Group #28488373 — subgroups

Subgroups — #28488373

Order 15104.

402 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16837807	C₂	1	1	yes	maximal, characteristic
2	#21425085	C₂	1	2	yes	maximal
3	#21422852	C₂	1	1	yes	maximal, characteristic
4	#21422780	C₂	1	2	yes	maximal
5	C₅₉×#1451	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273727	C₂²	1	2	yes	—
7	#16837807	—	2	2	no	—
8	#16837807	—	2	1	no	—
9	#16837807	—	2	1	no	—
10	C₂×#14273661	C₄	1	4	yes	—
11	#16837426	C₂²	1	2	yes	—
12	#16837170	—	2	4	no	—
13	#16837153	—	2	2	no	—
14	#16837149	—	2	4	no	—
15	C₅₉×#431	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#322	—	2	1	no	—
19	C₅₉×#322	—	2	1	no	—
20	C₂²×Dic₁₁₈	—	2	2	no	—
21	C₂×#11745664	—	2	2	no	—
22	#14273727	—	2	2	no	—
23	#14273727	—	2	2	no	—
24	#14273727	—	2	2	no	—
25	#14273727	—	2	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
28	#14273661	—	2	4	no	—
29	#14273661	—	2	4	no	—
30	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
31	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
32	C₂×M₁₆×C₅₉	D₄	1	2	yes	—
33	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
36	C₂²⋊C₈×C₅₉	—	2	1	no	—
37	#97×C₅₉	—	2	1	no	—
38	C₄×C₈×C₅₉	—	2	1	no	—
39	C₄×C₈×C₅₉	—	2	1	no	—
40	C₄×C₈×C₅₉	D₄	1	2	yes	—
41	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
46	C₂²×Dic₅₉	—	2	4	no	—
47	C₂×Dic₁₁₈	—	4	2	no	—
48	C₂×Dic₁₁₈	—	4	2	no	—
49	C₂×Dic₁₁₈	—	8	4	no	—
50	M₁₆×C₅₉	—	2	4	no	—
51	C₂×C₈×C₅₉	—	2	4	no	—
52	C₂×C₈×C₅₉	—	2	2	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₉	#41	1	1	yes	characteristic
60	C₄⋊Dic₅₉	—	2	4	no	—
61	C₄⋊Dic₅₉	—	2	4	no	—
62	C₄⋊Dic₅₉	—	4	2	no	—
63	C₄⋊Dic₅₉	—	4	2	no	—
64	#11745664	—	4	4	no	—
65	C₂×C₅₉⋊C₈	—	4	2	no	—
66	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	—	2	2	no	—
68	C₂×C₄×C₅₉	—	2	2	no	—
69	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
73	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#137	1	4	yes	—
76	C₂×Dic₅₉	—	4	4	no	—
77	C₂×Dic₅₉	—	8	4	no	—
78	Dic₁₁₈	—	8	4	no	—
79	Dic₁₁₈	—	8	4	no	—
80	C₈×C₅₉	—	2	2	no	—
81	C₈×C₅₉	—	4	2	no	—
82	C₈×C₅₉	—	4	2	no	—
83	C₈×C₅₉	#101	1	4	yes	—
84	C₅₉⋊C₈	—	8	2	no	—
85	#9810	—	59	1	no	maximal, Sylow
86	C₂²×C₅₉	—	2	1	no	—
87	C₂²×C₅₉	—	2	1	no	—
88	C₂²×C₅₉	#390	1	1	yes	characteristic
89	C₂²×C₅₉	#441	1	2	yes	—
90	C₄×C₅₉	—	2	1	no	—
91	C₄×C₅₉	—	2	1	no	—
92	C₄×C₅₉	C₂×#101	1	2	yes	—
93	C₄×C₅₉	#470	1	2	yes	—
94	C₄×C₅₉	#472	1	2	yes	—
95	C₄×C₅₉	#497	1	2	yes	—
96	Dic₅₉	—	8	4	no	—
97	C₂×#500	—	59	1	no	—
98	#1997	—	59	2	no	—
99	#1566	—	59	1	no	—
100	#1549	—	59	2	no	—
101	#1451	D₅₉	1	1	yes	characteristic
102	C₂×C₅₉	—	2	2	no	—
103	C₂×C₅₉	#1966	1	1	yes	characteristic
104	C₂×C₅₉	#1971	1	1	yes	characteristic
105	C₂×C₅₉	#1978	1	1	yes	characteristic
106	C₂×C₄⋊Q₈	—	59	2	no	—
107	#500	—	118	2	no	—
