Group #28507246 — subgroups

Subgroups — #28507246

Order 15104.

366 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×#16837800	C₂	1	1	yes	maximal, characteristic
2	#21423975	C₂	1	1	yes	maximal, characteristic
3	#21422202	C₂	1	2	yes	maximal
4	#21422195	C₂	1	2	yes	maximal
5	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273727	—	2	1	no	—
7	#16837800	—	2	2	no	—
8	#16837800	C₄	1	4	yes	complemented
9	C₂×#14273663	C₂²	1	1	yes	characteristic
10	#16837440	C₂²	1	1	yes	characteristic
11	C₂×#14273659	—	2	1	no	—
12	C₂×#14273659	C₂²	1	2	yes	—
13	#16837357	—	2	1	no	—
14	#16837162	—	2	1	no	—
15	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#364	—	2	2	no	—
18	C₂²×Dic₁₁₈	—	4	1	no	—
19	C₂×#11745664	—	2	1	no	—
20	#14273727	—	2	4	no	—
21	#14273727	—	4	2	no	—
22	#14273686	—	2	1	no	—
23	C₂×C₄⋊Dic₅₉	—	2	2	no	—
24	C₂×C₄⋊Dic₅₉	—	4	1	no	—
25	#14273663	—	2	1	no	—
26	#14273663	Q₈	1	2	yes	complemented
27	#14273661	Q₈	1	2	yes	complemented
28	#14273659	—	2	4	no	—
29	#14273659	—	2	2	no	—
30	#14273659	—	2	1	no	—
31	#14273659	C₂×C₄	1	4	yes	—
32	#14273637	—	4	1	no	—
33	C₈×Dic₅₉	—	4	1	no	—
34	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
37	M₃₂×C₅₉	—	2	2	no	—
38	M₃₂×C₅₉	—	2	2	no	—
39	C₂×C₁₆×C₅₉	—	2	2	no	—
40	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₉	D₄	1	2	yes	—
42	C₂²×C₄×C₅₉	—	2	1	no	—
43	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅₉	—	4	2	no	—
45	C₂×Dic₁₁₈	—	4	4	no	—
46	C₂×Dic₁₁₈	—	8	4	no	—
47	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
51	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
56	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
57	C₄²×C₅₉	—	2	1	no	—
58	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
59	C₂²⋊Dic₅₉	—	4	1	no	—
60	C₄⋊Dic₅₉	—	2	4	no	—
61	C₄⋊Dic₅₉	—	2	2	no	—
62	C₄⋊Dic₅₉	—	4	4	no	—
63	C₄⋊Dic₅₉	—	4	2	no	—
64	C₄×Dic₅₉	—	4	1	no	—
65	#11745664	—	2	2	no	—
66	#11745664	—	4	1	no	—
67	C₂×C₅₉⋊C₈	—	4	1	no	—
68	C₁₆×C₅₉	—	4	2	no	—
69	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	4	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×Dic₅₉	—	4	4	no	—
79	C₂×Dic₅₉	—	4	2	no	—
80	C₂×Dic₅₉	—	8	4	no	—
81	Dic₁₁₈	—	8	8	no	—
82	C₈×C₅₉	#103	1	2	yes	—
83	C₈×C₅₉	#103	1	2	yes	—
84	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
85	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₅₉	#124	1	1	yes	characteristic
87	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₅₉⋊C₈	—	4	2	no	—
89	#13272	—	59	1	no	maximal, Sylow
90	C₂²×C₅₉	—	2	2	no	—
91	C₂²×C₅₉	#418	1	1	yes	characteristic
92	C₂²×C₅₉	#476	1	1	yes	characteristic
93	C₂²×C₅₉	#477	1	1	yes	characteristic
94	C₄×C₅₉	—	2	4	no	—
95	C₄×C₅₉	#391	1	1	yes	characteristic
96	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
97	C₄×C₅₉	#476	1	1	yes	characteristic
98	C₄×C₅₉	#477	1	1	yes	characteristic
99	Dic₅₉	—	8	4	no	—
100	Dic₅₉	—	8	2	no	—
101	C₂×#498	—	59	1	no	—
102	C₂×#364	D₅₉	1	1	yes	characteristic
103	#1772	—	59	1	no	—
104	#1386	—	59	2	no	—
105	#1383	—	59	2	no	—
106	C₂×C₅₉	—	2	1	no	—
107	C₂×C₅₉	—	2	1	no	—
108	C₂×C₅₉	#1987	1	1	yes	characteristic
109	C₂×C₅₉	#2230	1	2	yes	—
110	C₂×C₄⋊Q₈	—	118	1	no	—
