Group #28744113 — subgroups

Subgroups — #28744113

Order 15104.

826 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 219 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×#16837834	C₂	1	4	yes	maximal, complemented
2	#21434524	C₂	1	2	yes	maximal, complemented
3	#21426367	C₂	1	4	yes	maximal, complemented
4	#21426362	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14273749	—	2	2	no	—
7	C₂×#14273749	C₂²	1	4	yes	—
8	#16837952	—	2	2	no	—
9	#16837834	—	2	16	no	—
10	C₂×Dic₄₇₂	C₂²	1	4	yes	—
11	C₂×#14273672	C₂²	1	8	yes	—
12	C₂×D₉₄₄	C₂²	1	4	yes	—
13	#16837786	—	2	2	no	—
14	#16837786	C₂²	1	4	yes	—
15	#16837782	C₂²	1	2	yes	—
16	#16837779	C₂²	1	2	yes	—
17	#16837775	—	2	2	no	—
18	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₅₉×#346	C₂²	1	4	yes	—
21	C₂×#11745685	—	2	4	no	—
22	#14273749	—	2	16	no	—
23	#14273749	—	4	8	no	—
24	C₂×Dic₂₃₆	—	2	2	no	—
25	C₂×Dic₂₃₆	C₂³	1	4	yes	—
26	C₂×#11745660	—	2	4	no	—
27	C₂×#11745660	—	2	4	no	—
28	C₂×D₄₇₂	—	2	2	no	—
29	C₂×D₄₇₂	C₂³	1	4	yes	—
30	#14273727	—	2	2	no	—
31	#14273724	—	2	2	no	—
32	#14273715	—	2	4	no	—
33	#14273701	—	4	4	no	—
34	C₄×D₂₃₆	—	4	4	no	—
35	Dic₄₇₂	—	2	8	no	—
36	#14273672	—	2	16	no	—
37	D₉₄₄	—	2	8	no	—
38	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₉	D₄	1	8	yes	complemented
42	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
43	C₄×C₈×C₅₉	D₄	1	2	yes	—
44	C₄×C₈×C₅₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₅₉	D₈	1	2	yes	—
46	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₅₉	—	4	4	no	—
48	#11745685	—	4	8	no	—
49	#11745685	—	4	8	no	—
50	C₂×D₂₃₆	—	2	4	no	—
51	C₂×D₂₃₆	—	4	4	no	—
52	C₂×C₄×D₅₉	—	4	4	no	—
53	C₂×Dic₁₁₈	—	2	4	no	—
54	C₂×C₈×C₅₉	D₈	1	4	yes	—
55	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
57	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
59	C₄²×C₅₉	D₈	1	2	yes	—
60	C₄²×C₅₉	C₂×D₄	1	2	yes	—
61	(C₂×D₅₉)⋊C₄	—	4	8	no	—
62	C₄⋊Dic₅₉	—	4	4	no	—
63	Dic₂₃₆	—	2	8	no	—
64	Dic₂₃₆	—	4	4	no	—
65	D₄₇₂	—	2	8	no	—
66	D₄₇₂	—	4	4	no	—
67	#11745660	—	4	8	no	—
68	#11745660	—	4	8	no	—
69	C₁₆×C₅₉	C₂×D₄	1	8	yes	—
70	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₅₉	—	4	4	no	—
72	C₂×C₄×C₅₉	—	4	2	no	—
73	C₂×C₄×C₅₉	#127	1	2	yes	—
74	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
78	C₂²⋊D₅₉	—	8	8	no	—
79	C₂×Dic₅₉	—	4	4	no	—
80	D₂₃₆	—	4	4	no	—
81	D₂₃₆	—	4	4	no	—
82	D₂₃₆	—	8	8	no	—
83	C₄×D₅₉	—	8	8	no	—
84	Dic₁₁₈	—	4	4	no	—
85	Dic₁₁₈	—	4	4	no	—
86	C₈×C₅₉	#127	1	2	yes	—
87	C₈×C₅₉	C₂×D₈	1	4	yes	—
88	C₈×C₅₉	C₂²×D₄	1	2	yes	—
89	#33716	—	59	1	no	maximal, Sylow
90	C₂²×C₅₉	—	2	2	no	—
91	C₂²×C₅₉	#493	1	2	yes	—
92	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
93	C₂×D₅₉	—	8	4	no	—
94	C₂×D₅₉	—	8	8	no	—
95	C₄×C₅₉	—	2	4	no	—
96	C₄×C₅₉	#493	1	2	yes	—
97	C₄×C₅₉	C₂×#127	1	1	yes	characteristic
98	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
99	Dic₅₉	—	8	4	no	—
100	C₂×#508	—	59	4	no	—
101	#3147	—	59	2	no	—
102	#2251	—	59	4	no	—
103	#2250	—	59	4	no	—
104	C₂×#346	D₅₉	1	1	yes	characteristic, complemented
105	C₂×C₅₉	—	2	2	no	—
106	C₂×C₅₉	C₂×#493	1	1	yes	characteristic
107	C₂×C₅₉	#3414	1	2	yes	—
108	D₅₉	—	16	4	no	—
109	C₂×#135	—	59	4	no	—
110	C₂×#135	—	118	2	no	—
111	#532	—	118	2	no	—
112	#508	—	118	16	no	—
113	C₂×Q₃₂	—	59	4	no	—
114	C₂×SD₃₂	—	59	8	no	—
115	C₂×D₁₆	—	59	4	no	—
116	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
117	#495	—	59	4	no	—
118	#495	—	118	2	no	—
119	#494	—	59	2	no	—
120	#493	—	59	2	no	—
121	#492	—	118	2	no	—
122	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
123	#346	C₂×D₅₉	1	4	yes	—
124	C₅₉	#33716	1	1	yes	characteristic, complemented, Sylow
125	C₂×#41	—	118	4	no	—
126	#135	—	118	16	no	—
127	#135	—	236	8	no	—
128	C₂×Q₁₆	—	59	4	no	—
129	C₂×Q₁₆	—	118	2	no	—
130	C₂×SD₁₆	—	118	4	no	—
131	C₂×SD₁₆	—	118	4	no	—
132	C₂×D₈	—	59	4	no	—
133	C₂×D₈	—	118	2	no	—
134	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
136	C₄⋊Q₈	—	118	2	no	—
137	#127	—	118	2	no	—
138	#124	—	118	4	no	—
139	#121	—	236	4	no	—
140	C₄×D₄	—	236	4	no	—
141	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
142	Q₃₂	—	118	8	no	—
143	SD₃₂	—	118	16	no	—
144	D₁₆	—	118	8	no	—
145	C₂×C₁₆	D₂₃₆	1	8	yes	complemented
146	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
147	C₄×C₈	D₂₃₆	1	2	yes	—
148	C₄×C₈	C₂²×D₅₉	1	2	yes	—
149	#41	—	236	8	no	—
150	#41	—	236	8	no	—
151	C₂×Q₈	—	118	4	no	—
152	C₂×D₄	—	118	4	no	—
153	C₂×D₄	—	236	4	no	—
154	C₂×D₄	—	236	4	no	—
155	C₂²×C₄	—	236	4	no	—
156	C₂²×C₄	D₄₇₂	1	2	yes	—
157	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
158	Q₁₆	—	118	8	no	—
159	Q₁₆	—	236	4	no	—
160	SD₁₆	—	236	8	no	—
161	SD₁₆	—	236	8	no	—
162	D₈	—	118	8	no	—
163	D₈	—	236	4	no	—
164	C₂×C₈	D₄₇₂	1	4	yes	—
165	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
166	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
167	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
168	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
169	C₄⋊C₄	—	236	4	no	—
170	C₂²⋊C₄	—	236	8	no	—
171	C₄²	D₄₇₂	1	2	yes	—
172	C₄²	C₂×D₂₃₆	1	2	yes	—
173	C₁₆	C₂×D₂₃₆	1	8	yes	—
174	C₂³	—	236	4	no	—
175	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
176	Q₈	—	236	4	no	—
177	Q₈	—	236	4	no	—
178	D₄	—	472	8	no	—
179	D₄	—	472	8	no	—
180	D₄	—	236	4	no	—
181	D₄	—	236	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	472	8	no	—
184	C₂×C₄	—	236	4	no	—
185	C₂×C₄	#14273724	1	2	yes	—
186	C₂×C₄	C₂×D₄₇₂	1	4	yes	—
187	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
188	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
189	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
190	C₈	#14273724	1	2	yes	—
191	C₈	C₂×D₄₇₂	1	4	yes	—
192	C₈	C₂²×D₂₃₆	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	472	8	no	—
195	C₂²	—	472	4	no	—
196	C₂²	#16837779	1	2	yes	—
197	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	472	4	no	—
200	C₄	#16837779	1	2	yes	—
201	C₄	C₂×#14273724	1	1	yes	characteristic
202	C₄	C₂²×D₄₇₂	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	944	4	no	—
205	C₂	C₂×#16837779	1	1	yes	characteristic
206	C₂	#21437232	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837834

