Group #28505169 — subgroups

Subgroups — #28505169

Order 15104.

370 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 135 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	#21425861	C₂	1	2	yes	maximal, complemented
2	#21425710	C₂	1	1	yes	maximal, characteristic
3	#21423896	C₂	1	1	yes	maximal, characteristic
4	#21422186	C₂	1	2	yes	maximal, complemented
5	C₅₉×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16837386	C₄	1	1	yes	characteristic
9	#16837376	C₄	1	1	yes	characteristic
10	#16837357	C₂²	1	2	yes	—
11	#16837222	—	2	4	no	—
12	C₂²⋊(C₅₉⋊C₁₆)	—	2	2	no	—
13	#16837173	C₄	1	2	yes	—
14	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#504	—	2	4	no	—
16	C₂×M₃₂×C₅₉	—	2	2	no	—
17	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
18	C₂×#11745664	C₂×C₄	1	2	yes	—
19	#14273686	C₂×C₄	1	2	yes	—
20	#14273667	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	—	2	1	no	—
22	C₂×C₅₉⋊C₁₆	—	2	1	no	—
23	C₂×C₅₉⋊C₁₆	—	2	2	no	—
24	#14273646	C₈	1	2	yes	—
25	#14273644	C₈	1	2	yes	—
26	#14273642	—	2	2	no	—
27	#14273637	—	2	2	no	—
28	C₈×Dic₅₉	—	2	2	no	—
29	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
30	#131×C₅₉	—	2	2	no	—
31	#131×C₅₉	—	2	2	no	—
32	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₅₉	Q₈	1	2	yes	—
34	C₂²×C₈×C₅₉	D₄	1	2	yes	—
35	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	M₃₂×C₅₉	—	2	4	no	—
37	M₃₂×C₅₉	—	2	2	no	—
38	M₃₂×C₅₉	D₄	1	4	yes	—
39	C₂×C₁₆×C₅₉	—	2	4	no	—
40	C₂×C₁₆×C₅₉	—	2	2	no	—
41	#41×C₅₉	—	2	2	no	—
42	#41×C₅₉	—	2	2	no	—
43	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
44	C₂×D₄×C₅₉	C₂×C₈	1	1	yes	characteristic
45	C₂×D₄×C₅₉	M₁₆	1	2	yes	—
46	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₅₉	—	2	2	no	—
49	M₁₆×C₅₉	—	2	4	no	—
50	C₂×C₈×C₅₉	—	2	2	no	—
51	C₂×C₈×C₅₉	—	2	2	no	—
52	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₅₉	#41	1	2	yes	—
57	C₂²⋊Dic₅₉	—	2	2	no	—
58	C₄⋊Dic₅₉	—	2	2	no	—
59	C₄×Dic₅₉	—	2	2	no	—
60	#11745664	—	2	2	no	—
61	#11745664	C₂×C₈	1	4	yes	—
62	C₂×C₅₉⋊C₈	—	2	2	no	—
63	C₁₆×C₅₉	—	2	4	no	—
64	C₅₉⋊C₁₆	—	4	1	no	—
65	C₅₉⋊C₁₆	—	4	1	no	—
66	C₂³×C₅₉	C₄⋊C₈	1	2	yes	—
67	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
68	Q₈×C₅₉	—	2	2	no	—
69	D₄×C₅₉	—	2	4	no	—
70	D₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	—	2	2	no	—
73	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#131	1	2	yes	—
78	C₂×Dic₅₉	—	4	2	no	—
79	C₈×C₅₉	—	2	2	no	—
80	C₈×C₅₉	#120	1	1	yes	characteristic
81	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
82	C₈×C₅₉	#123	1	1	yes	characteristic
83	C₅₉⋊C₈	—	2	4	no	—
84	#12916	—	59	1	no	maximal, Sylow
85	C₂²×C₅₉	—	2	2	no	—
86	C₂²×C₅₉	—	4	1	no	—
87	C₂²×C₅₉	—	4	2	no	—
88	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₅₉	#435	1	2	yes	—
90	C₄×C₅₉	—	2	2	no	—
91	C₄×C₅₉	#385	1	1	yes	characteristic
92	C₄×C₅₉	#424	1	1	yes	characteristic
