Group #28506855 — subgroups

Subgroups — #28506855

Order 15104.

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

normal subgroups only

This group has 442 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	#21432809	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837187	C₂	1	4	yes	maximal, complemented
3	C₂×#16837178	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14273733	C₂²	1	2	yes	—
6	#16837952	C₄	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
8	#16837481	—	2	2	no	—
9	C₈×D₂₃₆	—	2	2	no	—
10	#16837426	C₄	1	1	yes	characteristic
11	#16837368	C₄	1	2	yes	—
12	#16837187	—	2	8	no	—
13	#16837178	—	2	2	no	—
14	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×C₄⋊C₁₆	—	2	2	no	—
17	C₂×#11745685	C₂×C₄	1	2	yes	—
18	#14273733	—	2	4	no	—
19	#14273733	—	2	4	no	—
20	C₂×#11745664	C₂×C₄	1	2	yes	—
21	C₂×#11745659	—	2	2	no	—
22	C₂×C₈×D₅₉	—	2	2	no	—
23	#14273727	C₈	1	1	yes	characteristic
24	#14273724	C₈	1	1	yes	characteristic
25	#14273715	C₈	1	2	yes	—
26	#14273701	—	2	2	no	—
27	C₄×D₂₃₆	—	2	2	no	—
28	C₂×C₅₉⋊C₁₆	—	2	2	no	—
29	C₂×C₅₉⋊C₁₆	—	2	2	no	—
30	C₂×C₅₉⋊C₁₆	D₄	1	4	yes	complemented
31	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
32	(C₂×D₅₉)⋊C₈	—	2	4	no	—
33	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₅₉	—	2	4	no	—
37	C₂×C₁₆×C₅₉	—	2	2	no	—
38	C₄×C₈×C₅₉	D₄	1	2	yes	—
39	C₄×C₈×C₅₉	D₄	1	2	yes	—
40	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
41	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₅₉	—	2	2	no	—
43	#11745685	—	2	4	no	—
44	#11745685	—	2	4	no	—
45	C₂×D₂₃₆	—	2	2	no	—
46	C₂×D₂₃₆	C₂×C₈	1	2	yes	—
47	C₂×C₄×D₅₉	—	2	2	no	—
48	C₂×Dic₁₁₈	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₅₉	—	2	2	no	—
50	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	#41	1	2	yes	—
55	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
58	(C₂×D₅₉)⋊C₄	—	2	4	no	—
59	C₄⋊Dic₅₉	—	2	2	no	—
60	#11745664	—	2	4	no	—
61	C₂×C₅₉⋊C₈	—	2	2	no	—
62	#11745659	—	4	4	no	—
63	C₈×D₅₉	—	4	2	no	—
64	C₈×D₅₉	—	4	2	no	—
65	C₁₆×C₅₉	—	4	2	no	—
66	C₅₉⋊C₁₆	—	2	4	no	—
67	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×D₅₉	—	2	2	no	—
69	C₂×C₄×C₅₉	—	4	2	no	—
70	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
71	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#131	1	2	yes	—
76	C₂²⋊D₅₉	—	4	4	no	—
77	C₂×Dic₅₉	—	2	2	no	—
78	D₂₃₆	—	2	4	no	—
79	D₂₃₆	—	4	4	no	—
80	C₄×D₅₉	—	4	2	no	—
81	C₄×D₅₉	—	4	2	no	—
82	Dic₁₁₈	—	2	4	no	—
83	C₈×C₅₉	—	2	2	no	—
84	C₈×C₅₉	#121	1	2	yes	—
85	C₈×C₅₉	#124	1	1	yes	characteristic
86	C₈×C₅₉	#127	1	1	yes	characteristic
87	C₅₉⋊C₈	—	4	2	no	—
88	#13192	—	59	1	no	maximal, Sylow
89	C₂²×C₅₉	—	2	2	no	—
90	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂²×C₅₉	#436	1	2	yes	—
92	C₂×D₅₉	—	4	4	no	—
93	C₂×D₅₉	—	4	2	no	—
94	C₄×C₅₉	—	2	2	no	—
95	C₄×C₅₉	—	2	2	no	—
96	C₄×C₅₉	#390	1	1	yes	characteristic
97	C₄×C₅₉	#408	1	1	yes	characteristic
98	C₄×C₅₉	C₈×D₄	1	2	yes	—
99	Dic₅₉	—	4	2	no	—
100	#2931	—	59	1	no	—
101	#2150	D₅₉	1	1	yes	characteristic, complemented
102	C₂×#350	—	59	4	no	—
103	C₂×#346	—	59	1	no	—
104	C₂×C₅₉	—	2	2	no	—
105	C₂×C₅₉	#1954	1	1	yes	characteristic
106	C₂×C₅₉	#2117	1	2	yes	—