108	#500	—	118	1	no	—
109	#500	—	118	1	no	—
110	#431	C₂×D₅₉	1	1	yes	characteristic
111	C₂×#107	—	59	4	no	—
112	#423	—	59	2	no	—
113	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
115	#343	—	118	4	no	—
116	#334	—	118	2	no	—
117	#332	—	118	4	no	—
118	#322	—	2	1	no	—
119	#322	—	2	1	no	—
120	C₅₉	#9810	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	118	2	no	—
122	C₂×M₁₆	—	118	2	no	—
123	C₂×M₁₆	D₂₃₆	1	2	yes	—
124	C₂²×C₈	C₄×D₅₉	1	2	yes	—
125	C₄⋊Q₈	—	118	2	no	—
126	C₄⋊Q₈	—	118	2	no	—
127	C₄⋊Q₈	—	118	2	no	—
128	C₄⋊Q₈	—	118	2	no	—
129	C₂×C₄⋊C₄	—	59	4	no	—
130	C₂×C₄⋊C₄	—	118	2	no	—
131	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
132	#107	—	118	4	no	—
133	#107	—	118	4	no	—
134	C₄⋊C₈	—	118	2	no	—
135	C₄⋊C₈	—	118	2	no	—
136	C₄⋊C₈	D₂₃₆	1	2	yes	—
137	C₂²⋊C₈	—	2	1	no	—
138	#97	—	2	1	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	—	2	1	no	—
141	C₄×C₈	D₂₃₆	1	2	yes	—
142	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
144	C₂×Q₈	—	472	4	no	—
145	C₂×Q₈	—	236	2	no	—
146	C₂×Q₈	—	236	2	no	—
147	C₂²×C₄	—	118	4	no	—
148	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
150	C₂²×C₄	#11745685	1	1	yes	characteristic
151	M₁₆	—	2	4	no	—
152	M₁₆	—	236	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	236	2	no	—
157	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
158	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
159	C₄⋊C₄	—	118	4	no	—
160	C₄⋊C₄	—	118	4	no	—
161	C₄⋊C₄	—	236	2	no	—
162	C₄⋊C₄	—	236	2	no	—
163	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
164	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
165	C₄²	#11745685	1	1	yes	characteristic
166	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
167	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
168	Q₈	—	472	4	no	—
169	Q₈	—	472	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	472	4	no	—
173	C₂×C₄	—	236	4	no	—
174	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
176	C₂×C₄	#14273701	1	1	yes	characteristic
177	C₂×C₄	#14273701	1	1	yes	characteristic
178	C₂×C₄	#14273715	1	1	yes	characteristic
179	C₂×C₄	#14273724	1	1	yes	characteristic
180	C₂×C₄	#14273760	1	4	yes	—
181	C₈	—	2	2	no	—
182	C₈	—	4	2	no	—
183	C₈	—	4	2	no	—
184	C₈	—	472	2	no	—
185	C₈	#14273645	1	4	yes	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#16837304	1	1	yes	characteristic
189	C₂²	#16837510	1	2	yes	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	472	4	no	—
193	C₄	C₂×#14273645	1	2	yes	—
194	C₄	#16837660	1	2	yes	—
195	C₄	#16837672	1	2	yes	—
196	C₄	#16837796	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#21424940	1	1	yes	characteristic
199	C₂	#21424967	1	1	yes	characteristic
200	C₂	#21425006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837807

