Group #28502852 — subgroups

Subgroups — #28502852

Order 15104.

218 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 67 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	#21426238	C₂	1	1	yes	maximal, characteristic
2	#21423690	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#21422159	C₂	1	1	yes	maximal, characteristic
5	#21422150	C₂	1	1	yes	maximal, characteristic
6	#21422145	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#2209	C₂	1	1	yes	maximal, characteristic
8	#16837807	C₂²	1	1	yes	characteristic
9	#16837798	—	2	1	no	—
10	#16837754	—	2	1	no	—
11	Q₈×C₅₉⋊C₈	—	2	1	no	—
12	#16837229	—	2	1	no	—
13	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
14	#16837206	—	2	1	no	—
15	#16837155	C₂²	1	1	yes	characteristic
16	Q₈⋊(C₅₉⋊C₈)	—	2	1	no	—
17	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×Q₃₂×C₅₉	—	2	1	no	—
19	C₅₉×#494	C₂²	1	1	yes	characteristic
20	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₅₉×#358	—	2	1	no	—
23	#14273727	—	2	1	no	—
24	C₂×C₅₉⋊C₁₆	—	2	1	no	—
25	#14273661	—	2	1	no	—
26	#14273661	—	2	1	no	—
27	#14273658	—	4	2	no	—
28	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
29	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
30	#14273650	—	4	1	no	—
31	#14273650	—	4	1	no	—
32	C₄×C₅₉⋊C₈	—	4	1	no	—
33	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
34	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	C₂×Q₁₆×C₅₉	—	2	1	no	—
36	C₂×Q₁₆×C₅₉	—	2	1	no	—
37	C₂×Q₁₆×C₅₉	D₄	1	1	yes	characteristic
38	C₄⋊Q₈×C₅₉	—	2	1	no	—
39	C₄×Q₈×C₅₉	—	2	1	no	—
40	Q₃₂×C₅₉	—	4	2	no	—
41	C₂×C₁₆×C₅₉	—	2	1	no	—
42	#107×C₅₉	D₄	1	1	yes	characteristic
43	#103×C₅₉	—	2	1	no	—
44	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₅₉	—	2	1	no	—
46	C₂×Q₈×C₅₉	—	4	1	no	—
47	C₂×Dic₁₁₈	—	4	1	no	—
48	Q₁₆×C₅₉	—	2	1	no	—
49	Q₁₆×C₅₉	—	4	1	no	—
50	Q₁₆×C₅₉	—	4	1	no	—
51	Q₁₆×C₅₉	—	8	1	no	—
52	Q₁₆×C₅₉	SD₁₆	1	2	yes	—
53	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₅₉	—	2	1	no	—
56	C₄⋊C₄×C₅₉	—	4	1	no	—
57	C₄⋊C₄×C₅₉	—	4	1	no	—
58	C₄⋊C₄×C₅₉	—	4	1	no	—
59	C₄²×C₅₉	—	4	1	no	—
60	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₅₉	—	4	1	no	—
62	C₄⋊Dic₅₉	—	4	1	no	—
63	C₂×C₅₉⋊C₈	—	2	1	no	—
64	C₂×C₅₉⋊C₈	—	4	1	no	—
65	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
66	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
67	C₁₆×C₅₉	—	4	1	no	—
68	C₅₉⋊C₁₆	—	4	1	no	—
69	Q₈×C₅₉	—	2	2	no	—
70	Q₈×C₅₉	—	4	1	no	—
71	Q₈×C₅₉	—	8	1	no	—
72	Q₈×C₅₉	—	8	1	no	—
73	C₂×C₄×C₅₉	—	4	1	no	—
74	C₂×C₄×C₅₉	—	4	1	no	—
75	C₂×C₄×C₅₉	—	8	1	no	—
76	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×Dic₅₉	—	8	1	no	—
80	Dic₁₁₈	—	8	1	no	—
81	Dic₁₁₈	—	8	1	no	—
82	C₈×C₅₉	—	2	1	no	—
83	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
84	C₈×C₅₉	#135	1	1	yes	characteristic
85	C₅₉⋊C₈	—	2	2	no	—
86	C₅₉⋊C₈	—	2	1	no	—
87	C₅₉⋊C₈	—	2	1	no	—
88	C₅₉⋊C₈	—	8	1	no	—
89	#12423	—	59	1	no	maximal, Sylow
90	C₂²×C₅₉	#459	1	1	yes	characteristic
91	C₄×C₅₉	—	2	1	no	—
92	C₄×C₅₉	—	4	2	no	—
93	C₄×C₅₉	—	8	1	no	—
94	C₄×C₅₉	—	16	1	no	—
