Group #28746366 — subgroups

Subgroups — #28746366

Order 15104.

528 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 165 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	Q₈×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#21426319	C₂	1	3	yes	maximal, complemented
3	#21426243	C₂	1	3	yes	maximal, complemented
4	#21426133	C₂	1	3	yes	maximal, complemented
5	Q₈×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14273671	C₂	1	3	yes	maximal, complemented
7	C₅₉×#3391	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₁₈	—	2	1	no	—
9	C₂×#14273671	C₂²	1	3	yes	—
10	#16837807	C₂²	1	3	yes	—
11	#16837698	—	2	3	no	—
12	C₄×Dic₂₃₆	C₂²	1	3	yes	—
13	#16837229	C₂²	1	3	yes	—
14	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	3	yes	—
15	#16837205	C₂²	1	1	yes	characteristic
16	#16837203	C₂²	1	3	yes	—
17	#16837203	C₂²	1	6	yes	—
18	C₄×C₅₉⋊C₁₆	C₂²	1	3	yes	—
19	C₅₉×#550	—	2	1	no	—
20	C₅₉×#493	C₂²	1	3	yes	—
21	C₅₉×#459	—	2	3	no	—
22	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄×D₈×C₅₉	C₂²	1	3	yes	—
24	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
25	#14273727	—	2	3	no	—
26	#14273726	—	4	3	no	—
27	Q₈×Dic₅₉	—	4	1	no	—
28	C₄×Dic₁₁₈	—	2	3	no	—
29	#14273671	—	2	6	no	—
30	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
31	C₂×C₅₉⋊C₁₆	C₂³	1	3	yes	—
32	#14273661	—	2	3	no	—
33	#14273661	C₂³	1	3	yes	—
34	#14273652	—	2	3	no	—
35	C₂×#41×C₅₉	—	4	1	no	—
36	C₂×D₈×C₅₉	—	2	3	no	—
37	C₂×D₈×C₅₉	C₂³	1	3	yes	—
38	#127×C₅₉	—	2	3	no	—
39	#121×C₅₉	—	4	3	no	—
40	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
41	C₄×D₄×C₅₉	—	2	3	no	—
42	#107×C₅₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₉	D₄	1	3	yes	—
44	#102×C₅₉	—	2	3	no	—
45	C₄×C₈×C₅₉	C₂³	1	3	yes	—
46	#41×C₅₉	—	8	4	no	—
47	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
48	C₂×D₄×C₅₉	—	2	3	no	—
49	C₂×D₄×C₅₉	—	4	3	no	—
50	C₂×D₄×C₅₉	—	4	3	no	—
51	C₂²×C₄×C₅₉	—	4	3	no	—
52	C₂×Dic₁₁₈	—	2	1	no	—
53	C₂×Dic₁₁₈	—	4	3	no	—
54	D₈×C₅₉	—	2	3	no	—
55	D₈×C₅₉	—	2	3	no	—
56	D₈×C₅₉	—	8	3	no	—
57	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
58	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₅₉	—	2	1	no	—
60	C₄⋊C₄×C₅₉	D₈	1	3	yes	—
61	C₂²⋊C₄×C₅₉	—	4	3	no	—
62	C₄²×C₅₉	C₂×D₄	1	3	yes	—
63	C₄⋊Dic₅₉	—	2	3	no	—
64	C₄⋊Dic₅₉	—	4	3	no	—
65	Dic₅₉⋊C₄	—	4	3	no	—
66	C₄×Dic₅₉	—	4	3	no	—
67	Dic₂₃₆	—	2	1	no	—
68	Dic₂₃₆	#41	1	2	yes	complemented
69	C₅₉⋊C₁₆	—	2	3	no	—
70	C₅₉⋊C₁₆	—	2	1	no	—
71	C₂³×C₅₉	—	4	3	no	—
72	Q₈×C₅₉	SD₃₂	1	4	yes	complemented
73	D₄×C₅₉	—	4	3	no	—
74	D₄×C₅₉	—	4	3	no	—
75	D₄×C₅₉	—	8	6	no	—
76	D₄×C₅₉	—	16	3	no	—
77	C₂×C₄×C₅₉	—	4	1	no	—
78	C₂×C₄×C₅₉	—	8	6	no	—
79	C₂×C₄×C₅₉	C₂×D₈	1	3	yes	—
80	C₂×C₄×C₅₉	C₂×D₈	1	3	yes	—
81	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₅₉	—	4	1	no	—
83	C₂×Dic₅₉	—	4	3	no	—
84	Dic₁₁₈	—	2	2	no	—
85	Dic₁₁₈	—	4	1	no	—
86	Dic₁₁₈	—	16	3	no	—
87	C₈×C₅₉	—	2	3	no	—
88	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
89	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
90	#33929	—	59	1	no	maximal, Sylow
91	C₂²×C₅₉	—	8	3	no	—
92	C₂²×C₅₉	—	8	3	no	—
93	C₂²×C₅₉	—	8	3	no	—
94	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₅₉	—	2	3	no	—
