Group #28488549 — subgroups

Subgroups — #28488549

Order 15104.

510 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 195 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	#21425057	C₂	1	2	yes	maximal
3	#21424898	C₂	1	2	yes	maximal
4	#21424256	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273727	C₂²	1	2	yes	—
7	#16837800	—	2	2	no	—
8	#16837800	C₄	1	4	yes	complemented
9	C₂×#14273661	—	2	2	no	—
10	C₂×#14273659	—	2	2	no	—
11	C₂×#14273652	—	2	4	no	—
12	#16837272	C₂²	1	1	yes	characteristic
13	#16837272	C₂²	1	1	yes	characteristic
14	#16837165	—	2	4	no	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#322	—	2	2	no	—
18	C₂²×Dic₁₁₈	—	2	2	no	—
19	#14273727	—	2	2	no	—
20	#14273727	—	2	2	no	—
21	#14273727	—	2	2	no	—
22	#14273727	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₅₉	—	2	2	no	—
24	C₂×C₄⋊Dic₅₉	—	2	2	no	—
25	C₂×C₄⋊Dic₅₉	—	2	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄⋊Dic₅₉	—	2	1	no	—
28	C₂×C₄⋊Dic₅₉	—	2	1	no	—
29	#14273661	—	4	4	no	—
30	#14273659	—	2	8	no	—
31	#14273652	—	4	8	no	—
32	C₂²×C₈×C₅₉	D₄	1	4	yes	—
33	C₂²×C₈×C₅₉	D₄	1	2	yes	—
34	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₅₉	—	2	2	no	—
36	C₄×C₈×C₅₉	—	2	2	no	—
37	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₉	D₄	1	2	yes	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
41	C₂²×Dic₅₉	—	4	2	no	—
42	C₂²×Dic₅₉	—	4	1	no	—
43	C₂²×Dic₅₉	—	4	1	no	—
44	C₂×Dic₁₁₈	—	2	4	no	—
45	C₂×Dic₁₁₈	—	4	4	no	—
46	C₂×Dic₁₁₈	—	4	4	no	—
47	C₂×Dic₁₁₈	—	4	2	no	—
48	C₂×C₈×C₅₉	—	2	4	no	—
49	C₂×C₈×C₅₉	—	2	4	no	—
50	C₂×C₈×C₅₉	—	2	4	no	—
51	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
52	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
53	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
54	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₉	#41	1	2	yes	—
57	C₄⋊Dic₅₉	—	2	4	no	—
58	C₄⋊Dic₅₉	—	2	4	no	—
59	C₄⋊Dic₅₉	—	4	2	no	—
60	C₄⋊Dic₅₉	—	4	2	no	—
61	C₄⋊Dic₅₉	—	4	2	no	—
62	C₄⋊Dic₅₉	—	4	2	no	—
63	C₄⋊Dic₅₉	—	4	4	no	—
64	C₄⋊Dic₅₉	—	4	4	no	—
65	C₂³×C₅₉	#127	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₅₉	#121	1	2	yes	—
69	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
72	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
73	C₂×C₄×C₅₉	#137	1	4	yes	—
74	C₂×Dic₅₉	—	4	4	no	—
75	C₂×Dic₅₉	—	8	2	no	—
76	C₂×Dic₅₉	—	8	2	no	—
77	C₂×Dic₅₉	—	8	1	no	—
78	C₂×Dic₅₉	—	8	1	no	—
79	C₂×Dic₅₉	—	8	2	no	—
80	C₂×Dic₅₉	—	8	2	no	—
81	Dic₁₁₈	—	4	8	no	—
82	Dic₁₁₈	—	8	4	no	—
83	C₈×C₅₉	—	4	4	no	—
84	C₈×C₅₉	#103	1	8	yes	complemented
85	#9841	—	59	1	no	maximal, Sylow
86	C₂²×C₅₉	#390	1	1	yes	characteristic
87	C₂²×C₅₉	#492	1	1	yes	characteristic
88	C₂²×C₅₉	#494	1	1	yes	characteristic
89	C₂²×C₅₉	#496	1	2	yes	—
90	C₂²×C₅₉	#497	1	2	yes	—
91	C₄×C₅₉	C₂×#103	1	2	yes	—
92	C₄×C₅₉	#440	1	4	yes	—
93	C₄×C₅₉	#468	1	4	yes	—
94	C₄×C₅₉	#487	1	2	yes	—
95	Dic₅₉	—	8	4	no	—
96	Dic₅₉	—	16	1	no	—
97	Dic₅₉	—	16	1	no	—
98	C₂×#498	—	59	1	no	—
99	#1990	—	59	2	no	—
100	#1956	—	59	2	no	—
101	#1845	—	59	1	no	—
102	C₂×#322	D₅₉	1	1	yes	characteristic
