈×C₅₉	—	2	2	no	—
46	C₂×Q₈×C₅₉	—	2	2	no	—
47	C₂×Q₈×C₅₉	—	4	4	no	—
48	C₂×Q₈×C₅₉	—	4	1	no	—
49	C₂²×C₄×C₅₉	—	4	1	no	—
50	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	#41	1	2	yes	—
52	M₁₆×C₅₉	—	4	2	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₄⋊C₄×C₅₉	—	2	4	no	—
55	C₄⋊C₄×C₅₉	—	2	2	no	—
56	C₄⋊C₄×C₅₉	—	2	4	no	—
57	C₄⋊C₄×C₅₉	—	4	2	no	—
58	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	4	yes	—
59	C₂²⋊C₄×C₅₉	—	2	2	no	—
60	C₄²×C₅₉	—	2	2	no	—
61	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅₉	C₂×D₄	1	2	yes	—
64	#11745664	—	8	2	no	—
65	C₂×C₅₉⋊C₈	—	2	2	no	—
66	C₂×C₅₉⋊C₈	—	2	2	no	—
67	C₂×C₅₉⋊C₈	—	4	2	no	—
68	C₂³×C₅₉	#123	1	1	yes	characteristic
69	Q₈×C₅₉	—	4	4	no	—
70	Q₈×C₅₉	—	4	2	no	—
71	Q₈×C₅₉	—	4	2	no	—
72	Q₈×C₅₉	—	8	2	no	—
73	C₂×C₄×C₅₉	—	2	2	no	—
74	C₂×C₄×C₅₉	—	2	2	no	—
75	C₂×C₄×C₅₉	—	2	2	no	—
76	C₂×C₄×C₅₉	—	2	4	no	—
77	C₂×C₄×C₅₉	—	4	2	no	—
78	C₂×C₄×C₅₉	—	8	2	no	—
79	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
82	C₂×C₄×C₅₉	#121	1	2	yes	—
83	C₈×C₅₉	—	4	2	no	—
84	C₅₉⋊C₈	—	4	2	no	—
85	C₅₉⋊C₈	—	4	2	no	—
86	C₅₉⋊C₈	—	8	2	no	—
87	#9436	—	59	1	no	maximal, Sylow
88	C₂²×C₅₉	—	2	1	no	—
89	C₂²×C₅₉	—	2	2	no	—
90	C₂²×C₅₉	#386	1	1	yes	characteristic
91	C₄×C₅₉	—	2	1	no	—
92	C₄×C₅₉	—	2	2	no	—
93	C₄×C₅₉	—	4	4	no	—
94	C₄×C₅₉	—	8	2	no	—
95	C₄×C₅₉	#454	1	2	yes	—
96	C₄×C₅₉	#471	1	2	yes	—
97	C₄×C₅₉	#473	1	2	yes	—
98	#3139	D₅₉	1	1	yes	characteristic
99	#2986	—	59	1	no	—
100	#1561	—	59	1	no	—
101	#1549	—	59	2	no	—
102	#1299	—	59	2	no	—
103	C₂×C₅₉	—	4	1	no	—
104	C₂×C₅₉	#1888	1	1	yes	characteristic
105	C₂×C₅₉	#1908	1	2	yes	—
106	#533	C₂×D₅₉	1	1	yes	characteristic
107	C₂×C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
108	#491	Dic₅₉	1	4	yes	—
109	#487	—	2	2	no	—
110	#446	—	118	2	no	—
111	C₈×Q₈	—	118	2	no	—
112	#433	—	118	2	no	—
113	#431	—	59	2	no	—
114	#423	—	59	2	no	—
115	#419	—	2	2	no	—
116	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
117	#332	—	118	2	no	—
118	#332	—	118	2	no	—
119	#332	—	118	2	no	—
120	C₅₉	#9436	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	2	1	no	—
122	C₂×M₁₆	—	2	1	no	—
123	C₂×M₁₆	—	118	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	1	no	—
126	C₄⋊Q₈	C₂×Dic₅₉	1	2	yes	—
127	C₄⋊Q₈	C₂²⋊D₅₉	1	2	yes	—
128	#125	C₂×Dic₅₉	1	2	yes	—
129	C₂²⋊Q₈	—	2	2	no	—
130	C₄×Q₈	—	2	2	no	—
131	#117	C₂²⋊D₅₉	1	2	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
134	C₄⋊C₈	—	59	2	no	—
135	C₄⋊C₈	—	59	2	no	—
136	C₄⋊C₈	—	118	4	no	—
137	C₄⋊C₈	—	118	2	no	—
138	C₄⋊C₈	—	118	2	no	—
139	#103	—	2	8	no	—
140	C₂²⋊C₈	—	118	2	no	—
141	#97	—	118	2	no	—
142	#97	C₂×Dic₅₉	1	2	yes	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	118	2	no	—
145	C₄×C₈	—	118	2	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	4	4	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂²×C₄	—	4	1	no	—
151	C₂²×C₄	#11745687	1	2	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
153	M₁₆	—	4	2	no	—
154	M₁₆	—	472	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	118	2	no	—
157	C₂×C₈	—	118	2	no	—
158	C₂×C₈	—	236	2	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	C₂²⋊Dic₅₉	1	4	yes	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	D₄×D₅₉	1	2	yes	—
167	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
168	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
169	C₂³	#14273708	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	4	2	no	—
172	Q₈	—	4	2	no	—
173	Q₈	—	8	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
181	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
182	C₂×C₄	#14273695	1	1	yes	characteristic
183	C₂×C₄	#14273699	1	2	yes	—
184	C₈	—	4	2	no	—
185	C₈	—	472	2	no	—
186	C₈	—	236	2	no	—
187	C₈	—	236	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	#16837282	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	—	8	2	no	—
195	C₄	#16837574	1	2	yes	—
196	C₄	#16837667	1	2	yes	—
197	C₄	#16837675	1	2	yes	—
198	C₂	—	4	1	no	—
199	C₂	#21424500	1	1	yes	characteristic
200	C₂	#21424613	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21433198