96	C₄×C₅₉	—	8	1	no	—
97	C₄×C₅₉	C₂×SD₃₂	1	6	yes	—
98	C₄×C₅₉	#548	1	1	yes	characteristic
99	C₄×C₅₉	#577	1	1	yes	characteristic
100	Dic₅₉	—	4	2	no	—
101	Dic₅₉	—	8	3	no	—
102	#3391	D₅₉	1	1	yes	characteristic
103	Q₈×Q₁₆	—	59	1	no	—
104	#2240	—	59	3	no	—
105	#2226	—	59	3	no	—
106	#2207	—	59	3	no	—
107	Q₈×C₁₆	—	59	1	no	—
108	C₄×SD₃₂	—	59	3	no	—
109	C₂×C₅₉	—	16	3	no	—
110	C₂×C₅₉	#3358	1	1	yes	characteristic
111	C₂×C₅₉	C₂²×SD₃₂	1	1	yes	characteristic
112	C₂×C₅₉	#3415	1	1	yes	characteristic
113	Q₈²	—	118	1	no	—
114	#550	—	2	1	no	—
115	C₂×SD₃₂	—	59	3	no	—
116	#500	—	59	3	no	—
117	#493	C₂×D₅₉	1	3	yes	—
118	#477	—	118	3	no	—
119	#459	—	2	3	no	—
120	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
121	C₄×Q₁₆	—	59	3	no	—
122	C₄×D₈	C₂×D₅₉	1	3	yes	—
123	#367	—	59	3	no	—
124	C₄⋊C₁₆	—	59	3	no	—
125	#358	—	59	1	no	—
126	#357	—	59	3	no	—
127	#357	—	59	6	no	—
128	C₄×C₁₆	—	59	3	no	—
129	C₅₉	#33929	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	—	4	1	no	—
131	C₂×Q₁₆	—	59	1	no	—
132	C₂×D₈	—	2	3	no	—
133	C₂×D₈	C₂²×D₅₉	1	3	yes	—
134	C₄⋊Q₈	—	118	3	no	—
135	C₄⋊Q₈	—	236	3	no	—
136	#127	—	2	3	no	—
137	#121	—	4	3	no	—
138	C₄×Q₈	—	118	3	no	—
139	C₄×Q₈	—	236	1	no	—
140	C₄×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
141	C₄×D₄	—	2	3	no	—
142	SD₃₂	—	118	6	no	—
143	C₂×C₁₆	—	59	1	no	—
144	C₂×C₁₆	—	59	3	no	—
145	#107	—	59	3	no	—
146	#107	—	118	3	no	—
147	#107	C₂²×D₅₉	1	1	yes	characteristic
148	C₄⋊C₈	C₂²⋊D₅₉	1	3	yes	—
149	#103	—	118	3	no	—
150	#102	—	2	3	no	—
151	C₄×C₈	C₂²×D₅₉	1	3	yes	—
152	#41	—	8	4	no	—
153	C₂×Q₈	—	118	1	no	—
154	C₂×Q₈	—	236	3	no	—
155	C₂×Q₈	#11745669	1	1	yes	characteristic
156	C₂×D₄	—	2	3	no	—
157	C₂×D₄	—	4	3	no	—
158	C₂×D₄	—	4	3	no	—
159	C₂²×C₄	—	4	3	no	—
160	Q₁₆	—	59	2	no	—
161	Q₁₆	—	118	1	no	—
162	D₈	—	2	3	no	—
163	D₈	—	2	3	no	—
164	D₈	—	8	3	no	—
165	C₂×C₈	C₂×C₂²⋊D₅₉	1	3	yes	—
166	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	118	3	no	—
169	C₄⋊C₄	—	236	3	no	—
170	C₄⋊C₄	—	236	3	no	—
171	C₄⋊C₄	#11745669	1	3	yes	—
172	C₂²⋊C₄	—	4	3	no	—
173	C₄²	—	236	3	no	—
174	C₄²	C₂×C₂²⋊D₅₉	1	3	yes	—
175	C₁₆	—	118	3	no	—
176	C₁₆	—	118	1	no	—
177	C₂³	—	4	3	no	—
178	Q₈	—	118	2	no	—
179	Q₈	—	944	3	no	—
180	Q₈	—	236	1	no	—
181	Q₈	#14273671	1	4	yes	complemented
182	D₄	—	4	3	no	—
183	D₄	—	4	3	no	—
184	D₄	—	8	6	no	—
185	D₄	—	16	3	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	8	6	no	—
188	C₂×C₄	—	236	3	no	—
189	C₂×C₄	—	236	1	no	—
190	C₂×C₄	C₂×#11745669	1	3	yes	—
191	C₂×C₄	C₂×#11745669	1	3	yes	—
192	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
193	C₈	—	2	3	no	—
194	C₈	#41×D₅₉	1	1	yes	characteristic
195	C₈	#14273774	1	1	yes	characteristic
196	C₂²	—	8	3	no	—
197	C₂²	—	8	3	no	—
198	C₂²	—	8	3	no	—
199	C₂²	C₂²×#11745669	1	1	yes	characteristic
200	C₄	—	2	3	no	—
201	C₄	—	8	1	no	—
202	C₄	—	472	3	no	—
203	C₄	—	236	2	no	—
204	C₄	C₂×#14273671	1	6	yes	—
205	C₄	#16838059	1	1	yes	characteristic
206	C₄	#16838213	1	1	yes	characteristic
207	C₂	—	16	3	no	—
208	C₂	#21436727	1	1	yes	characteristic
209	C₂	C₂²×#14273671	1	1	yes	characteristic
210	C₂	#21437235	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₃₆