93	C₄×C₅₉	C₈×D₄	1	2	yes	—
94	Dic₅₉	—	8	2	no	—
95	#3407	D₅₉	1	1	yes	characteristic
96	#2153	—	59	2	no	—
97	#2113	—	59	1	no	—
98	#1757	—	59	1	no	—
99	#1379	—	59	2	no	—
100	C₂×C₅₉	—	2	2	no	—
101	C₂×C₅₉	—	2	1	no	—
102	C₂×C₅₉	—	4	2	no	—
103	C₂×C₅₉	#1850	1	1	yes	characteristic
104	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
105	#504	—	2	4	no	—
106	C₂×M₃₂	—	2	2	no	—
107	C₂×M₃₂	—	59	1	no	—
108	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
109	C₂²×C₁₆	—	59	1	no	—
110	#413	—	59	1	no	—
111	#410	—	59	1	no	—
112	#405	—	59	2	no	—
113	#364	—	118	4	no	—
114	C₂²⋊C₁₆	—	118	2	no	—
115	#344	—	59	2	no	—
116	C₅₉	#12916	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	C₄×D₅₉	1	1	yes	characteristic
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	C₂×M₁₆	—	59	2	no	—
121	C₂×M₁₆	Dic₁₁₈	1	2	yes	—
122	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
124	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
125	#117	—	59	2	no	—
126	M₃₂	—	2	4	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	—	118	2	no	—
129	M₃₂	C₂²⋊D₅₉	1	4	yes	—
130	C₂×C₁₆	—	2	4	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	118	2	no	—
133	C₂×C₁₆	—	118	1	no	—
134	C₂×C₁₆	—	118	1	no	—
135	#101	—	59	2	no	—
136	#100	—	59	2	no	—
137	#99	—	118	2	no	—
138	#97	—	118	2	no	—
139	C₄×C₈	—	118	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₈×D₅₉	1	1	yes	characteristic
143	C₂×D₄	C₈×D₅₉	1	1	yes	characteristic
144	C₂×D₄	#11745659	1	2	yes	—
145	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	59	4	no	—
150	M₁₆	—	118	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	118	2	no	—
154	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
155	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
156	C₂×C₈	#11745685	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
158	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
159	C₄⋊C₄	—	118	2	no	—
160	C₂²⋊C₄	—	118	2	no	—
161	C₄²	—	118	2	no	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	236	1	no	—
164	C₁₆	—	236	1	no	—
165	C₂³	Dic₅₉⋊C₈	1	2	yes	—
166	C₂³	C₂×#11745659	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	236	2	no	—
173	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
174	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
176	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
177	C₂×C₄	#14273733	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	118	4	no	—
180	C₈	#14273696	1	1	yes	characteristic
181	C₈	#14273706	1	2	yes	—
182	C₈	#14273710	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
187	C₂²	#16837473	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	472	2	no	—
190	C₄	#16837275	1	1	yes	characteristic
191	C₄	#16837428	1	1	yes	characteristic
192	C₄	C₈×C₂²⋊D₅₉	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#21424274	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425861

