Group #28425548 — subgroups

Subgroups — C₅₉×#13439 #28425548

Order 15104.

418 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₂²×D₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₉×#361	C₂	1	2	yes	maximal, complemented
4	C₅₉×#1978	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₅₉	C₂²	1	2	yes	—
7	C₂×D₁₆×C₅₉	—	2	4	no	—
8	C₂×D₁₆×C₅₉	—	2	2	no	—
9	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×#108×C₅₉	C₂²	1	2	yes	—
12	C₅₉×#418	—	2	2	no	—
13	C₂×#100×C₅₉	—	2	2	no	—
14	C₅₉×#405	C₂²	1	1	yes	characteristic
15	C₅₉×C₄⋊C₁₆	—	2	2	no	—
16	C₅₉×#361	—	2	4	no	—
17	C₂²×D₄×C₅₉	—	2	2	no	—
18	C₂×D₈×C₅₉	—	2	2	no	—
19	C₂×D₈×C₅₉	—	2	2	no	—
20	C₂×D₈×C₅₉	—	2	2	no	—
21	C₂×D₈×C₅₉	—	2	2	no	—
22	C₂×D₈×C₅₉	—	2	2	no	—
23	C₂×D₈×C₅₉	D₄	1	4	yes	—
24	C₂×M₁₆×C₅₉	—	2	2	no	—
25	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	#117×C₅₉	D₄	1	1	yes	characteristic
28	D₁₆×C₅₉	—	2	8	no	—
29	M₃₂×C₅₉	—	2	2	no	—
30	C₂×C₁₆×C₅₉	—	2	2	no	—
31	C₂×C₁₆×C₅₉	—	2	1	no	—
32	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
33	#108×C₅₉	—	2	2	no	—
34	#108×C₅₉	—	2	2	no	—
35	#102×C₅₉	—	4	4	no	—
36	#100×C₅₉	—	4	4	no	—
37	#97×C₅₉	—	2	1	no	—
38	C₄×C₈×C₅₉	—	2	1	no	—
39	C₂⁴×C₅₉	—	4	2	no	—
40	C₂×D₄×C₅₉	—	2	4	no	—
41	C₂×D₄×C₅₉	—	4	2	no	—
42	C₂×D₄×C₅₉	—	4	2	no	—
43	C₂×D₄×C₅₉	—	4	2	no	—
44	C₂×D₄×C₅₉	—	4	2	no	—
45	C₂×D₄×C₅₉	—	4	2	no	—
46	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	D₈×C₅₉	—	2	4	no	—
48	D₈×C₅₉	—	2	4	no	—
49	D₈×C₅₉	—	4	2	no	—
50	D₈×C₅₉	—	4	2	no	—
51	M₁₆×C₅₉	—	2	2	no	—
52	M₁₆×C₅₉	—	4	2	no	—
53	M₁₆×C₅₉	—	4	2	no	—
54	C₂×C₈×C₅₉	—	2	1	no	—
55	C₂×C₈×C₅₉	—	4	2	no	—
56	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
59	C₂×C₈×C₅₉	#41	1	2	yes	—
60	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
61	C₂²⋊C₄×C₅₉	—	2	1	no	—
62	C₄²×C₅₉	—	2	1	no	—
63	C₁₆×C₅₉	—	2	2	no	—
64	C₁₆×C₅₉	—	4	1	no	—
65	C₂³×C₅₉	—	4	4	no	—
66	C₂³×C₅₉	—	8	2	no	—
67	C₂³×C₅₉	—	8	2	no	—
68	C₂³×C₅₉	—	16	4	no	—
69	C₂³×C₅₉	#135	1	1	yes	characteristic
70	D₄×C₅₉	—	4	4	no	—
71	D₄×C₅₉	—	4	4	no	—
72	D₄×C₅₉	—	8	2	no	—
73	D₄×C₅₉	—	8	2	no	—
74	C₂×C₄×C₅₉	—	2	2	no	—
75	C₂×C₄×C₅₉	#121	1	2	yes	—
76	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
79	C₈×C₅₉	—	4	1	no	—
80	C₈×C₅₉	—	8	2	no	—
81	C₈×C₅₉	#120	1	1	yes	characteristic
82	C₈×C₅₉	#123	1	2	yes	—
83	C₈×C₅₉	#124	1	1	yes	characteristic
84	#13439	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₅₉	—	2	2	no	—
86	C₂²×C₅₉	—	8	4	no	—
87	C₂²×C₅₉	—	16	4	no	—
88	C₂²×C₅₉	—	16	4	no	—
89	C₂²×C₅₉	—	16	4	no	—
90	C₂²×C₅₉	#463	1	2	yes	—
91	C₂²×C₅₉	#466	1	1	yes	characteristic
92	C₄×C₅₉	—	4	2	no	—
93	C₄×C₅₉	#394	1	1	yes	characteristic
94	C₄×C₅₉	#459	1	2	yes	—
95	C₄×C₅₉	#469	1	1	yes	characteristic
96	C₂²×D₁₆	C₂×C₅₉	1	1	yes	characteristic
97	#2151	C₂×C₅₉	1	1	yes	characteristic, complemented
