Group #28488648 — subgroups

Subgroups — #28488648

Order 15104.

390 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 123 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	#21434610	C₂	1	1	yes	maximal, characteristic
2	#21424234	C₂	1	1	yes	maximal, characteristic
3	#21422552	C₂	1	2	yes	maximal
4	#21422552	C₂	1	2	yes	maximal
5	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic
6	#16837959	C₂²	1	2	yes	—
7	#16837807	C₄	1	1	yes	characteristic
8	#16837807	C₄	1	1	yes	characteristic
9	#16837804	—	2	2	no	—
10	#16837800	C₄	1	2	yes	—
11	#16837440	—	2	2	no	—
12	#16837426	C₂²	1	2	yes	—
13	#16837167	—	2	2	no	—
14	#16837134	—	2	4	no	—
15	#16837134	—	2	4	no	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#322	—	2	2	no	—
20	C₂×#11745664	—	2	2	no	—
21	#14273727	C₂×C₄	1	2	yes	—
22	#14273727	C₂×C₄	1	2	yes	—
23	#14273719	—	2	2	no	—
24	#14273706	—	2	2	no	—
25	#14273706	—	2	2	no	—
26	C₄×Dic₁₁₈	—	2	2	no	—
27	C₄×Dic₁₁₈	—	2	2	no	—
28	#14273686	—	2	2	no	—
29	#14273661	—	2	2	no	—
30	#14273661	—	2	2	no	—
31	#14273659	—	2	4	no	—
32	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
34	C₂²×C₈×C₅₉	—	2	1	no	—
35	C₂²×C₈×C₅₉	—	2	1	no	—
36	C₂²×C₈×C₅₉	D₄	1	2	yes	—
37	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₅₉	—	2	1	no	—
39	C₄×C₈×C₅₉	—	2	1	no	—
40	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₉	D₄	1	2	yes	—
43	C₄×C₈×C₅₉	D₄	1	2	yes	—
44	C₄×C₈×C₅₉	D₄	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
48	C₂×Dic₁₁₈	—	2	2	no	—
49	C₂×Dic₁₁₈	—	2	2	no	—
50	C₂×C₈×C₅₉	—	2	4	no	—
51	C₂×C₈×C₅₉	—	2	2	no	—
52	C₂×C₈×C₅₉	—	2	4	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₂×C₈×C₅₉	—	4	2	no	—
55	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂²⋊Dic₅₉	—	4	2	no	—
62	C₄⋊Dic₅₉	—	2	2	no	—
63	C₄⋊Dic₅₉	—	2	2	no	—
64	C₄⋊Dic₅₉	—	2	2	no	—
65	C₄⋊Dic₅₉	—	2	2	no	—
66	Dic₅₉⋊C₄	—	4	2	no	—
67	Dic₅₉⋊C₄	—	4	2	no	—
68	C₄×Dic₅₉	—	4	2	no	—
69	#11745664	—	4	4	no	—
70	C₂×C₅₉⋊C₈	—	4	2	no	—
71	C₂³×C₅₉	#124	1	1	yes	characteristic
72	C₂×C₄×C₅₉	—	2	2	no	—
73	C₂×C₄×C₅₉	—	2	2	no	—
74	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
78	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
79	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#137	1	4	yes	—
81	C₂×Dic₅₉	—	4	2	no	—
82	C₂×Dic₅₉	—	4	2	no	—
83	Dic₁₁₈	—	4	4	no	—
84	Dic₁₁₈	—	4	4	no	—
85	C₈×C₅₉	—	2	4	no	—
86	C₈×C₅₉	—	2	2	no	—
87	C₈×C₅₉	—	2	2	no	—
88	C₈×C₅₉	—	4	2	no	—
89	C₅₉⋊C₈	—	8	2	no	—
90	#9859	—	59	1	no	maximal, Sylow
91	C₂²×C₅₉	—	2	1	no	—
92	C₂²×C₅₉	—	2	1	no	—
93	C₂²×C₅₉	#390	1	1	yes	characteristic
94	C₂²×C₅₉	#491	1	2	yes	—
95	C₄×C₅₉	—	2	1	no	—
96	C₄×C₅₉	—	2	1	no	—
97	C₄×C₅₉	#419	1	2	yes	—
98	C₄×C₅₉	#443	1	2	yes	—
99	C₄×C₅₉	#444	1	2	yes	—
100	C₄×C₅₉	#497	1	2	yes	—
101	Dic₅₉	—	8	2	no	—
102	Dic₅₉	—	8	2	no	—
103	#3160	—	59	1	no	—
104	#1837	—	59	1	no	—
105	#1493	—	59	2	no	—
106	#1493	—	59	2	no	—
107	C₂×#322	D₅₉	1	1	yes	characteristic
108	C₂×C₅₉	—	2	2	no	—
109	C₂×C₅₉	#1960	1	1	yes	characteristic