C₂

yes

maximal, characteristic

#21422834

C₂

yes

maximal, characteristic

#21422781

C₂

yes

maximal

#21421986

C₂

yes

maximal

C₅₉×#3139

C₂

yes

maximal, characteristic

#16837533

—

Q₈×C₅₉⋊C₈

—

#16837463

—

#16837455

C₂²

yes

—

#16837426

C₂²

yes

—

#16837148

—

#16837148

—

#16837148

—

C₅₉×#533

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₅₉

C₂²

yes

characteristic

C₅₉×#491

C₄

yes

—

C₅₉×#487

—

C₅₉×#419

—

C₄×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂²×Q₈×C₅₉

—

C₂×M₁₆×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₂×C₄

yes

—

C₄⋊Q₈×C₅₉

D₄

yes

—

#125×C₅₉

C₂×C₄

yes

—

C₂²⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

#117×C₅₉

D₄

yes

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#103×C₅₉

—

#97×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#123

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×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

#120

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9436

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#386

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#454

yes

—

C₄×C₅₉

#471

yes

—

C₄×C₅₉

#473

yes

—

#3139

D₅₉

yes

characteristic

#2986

—

100

#1561

—

101

#1549

—

102

#1299

—

103

C₂×C₅₉

—

104

C₂×C₅₉

#1888

yes

characteristic

105

C₂×C₅₉

#1908

yes

—

106

#533

C₂×D₅₉

yes

characteristic

107

C₂×C₄⋊Q₈

C₂×D₅₉

yes

characteristic

108

#491

Dic₅₉

yes

—

109

#487

—

110

#446

—

118

—

111

C₈×Q₈

—

118

—

112

#433

—

118

—

113

#431

—

114

#423

—

115

#419

—

116

C₄×M₁₆

C₂×D₅₉

yes

characteristic

117

#332

—

118

—

118

#332

—

118

—

119

#332

—

118

—

120

C₅₉

#9436

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

118

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

C₂×Dic₅₉

yes

—

127

C₄⋊Q₈

C₂²⋊D₅₉

yes

—

128

#125

C₂×Dic₅₉

yes

—

129

C₂²⋊Q₈

—

130

C₄×Q₈

—

131

#117

C₂²⋊D₅₉

yes

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²×D₅₉

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

118

—

137

C₄⋊C₈

—

118

—

138

C₄⋊C₈

—

118

—

139

#103

—

140

C₂²⋊C₈

—

118

—

141

#97

—

118

—

142

#97

C₂×Dic₅₉

yes

—

143

C₄×C₈

—

144

C₄×C₈

—

118

—

145

C₄×C₈

—

118

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂²×C₄

—

151

C₂²×C₄

#11745687

yes

—

152

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

153

M₁₆

—

154

M₁₆

—

472

—

155

C₂×C₈

—

156

C₂×C₈

—

118

—

157

C₂×C₈

—

118

—

158

C₂×C₈

—

236

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

C₂²⋊Dic₅₉

yes

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

D₄×D₅₉

yes

—

167

C₄²

C₂²×Dic₅₉

yes

characteristic

168

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

169

C₂³

#14273708

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

181

C₂×C₄

D₄×Dic₅₉

yes

—

182

C₂×C₄

#14273695

yes

characteristic

183

C₂×C₄

#14273699

yes

—

184

C₈

—

185

C₈

—

472

—

186

C₈

—

236

—

187

C₈

—

236

—

188

C₂²

—

189

C₂²

—

190

C₂²

#16837282

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

#16837574

yes

—

196

C₄

#16837667

yes

—

197

C₄

#16837675

yes

—

198

C₂

—

199

C₂

#21424500

yes

characteristic

200

C₂

#21424613

yes

—

	subgroup	G/H
center	C₂²	#16837282
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#3139	C₂
socle	C₂²×C₅₉	#386
prime radical	C₂³×C₅₉	#123
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#3139	D₅₉
hypocenter	C₅₉	#9436
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3139

subgroup

G/H

center

C₂²

#16837282

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#3139

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3139

N	H ≅ G/N	type
120. C₅₉	#9436	semidirect

H ≅ G/N

type

120. C₅₉

#9436

semidirect

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

subgroup	index
#21433198	2
#21422834	2
#21422781	2
#21421986	2
C₅₉×#3139	2
#9436	59

subgroup

index

C₅₉×#3139

Subgroups — #28486294

Conjugacy classes of subgroups

Complements

Maximal subgroups