Group #28488818 — subgroups

Subgroups — #28488818

Order 15104.

430 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 163 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	#21432459	C₂	1	1	yes	maximal, characteristic
2	#21424213	C₂	1	1	yes	maximal, characteristic
3	#21421980	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273719	—	2	1	no	—
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	#16837300	—	2	2	no	—
8	#16837300	C₄	1	4	yes	—
9	#16837272	C₂²	1	2	yes	—
10	#16837268	—	2	1	no	—
11	#16837157	—	2	2	no	—
12	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₉	—	2	1	no	—
14	C₄³×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#336	—	2	1	no	—
16	C₂²×C₅₉⋊C₈	—	2	2	no	—
17	#14273719	—	4	2	no	—
18	#14273719	—	4	2	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
22	C₂×Dic₅₉⋊C₄	—	2	4	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
24	C₅₉⋊#95	—	2	4	no	—
25	C₂²×C₈×C₅₉	—	2	2	no	—
26	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₉	D₄	1	2	yes	—
28	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	—	2	2	no	—
32	C₄⋊C₈×C₅₉	Q₈	1	4	yes	—
33	C₄×C₈×C₅₉	—	4	2	no	—
34	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅₉	—	2	4	no	—
40	C₂×C₈×C₅₉	—	2	4	no	—
41	C₂×C₈×C₅₉	—	4	2	no	—
42	C₂×C₈×C₅₉	—	4	2	no	—
43	C₄²×C₅₉	—	2	4	no	—
44	C₄²×C₅₉	—	2	2	no	—
45	C₄²×C₅₉	—	4	2	no	—
46	C₄²×C₅₉	D₈	1	2	yes	—
47	C₄²×C₅₉	SD₁₆	1	2	yes	—
48	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
49	C₄²×C₅₉	#41	1	2	yes	—
50	C₄⋊Dic₅₉	—	4	2	no	—
51	C₄⋊Dic₅₉	—	4	2	no	—
52	C₄⋊Dic₅₉	—	4	4	no	—
53	C₄⋊Dic₅₉	—	4	4	no	—
54	Dic₅₉⋊C₄	—	4	8	no	—
55	C₂×C₅₉⋊C₈	—	4	2	no	—
56	C₂×C₅₉⋊C₈	—	4	4	no	—
57	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	—	2	2	no	—
62	C₂×C₄×C₅₉	—	2	2	no	—
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	#102	1	4	yes	—
65	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
66	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#135	1	4	yes	—
72	C₂×C₄×C₅₉	#137	1	2	yes	—
73	C₂×Dic₅₉	—	4	4	no	—
74	C₂×Dic₅₉	—	4	4	no	—
75	C₂×Dic₅₉	—	4	4	no	—
76	C₈×C₅₉	—	4	4	no	—
77	C₅₉⋊C₈	—	8	2	no	—
78	#9899	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	#391	1	1	yes	characteristic
80	C₂²×C₅₉	#413	1	2	yes	—
81	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
82	C₂²×C₅₉	#416	1	2	yes	—
83	C₂²×C₅₉	#419	1	1	yes	characteristic
84	C₄×C₅₉	—	2	4	no	—
85	C₄×C₅₉	—	2	2	no	—
86	C₄×C₅₉	—	4	2	no	—
87	C₄×C₅₉	#474	1	2	yes	—
88	C₄×C₅₉	#475	1	2	yes	—
89	C₄×C₅₉	#488	1	2	yes	—
90	C₄×C₅₉	#499	1	2	yes	—
91	Dic₅₉	—	8	4	no	—
92	#2859	—	59	1	no	—
93	#1831	—	59	1	no	—