C₂

yes

maximal, complemented

#21434524

C₂

yes

maximal, complemented

#21426367

C₂

yes

maximal, complemented

#21426362

C₂

yes

maximal, complemented

C₂×C₅₉×#346

C₂

yes

maximal, characteristic, complemented

C₂×#14273749

—

C₂×#14273749

C₂²

yes

—

#16837952

—

#16837834

—

C₂×Dic₄₇₂

C₂²

yes

—

C₂×#14273672

C₂²

yes

—

C₂×D₉₄₄

C₂²

yes

—

#16837786

—

#16837786

C₂²

yes

—

#16837782

C₂²

yes

—

#16837779

C₂²

yes

—

#16837775

—

C₂²×C₁₆×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#346

C₂²

yes

—

C₂×#11745685

—

#14273749

—

#14273749

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

C₂³

yes

—

C₂×#11745660

—

C₂×#11745660

—

C₂×D₄₇₂

—

C₂×D₄₇₂

C₂³

yes

—

—

—

—

—

C₄×D₂₃₆

—

Dic₄₇₂

—

#14273672

—

D₉₄₄

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

complemented

C₂×C₁₆×C₅₉

C₂³

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

D₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₅₉

—

#11745685

—

#11745685

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂⁴

yes

characteristic

C₄²×C₅₉

D₈

yes

—

C₄²×C₅₉

C₂×D₄

yes

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

D₄₇₂

—

D₄₇₂

—

#11745660

—

#11745660

—

C₁₆×C₅₉

C₂×D₄

yes

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#127

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂²×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

