Group #28493433 — subgroups

Subgroups — #28493433

Order 15104.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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	#21424224	C₂	1	1	yes	maximal, characteristic
2	#21424101	C₂	1	2	yes	maximal
3	#21421980	C₂	1	2	yes	maximal
4	#21421931	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14273651	—	2	2	no	—
8	#16837272	C₂²	1	2	yes	—
9	#16837165	—	2	2	no	—
10	#16837164	—	2	2	no	—
11	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#336	—	2	2	no	—
15	C₂²×C₅₉⋊C₈	—	2	2	no	—
16	C₂×C₄⋊Dic₅₉	—	2	2	no	—
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
20	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
21	#14273651	—	4	4	no	—
22	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	—	2	1	no	—
24	C₂²×C₈×C₅₉	D₄	1	2	yes	—
25	C₄×Q₈×C₅₉	—	2	2	no	—
26	C₄×Q₈×C₅₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₉	—	2	1	no	—
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	—	2	2	no	—
32	C₄×C₈×C₅₉	—	2	2	no	—
33	C₂×Q₈×C₅₉	—	2	1	no	—
34	C₂×Q₈×C₅₉	—	2	2	no	—
35	C₂×Q₈×C₅₉	—	2	2	no	—
36	C₂×Q₈×C₅₉	D₈	1	2	yes	—
37	C₂²×C₄×C₅₉	—	2	1	no	—
38	C₂²×C₄×C₅₉	—	2	1	no	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
42	C₂²×Dic₅₉	—	4	2	no	—
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₂×C₈×C₅₉	—	2	2	no	—
46	C₂×C₈×C₅₉	—	4	1	no	—
47	C₂×C₈×C₅₉	—	4	2	no	—
48	C₄⋊C₄×C₅₉	—	2	2	no	—
49	C₄⋊C₄×C₅₉	—	4	2	no	—
50	C₄²×C₅₉	—	4	1	no	—
51	C₄²×C₅₉	—	4	1	no	—
52	C₄²×C₅₉	#41	1	2	yes	—
53	C₄²×C₅₉	#41	1	2	yes	—
54	C₄⋊Dic₅₉	—	4	2	no	—
55	C₄⋊Dic₅₉	—	4	2	no	—
56	C₄⋊Dic₅₉	—	4	2	no	—
57	C₄⋊Dic₅₉	—	4	2	no	—
58	C₄⋊Dic₅₉	—	8	2	no	—
59	C₂×C₅₉⋊C₈	—	4	2	no	—
60	C₂×C₅₉⋊C₈	—	4	2	no	—
61	C₂×C₅₉⋊C₈	—	4	2	no	—
62	C₂³×C₅₉	#121	1	1	yes	characteristic
63	Q₈×C₅₉	—	2	4	no	—
64	Q₈×C₅₉	—	4	2	no	—
65	C₂×C₄×C₅₉	—	2	2	no	—
66	C₂×C₄×C₅₉	—	2	2	no	—
67	C₂×C₄×C₅₉	—	4	1	no	—
68	C₂×C₄×C₅₉	—	4	1	no	—
69	C₂×C₄×C₅₉	—	4	1	no	—
70	C₂×C₄×C₅₉	—	4	2	no	—
71	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#123	1	2	yes	—
73	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#126	1	2	yes	—
75	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#135	1	2	yes	—
77	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
78	C₂×C₄×C₅₉	#137	1	2	yes	—
79	C₂×C₄×C₅₉	#137	1	2	yes	—
80	C₂×Dic₅₉	—	8	2	no	—
81	C₂×Dic₅₉	—	8	2	no	—
82	C₂×Dic₅₉	—	8	2	no	—
83	C₈×C₅₉	—	4	2	no	—
84	C₈×C₅₉	—	8	1	no	—
85	C₅₉⋊C₈	—	8	2	no	—
86	#10623	—	59	1	no	maximal, Sylow
87	C₂²×C₅₉	#397	1	1	yes	characteristic
88	C₂²×C₅₉	#464	1	2	yes	—
89	C₂²×C₅₉	#466	1	1	yes	characteristic
90	C₂²×C₅₉	#470	1	1	yes	characteristic
91	C₂²×C₅₉	#471	1	2	yes	—
92	C₄×C₅₉	—	2	2	no	—
93	C₄×C₅₉	—	4	2	no	—
94	C₄×C₅₉	—	8	1	no	—
95	C₄×C₅₉	#480	1	2	yes	—
96	C₄×C₅₉	#485	1	2	yes	—
97	C₄×C₅₉	#485	1	2	yes	—
98	C₄×C₅₉	#488	1	1	yes	characteristic
99	C₄×C₅₉	#491	1	1	yes	characteristic
100	Dic₅₉	—	16	2	no	—
101	#1984	D₅₉	1	1	yes	characteristic