111	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
112	#498	—	59	4	no	—
113	#498	—	118	2	no	—
114	C₂×#108	—	59	1	no	—
115	#427	—	59	1	no	—
116	C₂×#106	—	59	2	no	—
117	C₂×#106	—	118	1	no	—
118	#405	—	118	1	no	—
119	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
120	#364	—	2	2	no	—
121	#339	—	118	1	no	—
122	C₅₉	#13272	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	—	236	1	no	—
124	C₂×M₁₆	—	118	1	no	—
125	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
127	C₄⋊Q₈	—	118	4	no	—
128	C₄⋊Q₈	—	236	2	no	—
129	#117	—	118	1	no	—
130	C₂×C₄⋊C₄	—	118	2	no	—
131	C₂×C₄⋊C₄	—	236	1	no	—
132	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
133	M₃₂	—	2	2	no	—
134	M₃₂	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	#108	—	59	2	no	—
137	#108	—	118	1	no	—
138	#107	—	59	2	no	—
139	#106	—	59	4	no	—
140	#106	—	118	4	no	—
141	#106	—	118	1	no	—
142	#106	—	118	2	no	—
143	#97	—	236	1	no	—
144	C₄×C₈	—	236	1	no	—
145	C₄×C₈	C₄×D₅₉	1	2	yes	—
146	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
147	C₂×Q₈	—	472	4	no	—
148	C₂×Q₈	—	236	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	236	2	no	—
151	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
152	M₁₆	—	118	2	no	—
153	M₁₆	—	236	1	no	—
154	C₂×C₈	—	236	1	no	—
155	C₂×C₈	#11745660	1	1	yes	characteristic
156	C₂×C₈	#11745660	1	1	yes	characteristic
157	C₂×C₈	Dic₂₃₆	1	2	yes	—
158	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
159	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
160	C₂×C₈	#11745687	1	1	yes	characteristic
161	C₂×C₈	#11745687	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
164	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
165	C₄⋊C₄	—	118	4	no	—
166	C₄⋊C₄	—	118	2	no	—
167	C₄⋊C₄	—	236	4	no	—
168	C₄⋊C₄	—	236	2	no	—
169	C₂²⋊C₄	—	236	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	236	1	no	—
172	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
173	C₁₆	—	4	2	no	—
174	C₂³	#14273755	1	1	yes	characteristic
175	Q₈	—	472	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	472	4	no	—
179	C₂×C₄	—	236	4	no	—
180	C₂×C₄	—	236	2	no	—
181	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
182	C₂×C₄	#14273707	1	1	yes	characteristic
183	C₂×C₄	#14273707	1	1	yes	characteristic
184	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
186	C₂×C₄	#14273755	1	1	yes	characteristic
187	C₈	—	236	2	no	—
188	C₈	#14273652	1	2	yes	—
189	C₈	#14273652	1	2	yes	—
190	C₈	#14273692	1	1	yes	characteristic
191	C₈	#14273692	1	1	yes	characteristic
192	C₈	#14273714	1	1	yes	characteristic
193	C₈	#14273725	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	#16837408	1	1	yes	characteristic
196	C₂²	#16837693	1	1	yes	characteristic
197	C₂²	#16837698	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	472	4	no	—
200	C₄	—	472	2	no	—
201	C₄	#16837307	1	1	yes	characteristic
202	C₄	C₂×#14273652	1	1	yes	characteristic
203	C₄	#16837693	1	1	yes	characteristic
204	C₄	#16837698	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#21425043	1	1	yes	characteristic
208	C₂	#21426268	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837800