103	C₂×C₅₉	#1715	1	1	yes	characteristic
104	C₂×C₅₉	#1716	1	1	yes	characteristic
105	C₂×C₅₉	#1723	1	2	yes	—
106	C₂×C₅₉	#1956	1	1	yes	characteristic
107	C₂×C₅₉	#1973	1	2	yes	—
108	C₂×C₄⋊Q₈	—	59	2	no	—
109	#498	—	59	4	no	—
110	#498	—	118	2	no	—
111	C₂×#107	—	118	2	no	—
112	C₂×#106	—	118	2	no	—
113	C₂×#103	—	118	4	no	—
114	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
116	#384	—	59	1	no	—
117	#384	—	59	1	no	—
118	#340	—	118	4	no	—
119	#322	—	2	2	no	—
120	C₅₉	#9841	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	118	2	no	—
122	C₂²×C₈	D₂₃₆	1	4	yes	—
123	C₂²×C₈	D₂₃₆	1	2	yes	—
124	C₄⋊Q₈	—	59	4	no	—
125	C₄⋊Q₈	—	118	2	no	—
126	C₄⋊Q₈	—	118	2	no	—
127	C₄⋊Q₈	—	118	2	no	—
128	C₂×C₄⋊C₄	—	118	2	no	—
129	C₂×C₄⋊C₄	—	118	2	no	—
130	C₂×C₄⋊C₄	—	118	2	no	—
131	C₂×C₄⋊C₄	—	118	2	no	—
132	C₂×C₄⋊C₄	—	118	1	no	—
133	C₂×C₄⋊C₄	—	118	1	no	—
134	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
135	#107	—	236	4	no	—
136	#106	—	118	8	no	—
137	#103	—	236	8	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₄×D₅₉	1	2	yes	—
141	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
142	C₂×Q₈	—	118	4	no	—
143	C₂×Q₈	—	236	4	no	—
144	C₂×Q₈	—	236	4	no	—
145	C₂×Q₈	—	236	2	no	—
146	C₂²×C₄	—	236	1	no	—
147	C₂²×C₄	—	236	1	no	—
148	C₂²×C₄	—	236	2	no	—
149	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	#11745660	1	4	yes	—
155	C₂×C₈	Dic₂₃₆	1	4	yes	—
156	C₂×C₈	(C₂×D₅₉)⋊C₄	1	4	yes	—
157	C₄⋊C₄	—	118	4	no	—
158	C₄⋊C₄	—	118	4	no	—
159	C₄⋊C₄	—	236	2	no	—
160	C₄⋊C₄	—	236	2	no	—
161	C₄⋊C₄	—	236	4	no	—
162	C₄⋊C₄	—	236	4	no	—
163	C₄⋊C₄	—	236	2	no	—
164	C₄⋊C₄	—	236	2	no	—
165	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
166	C₄²	#11745685	1	2	yes	—
167	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
168	C₂³	#14273724	1	1	yes	characteristic
169	Q₈	—	472	4	no	—
170	Q₈	—	236	8	no	—
171	C₂×C₄	—	472	1	no	—
172	C₂×C₄	—	472	1	no	—
173	C₂×C₄	—	472	2	no	—
174	C₂×C₄	—	472	2	no	—
175	C₂×C₄	—	472	2	no	—
176	C₂×C₄	—	472	2	no	—
177	C₂×C₄	—	236	4	no	—
178	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
179	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
180	C₂×C₄	#14273701	1	2	yes	—
181	C₂×C₄	#14273715	1	1	yes	characteristic
182	C₂×C₄	C₂×#11745660	1	2	yes	—
183	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
184	C₂×C₄	#14273755	1	4	yes	—
185	C₂×C₄	#14273760	1	4	yes	—
186	C₈	—	4	4	no	—
187	C₈	#14273652	1	8	yes	complemented
188	C₂²	#16837304	1	1	yes	characteristic
189	C₂²	#16837775	1	1	yes	characteristic
190	C₂²	#16837782	1	1	yes	characteristic
191	C₂²	#16837791	1	2	yes	—
192	C₂²	#16837796	1	2	yes	—
193	C₄	—	944	1	no	—
194	C₄	—	944	1	no	—
195	C₄	—	472	4	no	—
196	C₄	C₂×#14273652	1	2	yes	—
197	C₄	#16837504	1	4	yes	—
198	C₄	#16837648	1	4	yes	—
199	C₄	#16837753	1	2	yes	—
200	C₂	#21423724	1	1	yes	characteristic
201	C₂	#21423729	1	1	yes	characteristic
202	C₂	#21423773	1	2	yes	—
203	C₂	#21424898	1	1	yes	characteristic
204	C₂	#21424977	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837800

