Group #28508293 — subgroups

Subgroups — #28508293

Order 15104.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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×Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837206	C₂	1	2	yes	maximal
3	#21424958	C₂	1	2	yes	maximal
4	#21422188	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273661	C₂²	1	2	yes	—
8	C₂×#14273651	—	2	2	no	—
9	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837309	—	2	2	no	—
11	Dic₅₉⋊C₁₆	—	2	2	no	—
12	#16837206	—	2	4	no	—
13	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
14	C₂×Q₃₂×C₅₉	—	2	4	no	—
15	C₂×Q₃₂×C₅₉	—	2	2	no	—
16	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	—	2	1	no	—
22	#14273661	—	2	2	no	—
23	#14273661	—	2	2	no	—
24	#14273651	—	4	4	no	—
25	C₈×Dic₅₉	—	2	1	no	—
26	C₈×Dic₅₉	—	2	1	no	—
27	C₅₉⋊#95	—	4	2	no	—
28	C₅₉⋊#95	—	4	2	no	—
29	C₂²×Q₈×C₅₉	—	2	2	no	—
30	C₂×Q₁₆×C₅₉	—	2	2	no	—
31	C₂×Q₁₆×C₅₉	—	2	2	no	—
32	C₂×Q₁₆×C₅₉	—	2	2	no	—
33	C₂×Q₁₆×C₅₉	—	2	2	no	—
34	C₂×Q₁₆×C₅₉	—	2	2	no	—
35	C₂×Q₁₆×C₅₉	D₄	1	4	yes	—
36	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
37	Q₃₂×C₅₉	—	2	8	no	—
38	C₂×C₁₆×C₅₉	—	2	2	no	—
39	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
40	C₂×Q₈×C₅₉	—	2	4	no	—
41	C₂×Q₈×C₅₉	—	4	2	no	—
42	C₂×Q₈×C₅₉	—	4	2	no	—
43	C₂×Q₈×C₅₉	—	4	2	no	—
44	C₂×Q₈×C₅₉	—	4	2	no	—
45	C₂×Q₈×C₅₉	—	4	2	no	—
46	C₂²×C₄×C₅₉	—	4	2	no	—
47	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₅₉	—	4	2	no	—
49	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
50	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
51	Q₁₆×C₅₉	—	2	4	no	—
52	Q₁₆×C₅₉	—	2	4	no	—
53	Q₁₆×C₅₉	—	4	2	no	—
54	Q₁₆×C₅₉	—	4	2	no	—
55	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
58	C₂×C₈×C₅₉	#41	1	2	yes	—
59	C₄⋊Dic₅₉	—	4	2	no	—
60	C₄⋊Dic₅₉	—	4	2	no	—
61	C₄×Dic₅₉	—	2	1	no	—
62	C₄×Dic₅₉	—	2	1	no	—
63	C₂×C₅₉⋊C₈	—	2	1	no	—
64	C₂×C₅₉⋊C₈	—	2	2	no	—
65	C₁₆×C₅₉	—	2	2	no	—
66	C₅₉⋊C₁₆	—	4	1	no	—
67	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₅₉	—	4	4	no	—
69	Q₈×C₅₉	—	4	4	no	—
70	Q₈×C₅₉	—	8	2	no	—
71	Q₈×C₅₉	—	8	2	no	—
72	C₂×C₄×C₅₉	—	4	4	no	—
73	C₂×C₄×C₅₉	—	8	2	no	—
74	C₂×C₄×C₅₉	—	8	2	no	—
75	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#121	1	2	yes	—
77	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
78	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×Dic₅₉	—	2	1	no	—
80	C₂×Dic₅₉	—	2	1	no	—
81	C₂×Dic₅₉	—	2	1	no	—
82	C₂×Dic₅₉	—	2	2	no	—
83	C₂×Dic₅₉	—	8	2	no	—
84	C₂×Dic₅₉	—	8	2	no	—
85	C₂×Dic₅₉	—	8	2	no	—
86	C₂×Dic₅₉	Q₃₂	1	2	yes	complemented
87	C₈×C₅₉	#120	1	1	yes	characteristic
88	C₈×C₅₉	#123	1	2	yes	—
89	C₈×C₅₉	#124	1	1	yes	characteristic
90	C₅₉⋊C₈	—	4	1	no	—
91	#13450	—	59	1	no	maximal, Sylow
92	C₂²×C₅₉	#459	1	2	yes	—
93	C₂²×C₅₉	#466	1	1	yes	characteristic
94	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
95	C₂²×C₅₉	#508	1	1	yes	characteristic
96	C₂²×C₅₉	#510	1	2	yes	—
97	C₄×C₅₉	—	8	4	no	—
98	C₄×C₅₉	#394	1	1	yes	characteristic
99	C₄×C₅₉	#463	1	2	yes	—
100	C₄×C₅₉	#469	1	1	yes	characteristic
101	Dic₅₉	—	2	2	no	—
102	Dic₅₉	—	4	1	no	—
103	Dic₅₉	—	16	2	no	—