C₂

yes

maximal, characteristic

#21423975

C₂

yes

maximal, characteristic

#21422202

C₂

yes

maximal

#21422195

C₂

yes

maximal

C₂×C₅₉×#364

C₂

yes

maximal, characteristic

C₂×#14273727

—

#16837800

—

#16837800

C₄

yes

complemented

C₂×#14273663

C₂²

yes

characteristic

#16837440

C₂²

yes

characteristic

C₂×#14273659

—

C₂×#14273659

C₂²

yes

—

#16837357

—

#16837162

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#364

—

C₂²×Dic₁₁₈

—

—

—

—

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273663

—

#14273663

Q₈

yes

complemented

#14273661

Q₈

yes

—

—

—

C₂×C₄

yes

—

#14273637

—

C₈×Dic₅₉

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

characteristic

C₂×C₈×C₅₉

SD₁₆

yes

characteristic

C₂×C₈×C₅₉

Q₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

C₂×C₈×C₅₉

yes

C₂×C₈×C₅₉

yes

C₄²×C₅₉

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₂³×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

#103

yes

—

C₈×C₅₉

#103

yes

—

C₈×C₅₉

C₄×Q₈

yes

characteristic

C₈×C₅₉

C₄×Q₈

yes

characteristic

C₈×C₅₉

#124

yes

characteristic

C₈×C₅₉

C₄⋊Q₈

yes

characteristic

C₅₉⋊C₈

—

#13272

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

100

Dic₅₉

—

101

C₂×#498

—

102

C₂×#364

D₅₉

yes

characteristic

103

#1772

—

104

#1386

—

105

#1383

—

106

C₂×C₅₉

—

107

C₂×C₅₉

—

108

C₂×C₅₉

#1987

yes

characteristic

109

C₂×C₅₉

#2230

yes

—

110

C₂×C₄⋊Q₈

—

118

—

111

C₂×M₃₂

C₂×D₅₉

yes

—

112

#498

—

113

#498

—

118

—

114

C₂×#108

—

115

#427

—

116

C₂×#106

—

117

C₂×#106

—

118

—

118

#405

—

118

—

119

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

120

#364

—

121

#339

—

118

—

122

C₅₉

#13272

yes

characteristic, complemented, Sylow

123

C₂²×Q₈

—

236

—

124

C₂×M₁₆

—

118

—

125

C₂²×C₈

D₂₃₆

yes

characteristic

126

C₂²×C₈

C₂²×D₅₉

yes

characteristic

127

C₄⋊Q₈

—

118

—

128

C₄⋊Q₈

—

236

—

129

#117

—

118

—

130

C₂×C₄⋊C₄

—

118

—

131

C₂×C₄⋊C₄

—

236

—

132

C₂×C₄²

D₂₃₆

yes

characteristic

133

M₃₂

—

134

M₃₂

—

135

C₂×C₁₆

—

136

#108

—

137

#108

—

118

—

138

#107

—

139

#106

—

140

#106

—

118

—

141

#106

—

118

—

142

#106

—

118

—

143

#97

—

236

—

144

C₄×C₈

—

236

—

145

C₄×C₈

C₄×D₅₉

yes

—

146

C₄×C₈

C₂²⋊D₅₉

yes

—

147

C₂×Q₈

—

472

—

148

C₂×Q₈

—

236

—

149

C₂²×C₄

—

150

C₂²×C₄

—

236

—

151

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

152

M₁₆

—

118

—

153

M₁₆

—

236

—

154

C₂×C₈

—

236

—

155

C₂×C₈

#11745660

yes

characteristic

156

C₂×C₈

#11745660

yes

characteristic

157

C₂×C₈

Dic₂₃₆

yes

—

158

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

159

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

160

C₂×C₈

#11745687

yes

characteristic

161

C₂×C₈

#11745687

yes

characteristic

162

C₂×C₈

Q₈×D₅₉

yes

characteristic

163

C₂×C₈

Q₈×D₅₉

yes

characteristic

164

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

165

C₄⋊C₄

—

118

—

166

C₄⋊C₄

—

118

—

167

C₄⋊C₄

—

236

—

168

C₄⋊C₄

—

236

—

169

C₂²⋊C₄

—

236

—

170

C₄²

—

171

C₄²

—

236

—

172

C₄²

(C₂×D₅₉)⋊C₄

yes

—

173

C₁₆

—

174

C₂³

#14273755

yes

characteristic

175

Q₈

—

472

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

472

—

179

C₂×C₄

—

236

—

180

C₂×C₄

—

236

—

181

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

182

C₂×C₄

#14273707

yes

characteristic

183

C₂×C₄

#14273707

yes

characteristic

184

C₂×C₄

C₂×#11745660

yes

characteristic

185

C₂×C₄

C₂×Dic₂₃₆

yes

characteristic

186

C₂×C₄

#14273755

yes

characteristic

187

C₈

—

236

—

188

C₈

#14273652

yes

—

189

C₈

#14273652

yes

—

190

C₈

#14273692

yes

characteristic

191

C₈

#14273692

yes

characteristic

192

C₈

#14273714

yes

characteristic

193

C₈

#14273725

yes

characteristic

194

C₂²

—

195

C₂²

#16837408

yes

characteristic

196

C₂²

#16837693

yes

characteristic

197

C₂²

#16837698

yes

characteristic

198

C₄

—

199

C₄

—

472

—

200

C₄

—

472

—

201

C₄

#16837307

yes

characteristic

202

C₄

C₂×#14273652

yes

characteristic

203

C₄

#16837693

yes

characteristic

204

C₄

#16837698

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

#21425043

yes

characteristic

208

C₂

#21426268

yes

—

	subgroup	G/H
center	C₂²	#16837693
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂²×C₅₉	#476
prime radical	C₂³×C₅₉	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	#13272
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

subgroup

G/H

center

C₂²

#16837693

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#364

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

N	H ≅ G/N	type
8. #16837800	C₄	semidirect
26. #14273663	Q₈	semidirect
27. #14273661	Q₈	semidirect
122. C₅₉	#13272	semidirect

H ≅ G/N

type

8. #16837800

C₄

semidirect

26. #14273663

Q₈

semidirect

27. #14273661

Q₈

semidirect

122. C₅₉

#13272

semidirect

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

subgroup	index
C₂×#16837800	2
#21423975	2
#21422202	2
#21422195	2
C₂×C₅₉×#364	2
#13272	59

subgroup

index

C₂×C₅₉×#364

Subgroups — #28507246

Conjugacy classes of subgroups

Complements

Maximal subgroups