C₂

yes

maximal, characteristic, complemented

#21426319

C₂

yes

maximal, complemented

#21426243

C₂

yes

maximal, complemented

#21426133

C₂

yes

maximal, complemented

Q₈×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄×#14273671

C₂

yes

maximal, complemented

C₅₉×#3391

C₂

yes

maximal, characteristic

Q₈×Dic₁₁₈

—

C₂×#14273671

C₂²

yes

—

#16837807

C₂²

yes

—

#16837698

—

C₄×Dic₂₃₆

C₂²

yes

—

#16837229

C₂²

yes

—

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

—

#16837205

C₂²

yes

characteristic

#16837203

C₂²

yes

—

#16837203

C₂²

yes

—

C₄×C₅₉⋊C₁₆

C₂²

yes

—

C₅₉×#550

—

C₅₉×#493

C₂²

yes

—

C₅₉×#459

—

C₈×Q₈×C₅₉

C₂²

yes

characteristic

C₄×D₈×C₅₉

C₂²

yes

—

C₂×Dic₂₃₆

C₂³

yes

characteristic

#14273727

—

#14273726

—

Q₈×Dic₅₉

—

C₄×Dic₁₁₈

—

#14273671

—

C₂×C₅₉⋊C₁₆

C₂³

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂³

yes

—

#14273661

—

#14273661

C₂³

yes

—

#14273652

—

C₂×#41×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

C₂³

yes

—

#127×C₅₉

—

#121×C₅₉

—

C₄×Q₈×C₅₉

D₄

yes

characteristic

C₄×D₄×C₅₉

—

#107×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

D₄

yes

—

#102×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

—

#41×C₅₉

—

C₂×Q₈×C₅₉

D₈

yes

characteristic

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

D₈×C₅₉

—

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₈

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

#41

yes

complemented

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

Q₈×C₅₉

SD₃₂

yes

complemented

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂²×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×#41