110	C₂×C₅₉	#1974	1	1	yes	characteristic
111	C₂×C₅₉	#1975	1	1	yes	characteristic
112	#533	—	59	2	no	—
113	#500	—	59	1	no	—
114	#500	—	59	1	no	—
115	#499	—	118	2	no	—
116	#498	—	59	2	no	—
117	#427	—	118	2	no	—
118	#423	—	59	2	no	—
119	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
122	#341	—	118	2	no	—
123	Q₈⋊C₈	—	118	4	no	—
124	Q₈⋊C₈	—	118	4	no	—
125	#322	—	2	2	no	—
126	C₅₉	#9859	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	118	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	D₂₃₆	1	2	yes	—
131	C₄⋊Q₈	—	59	2	no	—
132	C₄⋊Q₈	—	59	2	no	—
133	#125	—	118	2	no	—
134	C₂²⋊Q₈	—	118	2	no	—
135	C₂²⋊Q₈	—	118	2	no	—
136	C₄×Q₈	—	118	2	no	—
137	C₄×Q₈	—	118	2	no	—
138	#117	—	118	2	no	—
139	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
140	#107	—	118	2	no	—
141	#107	—	118	2	no	—
142	#106	—	118	4	no	—
143	C₄⋊C₈	—	118	2	no	—
144	C₄⋊C₈	—	118	2	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
148	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
149	C₄×C₈	D₂₃₆	1	2	yes	—
150	C₄×C₈	D₂₃₆	1	2	yes	—
151	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
152	C₂×Q₈	—	118	2	no	—
153	C₂×Q₈	—	118	2	no	—
154	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
155	C₂²×C₄	#11745685	1	1	yes	characteristic
156	C₂²×C₄	#11745685	1	1	yes	characteristic
157	M₁₆	—	236	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	236	2	no	—
164	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
165	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
166	C₄⋊C₄	—	118	2	no	—
167	C₄⋊C₄	—	118	2	no	—
168	C₄⋊C₄	—	118	2	no	—
169	C₄⋊C₄	—	118	2	no	—
170	C₄⋊C₄	—	236	2	no	—
171	C₄⋊C₄	—	236	2	no	—
172	C₂²⋊C₄	—	236	2	no	—
173	C₄²	—	236	2	no	—
174	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
175	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
176	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
177	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
178	C₂³	#14273715	1	1	yes	characteristic
179	Q₈	—	236	4	no	—
180	Q₈	—	236	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	236	2	no	—
184	C₂×C₄	—	236	2	no	—
185	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
186	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
187	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
188	C₂×C₄	#14273701	1	1	yes	characteristic
189	C₂×C₄	#14273701	1	1	yes	characteristic
190	C₂×C₄	#14273724	1	1	yes	characteristic
191	C₂×C₄	#14273760	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	472	2	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#16837304	1	1	yes	characteristic
200	C₂²	#16837771	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	472	2	no	—
204	C₄	—	472	2	no	—
205	C₄	#16837412	1	2	yes	—
206	C₄	#16837520	1	2	yes	—
207	C₄	#16837525	1	2	yes	—
208	C₄	#16837796	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	#21424916	1	1	yes	characteristic
211	C₂	#21424984	1	1	yes	characteristic
212	C₂	#21424989	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21434610