104	C₂²×Q₃₂	D₅₉	1	1	yes	characteristic
105	C₂×C₄⋊C₁₆	—	59	1	no	—
106	C₂×#358	—	59	2	no	—
107	#1969	—	59	2	no	—
108	#1380	—	59	1	no	—
109	C₂×C₅₉	#2054	1	1	yes	characteristic
110	C₂×C₅₉	#2209	1	2	yes	—
111	C₂×C₅₉	#2211	1	2	yes	—
112	C₂×C₅₉	#2216	1	1	yes	characteristic
113	C₂×C₅₉	#2219	1	1	yes	characteristic
114	C₂²×Q₁₆	C₂×D₅₉	1	2	yes	—
115	C₂×Q₃₂	—	2	4	no	—
116	C₂×Q₃₂	—	2	2	no	—
117	C₂²×C₁₆	—	59	1	no	—
118	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
119	C₂×#107	—	59	2	no	—
120	C₂×#103	—	118	2	no	—
121	C₂×C₄×C₈	—	59	1	no	—
122	#391	—	118	2	no	—
123	C₄⋊C₁₆	—	118	2	no	—
124	#358	—	118	4	no	—
125	C₅₉	#13450	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	2	2	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	C₂²⋊D₅₉	1	4	yes	—
133	C₂²×C₈	—	59	1	no	—
134	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	118	2	no	—
136	C₂×C₄²	—	59	1	no	—
137	Q₃₂	—	2	8	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	118	2	no	—
140	C₂×C₁₆	—	118	1	no	—
141	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
142	#107	—	118	2	no	—
143	#107	—	118	2	no	—
144	#103	—	236	4	no	—
145	C₄×C₈	—	118	1	no	—
146	C₄×C₈	—	118	1	no	—
147	#95	—	236	2	no	—
148	#95	—	236	2	no	—
149	C₂×Q₈	—	2	4	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	59	1	no	—
157	C₂²×C₄	—	59	1	no	—
158	C₂²×C₄	—	236	2	no	—
159	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
160	Q₁₆	—	2	4	no	—
161	Q₁₆	—	2	4	no	—
162	Q₁₆	—	4	2	no	—
163	Q₁₆	—	4	2	no	—
164	C₂×C₈	—	118	1	no	—
165	C₂×C₈	—	118	2	no	—
166	C₂×C₈	#11745687	1	1	yes	characteristic
167	C₂×C₈	#11745687	1	2	yes	—
168	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
169	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
170	C₄⋊C₄	—	236	2	no	—
171	C₄⋊C₄	—	236	2	no	—
172	C₄²	—	118	1	no	—
173	C₄²	—	118	1	no	—
174	C₁₆	—	2	2	no	—
175	C₁₆	—	236	1	no	—
176	C₂³	D₈×D₅₉	1	1	yes	characteristic
177	Q₈	—	4	4	no	—
178	Q₈	—	4	4	no	—
179	Q₈	—	8	2	no	—
180	Q₈	—	8	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	59	2	no	—
185	C₂×C₄	—	118	1	no	—
186	C₂×C₄	—	118	1	no	—
187	C₂×C₄	—	118	1	no	—
188	C₂×C₄	—	118	2	no	—
189	C₂×C₄	—	472	2	no	—
190	C₂×C₄	—	472	2	no	—
191	C₂×C₄	—	472	2	no	—
192	C₂×C₄	#14273698	1	1	yes	characteristic
193	C₂×C₄	#14273699	1	2	yes	—
194	C₂×C₄	#14273746	1	1	yes	characteristic
195	C₂×C₄	#14273751	1	2	yes	—
196	C₈	—	236	1	no	—
197	C₈	#14273696	1	1	yes	characteristic
198	C₈	#14273708	1	2	yes	—
199	C₈	#14273714	1	1	yes	characteristic
200	C₂²	#16837595	1	2	yes	—
201	C₂²	#16837635	1	1	yes	characteristic
202	C₂²	Q₃₂×D₅₉	1	1	yes	characteristic
203	C₂²	#16837835	1	1	yes	characteristic
204	C₂²	#16837844	1	2	yes	—
205	C₄	—	8	4	no	—
206	C₄	—	118	2	no	—
207	C₄	—	944	2	no	—
208	C₄	—	236	1	no	—
209	C₄	#16837316	1	1	yes	characteristic
210	C₄	#16837615	1	2	yes	—
211	C₄	#16837652	1	1	yes	characteristic
212	C₂	#21425404	1	1	yes	characteristic
213	C₂	#21426144	1	2	yes	—
214	C₂	#21426160	1	2	yes	—
215	C₂	#21426184	1	1	yes	characteristic
216	C₂	#21426201	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₅₉⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#16837206