94	C₄×C₄⋊C₈	D₅₉	1	1	yes	characteristic
95	#1296	—	59	4	no	—
96	C₂×C₅₉	#1540	1	2	yes	—
97	C₂×C₅₉	#1554	1	2	yes	—
98	C₂×C₅₉	#1988	1	1	yes	characteristic
99	C₂×C₅₉	#1991	1	1	yes	characteristic
100	C₂×C₅₉	#1998	1	1	yes	characteristic
101	C₂×#125	—	118	1	no	—
102	C₂×C₄⋊C₈	—	59	2	no	—
103	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
104	C₂×C₄×C₈	—	2	1	no	—
105	#389	—	59	4	no	—
106	#389	—	118	2	no	—
107	#384	—	59	2	no	—
108	#382	—	118	1	no	—
109	C₄³	C₂×D₅₉	1	1	yes	characteristic
110	#336	—	2	1	no	—
111	#336	—	118	2	no	—
112	C₅₉	#9899	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	118	2	no	—
115	#125	—	236	2	no	—
116	#125	—	236	2	no	—
117	C₂×C₄⋊C₄	—	59	4	no	—
118	C₂×C₄⋊C₄	—	118	4	no	—
119	C₂×C₄⋊C₄	—	118	2	no	—
120	C₂×C₄⋊C₄	—	118	2	no	—
121	C₂×C₄²	C₄×D₅₉	1	2	yes	—
122	C₂×C₄²	D₂₃₆	1	2	yes	—
123	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
124	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	118	4	no	—
128	C₄⋊C₈	Dic₁₁₈	1	4	yes	—
129	C₄×C₈	—	4	2	no	—
130	#95	—	118	4	no	—
131	C₂²×C₄	—	118	4	no	—
132	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
133	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	236	2	no	—
141	C₂×C₈	—	236	4	no	—
142	C₄⋊C₄	—	236	4	no	—
143	C₄⋊C₄	—	236	8	no	—
144	C₄⋊C₄	—	236	2	no	—
145	C₄⋊C₄	—	236	2	no	—
146	C₄⋊C₄	—	236	4	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	4	2	no	—
150	C₄²	#11745669	1	2	yes	—
151	C₄²	#11745671	1	2	yes	—
152	C₄²	C₂×Dic₁₁₈	1	2	yes	—
153	C₄²	#11745685	1	2	yes	—
154	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	236	4	no	—
162	C₂×C₄	—	236	4	no	—
163	C₂×C₄	—	236	4	no	—
164	C₂×C₄	#14273647	1	4	yes	—
165	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
166	C₂×C₄	#14273706	1	2	yes	—
167	C₂×C₄	#14273715	1	1	yes	characteristic
168	C₂×C₄	#14273727	1	1	yes	characteristic
169	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
170	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
171	C₂×C₄	#14273749	1	4	yes	—
172	C₂×C₄	#14273757	1	2	yes	—
173	C₈	—	4	4	no	—
174	C₈	—	472	2	no	—
175	C₂²	#16837308	1	1	yes	characteristic
176	C₂²	#16837384	1	2	yes	—
177	C₂²	C₂×#14273647	1	1	yes	characteristic
178	C₂²	#16837399	1	2	yes	—
179	C₂²	#16837410	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	472	4	no	—
184	C₄	#16837682	1	2	yes	—
185	C₄	#16837687	1	2	yes	—
186	C₄	#16837757	1	2	yes	—
187	C₄	#16837804	1	2	yes	—
188	C₂	#21422746	1	2	yes	—
189	C₂	#21422804	1	2	yes	—
190	C₂	#21425048	1	1	yes	characteristic
191	C₂	#21425061	1	1	yes	characteristic
192	C₂	#21425089	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21432459