C₂

yes

maximal, characteristic

#21425057

C₂

yes

maximal

#21424898

C₂

yes

maximal

#21424256

C₂

yes

maximal, characteristic

C₂×C₅₉×#322

C₂

yes

maximal, characteristic

C₂×#14273727

C₂²

yes

—

#16837800

—

#16837800

C₄

yes

—

—

—

C₂²

yes

characteristic

#16837272

C₂²

yes

characteristic

#16837165

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

—

C₂²×Dic₁₁₈

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

#14273661

—

#14273659

—

#14273652

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂³×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

yes

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#497

yes

—

C₄×C₅₉

C₂×#103

yes

—

C₄×C₅₉

#440

yes

—

C₄×C₅₉

#468

yes

—

C₄×C₅₉

#487

yes

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

C₂×#498

—

#1990

—

100

#1956

—

101

#1845

—

102

C₂×#322

D₅₉

yes

characteristic

103

C₂×C₅₉

#1715

yes

characteristic

104

C₂×C₅₉

#1716

yes

characteristic

105

C₂×C₅₉

#1723

yes

—

106

C₂×C₅₉

#1956

yes

characteristic

107

C₂×C₅₉

#1973

yes

—

108

C₂×C₄⋊Q₈

—

109

#498

—

110

#498

—

118

—

111

C₂×#107

—

118

—

112

C₂×#106

—

118

—

113

C₂×#103

—

118

—

114

C₂×C₄×C₈

C₂×D₅₉

yes

—

115

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

116

#384

—

117

#384

—

118

#340

—

118

—

119

#322

—

120

C₅₉

#9841

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

118

—

122

C₂²×C₈

D₂₃₆

yes

—

123

C₂²×C₈

D₂₃₆

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

118

—

126

C₄⋊Q₈

—

118

—

127

C₄⋊Q₈

—

118

—

128

C₂×C₄⋊C₄

—

118

—

129

C₂×C₄⋊C₄

—

118

—

130

C₂×C₄⋊C₄

—

118

—

131

C₂×C₄⋊C₄

—

118

—

132

C₂×C₄⋊C₄

—

118

—

133

C₂×C₄⋊C₄

—

118

—

134

C₂×C₄²

C₂²×D₅₉

yes

characteristic

135

#107

—

236

—

136

#106

—

118

—

137

#103

—

236

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

C₄×D₅₉

yes

—

141

C₄×C₈

C₂²⋊D₅₉

yes

—

142

C₂×Q₈

—

118

—

143

C₂×Q₈

—

236

—

144

C₂×Q₈

—

236

—

145

C₂×Q₈

—

236

—

146

C₂²×C₄

—

236

—

147

C₂²×C₄

—

236

—

148

C₂²×C₄

—

236

—

149

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

150

C₂²×C₄

C₂×D₂₃₆

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

#11745660

yes

—

155

C₂×C₈

Dic₂₃₆

yes

—

156

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

157

C₄⋊C₄

—

118

—

158

C₄⋊C₄

—

118

—

159

C₄⋊C₄

—

236

—

160

C₄⋊C₄

—

236

—

161

C₄⋊C₄

—

236

—

162

C₄⋊C₄

—

236

—

163

C₄⋊C₄

—

236

—

164

C₄⋊C₄

—

236

—

165

C₄²

C₂×C₄×D₅₉

yes

characteristic

166

C₄²

#11745685

yes

—

167

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

168

C₂³

#14273724

yes

characteristic

169

Q₈

—

472

—

170

Q₈

—

236

—

171

C₂×C₄

—

472

—

172

C₂×C₄

—

472

—

173

C₂×C₄

—

472

—

174

C₂×C₄

—

472

—

175

C₂×C₄

—

472

—

176

C₂×C₄

—

472

—

177

C₂×C₄

—

236

—

178

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

179

C₂×C₄

C₄×D₂₃₆

yes

—

180

C₂×C₄

#14273701

yes

—

181

C₂×C₄

#14273715

yes

characteristic

182

C₂×C₄

C₂×#11745660

yes

—

183

C₂×C₄

C₂×Dic₂₃₆

yes

—

184

C₂×C₄

#14273755

yes

—

185

C₂×C₄

#14273760

yes

—

186

C₈

—

187

C₈

#14273652

yes

complemented

188

C₂²

#16837304

yes

characteristic

189

C₂²

#16837775

yes

characteristic

190

C₂²

#16837782

yes

characteristic

191

C₂²

#16837791

yes

—

192

C₂²

#16837796

yes

—

193

C₄

—

944

—

194

C₄

—

944

—

195

C₄

—

472

—

196

C₄

C₂×#14273652

yes

—

197

C₄

#16837504

yes

—

198

C₄

#16837648

yes

—

199

C₄

#16837753

yes

—

200

C₂

#21423724

yes

characteristic

201

C₂

#21423729

yes

characteristic

202

C₂

#21423773

yes

—

203

C₂

#21424898

yes

characteristic

204

C₂

#21424977

yes

—

	subgroup	G/H
center	C₂³	#14273724
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#322	C₂
socle, prime radical	C₂³×C₅₉	#127
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	#9841
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#14273724

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₅₉×#322

C₂

socle, prime radical

C₂³×C₅₉

#127

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
8. #16837800	C₄	semidirect
84. C₈×C₅₉	#103	semidirect
120. C₅₉	#9841	semidirect
187. C₈	#14273652	semidirect

H ≅ G/N

type

8. #16837800

C₄

semidirect

84. C₈×C₅₉

#103

semidirect

120. C₅₉

#9841

semidirect

187. C₈

#14273652

semidirect

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

subgroup	index
C₂×#16837800	2
#21425057	2
#21424898	2
#21424256	2
C₂×C₅₉×#322	2
#9841	59

subgroup

index

C₂×C₅₉×#322

Subgroups — #28488549

Conjugacy classes of subgroups

Complements

Maximal subgroups