102	#1834	—	59	1	no	—
103	#1803	—	59	2	no	—
104	#1296	—	59	2	no	—
105	#1277	—	59	1	no	—
106	C₂×C₅₉	#1682	1	2	yes	—
107	C₂×C₅₉	#1704	1	2	yes	—
108	C₂×C₅₉	#2074	1	1	yes	characteristic
109	C₂×C₅₉	#2075	1	1	yes	characteristic
110	C₂×C₅₉	#2079	1	1	yes	characteristic
111	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
112	C₂×C₄⋊C₈	—	59	2	no	—
113	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
114	C₂×#103	—	118	2	no	—
115	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
116	#384	—	59	2	no	—
117	#340	—	118	2	no	—
118	#340	—	118	2	no	—
119	#336	—	2	2	no	—
120	C₅₉	#10623	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	D₂₃₆	1	1	yes	characteristic
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	118	2	no	—
124	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
125	C₄×Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	118	2	no	—
129	C₂×C₄⋊C₄	—	118	2	no	—
130	C₂×C₄⋊C₄	—	118	2	no	—
131	C₂×C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	118	2	no	—
136	C₄⋊C₈	—	118	2	no	—
137	#103	—	236	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	D₄₇₂	1	2	yes	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	236	2	no	—
146	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
147	C₂²×C₄	#11745685	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	236	2	no	—
155	C₂×C₈	—	236	2	no	—
156	C₂×C₈	—	236	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	472	2	no	—
160	C₄⋊C₄	—	236	2	no	—
161	C₄⋊C₄	—	236	2	no	—
162	C₄⋊C₄	—	236	2	no	—
163	C₄⋊C₄	—	236	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	#11745687	1	2	yes	—
167	C₄²	#11745687	1	2	yes	—
168	C₂³	#14273701	1	1	yes	characteristic
169	Q₈	—	2	4	no	—
170	Q₈	—	4	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	472	2	no	—
178	C₂×C₄	—	472	2	no	—
179	C₂×C₄	—	472	2	no	—
180	C₂×C₄	#14273708	1	1	yes	characteristic
181	C₂×C₄	#14273711	1	2	yes	—
182	C₂×C₄	#14273714	1	1	yes	characteristic
183	C₂×C₄	#14273720	1	2	yes	—
184	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
185	C₂×C₄	#14273747	1	2	yes	—
186	C₂×C₄	#14273749	1	1	yes	characteristic
187	C₂×C₄	#14273758	1	2	yes	—
188	C₂×C₄	#14273760	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	1	no	—
191	C₈	—	472	2	no	—
192	C₂²	#16837327	1	1	yes	characteristic
193	C₂²	#16837624	1	2	yes	—
194	C₂²	#16837638	1	1	yes	characteristic
195	C₂²	#16837660	1	1	yes	characteristic
196	C₂²	#16837665	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	944	2	no	—
201	C₄	#16837716	1	2	yes	—
202	C₄	#16837740	1	2	yes	—
203	C₄	#16837744	1	2	yes	—
204	C₄	#16837756	1	1	yes	characteristic
205	C₄	#16837770	1	1	yes	characteristic
206	C₂	#21423540	1	2	yes	—
207	C₂	#21423678	1	2	yes	—
208	C₂	#21425514	1	1	yes	characteristic
209	C₂	#21425519	1	1	yes	characteristic
210	C₂	#21425541	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424224