95	C₄×C₅₉	#465	1	1	yes	characteristic
96	C₄×C₅₉	#471	1	1	yes	characteristic
97	C₄×C₅₉	#508	1	1	yes	characteristic
98	C₄×C₅₉	#510	1	1	yes	characteristic
99	Dic₅₉	—	16	1	no	—
100	#2225	—	59	1	no	—
101	#2209	D₅₉	1	1	yes	characteristic
102	#1707	—	59	1	no	—
103	C₈×Q₁₆	—	59	1	no	—
104	#1367	—	59	1	no	—
105	#1364	—	59	1	no	—
106	#1362	—	59	1	no	—
107	C₂×C₅₉	#1665	1	1	yes	characteristic
108	C₂×C₅₉	#2211	1	1	yes	characteristic
109	C₂×C₅₉	#2214	1	1	yes	characteristic
110	C₂×Q₃₂	—	2	1	no	—
111	#500	—	59	1	no	—
112	#498	—	118	1	no	—
113	#494	C₂×D₅₉	1	1	yes	characteristic
114	#487	—	118	1	no	—
115	C₈×Q₈	—	118	1	no	—
116	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
117	#367	—	118	1	no	—
118	C₄⋊C₁₆	—	59	1	no	—
119	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
120	#358	—	2	1	no	—
121	#358	—	118	1	no	—
122	#335	—	59	1	no	—
123	Q₈⋊C₈	—	118	1	no	—
124	C₈²	—	59	1	no	—
125	C₅₉	#12423	1	1	yes	characteristic, complemented, Sylow
126	C₂×Q₁₆	—	2	1	no	—
127	C₂×Q₁₆	—	2	1	no	—
128	C₂×Q₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	1	no	—
130	C₄⋊Q₈	—	118	1	no	—
131	C₄×Q₈	—	2	1	no	—
132	Q₃₂	—	4	2	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	—	118	1	no	—
135	#107	—	118	1	no	—
136	#107	—	118	1	no	—
137	#107	C₂²⋊D₅₉	1	1	yes	characteristic
138	#106	—	236	2	no	—
139	C₄⋊C₈	—	118	1	no	—
140	C₄⋊C₈	—	236	1	no	—
141	#103	—	2	1	no	—
142	#103	—	236	1	no	—
143	#103	—	236	1	no	—
144	C₄×C₈	—	59	1	no	—
145	C₄×C₈	—	59	1	no	—
146	C₄×C₈	—	236	1	no	—
147	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂×Q₈	—	236	1	no	—
151	Q₁₆	—	2	1	no	—
152	Q₁₆	—	4	1	no	—
153	Q₁₆	—	4	1	no	—
154	Q₁₆	—	8	1	no	—
155	Q₁₆	#11745670	1	2	yes	—
156	C₂×C₈	—	59	1	no	—
157	C₂×C₈	—	59	1	no	—
158	C₂×C₈	—	118	1	no	—
159	C₂×C₈	—	236	1	no	—
160	C₂×C₈	#11745687	1	1	yes	characteristic
161	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄⋊C₄	—	4	1	no	—
166	C₄⋊C₄	—	236	1	no	—
167	C₄⋊C₄	—	236	1	no	—
168	C₄²	—	4	1	no	—
169	C₄²	D₄×D₅₉	1	1	yes	characteristic
170	C₁₆	—	4	1	no	—
171	C₁₆	—	236	1	no	—
172	Q₈	—	2	2	no	—
173	Q₈	—	4	1	no	—
174	Q₈	—	8	1	no	—
175	Q₈	—	8	1	no	—
176	Q₈	—	472	1	no	—
177	Q₈	—	472	1	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	8	1	no	—
181	C₂×C₄	—	472	1	no	—
182	C₂×C₄	#14273699	1	1	yes	characteristic
183	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
184	C₂×C₄	#14273751	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	118	2	no	—
187	C₈	—	118	1	no	—
188	C₈	—	118	1	no	—
189	C₈	—	472	1	no	—
190	C₈	C₂×#11745670	1	1	yes	characteristic
191	C₈	#14273747	1	1	yes	characteristic
192	C₂²	#16837595	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	4	2	no	—
195	C₄	—	8	1	no	—
196	C₄	—	16	1	no	—
197	C₄	—	944	1	no	—
198	C₄	#16837629	1	1	yes	characteristic
199	C₄	#16837663	1	1	yes	characteristic
200	C₄	#16837835	1	1	yes	characteristic
201	C₄	#16837844	1	1	yes	characteristic
202	C₂	#21423435	1	1	yes	characteristic
203	C₂	#21426160	1	1	yes	characteristic
204	C₂	#21426175	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426238

