Group #28746330 — subgroups

Subgroups — #28746330

Order 15104.

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

normal subgroups only

This group has 562 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×D₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#21426319	C₂	1	3	yes	maximal, complemented
3	#21426233	C₂	1	3	yes	maximal, complemented
4	#21426128	C₂	1	3	yes	maximal, complemented
5	Q₈×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14273668	C₂	1	3	yes	maximal, complemented
7	C₅₉×#3391	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×D₂₃₆	—	2	1	no	—
9	C₂×#14273668	C₂²	1	3	yes	—
10	#16837807	C₂²	1	3	yes	—
11	#16837683	—	2	3	no	—
12	C₄×D₄₇₂	C₂²	1	3	yes	—
13	#16837227	C₂²	1	3	yes	—
14	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	3	yes	—
15	#16837203	C₂²	1	3	yes	—
16	#16837203	C₂²	1	6	yes	—
17	#16837202	C₂²	1	1	yes	characteristic
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×Q₈×D₅₉	—	4	1	no	—
25	C₂×D₄₇₂	C₂³	1	1	yes	characteristic
26	#14273727	—	2	3	no	—
27	#14273707	—	4	3	no	—
28	C₄×D₂₃₆	—	2	3	no	—
29	#14273668	—	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	#14273649	—	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	Q₈×D₅₉	—	8	4	no	—
53	C₂×D₂₃₆	—	2	1	no	—
54	C₂×C₄×D₅₉	—	4	3	no	—
55	C₂×Dic₁₁₈	—	4	3	no	—
56	D₈×C₅₉	—	2	3	no	—
57	D₈×C₅₉	—	2	3	no	—
58	D₈×C₅₉	—	8	3	no	—
59	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
60	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₅₉	—	2	1	no	—
62	C₄⋊C₄×C₅₉	D₈	1	3	yes	—
63	C₂²⋊C₄×C₅₉	—	4	3	no	—
64	C₄²×C₅₉	C₂×D₄	1	3	yes	—
65	(C₂×D₅₉)⋊C₄	—	4	3	no	—
66	C₄⋊Dic₅₉	—	2	3	no	—
67	C₄⋊Dic₅₉	—	4	3	no	—
68	D₄₇₂	—	2	1	no	—
69	D₄₇₂	#41	1	2	yes	complemented
70	C₅₉⋊C₁₆	—	2	3	no	—
71	C₅₉⋊C₁₆	—	2	1	no	—
72	C₂³×C₅₉	—	4	3	no	—
73	C₂²×D₅₉	—	4	1	no	—
74	Q₈×C₅₉	D₁₆	1	4	yes	complemented
75	D₄×C₅₉	—	4	3	no	—
76	D₄×C₅₉	—	4	3	no	—
77	D₄×C₅₉	—	8	6	no	—
78	D₄×C₅₉	—	16	3	no	—
79	C₂×C₄×C₅₉	—	4	1	no	—
80	C₂×C₄×C₅₉	—	8	6	no	—
81	C₂×C₄×C₅₉	C₂×D₈	1	3	yes	—
82	C₂×C₄×C₅₉	C₂×D₈	1	3	yes	—
83	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₅₉	—	4	3	no	—
85	D₂₃₆	—	2	2	no	—
86	D₂₃₆	—	4	1	no	—
87	C₄×D₅₉	—	8	6	no	—
88	Dic₁₁₈	—	8	6	no	—
89	C₈×C₅₉	—	2	3	no	—
90	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
91	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
92	#33925	—	59	1	no	maximal, Sylow
93	C₂²×C₅₉	—	8	3	no	—
94	C₂²×C₅₉	—	8	3	no	—
95	C₂²×C₅₉	—	8	3	no	—
96	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
97	C₂×D₅₉	—	4	2	no	—
98	C₂×D₅₉	—	8	1	no	—
99	C₂×D₅₉	—	8	1	no	—
100	C₄×C₅₉	—	2	3	no	—
101	C₄×C₅₉	—	8	1	no	—
102	C₄×C₅₉	C₂×D₁₆	1	6	yes	—
103	C₄×C₅₉	#548	1	1	yes	characteristic
104	C₄×C₅₉	#577	1	1	yes	characteristic
105	Dic₅₉	—	8	3	no	—
106	#3391	D₅₉	1	1	yes	characteristic, complemented
107	Q₈×D₈	—	59	1	no	—
108	#2240	—	59	3	no	—
109	#2224	—	59	3	no	—
110	#2206	—	59	3	no	—
111	Q₈×C₁₆	—	59	1	no	—