C₂

yes

maximal, characteristic

#21424234

C₂

yes

maximal, characteristic

#21422552

C₂

yes

maximal

#21422552

C₂

yes

maximal

C₂×C₅₉×#322

C₂

yes

maximal, characteristic

#16837959

C₂²

yes

—

#16837807

C₄

yes

characteristic

#16837807

C₄

yes

characteristic

#16837804

—

#16837800

C₄

yes

—

#16837440

—

#16837426

C₂²

yes

—

#16837167

—

#16837134

—

#16837134

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

—

C₂×#11745664

—

#14273727

C₂×C₄

yes

—

#14273727

C₂×C₄

yes

—

#14273719

—

#14273706

—

#14273706

—

C₄×Dic₁₁₈

—

C₄×Dic₁₁₈

—

—

—

—

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

C₂²×C₄×C₅₉

yes

C₂²×C₄×C₅₉

yes

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

#9859

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#491

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#419

yes

—

C₄×C₅₉

#443

yes

—

C₄×C₅₉

#444

yes

—

100

C₄×C₅₉

#497

yes

—

101

Dic₅₉

—

102

Dic₅₉

—

103

#3160

—

104

#1837

—

105

#1493

—

106

#1493

—

107

C₂×#322

D₅₉

yes

characteristic

108

C₂×C₅₉

—

109

C₂×C₅₉

#1960

yes

characteristic

110

C₂×C₅₉

#1974

yes

characteristic

111

C₂×C₅₉

#1975

yes

characteristic

112

#533

—

113

#500

—

114

#500

—

115

#499

—

118

—

116

#498

—

117

#427

—

118

—

118

#423

—

119

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

120

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

121

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

122

#341

—

118

—

123

Q₈⋊C₈

—

118

—

124

Q₈⋊C₈

—

118

—

125

#322

—

126

C₅₉

#9859

yes

characteristic, complemented, Sylow

127

C₂×M₁₆

—

118

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

D₂₃₆

yes

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

133

#125

—

118

—

134

C₂²⋊Q₈

—

118

—

135

C₂²⋊Q₈

—

118

—

136

C₄×Q₈

—

118

—

137

C₄×Q₈

—

118

—

138

#117

—

118

—

139

C₂×C₄²

C₂²×D₅₉

yes

characteristic

140

#107

—

118

—

141

#107

—

118

—

142

#106

—

118

—

143

C₄⋊C₈

—

118

—

144

C₄⋊C₈

—

118

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

C₄×D₅₉

yes

characteristic

148

C₄×C₈

C₄×D₅₉

yes

characteristic

149

C₄×C₈

D₂₃₆

yes

—

150

C₄×C₈

D₂₃₆

yes

—

151

C₄×C₈

C₂²⋊D₅₉

yes

—

152

C₂×Q₈

—

118

—

153

C₂×Q₈

—

118

—

154

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

155

C₂²×C₄

#11745685

yes

characteristic

156

C₂²×C₄

#11745685

yes

characteristic

157

M₁₆

—

236

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

236

—

164

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

165

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

166

C₄⋊C₄

—

118

—

167

C₄⋊C₄

—

118

—

168

C₄⋊C₄

—

118

—

169

C₄⋊C₄

—

118

—

170

C₄⋊C₄

—

236

—

171

C₄⋊C₄

—

236

—

172

C₂²⋊C₄

—

236

—

173

C₄²

—

236

—

174

C₄²

C₂×C₄×D₅₉

yes

characteristic

175

C₄²

C₂×D₂₃₆

yes

characteristic

176

C₄²

C₂×D₂₃₆

yes

characteristic

177

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

178

C₂³

#14273715

yes

characteristic

179

Q₈

—

236

—

180

Q₈

—

236

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

236

—

184

C₂×C₄

—

236

—

185

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

186

C₂×C₄

C₄×D₂₃₆

yes

characteristic

187

C₂×C₄

C₄×D₂₃₆

yes

characteristic

188

C₂×C₄

#14273701

yes

characteristic

189

C₂×C₄

#14273701

yes

characteristic

190

C₂×C₄

#14273724

yes

characteristic

191

C₂×C₄

#14273760

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

472

—

197

C₂²

—

198

C₂²

—

199

C₂²

#16837304

yes

characteristic

200

C₂²

#16837771

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

472

—

204

C₄

—

472

—

205

C₄

#16837412

yes

—

206

C₄

#16837520

yes

—

207

C₄

#16837525

yes

—

208

C₄

#16837796

yes

—

209

C₂

—

210

C₂

#21424916

yes

characteristic

211

C₂

#21424984

yes

characteristic

212

C₂

#21424989

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14273724
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#322	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂³×C₅₉	#124
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	#9859
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14273724

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₂×C₅₉×#322

C₂

C₂²×C₅₉

C₂³×C₅₉

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
126. C₅₉	#9859	semidirect

H ≅ G/N

type

126. C₅₉

#9859

semidirect

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

subgroup	index
#21434610	2
#21424234	2
#21422552	2
C₂×C₅₉×#322	2
#9859	59

subgroup

index

C₂×C₅₉×#322

Subgroups — #28488648

Conjugacy classes of subgroups

Complements

Maximal subgroups