C₂

yes

maximal, characteristic

#21424101

C₂

yes

maximal

#21421980

C₂

yes

maximal

#21421931

C₂

yes

maximal, characteristic

C₅₉×#1984

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂×#14273651

—

#16837272

C₂²

yes

—

#16837165

—

#16837164

—

C₂×C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#336

—

C₂²×C₅₉⋊C₈

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

#14273651

—

C₂²×Q₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₄×Q₈×C₅₉

—

C₄×Q₈×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

D₈

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

#41

yes

—

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#121

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#123

yes

characteristic

C₂×C₄×C₅₉

#123

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

#126

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×C₄×C₅₉

#137

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#480

yes

—

C₄×C₅₉

#485

yes

—

C₄×C₅₉

#485

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

100

Dic₅₉

—

101

#1984

D₅₉

yes

characteristic

102

#1834

—

103

#1803

—

104

#1296

—

105

#1277

—

106

C₂×C₅₉

#1682

yes

—

107

C₂×C₅₉

#1704

yes

—

108

C₂×C₅₉

#2074

yes

characteristic

109

C₂×C₅₉

#2075

yes

characteristic

110

C₂×C₅₉

#2079

yes

characteristic

111

C₂×C₄×Q₈

C₂×D₅₉

yes

characteristic

112

C₂×C₄⋊C₈

—

113

C₂×C₄⋊C₈

C₂×D₅₉

yes

characteristic

114

C₂×#103

—

118

—

115

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

116

#384

—

117

#340

—

118

—

118

#340

—

118

—

119

#336

—

120

C₅₉

#10623

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

D₂₃₆

yes

characteristic

122

C₂²×C₈

—

123

C₂²×C₈

—

118

—

124

C₂²×C₈

C₂²⋊D₅₉

yes

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

118

—

129

C₂×C₄⋊C₄

—

118

—

130

C₂×C₄⋊C₄

—

118

—

131

C₂×C₄⋊C₄

D₂₃₆

yes

characteristic

132

C₂×C₄²

—

133

C₂×C₄²

C₂²×D₅₉

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

118

—

136

C₄⋊C₈

—

118

—

137

#103

—

236

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

D₄₇₂

yes

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

236

—

146

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

147

C₂²×C₄

#11745685

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

236

—

155

C₂×C₈

—

236

—

156

C₂×C₈

—

236

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

472

—

160

C₄⋊C₄

—

236

—

161

C₄⋊C₄

—

236

—

162

C₄⋊C₄

—

236

—

163

C₄⋊C₄

—

236

—

164

C₄²

—

165

C₄²

—

166

C₄²

#11745687

yes

—

167

C₄²

#11745687

yes

—

168

C₂³

#14273701

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

472

—

178

C₂×C₄

—

472

—

179

C₂×C₄

—

472

—

180

C₂×C₄

#14273708

yes

characteristic

181

C₂×C₄

#14273711

yes

—

182

C₂×C₄

#14273714

yes

characteristic

183

C₂×C₄

#14273720

yes

—

184

C₂×C₄

C₂×D₄₇₂

yes

characteristic

185

C₂×C₄

#14273747

yes

—

186

C₂×C₄

#14273749

yes

characteristic

187

C₂×C₄

#14273758

yes

—

188

C₂×C₄

#14273760

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

472

—

192

C₂²

#16837327

yes

characteristic

193

C₂²

#16837624

yes

—

194

C₂²

#16837638

yes

characteristic

195

C₂²

#16837660

yes

characteristic

196

C₂²

#16837665

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

944

—

201

C₄

#16837716

yes

—

202

C₄

#16837740

yes

—

203

C₄

#16837744

yes

—

204

C₄

#16837756

yes

characteristic

205

C₄

#16837770

yes

characteristic

206

C₂

#21423540

yes

—

207

C₂

#21423678

yes

—

208

C₂

#21425514

yes

characteristic

209

C₂

#21425519

yes

characteristic

210

C₂

#21425541

yes

characteristic

	subgroup	G/H
center	C₂³	#14273701
commutator, cosocle	C₂×C₄²×C₅₉	C₂³
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1984	C₂
socle, prime radical	C₂³×C₅₉	#121
hypercenter	#1984	D₅₉
hypocenter	C₅₉	#10623
LCS	G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

subgroup

G/H

center

C₂³

#14273701

commutator, cosocle

C₂×C₄²×C₅₉

C₂³

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1984

C₂

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

N	H ≅ G/N	type
120. C₅₉	#10623	semidirect

H ≅ G/N

type

120. C₅₉

#10623

semidirect

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

subgroup	index
#21424224	2
#21424101	2
#21421980	2
#21421931	2
C₅₉×#1984	2
#10623	59

subgroup

index

C₅₉×#1984

Subgroups — #28493433

Conjugacy classes of subgroups

Complements

Maximal subgroups