112	C₄×D₁₆	—	59	3	no	—
113	C₂×C₅₉	—	16	3	no	—
114	C₂×C₅₉	#3358	1	1	yes	characteristic
115	C₂×C₅₉	C₂²×D₁₆	1	1	yes	characteristic
116	C₂×C₅₉	#3416	1	1	yes	characteristic
117	D₅₉	—	8	2	no	—
118	#550	—	2	1	no	—
119	D₄×Q₈	—	118	1	no	—
120	C₂×D₁₆	—	59	3	no	—
121	#500	—	59	3	no	—
122	#493	C₂×D₅₉	1	3	yes	—
123	#474	—	118	3	no	—
124	#459	—	2	3	no	—
125	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
126	C₄×D₈	—	59	3	no	—
127	C₄×D₈	C₂×D₅₉	1	3	yes	—
128	#366	—	59	3	no	—
129	C₄⋊C₁₆	—	59	3	no	—
130	#357	—	59	3	no	—
131	#357	—	59	6	no	—
132	#357	—	59	1	no	—
133	C₄×C₁₆	—	59	3	no	—
134	C₅₉	#33925	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	—	4	1	no	—
136	C₂²×Q₈	—	236	1	no	—
137	C₂×D₈	—	2	3	no	—
138	C₂×D₈	—	59	1	no	—
139	C₂×D₈	C₂²×D₅₉	1	3	yes	—
140	C₄⋊Q₈	—	118	3	no	—
141	#127	—	2	3	no	—
142	C₂²⋊Q₈	—	236	3	no	—
143	#121	—	4	3	no	—
144	C₄×Q₈	C₂²⋊D₅₉	1	1	yes	characteristic
145	C₄×D₄	—	2	3	no	—
146	C₄×D₄	—	118	3	no	—
147	D₁₆	—	118	6	no	—
148	C₂×C₁₆	—	59	1	no	—
149	C₂×C₁₆	—	59	3	no	—
150	#107	—	59	3	no	—
151	#107	—	118	3	no	—
152	#107	C₂²×D₅₉	1	1	yes	characteristic
153	C₄⋊C₈	C₂²⋊D₅₉	1	3	yes	—
154	#102	—	2	3	no	—
155	#102	—	118	3	no	—
156	C₄×C₈	C₂²×D₅₉	1	3	yes	—
157	#41	—	8	4	no	—
158	C₂×Q₈	—	472	4	no	—
159	C₂×Q₈	—	236	3	no	—
160	C₂×Q₈	#11745669	1	1	yes	characteristic
161	C₂×D₄	—	2	3	no	—
162	C₂×D₄	—	4	3	no	—
163	C₂×D₄	—	4	3	no	—
164	C₂×D₄	—	118	1	no	—
165	C₂²×C₄	—	4	3	no	—
166	C₂²×C₄	—	236	3	no	—
167	D₈	—	2	3	no	—
168	D₈	—	2	3	no	—
169	D₈	—	8	3	no	—
170	D₈	—	59	2	no	—
171	D₈	—	118	1	no	—
172	C₂×C₈	C₂×C₂²⋊D₅₉	1	3	yes	—
173	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	118	3	no	—
176	C₄⋊C₄	—	236	3	no	—
177	C₄⋊C₄	#11745669	1	3	yes	—
178	C₂²⋊C₄	—	4	3	no	—
179	C₂²⋊C₄	—	236	3	no	—
180	C₄²	C₂×C₂²⋊D₅₉	1	3	yes	—
181	C₁₆	—	118	3	no	—
182	C₁₆	—	118	1	no	—
183	C₂³	—	4	3	no	—
184	C₂³	—	236	1	no	—
185	Q₈	—	472	6	no	—
186	Q₈	#14273668	1	4	yes	complemented
187	D₄	—	4	3	no	—
188	D₄	—	4	3	no	—
189	D₄	—	8	6	no	—
190	D₄	—	16	3	no	—
191	D₄	—	118	2	no	—
192	D₄	—	236	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	8	6	no	—
195	C₂×C₄	—	472	6	no	—
196	C₂×C₄	—	236	3	no	—
197	C₂×C₄	C₂×#11745669	1	3	yes	—
198	C₂×C₄	C₂×#11745669	1	3	yes	—
199	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
200	C₈	—	2	3	no	—
201	C₈	#41×D₅₉	1	1	yes	characteristic
202	C₈	#14273774	1	1	yes	characteristic
203	C₂²	—	8	3	no	—
204	C₂²	—	8	3	no	—
205	C₂²	—	8	3	no	—
206	C₂²	—	472	1	no	—
207	C₂²	—	472	1	no	—
208	C₂²	—	236	2	no	—
209	C₂²	C₂²×#11745669	1	1	yes	characteristic
210	C₄	—	2	3	no	—
211	C₄	—	8	1	no	—
212	C₄	—	472	3	no	—
213	C₄	C₂×#14273668	1	6	yes	—
214	C₄	#16838059	1	1	yes	characteristic
215	C₄	#16838213	1	1	yes	characteristic
216	C₂	—	16	3	no	—
217	C₂	—	472	2	no	—
218	C₂	#21436727	1	1	yes	characteristic
219	C₂	C₂²×#14273668	1	1	yes	characteristic
220	C₂	#21437243	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₄₇₂