yes

characteristic

C₈×C₅₉

2₋¹⁺⁴

yes

characteristic

#33929

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×SD₃₂

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

100

Dic₅₉

—

101

Dic₅₉

—

102

#3391

D₅₉

yes

characteristic

103

Q₈×Q₁₆

—

104

#2240

—

105

#2226

—

106

#2207

—

107

Q₈×C₁₆

—

108

C₄×SD₃₂

—

109

C₂×C₅₉

—

110

C₂×C₅₉

#3358

yes

characteristic

111

C₂×C₅₉

C₂²×SD₃₂

yes

characteristic

112

C₂×C₅₉

#3415

yes

characteristic

113

Q₈²

—

118

—

114

#550

—

115

C₂×SD₃₂

—

116

#500

—

117

#493

C₂×D₅₉

yes

—

118

#477

—

118

—

119

#459

—

120

C₈×Q₈

C₂×D₅₉

yes

characteristic

121

C₄×Q₁₆

—

122

C₄×D₈

C₂×D₅₉

yes

—

123

#367

—

124

C₄⋊C₁₆

—

125

#358

—

126

#357

—

127

#357

—

128

C₄×C₁₆

—

129

C₅₉

#33929

yes

characteristic, complemented, Sylow

130

C₂×#41

—

131

C₂×Q₁₆

—

132

C₂×D₈

—

133

C₂×D₈

C₂²×D₅₉

yes

—

134

C₄⋊Q₈

—

118

—

135

C₄⋊Q₈

—

236

—

136

#127

—

137

#121

—

138

C₄×Q₈

—

118

—

139

C₄×Q₈

—

236

—

140

C₄×Q₈

C₂²⋊D₅₉

yes

characteristic

141

C₄×D₄

—

142

SD₃₂

—

118

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

145

#107

—

146

#107

—

118

—

147

#107

C₂²×D₅₉

yes

characteristic

148

C₄⋊C₈

C₂²⋊D₅₉

yes

—

149

#103

—

118

—

150

#102

—

151

C₄×C₈

C₂²×D₅₉

yes

—

152

#41

—

153

C₂×Q₈

—

118

—

154

C₂×Q₈

—

236

—

155

C₂×Q₈

#11745669

yes

characteristic

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂²×C₄

—

160

Q₁₆

—

161

Q₁₆

—

118

—

162

D₈

—

163

D₈

—

164

D₈

—

165

C₂×C₈

C₂×C₂²⋊D₅₉

yes

—

166

C₂×C₈

C₂³×D₅₉

yes

characteristic

167

C₄⋊C₄

—

168

C₄⋊C₄

—

118

—

169

C₄⋊C₄

—

236

—

170

C₄⋊C₄

—

236

—

171

C₄⋊C₄

#11745669

yes

—

172

C₂²⋊C₄

—

173

C₄²

—

236

—

174

C₄²

C₂×C₂²⋊D₅₉

yes

—

175

C₁₆

—

118

—

176

C₁₆

—

118

—

177

C₂³

—

178

Q₈

—

118

—

179

Q₈

—

944

—

180

Q₈

—

236

—

181

Q₈

#14273671

yes

complemented

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

236

—

189

C₂×C₄

—

236

—

190

C₂×C₄

C₂×#11745669

yes

—

191

C₂×C₄

C₂×#11745669

yes

—

192

C₂×C₄

C₂²×C₂²⋊D₅₉

yes

characteristic

193

C₈

—

194

C₈

#41×D₅₉

yes

characteristic

195

C₈

#14273774

yes

characteristic

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

C₂²×#11745669

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

472

—

203

C₄

—

236

—

204

C₄

C₂×#14273671

yes

—

205

C₄

#16838059

yes

characteristic

206

C₄

#16838213

yes

characteristic

207

C₂

—

208

C₂

#21436727

yes

characteristic

209

C₂

C₂²×#14273671

yes

characteristic

210

C₂

#21437235

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting, prime radical	C₅₉×#3391	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	#3391	D₅₉
hypocenter	C₅₉	#33929
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

subgroup

G/H

center

C₂²

C₂²×#11745669

commutator, cosocle

C₂×C₈×C₅₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₉

Fitting, prime radical

C₅₉×#3391

C₂

socle

C₂²×C₅₉

C₂²×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

N	H ≅ G/N	type
1. Q₈×Dic₂₃₆	C₂	semidirect
2. #21426319	C₂	semidirect
3. #21426243	C₂	semidirect
4. #21426133	C₂	semidirect
5. Q₈×C₅₉⋊C₁₆	C₂	semidirect
6. C₄×#14273671	C₂	semidirect
68. Dic₂₃₆	#41	semidirect
72. Q₈×C₅₉	SD₃₂	semidirect
129. C₅₉	#33929	semidirect
181. Q₈	#14273671	semidirect

H ≅ G/N

type

1. Q₈×Dic₂₃₆

C₂

semidirect

2. #21426319

C₂

semidirect

3. #21426243

C₂

semidirect

4. #21426133

C₂

semidirect

5. Q₈×C₅₉⋊C₁₆

C₂

semidirect

6. C₄×#14273671

C₂

semidirect

68. Dic₂₃₆

#41

semidirect

72. Q₈×C₅₉

SD₃₂

semidirect

129. C₅₉

#33929

semidirect

181. Q₈

#14273671

semidirect

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

subgroup	index
Q₈×Dic₂₃₆	2
#21426319	2
#21426243	2
#21426133	2
Q₈×C₅₉⋊C₁₆	2
C₄×#14273671	2
C₅₉×#3391	2
#33929	59

subgroup

index

Q₈×Dic₂₃₆

Q₈×C₅₉⋊C₁₆

C₅₉×#3391

Subgroups — #28746366

Conjugacy classes of subgroups

Complements

Maximal subgroups