98	C₂×#361	C₂×C₅₉	1	2	yes	complemented
99	#1978	C₂×C₅₉	1	2	yes	complemented
100	#1381	C₂×C₅₉	1	1	yes	characteristic, complemented
101	C₂×C₅₉	—	2	2	no	—
102	C₂×C₅₉	—	16	4	no	—
103	C₂×C₅₉	#2054	1	1	yes	characteristic
104	C₂×C₅₉	#2221	1	2	yes	—
105	C₂²×D₈	C₂²×C₅₉	1	2	yes	—
106	C₂×D₁₆	—	2	4	no	—
107	C₂×D₁₆	—	2	2	no	—
108	C₂×M₃₂	C₂²×C₅₉	1	1	yes	characteristic
109	C₂²×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
110	C₂×#108	C₂²×C₅₉	1	2	yes	—
111	#418	—	2	2	no	—
112	C₂×#100	—	2	2	no	—
113	#405	C₂²×C₅₉	1	1	yes	characteristic
114	C₄⋊C₁₆	—	2	2	no	—
115	#361	—	2	4	no	—
116	C₅₉	#13439	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×D₄	—	2	2	no	—
118	C₂×D₈	—	2	2	no	—
119	C₂×D₈	—	2	2	no	—
120	C₂×D₈	—	2	2	no	—
121	C₂×D₈	—	2	2	no	—
122	C₂×D₈	—	2	2	no	—
123	C₂×D₈	D₄×C₅₉	1	4	yes	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂×M₁₆	D₄×C₅₉	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
127	#117	D₄×C₅₉	1	1	yes	characteristic
128	D₁₆	—	2	8	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	D₄×C₅₉	1	2	yes	—
133	#108	—	2	2	no	—
134	#108	—	2	2	no	—
135	#102	—	4	4	no	—
136	#100	—	4	4	no	—
137	#97	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₂⁴	—	4	2	no	—
140	C₂×D₄	—	2	4	no	—
141	C₂×D₄	—	4	2	no	—
142	C₂×D₄	—	4	2	no	—
143	C₂×D₄	—	4	2	no	—
144	C₂×D₄	—	4	2	no	—
145	C₂×D₄	—	4	2	no	—
146	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
147	D₈	—	2	4	no	—
148	D₈	—	2	4	no	—
149	D₈	—	4	2	no	—
150	D₈	—	4	2	no	—
151	M₁₆	—	2	2	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
157	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
158	C₂×C₈	#41×C₅₉	1	1	yes	characteristic
159	C₂×C₈	#41×C₅₉	1	2	yes	—
160	C₄⋊C₄	D₈×C₅₉	1	2	yes	—
161	C₂²⋊C₄	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₁₆	—	2	2	no	—
164	C₁₆	—	4	1	no	—
165	C₂³	—	4	4	no	—
166	C₂³	—	8	2	no	—
167	C₂³	—	8	2	no	—
168	C₂³	—	16	4	no	—
169	C₂³	#135×C₅₉	1	1	yes	characteristic
170	D₄	—	4	4	no	—
171	D₄	—	4	4	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#121×C₅₉	1	2	yes	—
176	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
177	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
178	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
179	C₈	—	4	1	no	—
180	C₈	—	8	2	no	—
181	C₈	#120×C₅₉	1	1	yes	characteristic
182	C₈	#123×C₅₉	1	2	yes	—
183	C₈	#124×C₅₉	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	—	16	4	no	—
188	C₂²	—	16	4	no	—
189	C₂²	C₅₉×#463	1	2	yes	—
190	C₂²	C₅₉×#466	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	C₅₉×#394	1	1	yes	characteristic
193	C₄	C₅₉×#459	1	2	yes	—
194	C₄	C₅₉×#469	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	4	no	—
197	C₂	C₅₉×#2054	1	1	yes	characteristic
198	C₂	C₅₉×#2221	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₆×C₅₉