C₂

yes

maximal, characteristic, complemented

#21426319

C₂

yes

maximal, complemented

#21426233

C₂

yes

maximal, complemented

#21426128

C₂

yes

maximal, complemented

Q₈×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄×#14273668

C₂

yes

maximal, complemented

C₅₉×#3391

C₂

yes

maximal, characteristic, complemented

Q₈×D₂₃₆

—

C₂×#14273668

C₂²

yes

—

#16837807

C₂²

yes

—

#16837683

—

C₄×D₄₇₂

C₂²

yes

—

#16837227

C₂²

yes

—

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

—

#16837203

C₂²

yes

—

#16837203

C₂²

yes

—

#16837202

C₂²

yes

characteristic

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×Q₈×D₅₉

—

C₂×D₄₇₂

C₂³

yes

characteristic

#14273727

—

#14273707

—

C₄×D₂₃₆

—

#14273668

—

C₂×C₅₉⋊C₁₆

C₂³

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂³

yes

—

#14273661

—

#14273661

C₂³

yes

—

#14273649

—

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

—

Q₈×D₅₉

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

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

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

D₄₇₂

—

D₄₇₂

#41

yes

complemented

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

C₂²×D₅₉

—

Q₈×C₅₉

D₁₆

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

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×#41

yes

characteristic

C₈×C₅₉

2₋¹⁺⁴

yes

characteristic

#33925

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

100

C₄×C₅₉

—

101

C₄×C₅₉

—

102

C₄×C₅₉

C₂×D₁₆

yes

—

103

C₄×C₅₉

#548

yes

characteristic

104

C₄×C₅₉

#577

yes

characteristic

105

Dic₅₉

—

106

#3391

D₅₉

yes

characteristic, complemented

107

Q₈×D₈

—

108

#2240

—

109

#2224

—

110

#2206

—

111

Q₈×C₁₆

—

112

C₄×D₁₆

—

113

C₂×C₅₉

—

114

C₂×C₅₉

#3358

yes

characteristic

115

C₂×C₅₉

C₂²×D₁₆

yes

characteristic

116

C₂×C₅₉

#3416

yes

characteristic

117

D₅₉

—

118

#550

—

119

D₄×Q₈

—

118

—

120

C₂×D₁₆

—

121

#500

—

122

#493

C₂×D₅₉

yes

—

123

#474

—

118

—

124

#459

—

125

C₈×Q₈

C₂×D₅₉

yes

characteristic

126

C₄×D₈

—

127

C₄×D₈

C₂×D₅₉

yes

—

128

#366

—

129

C₄⋊C₁₆

—

130

#357

—

131

#357

—

132

#357

—

133

C₄×C₁₆

—

134

C₅₉

#33925

yes

characteristic, complemented, Sylow

135

C₂×#41

—

136

C₂²×Q₈

—

236

—

137

C₂×D₈

—

138

C₂×D₈

—

139

C₂×D₈

C₂²×D₅₉

yes

—

140

C₄⋊Q₈

—

118

—

141

#127

—

142

C₂²⋊Q₈

—

236

—

143

#121

—

144

C₄×Q₈

C₂²⋊D₅₉

yes

characteristic

145

C₄×D₄

—

146

C₄×D₄

—

118

—

147

D₁₆

—

118

—

148

C₂×C₁₆

—

149

C₂×C₁₆

—

150

#107

—

151

#107

—

118

—

152

#107

C₂²×D₅₉

yes

characteristic

153

C₄⋊C₈

C₂²⋊D₅₉

yes

—

154

#102

—

155

#102

—

118

—

156

C₄×C₈