C₂

yes

maximal, characteristic

#21425085

C₂

yes

maximal

#21422852

C₂

yes

maximal, characteristic

#21422780

C₂

yes

maximal

C₅₉×#1451

C₂

yes

maximal, characteristic

C₂×#14273727

C₂²

yes

—

—

—

—

C₄

yes

—

#16837426

C₂²

yes

—

#16837170

—

#16837153

—

#16837149

—

C₅₉×#431

C₂²

yes

characteristic

C₄×M₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

—

C₅₉×#322

—

C₂²×Dic₁₁₈

—

—

—

—

—

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂×C₄

yes

—

#14273661

—

#14273661

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₂×M₁₆×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

D₄

yes

—

C₂²⋊C₈×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

#41

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

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×C₄×C₅₉

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#101

yes

—

C₅₉⋊C₈

—

#9810

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#441

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#101

yes

—

C₄×C₅₉

#470

yes

—

C₄×C₅₉

#472

yes

—

C₄×C₅₉

#497

yes

—

Dic₅₉

—

C₂×#500

—

#1997

—

#1566

—

100

#1549

—

101

#1451

D₅₉

yes

characteristic

102

C₂×C₅₉

—

103

C₂×C₅₉

#1966

yes

characteristic

104

C₂×C₅₉

#1971

yes

characteristic

105

C₂×C₅₉

#1978

yes

characteristic

106

C₂×C₄⋊Q₈

—

107

#500

—

118

—

108

#500

—

118

—

109

#500

—

118

—

110

#431

C₂×D₅₉

yes

characteristic

111

C₂×#107

—

112

#423

—

113

C₄×M₁₆

C₂×D₅₉

yes

characteristic

114

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

115

#343

—

118

—

116

#334

—

118

—

117

#332

—

118

—

118

#322

—

119

#322

—

120

C₅₉

#9810

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

118

—

122

C₂×M₁₆

—

118

—

123

C₂×M₁₆

D₂₃₆

yes

—

124

C₂²×C₈

C₄×D₅₉

yes

—

125

C₄⋊Q₈

—

118

—

126

C₄⋊Q₈

—

118

—

127

C₄⋊Q₈

—

118

—

128

C₄⋊Q₈

—

118

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

118

—

131

C₂×C₄²

C₂²×D₅₉

yes

characteristic

132

#107

—

118

—

133

#107

—

118

—

134

C₄⋊C₈

—

118

—

135

C₄⋊C₈

—

118

—

136

C₄⋊C₈

D₂₃₆

yes

—

137

C₂²⋊C₈

—

138

#97

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

D₂₃₆

yes

—

142

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

143

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

144

C₂×Q₈

—

472

—

145

C₂×Q₈

—

236

—

146

C₂×Q₈

—

236

—

147

C₂²×C₄

—

118

—

148

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

149

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

150

C₂²×C₄

#11745685

yes

characteristic

151

M₁₆

—

152

M₁₆

—

236

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

236

—

157

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

158

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

159

C₄⋊C₄

—

118

—

160

C₄⋊C₄

—

118

—

161

C₄⋊C₄

—

236

—

162

C₄⋊C₄

—

236

—

163

C₄²

C₂×D₂₃₆

yes

characteristic

164

C₄²

C₂×D₂₃₆

yes

characteristic

165

C₄²

#11745685

yes

characteristic

166

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

167

C₂³

C₄×D₂₃₆

yes

characteristic

168

Q₈

—

472

—

169

Q₈

—

472

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

472

—

173

C₂×C₄

—

236

—

174

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

175

C₂×C₄

C₄×D₂₃₆

yes

characteristic

176

C₂×C₄

#14273701

yes

characteristic

177

C₂×C₄

#14273701

yes

characteristic

178

C₂×C₄

#14273715

yes

characteristic

179

C₂×C₄

#14273724

yes

characteristic

180

C₂×C₄

#14273760

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

472

—

185

C₈

#14273645

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

#16837304

yes

characteristic

189

C₂²

#16837510

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

472

—

193

C₄

C₂×#14273645

yes

—

194

C₄

#16837660

yes

—

195

C₄

#16837672

yes

—

196

C₄

#16837796

yes

—

197

C₂

—

198

C₂

#21424940

yes

characteristic

199

C₂

#21424967

yes

characteristic

200

C₂

#21425006

yes

characteristic

	subgroup	G/H
center	C₂²	#16837304
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1451	C₂
socle	C₂²×C₅₉	#390
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1451	D₅₉
hypocenter	C₅₉	#9810
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451

subgroup

G/H

center

C₂²

#16837304

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1451

C₂

socle

C₂²×C₅₉

#390

prime radical

C₂³×C₅₉

C₄×D₄

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451

N	H ≅ G/N	type
120. C₅₉	#9810	semidirect

H ≅ G/N

type

120. C₅₉

#9810

semidirect

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

subgroup	index
C₂×#16837807	2
#21425085	2
#21422852	2
#21422780	2
C₅₉×#1451	2
#9810	59

subgroup

index

C₅₉×#1451

Subgroups — #28488373

Conjugacy classes of subgroups

Complements

Maximal subgroups