C₂

yes

maximal, characteristic

C₅₉×#2151

C₂

yes

maximal, characteristic, complemented

C₂×C₅₉×#361

C₂

yes

maximal, complemented

C₅₉×#1978

C₂

yes

maximal, complemented

C₅₉×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₅₉

C₂²

yes

—

C₂×D₁₆×C₅₉

—

C₂×D₁₆×C₅₉

—

C₂×M₃₂×C₅₉

C₂²

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#108×C₅₉

C₂²

yes

—

C₅₉×#418

—

C₂×#100×C₅₉

—

C₅₉×#405

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₅₉×#361

—

C₂²×D₄×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

—

C₂×D₈×C₅₉

D₄

yes

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

D₄

yes

characteristic

D₁₆×C₅₉

—

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

—

#108×C₅₉

—

#108×C₅₉

—

#102×C₅₉

—

#100×C₅₉

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₂⁴×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

D₈×C₅₉

—

D₈×C₅₉

—

D₈×C₅₉

—

D₈×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

C₂×C₈×C₅₉

yes

C₂×C₈×C₅₉

yes

—

C₄⋊C₄×C₅₉

D₈

yes

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

#135

yes

characteristic

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#120

yes

characteristic

C₈×C₅₉

#123

yes

—

C₈×C₅₉

#124

yes

characteristic

#13439

C₅₉

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#463

yes

—

C₂²×C₅₉

#466

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#394

yes

characteristic

C₄×C₅₉

#459

yes

—

C₄×C₅₉

#469

yes

characteristic

C₂²×D₁₆

C₂×C₅₉

yes

characteristic

#2151

C₂×C₅₉

yes

characteristic, complemented

C₂×#361

C₂×C₅₉

yes

complemented

#1978

C₂×C₅₉

yes

complemented

100

#1381

C₂×C₅₉

yes

characteristic, complemented

101

C₂×C₅₉

—

102

C₂×C₅₉

—

103

C₂×C₅₉

#2054

yes

characteristic

104

C₂×C₅₉

#2221

yes

—

105

C₂²×D₈

C₂²×C₅₉

yes

—

106

C₂×D₁₆

—

107

C₂×D₁₆

—

108

C₂×M₃₂

C₂²×C₅₉

yes

characteristic

109

C₂²×C₁₆

C₂²×C₅₉

yes

characteristic

110

C₂×#108

C₂²×C₅₉

yes

—

111

#418

—

112

C₂×#100

—

113

#405

C₂²×C₅₉

yes

characteristic

114

C₄⋊C₁₆

—

115

#361

—

116

C₅₉

#13439

yes

characteristic, direct product factor, Sylow

117

C₂²×D₄

—

118

C₂×D₈

—

119

C₂×D₈

—

120

C₂×D₈

—

121

C₂×D₈

—

122

C₂×D₈

—

123

C₂×D₈

D₄×C₅₉

yes

—

124

C₂×M₁₆

—

125

C₂×M₁₆

D₄×C₅₉

yes

characteristic

126

C₂²×C₈

C₂³×C₅₉

yes

characteristic

127

#117

D₄×C₅₉

yes

characteristic

128

D₁₆

—

129

M₃₂

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

D₄×C₅₉

yes

—

133

#108

—

134

#108

—

135

#102

—

136

#100

—

137

#97

—

138

C₄×C₈

—