C₂²×D₅₉

yes

—

157

#41

—

158

C₂×Q₈

—

472

—

159

C₂×Q₈

—

236

—

160

C₂×Q₈

#11745669

yes

characteristic

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂×D₄

—

164

C₂×D₄

—

118

—

165

C₂²×C₄

—

166

C₂²×C₄

—

236

—

167

D₈

—

168

D₈

—

169

D₈

—

170

D₈

—

171

D₈

—

118

—

172

C₂×C₈

C₂×C₂²⋊D₅₉

yes

—

173

C₂×C₈

C₂³×D₅₉

yes

characteristic

174

C₄⋊C₄

—

175

C₄⋊C₄

—

118

—

176

C₄⋊C₄

—

236

—

177

C₄⋊C₄

#11745669

yes

—

178

C₂²⋊C₄

—

179

C₂²⋊C₄

—

236

—

180

C₄²

C₂×C₂²⋊D₅₉

yes

—

181

C₁₆

—

118

—

182

C₁₆

—

118

—

183

C₂³

—

184

C₂³

—

236

—

185

Q₈

—

472

—

186

Q₈

#14273668

yes

complemented

187

D₄

—

188

D₄

—

189

D₄

—

190

D₄

—

191

D₄

—

118

—

192

D₄

—

236

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

472

—

196

C₂×C₄

—

236

—

197

C₂×C₄

C₂×#11745669

yes

—

198

C₂×C₄

C₂×#11745669

yes

—

199

C₂×C₄

C₂²×C₂²⋊D₅₉

yes

characteristic

200

C₈

—

201

C₈

#41×D₅₉

yes

characteristic

202

C₈

#14273774

yes

characteristic

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

472

—

207

C₂²

—

472

—

208

C₂²

—

236

—

209

C₂²

C₂²×#11745669

yes

characteristic

210

C₄

—

211

C₄

—

212

C₄

—

472

—

213

C₄

C₂×#14273668

yes

—

214

C₄

#16838059

yes

characteristic

215

C₄

#16838213

yes

characteristic

216

C₂

—

217

C₂

—

472

—

218

C₂

#21436727

yes

characteristic

219

C₂

C₂²×#14273668

yes

characteristic

220

C₂

#21437243

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3391	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	#3391	D₅₉
hypocenter	C₅₉	#33925
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

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×D₄₇₂	C₂	semidirect
2. #21426319	C₂	semidirect
3. #21426233	C₂	semidirect
4. #21426128	C₂	semidirect
5. Q₈×C₅₉⋊C₁₆	C₂	semidirect
6. C₄×#14273668	C₂	semidirect
7. C₅₉×#3391	C₂	semidirect
69. D₄₇₂	#41	semidirect
74. Q₈×C₅₉	D₁₆	semidirect
106. #3391	D₅₉	semidirect
134. C₅₉	#33925	semidirect
186. Q₈	#14273668	semidirect

H ≅ G/N

type

1. Q₈×D₄₇₂

C₂

semidirect

2. #21426319

C₂

semidirect

3. #21426233

C₂

semidirect

4. #21426128

C₂

semidirect

5. Q₈×C₅₉⋊C₁₆

C₂

semidirect

6. C₄×#14273668

C₂

semidirect

7. C₅₉×#3391

C₂

semidirect

69. D₄₇₂

#41

semidirect

74. Q₈×C₅₉

D₁₆

semidirect

106. #3391

D₅₉

semidirect

134. C₅₉

#33925

semidirect

186. Q₈

#14273668

semidirect

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

subgroup	index
Q₈×D₄₇₂	2
#21426319	2
#21426233	2
#21426128	2
Q₈×C₅₉⋊C₁₆	2
C₄×#14273668	2
C₅₉×#3391	2
#33925	59

subgroup

index

Q₈×D₄₇₂

Q₈×C₅₉⋊C₁₆

C₅₉×#3391

Subgroups — #28746330

Conjugacy classes of subgroups

Complements

Maximal subgroups