C₂

yes

maximal

#21424958

C₂

yes

maximal

#21422188

C₂

yes

maximal, characteristic

C₂²×Q₃₂×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14273661

C₂²

yes

—

C₂×#14273651

—

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

#16837309

—

Dic₅₉⋊C₁₆

—

#16837206

—

C₂²×Q₁₆×C₅₉

C₂²

yes

—

C₂×Q₃₂×C₅₉

—

C₂×Q₃₂×C₅₉

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₅₉

—

C₂×C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

#14273661

—

#14273661

—

#14273651

—

C₈×Dic₅₉

—

C₈×Dic₅₉

—

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂²×Q₈×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

Q₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₈

yes

characteristic

C₂²×Dic₅₉

D₈

yes

characteristic

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

C₂×C₈×C₅₉

yes

C₂×C₈×C₅₉

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

Q₃₂

yes

C₈×C₅₉

yes

C₈×C₅₉

yes

—

C₈×C₅₉

#124

yes

characteristic

C₅₉⋊C₈

—

#13450

—

maximal, Sylow

C₂²×C₅₉

#459

yes

—

C₂²×C₅₉

#466

yes

characteristic

C₂²×C₅₉

C₂×Q₃₂

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

#394

yes

characteristic

C₄×C₅₉

#463

yes

—

100

C₄×C₅₉

#469

yes

characteristic

101

Dic₅₉

—

102

Dic₅₉

—

103

Dic₅₉

—

104

C₂²×Q₃₂

D₅₉

yes

characteristic

105

C₂×C₄⋊C₁₆

—

106

C₂×#358

—

107

#1969

—

108

#1380

—

109

C₂×C₅₉

#2054

yes

characteristic

110

C₂×C₅₉

#2209

yes

—

111

C₂×C₅₉

#2211

yes

—

112

C₂×C₅₉

#2216

yes

characteristic

113

C₂×C₅₉

#2219

yes

characteristic

114

C₂²×Q₁₆

C₂×D₅₉

yes

—

115

C₂×Q₃₂

—

116

C₂×Q₃₂

—

117

C₂²×C₁₆

—

118

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

119

C₂×#107

—

120

C₂×#103

—

118

—

121

C₂×C₄×C₈

—

122

#391

—

118

—

123

C₄⋊C₁₆

—

118

—

124

#358

—

118

—

125

C₅₉

#13450

yes

characteristic, complemented, Sylow

126

C₂²×Q₈

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

C₂²⋊D₅₉

yes

—

133

C₂²×C₈

—

134

C₂²×C₈

C₂²×D₅₉

yes

characteristic

135

C₂×C₄⋊C₄

—

118

—

136

C₂×C₄²

—

137

Q₃₂

—

138

C₂×C₁₆

—

139

C₂×C₁₆

—

118

—

140

C₂×C₁₆

—

118

—

141

C₂×C₁₆

C₂²⋊D₅₉

yes

—

142

#107

—

118

—

143

#107

—

118

—

144

#103

—

236

—

145

C₄×C₈

—

118

—

146

C₄×C₈

—

118

—

147

#95

—

236

—

148

#95

—

236

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

236

—

159

C₂²×C₄

D₄×D₅₉

yes

characteristic

160