C₂

yes

maximal, characteristic

#21423690

C₂

yes

maximal, characteristic

Q₁₆×C₅₉⋊C₈

C₂

yes

maximal, characteristic

#21422159

C₂

yes

maximal, characteristic

#21422150

C₂

yes

maximal, characteristic

#21422145

C₂

yes

maximal, characteristic

C₅₉×#2209

C₂

yes

maximal, characteristic

#16837807

C₂²

yes

characteristic

#16837798

—

#16837754

—

Q₈×C₅₉⋊C₈

—

#16837229

—

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

characteristic

#16837206

—

#16837155

C₂²

yes

characteristic

Q₈⋊(C₅₉⋊C₈)

—

C₈×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₅₉

—

C₅₉×#494

C₂²

yes

characteristic

C₄×Q₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

C₂²

yes

characteristic

C₅₉×#358

—

#14273727

—

C₂×C₅₉⋊C₁₆

—

#14273661

—

#14273661

—

#14273658

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

#14273650

—

#14273650

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

D₄

yes

characteristic

C₄×C₅₉⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

D₄

yes

characteristic

C₄⋊Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

Q₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

#107×C₅₉

D₄

yes

characteristic

#103×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

D₈

yes

characteristic

C₂×C₅₉⋊C₈

D₈

yes

characteristic

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×SD₁₆

yes

characteristic

C₈×C₅₉

#135

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

100

#2225

—

101

#2209

D₅₉

yes

characteristic

102

#1707

—

103

C₈×Q₁₆

—

104

#1367

—

105

#1364

—

106

#1362

—

107

C₂×C₅₉

#1665

yes

characteristic

108

C₂×C₅₉

#2211

yes

characteristic

109

C₂×C₅₉

#2214

yes

characteristic

110

C₂×Q₃₂

—

111

#500

—

112

#498

—

118

—

113

#494

C₂×D₅₉

yes

characteristic

114

#487

—

118

—

115

C₈×Q₈

—

118

—

116

C₄×Q₁₆

C₂×D₅₉

yes

characteristic

117

#367

—

118

—

118

C₄⋊C₁₆

—

119

C₄⋊C₁₆

C₂×D₅₉

yes

characteristic

120

#358

—

121

#358

—

118

—

122

#335

—

123

Q₈⋊C₈

—

118

—

124

C₈²

—

125

C₅₉

#12423

yes

characteristic, complemented, Sylow

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

C₂²⋊D₅₉

yes

characteristic

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

118

—

131

C₄×Q₈

—

132

Q₃₂

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

118

—

135

#107

—

118

—

136

#107

—

118

—

137

#107

C₂²⋊D₅₉

yes

characteristic

138

#106

—

236

—

139

C₄⋊C₈

—

118

—

140

C₄⋊C₈

—

236

—

141

#103

—

142

#103

—

236

—

143

#103

—

236

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

236

—

147

C₄×C₈

C₂²×D₅₉

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

236

—

151

Q₁₆

—

152

Q₁₆

—

153

Q₁₆

—

154

Q₁₆

—

155

Q₁₆

#11745670

yes

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

118

—

159

C₂×C₈

—

236

—

160

C₂×C₈

#11745687

yes

characteristic

161

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

236

—

167

C₄⋊C₄

—

236

—

168

C₄²

—

169

C₄²

D₄×D₅₉

yes

characteristic

170

C₁₆

—

171

C₁₆

—

236

—

172

Q₈

—

173

Q₈

—

174

Q₈

—

175

Q₈

—

176

Q₈

—

472

—

177

Q₈

—

472

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

472

—

182

C₂×C₄

#14273699

yes

characteristic

183

C₂×C₄

D₈×D₅₉

yes

characteristic

184

C₂×C₄

#14273751

yes

characteristic

185

C₈

—

186

C₈

—

118

—

187

C₈

—

118

—

188

C₈

—

118

—

189

C₈

—

472

—

190

C₈

C₂×#11745670

yes

characteristic

191

C₈

#14273747

yes

characteristic

192

C₂²

#16837595

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

944

—

198

C₄

#16837629

yes

characteristic

199

C₄

#16837663

yes

characteristic

200

C₄

#16837835

yes

characteristic

201

C₄

#16837844

yes

characteristic

202

C₂

#21423435

yes

characteristic

203

C₂

#21426160

yes

characteristic

204

C₂

#21426175

yes

characteristic

	subgroup	G/H
center	C₂²	#16837595
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2209	C₂
socle, prime radical	C₂²×C₅₉	#459
hypercenter	#2209	D₅₉
hypocenter	C₅₉	#12423
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

subgroup

G/H

center

C₂²

#16837595

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#2209

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

N	H ≅ G/N	type
125. C₅₉	#12423	semidirect

H ≅ G/N

type

125. C₅₉

#12423

semidirect

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

subgroup	index
#21426238	2
#21423690	2
Q₁₆×C₅₉⋊C₈	2
#21422159	2
#21422150	2
#21422145	2
C₅₉×#2209	2
#12423	59

subgroup

index

Q₁₆×C₅₉⋊C₈

C₅₉×#2209

Subgroups — #28502852

Conjugacy classes of subgroups

Complements

Maximal subgroups