C₂

yes

maximal, complemented

#21425710

C₂

yes

maximal, characteristic

#21423896

C₂

yes

maximal, characteristic

#21422186

C₂

yes

maximal, complemented

C₅₉×#3407

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

#16837386

C₄

yes

characteristic

#16837376

C₄

yes

characteristic

#16837357

C₂²

yes

—

#16837222

—

C₂²⋊(C₅₉⋊C₁₆)

—

#16837173

C₄

yes

—

C₂×#131×C₅₉

C₂²

yes

characteristic

C₅₉×#504

—

C₂×M₃₂×C₅₉

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×#11745664

C₂×C₄

yes

—

#14273686

C₂×C₄

yes

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

#14273646

C₈

yes

—

#14273644

C₈

yes

—

#14273642

—

#14273637

—

C₈×Dic₅₉

—

C₂×#41×C₅₉

C₂×C₄

yes

characteristic

#131×C₅₉

—

#131×C₅₉

—

C₂×M₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₅₉

Q₈

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

M₃₂×C₅₉

—

M₃₂×C₅₉

—

M₃₂×C₅₉

D₄

yes

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

C₂×C₈

yes

characteristic

C₂×D₄×C₅₉

C₂×C₈

yes

characteristic

C₂×D₄×C₅₉

M₁₆

yes

—

C₂²×C₄×C₅₉

C₄⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

C₂×C₈

yes

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₄⋊C₈

yes

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

#131

yes

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#120

yes

characteristic

C₈×C₅₉

C₂²⋊Q₈

yes

—

C₈×C₅₉

#123

yes

characteristic

C₅₉⋊C₈

—

#12916

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₉

#435

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₈×D₄

yes

—

Dic₅₉

—

#3407

D₅₉

yes

—

—

—

—

100

C₂×C₅₉

—

101

C₂×C₅₉

—

102

C₂×C₅₉

—

103

C₂×C₅₉

#1850

yes

characteristic

104

C₂×#131

C₂×D₅₉

yes

characteristic

105

#504

—

106

C₂×M₃₂

—

107

C₂×M₃₂

—

108

C₂×M₃₂

C₂×D₅₉

yes

—

109

C₂²×C₁₆

—

110

#413

—

111

#410

—

112

#405

—

113

#364

—

118

—

114

C₂²⋊C₁₆

—

118

—

115

#344

—

116

C₅₉

#12916

yes

characteristic, complemented, Sylow

117

C₂×#41

C₄×D₅₉

yes

characteristic

118

#131

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

Dic₁₁₈

yes

—

122

C₂×M₁₆

C₄×D₅₉

yes

characteristic

123

C₂²×C₈

C₂²⋊D₅₉

yes

—

124

C₂²×C₈

C₂²×D₅₉

yes

characteristic

125

#117

—

126

M₃₂

—

127

M₃₂

—

128

M₃₂

—

118

—

129

M₃₂

C₂²⋊D₅₉

yes

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

118

—

133

C₂×C₁₆

—

118

—

134

C₂×C₁₆

—

118

—

135

#101

—

136

#100

—

137

#99

—

118

—

138

#97

—

118

—

139

C₄×C₈

—

118

—

140

#41

—

141

#41

—

142

C₂×Q₈

C₈×D₅₉

yes

characteristic

143

C₂×D₄

C₈×D₅₉

yes

characteristic

144

C₂×D₄

#11745659

yes

—

145

C₂²×C₄

Dic₅₉⋊C₄

yes

—

146

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

118

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

118

—

154

C₂×C₈

Dic₅₉⋊C₄

yes

—

155

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

156

C₂×C₈

#11745685

yes

—

157

C₂×C₈

C₂×C₂²⋊D₅₉

yes

—

158

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

159

C₄⋊C₄

—

118

—

160

C₂²⋊C₄

—

118

—

161

C₄²

—

118

—

162

C₁₆

—

163

C₁₆

—

236

—

164

C₁₆

—

236

—

165

C₂³

Dic₅₉⋊C₈

yes

—

166

C₂³

C₂×#11745659

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

236

—

173

C₂×C₄

Dic₅₉⋊C₈

yes

—

174

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

175

C₂×C₄

C₄×C₂²⋊D₅₉

yes

—

176

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

177

C₂×C₄

#14273733

yes

—

178

C₈

—

179

C₈

—

118

—

180

C₈

#14273696

yes

characteristic

181

C₈

#14273706

yes

—

182

C₈

#14273710

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×Dic₅₉⋊C₈

yes

characteristic

187

C₂²

#16837473

yes

—

188

C₄

—

189

C₄

—

472

—

190

C₄

#16837275

yes

characteristic

191

C₄

#16837428

yes

characteristic

192

C₄

C₈×C₂²⋊D₅₉

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#21424274

yes

characteristic

	subgroup	G/H
center	C₈	#14273696
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#3407	C₂
socle	C₂×C₅₉	#1850
prime radical	C₂×#41×C₅₉	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#3407	D₅₉
hypocenter	C₅₉	#12916
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

subgroup

G/H

center

C₈

#14273696

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#3407

C₂

socle

C₂×C₅₉

#1850

prime radical

C₂×#41×C₅₉

C₂×C₄

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

N	H ≅ G/N	type
1. #21425861	C₂	semidirect
4. #21422186	C₂	semidirect
116. C₅₉	#12916	semidirect

H ≅ G/N

type

1. #21425861

C₂

semidirect

4. #21422186

C₂

semidirect

116. C₅₉

#12916

semidirect

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

subgroup	index
#21425861	2
#21425710	2
#21423896	2
#21422186	2
C₅₉×#3407	2
#12916	59

subgroup

index

C₅₉×#3407

Subgroups — #28505169

Conjugacy classes of subgroups

Complements

Maximal subgroups