Q₁₆

—

161

Q₁₆

—

162

Q₁₆

—

163

Q₁₆

—

164

C₂×C₈

—

118

—

165

C₂×C₈

—

118

—

166

C₂×C₈

#11745687

yes

characteristic

167

C₂×C₈

#11745687

yes

—

168

C₂×C₈

C₂×C₂²⋊D₅₉

yes

—

169

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

170

C₄⋊C₄

—

236

—

171

C₄⋊C₄

—

236

—

172

C₄²

—

118

—

173

C₄²

—

118

—

174

C₁₆

—

175

C₁₆

—

236

—

176

C₂³

D₈×D₅₉

yes

characteristic

177

Q₈

—

178

Q₈

—

179

Q₈

—

180

Q₈

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

118

—

186

C₂×C₄

—

118

—

187

C₂×C₄

—

118

—

188

C₂×C₄

—

118

—

189

C₂×C₄

—

472

—

190

C₂×C₄

—

472

—

191

C₂×C₄

—

472

—

192

C₂×C₄

#14273698

yes

characteristic

193

C₂×C₄

#14273699

yes

—

194

C₂×C₄

#14273746

yes

characteristic

195

C₂×C₄

#14273751

yes

—

196

C₈

—

236

—

197

C₈

#14273696

yes

characteristic

198

C₈

#14273708

yes

—

199

C₈

#14273714

yes

characteristic

200

C₂²

#16837595

yes

—

201

C₂²

#16837635

yes

characteristic

202

C₂²

Q₃₂×D₅₉

yes

characteristic

203

C₂²

#16837835

yes

characteristic

204

C₂²

#16837844

yes

—

205

C₄

—

206

C₄

—

118

—

207

C₄

—

944

—

208

C₄

—

236

—

209

C₄

#16837316

yes

characteristic

210

C₄

#16837615

yes

—

211

C₄

#16837652

yes

characteristic

212

C₂

#21425404

yes

characteristic

213

C₂

#21426144

yes

—

214

C₂

#21426160

yes

—

215

C₂

#21426184

yes

characteristic

216

C₂

#21426201

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₉
commutator, cosocle	C₂²×C₈×C₅₉	C₂³
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂²×Q₃₂×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	C₂²×Q₃₂	D₅₉
hypocenter	C₅₉	#13450
LCS	G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂³

D₈×D₅₉

commutator, cosocle

C₂²×C₈×C₅₉

C₂³

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂²×Q₃₂×C₅₉

C₂

socle, prime radical

C₂³×C₅₉

C₂×D₈

hypercenter

C₂²×Q₃₂

D₅₉

hypocenter

C₅₉

#13450

LCS

G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
86. C₂×Dic₅₉	Q₃₂	semidirect
125. C₅₉	#13450	semidirect

H ≅ G/N

type

86. C₂×Dic₅₉

Q₃₂

semidirect

125. C₅₉

#13450

semidirect

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

subgroup	index
C₂×Dic₅₉⋊C₁₆	2
C₂×#16837206	2
#21424958	2
#21422188	2
C₂²×Q₃₂×C₅₉	2
#13450	59

subgroup

index

C₂×Dic₅₉⋊C₁₆

C₂²×Q₃₂×C₅₉

Subgroups — #28508293

Conjugacy classes of subgroups

Complements

Maximal subgroups