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂²×C₄

C₂×D₄×C₅₉

yes

characteristic

147

D₈

—

148

D₈

—

149

D₈

—

150

D₈

—

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂×D₄×C₅₉

yes

—

157

C₂×C₈

C₂×D₄×C₅₉

yes

characteristic

158

C₂×C₈

#41×C₅₉

yes

characteristic

159

C₂×C₈

#41×C₅₉

yes

—

160

C₄⋊C₄

D₈×C₅₉

yes

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₁₆

—

164

C₁₆

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

#135×C₅₉

yes

characteristic

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

174

C₂×C₄

—

175

C₂×C₄

#121×C₅₉

yes

—

176

C₂×C₄

#121×C₅₉

yes

characteristic

177

C₂×C₄

C₂×D₈×C₅₉

yes

characteristic

178

C₂×C₄

AGL(1,ℤ/8)×C₅₉

yes

—

179

C₈

—

180

C₈

—

181

C₈

#120×C₅₉

yes

characteristic

182

C₈

#123×C₅₉

yes

—

183

C₈

#124×C₅₉

yes

characteristic

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

C₅₉×#463

yes

—

190

C₂²

C₅₉×#466

yes

characteristic

191

C₄

—

192

C₄

C₅₉×#394

yes

characteristic

193

C₄

C₅₉×#459

yes

—

194

C₄

C₅₉×#469

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

C₅₉×#2054

yes

characteristic

198

C₂

C₅₉×#2221

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₉	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₉
prime radical	C₂²×D₁₆×C₅₉	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₅₉×#405 ◁ G

subgroup

G/H

center, socle

C₂²×C₅₉

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₅₉

prime radical

C₂²×D₁₆×C₅₉

C₂

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₅₉×#405 ◁ G

N	H ≅ G/N	type
2. C₅₉×#2151	C₂	semidirect
3. C₂×C₅₉×#361	C₂	semidirect
4. C₅₉×#1978	C₂	semidirect
5. C₅₉×#1381	C₂	semidirect
84. #13439	C₅₉	direct (N × H)
97. #2151	C₂×C₅₉	semidirect
98. C₂×#361	C₂×C₅₉	semidirect
99. #1978	C₂×C₅₉	semidirect
100. #1381	C₂×C₅₉	semidirect
116. C₅₉	#13439	direct (N × H)

H ≅ G/N

type

2. C₅₉×#2151

C₂

semidirect

3. C₂×C₅₉×#361

C₂

semidirect

4. C₅₉×#1978

C₂

semidirect

5. C₅₉×#1381

C₂

semidirect

84. #13439

C₅₉

direct (N × H)

97. #2151

C₂×C₅₉

semidirect

98. C₂×#361

C₂×C₅₉

semidirect

99. #1978

C₂×C₅₉

semidirect

100. #1381

C₂×C₅₉

semidirect

116. C₅₉

#13439

direct (N × H)

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

subgroup	index
C₂²×D₁₆×C₅₉	2
C₅₉×#2151	2
C₂×C₅₉×#361	2
C₅₉×#1978	2
C₅₉×#1381	2
#13439	59

subgroup

index

C₂²×D₁₆×C₅₉

C₅₉×#2151

C₂×C₅₉×#361

C₅₉×#1978

C₅₉×#1381

#13439

Subgroups — C₅₉×#13439 #28425548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C59×#13439 #28425548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#13439 #28425548