C₂

yes

maximal, characteristic

#21424213

C₂

yes

maximal, characteristic

#21421980

C₂

yes

maximal

C₄×C₄⋊C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#14273719

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#16837300

—

#16837300

C₄

yes

—

#16837272

C₂²

yes

—

#16837268

—

#16837157

—

C₂×C₄⋊C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

—

C₄³×C₅₉

C₂²

yes

characteristic

C₅₉×#336

—

C₂²×C₅₉⋊C₈

—

#14273719

—

#14273719

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂×C₄

yes

—

C₂×Dic₅₉⋊C₄

—

C₄⋊(C₅₉⋊C₈)

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

Q₈

yes

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

D₈

yes

—

C₄²×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂×Q₈

yes

—

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

C₄×Q₈

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

C₂×#102

yes

characteristic

C₂²×C₅₉

#416

yes

—

C₂²×C₅₉

#419

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#474

yes

—

C₄×C₅₉

#475

yes

—

C₄×C₅₉

#488

yes

—

C₄×C₅₉

#499

yes

—

Dic₅₉

—

#2859

—

#1831

—

C₄×C₄⋊C₈

D₅₉

yes

characteristic

#1296

—

C₂×C₅₉

#1540

yes

—

C₂×C₅₉

#1554

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

100

C₂×C₅₉

#1998

yes

characteristic

101

C₂×#125

—

118

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄⋊C₈

C₂×D₅₉

yes

—

104

C₂×C₄×C₈

—

105

#389

—

106

#389

—

118

—

107

#384

—

108

#382

—

118

—

109

C₄³

C₂×D₅₉

yes

characteristic

110

#336

—

111

#336

—

118

—

112

C₅₉

#9899

yes

characteristic, complemented, Sylow

113

C₂²×C₈

—

114

C₂²×C₈

—

118

—

115

#125

—

236

—

116

#125

—

236

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

118

—

119

C₂×C₄⋊C₄

—

118

—

120

C₂×C₄⋊C₄

—

118

—

121

C₂×C₄²

C₄×D₅₉

yes

—

122

C₂×C₄²

D₂₃₆

yes

—

123

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

124

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

125

C₂×C₄²

C₂²×D₅₉

yes

characteristic

126

C₄⋊C₈

—

127

C₄⋊C₈

—

118

—

128

C₄⋊C₈

Dic₁₁₈

yes

—

129

C₄×C₈

—

130

#95

—

118

—

131

C₂²×C₄

—

118

—

132

C₂²×C₄

(C₂×D₅₉)⋊C₄

yes

—

133

C₂²×C₄

(C₂×D₅₉)⋊C₄

yes

—

134

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

135

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

136

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

236

—

141

C₂×C₈

—

236

—

142

C₄⋊C₄

—

236

—

143

C₄⋊C₄

—

236

—

144

C₄⋊C₄

—

236

—

145

C₄⋊C₄

—

236

—

146

C₄⋊C₄

—

236

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

#11745669

yes

—

151

C₄²

#11745671

yes

—

152

C₄²

C₂×Dic₁₁₈

yes

—

153

C₄²

#11745685

yes

—

154

C₂³

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

236

—

162

C₂×C₄

—

236

—

163

C₂×C₄

—

236

—

164

C₂×C₄

#14273647

yes

—

165

C₂×C₄

C₄×Dic₁₁₈

yes

—

166

C₂×C₄

#14273706

yes

—

167

C₂×C₄

#14273715

yes

characteristic

168

C₂×C₄

#14273727

yes

characteristic

169

C₂×C₄

C₂×#11745669

yes

characteristic

170

C₂×C₄

C₂×#11745671

yes

characteristic

171

C₂×C₄

#14273749

yes

—

172

C₂×C₄

#14273757

yes

—

173

C₈

—

174

C₈

—

472

—

175

C₂²

#16837308

yes

characteristic

176

C₂²

#16837384

yes

—

177

C₂²

C₂×#14273647

yes

characteristic

178

C₂²

#16837399

yes

—

179

C₂²

#16837410

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

472

—

184

C₄

#16837682

yes

—

185

C₄

#16837687

yes

—

186

C₄

#16837757

yes

—

187

C₄

#16837804

yes

—

188

C₂

#21422746

yes

—

189

C₂

#21422804

yes

—

190

C₂

#21425048

yes

characteristic

191

C₂

#21425061

yes

characteristic

192

C₂

#21425089

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₅₉)⋊C₄
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×C₄⋊C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#9899
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×(C₂×D₅₉)⋊C₄

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₄×C₄⋊C₈×C₅₉

C₂

socle, prime radical

C₂³×C₅₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₉

C₂³

hypercenter

C₄×C₄⋊C₈

D₅₉

hypocenter

C₅₉

#9899

LCS

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

N	H ≅ G/N	type
112. C₅₉	#9899	semidirect

H ≅ G/N

type

112. C₅₉

#9899

semidirect

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

subgroup	index
#21432459	2
#21424213	2
#21421980	2
C₄×C₄⋊C₈×C₅₉	2
#9899	59

subgroup

index

C₄×C₄⋊C₈×C₅₉

Subgroups — #28488818

Conjugacy classes of subgroups

Complements

Maximal subgroups