107	D₅₉	—	8	2	no	—
108	C₂×#131	—	59	2	no	—
109	#532	—	59	1	no	—
110	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
111	C₂²×C₁₆	—	59	2	no	—
112	#436	—	118	2	no	—
113	C₈×D₄	—	118	2	no	—
114	#423	—	59	1	no	—
115	#408	—	59	2	no	—
116	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
117	C₄⋊C₁₆	—	2	2	no	—
118	#350	—	118	8	no	—
119	#346	—	118	2	no	—
120	C₅₉	#13192	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	59	2	no	—
122	#131	—	118	4	no	—
123	#131	—	118	4	no	—
124	C₂×M₁₆	—	59	2	no	—
125	C₂×M₁₆	—	118	2	no	—
126	C₂²×C₈	—	118	2	no	—
127	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
129	C₄⋊Q₈	—	59	1	no	—
130	#127	—	59	1	no	—
131	#124	—	59	2	no	—
132	#121	—	118	2	no	—
133	C₄×D₄	—	118	2	no	—
134	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
135	M₃₂	—	2	4	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	59	4	no	—
138	C₂×C₁₆	—	118	2	no	—
139	C₂×C₁₆	—	118	2	no	—
140	C₄⋊C₈	—	118	2	no	—
141	C₂²⋊C₈	—	118	4	no	—
142	C₄×C₈	D₂₃₆	1	2	yes	—
143	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
144	#41	—	118	4	no	—
145	#41	—	118	4	no	—
146	C₂×Q₈	—	59	2	no	—
147	C₂×D₄	—	59	2	no	—
148	C₂×D₄	—	118	2	no	—
149	C₂×D₄	—	118	2	no	—
150	C₂²×C₄	—	118	2	no	—
151	C₂²×C₄	#11745659	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
153	M₁₆	—	118	4	no	—
154	M₁₆	—	236	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	118	2	no	—
157	C₂×C₈	—	236	2	no	—
158	C₂×C₈	—	236	2	no	—
159	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
160	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
161	C₂×C₈	D₄×D₅₉	1	2	yes	—
162	C₂×C₈	#11745689	1	2	yes	—
163	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
164	C₄⋊C₄	—	118	2	no	—
165	C₂²⋊C₄	—	118	4	no	—
166	C₄²	C₈×D₅₉	1	1	yes	characteristic
167	C₄²	C₈×D₅₉	1	1	yes	characteristic
168	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
169	C₁₆	—	4	2	no	—
170	C₁₆	—	118	4	no	—
171	C₂³	—	118	2	no	—
172	C₂³	C₂×#11745659	1	1	yes	characteristic
173	Q₈	—	118	4	no	—
174	D₄	—	118	4	no	—
175	D₄	—	236	4	no	—
176	D₄	—	236	4	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	118	2	no	—
179	C₂×C₄	—	236	2	no	—
180	C₂×C₄	—	236	2	no	—
181	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
182	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
183	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
184	C₂×C₄	#14273689	1	2	yes	—
185	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
186	C₂×C₄	#14273735	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	236	2	no	—
189	C₈	#14273702	1	2	yes	—
190	C₈	#14273716	1	1	yes	characteristic
191	C₈	#14273723	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	236	4	no	—
194	C₂²	—	236	2	no	—
195	C₂²	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
196	C₂²	#16837479	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	236	2	no	—
200	C₄	#16837303	1	1	yes	characteristic
201	C₄	#16837370	1	1	yes	characteristic
202	C₄	#16837467	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	472	2	no	—
205	C₂	#21424889	1	1	yes	characteristic
206	C₂	#21425725	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21432809

C₂

yes

maximal, characteristic

C₂×#16837187

C₂

yes

maximal, complemented

C₂×#16837178

C₂

yes

maximal, characteristic, complemented

C₅₉×#2150

C₂

yes

maximal, characteristic, complemented

C₂×#14273733

C₂²

yes

—

#16837952

C₄

yes

characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

—

#16837481

—

C₈×D₂₃₆

—

#16837426

C₄

yes

characteristic

#16837368

C₄

yes

—

#16837187

—

#16837178

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×C₄⋊C₁₆

—

C₂×#11745685

C₂×C₄

yes

—

#14273733

—

#14273733

—

C₂×#11745664

C₂×C₄

yes

—

C₂×#11745659

—

C₂×C₈×D₅₉

—

#14273727

C₈

yes

characteristic

#14273724

C₈

yes

characteristic

#14273715

C₈

yes

—

#14273701

—

C₄×D₂₃₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

D₄

yes

complemented

C₄⋊(C₅₉⋊C₈)

—

(C₂×D₅₉)⋊C₈

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

M₁₆

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₂²⋊D₅₉

—

#11745685

—

#11745685

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂×C₈

yes

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

C₂×C₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₄²×C₅₉

C₂×C₈

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

#11745659

—

C₈×D₅₉

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₈

yes

—

C₂×C₄×C₅₉

C₂²⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

#131

yes

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#121

yes

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

#13192

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₉

#436

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₈×D₄

yes

—

Dic₅₉

—

100

#2931

—

101

#2150

D₅₉

yes

characteristic, complemented

102

C₂×#350

—

103

C₂×#346

—

104

C₂×C₅₉

—

105

C₂×C₅₉

#1954

yes

characteristic

106

C₂×C₅₉

#2117

yes

—

107

D₅₉

—

108

C₂×#131

—

109

#532

—

110

C₂×M₃₂

C₂×D₅₉

yes

—

111

C₂²×C₁₆

—

112

#436

—

118

—

113

C₈×D₄

—

118

—

114

#423

—

115

#408

—

116

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

117

C₄⋊C₁₆

—

118

#350

—

118

—

119

#346

—

118

—

120

C₅₉

#13192

yes

characteristic, complemented, Sylow

121

C₂×#41

—

122

#131

—

118

—

123

#131

—

118

—

124

C₂×M₁₆

—

125

C₂×M₁₆

—

118

—

126

C₂²×C₈

—

118

—

127

C₂²×C₈

C₄×D₅₉

yes

characteristic

128

C₂²×C₈

C₂²×D₅₉

yes

characteristic

129

C₄⋊Q₈

—

130

#127

—

131

#124

—

132

#121

—

118

—

133

C₄×D₄

—

118

—

134

C₂×C₄²

C₄×D₅₉

yes

characteristic

135

M₃₂

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

118

—

139

C₂×C₁₆

—

118

—

140

C₄⋊C₈

—

118

—

141

C₂²⋊C₈

—

118

—

142

C₄×C₈

D₂₃₆

yes

—

143

C₄×C₈

C₂²⋊D₅₉

yes

—

144

#41

—

118

—

145

#41

—

118

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

118

—

149

C₂×D₄

—

118

—

150

C₂²×C₄

—

118

—

151

C₂²×C₄

#11745659

yes

—

152

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

153

M₁₆

—

118

—

154

M₁₆

—

236

—

155

C₂×C₈

—

156

C₂×C₈

—

118

—

157

C₂×C₈

—

236

—

158

C₂×C₈

—

236

—

159

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

160

C₂×C₈

C₂×D₂₃₆

yes

characteristic

161

C₂×C₈

D₄×D₅₉

yes

—

162

C₂×C₈

#11745689

yes

—

163

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

164

C₄⋊C₄

—

118

—

165

C₂²⋊C₄

—

118

—

166

C₄²

C₈×D₅₉

yes

characteristic

167

C₄²

C₈×D₅₉

yes

characteristic

168

C₄²

(C₂×D₅₉)⋊C₄

yes

—

169

C₁₆

—

170

C₁₆

—

118

—

171

C₂³

—

118

—

172

C₂³

C₂×#11745659

yes

characteristic

173

Q₈

—

118

—

174

D₄

—

118

—

175

D₄

—

236

—

176

D₄

—

236

—

177

C₂×C₄

—

178

C₂×C₄

—

118

—

179

C₂×C₄

—

236

—

180

C₂×C₄

—

236

—

181

C₂×C₄

(C₂×D₅₉)⋊C₈

yes

—

182

C₂×C₄

(C₂×D₅₉)⋊C₈

yes

—

183

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

184

C₂×C₄

#14273689

yes

—

185

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

186

C₂×C₄

#14273735

yes

—

187

C₈

—

188

C₈

—

236

—

189

C₈

#14273702

yes

—

190

C₈

#14273716

yes

characteristic

191

C₈

#14273723

yes

characteristic

192

C₂²

—

193

C₂²

—

236

—

194

C₂²

—

236

—

195

C₂²

C₂×(C₂×D₅₉)⋊C₈

yes

characteristic

196

C₂²

#16837479

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

236

—

200

C₄

#16837303

yes

characteristic

201

C₄

#16837370

yes

characteristic

202

C₄

#16837467

yes

—

203

C₂

—

204

C₂

—

472

—

205

C₂

#21424889

yes

characteristic

206

C₂

#21425725

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₈
prime radical	#16837952	C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#13192
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#2150

C₂

socle

C₂²×C₅₉

C₂×C₂²⋊C₈

prime radical

#16837952

C₄

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
2. C₂×#16837187	C₂	semidirect
3. C₂×#16837178	C₂	semidirect
4. C₅₉×#2150	C₂	semidirect
30. C₂×C₅₉⋊C₁₆	D₄	semidirect
101. #2150	D₅₉	semidirect
120. C₅₉	#13192	semidirect

H ≅ G/N

type

2. C₂×#16837187

C₂

semidirect

3. C₂×#16837178

C₂

semidirect

4. C₅₉×#2150

C₂

semidirect

30. C₂×C₅₉⋊C₁₆

D₄

semidirect

101. #2150

D₅₉

semidirect

120. C₅₉

#13192

semidirect

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

subgroup	index
#21432809	2
C₂×#16837187	2
C₂×#16837178	2
C₅₉×#2150	2
#13192	59

subgroup

index

C₅₉×#2150

Subgroups — #28506855

Conjugacy classes of subgroups

Complements

Maximal subgroups