#127

yes

—

C₈×C₅₉

C₂×D₈

yes

—

C₈×C₅₉

C₂²×D₄

yes

—

#33716

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#493

yes

—

C₂²×C₅₉

C₂²×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#493

yes

—

C₄×C₅₉

C₂×#127

yes

characteristic

C₄×C₅₉

C₂²×D₈

yes

characteristic

Dic₅₉

—

100

C₂×#508

—

101

#3147

—

102

#2251

—

103

#2250

—

104

C₂×#346

D₅₉

yes

characteristic, complemented

105

C₂×C₅₉

—

106

C₂×C₅₉

C₂×#493

yes

characteristic

107

C₂×C₅₉

#3414

yes

—

108

D₅₉

—

109

C₂×#135

—

110

C₂×#135

—

118

—

111

#532

—

118

—

112

#508

—

118

—

113

C₂×Q₃₂

—

114

C₂×SD₃₂

—

115

C₂×D₁₆

—

116

C₂²×C₁₆

C₂×D₅₉

yes

—

117

#495

—

118

#495

—

118

—

119

#494

—

120

#493

—

121

#492

—

118

—

122

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

123

#346

C₂×D₅₉

yes

—

124

C₅₉

#33716

yes

characteristic, complemented, Sylow

125

C₂×#41

—

118

—

126

#135

—

118

—

127

#135

—

236

—

128

C₂×Q₁₆

—

129

C₂×Q₁₆

—

118

—

130

C₂×SD₁₆

—

118

—

131

C₂×SD₁₆

—

118

—

132

C₂×D₈

—

133

C₂×D₈

—

118

—

134

C₂²×C₈

D₂₃₆

yes

characteristic

135

C₂²×C₈

C₂²×D₅₉

yes

characteristic

136

C₄⋊Q₈

—

118

—

137

#127

—

118

—

138

#124

—

118

—

139

#121

—

236

—

140

C₄×D₄

—

236

—

141

C₂×C₄²

D₂₃₆

yes

characteristic

142

Q₃₂

—

118

—

143

SD₃₂

—

118

—

144

D₁₆

—

118

—

145

C₂×C₁₆

D₂₃₆

yes

complemented

146

C₂×C₁₆

C₂²×D₅₉

yes

—

147

C₄×C₈

D₂₃₆

yes

—

148

C₄×C₈

C₂²×D₅₉

yes

—

149

#41

—

236

—

150

#41

—

236

—

151

C₂×Q₈

—

118

—

152

C₂×D₄

—

118

—

153

C₂×D₄

—

236

—

154

C₂×D₄

—

236

—

155

C₂²×C₄

—

236

—

156

C₂²×C₄

D₄₇₂

yes

—

157

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

158

Q₁₆

—

118

—

159

Q₁₆

—

236

—

160

SD₁₆

—

236

—

161

SD₁₆

—

236

—

162

D₈

—

118

—

163

D₈

—

236

—

164

C₂×C₈

D₄₇₂

yes

—

165

C₂×C₈

C₂×D₂₃₆

yes

characteristic

166

C₂×C₈

C₂×D₂₃₆

yes

—

167

C₂×C₈

C₂×D₂₃₆

yes

—

168

C₂×C₈

C₂³×D₅₉

yes

characteristic

169

C₄⋊C₄

—

236

—

170

C₂²⋊C₄

—

236

—

171

C₄²

D₄₇₂

yes

—

172

C₄²

C₂×D₂₃₆

yes

—

173

C₁₆

C₂×D₂₃₆

yes

—

174

C₂³

—

236

—

175

C₂³

C₂×D₄₇₂

yes

characteristic

176

Q₈

—

236

—

177

Q₈

—

236

—

178

D₄

—

472

—

179

D₄

—

472

—

180

D₄

—

236

—

181

D₄

—

236

—

182

C₂×C₄

—

183

C₂×C₄

—

472

—

184

C₂×C₄

—

236

—

185

C₂×C₄

#14273724

yes

—

186

C₂×C₄

C₂×D₄₇₂

yes

—

187

C₂×C₄

C₂×D₄₇₂

yes

—

188

C₂×C₄

C₂×D₄₇₂

yes

characteristic

189

C₂×C₄

C₂²×D₂₃₆

yes

characteristic

190

C₈

#14273724

yes

—

191

C₈

C₂×D₄₇₂

yes

—

192

C₈

C₂²×D₂₃₆

yes

—

193

C₂²

—

194

C₂²

—

472

—

195

C₂²

—

472

—

196

C₂²

#16837779

yes

—

197

C₂²

C₂²×D₄₇₂

yes

characteristic

198

C₄

—

199

C₄

—

472

—

200

C₄

#16837779

yes

—

201

C₄

C₂×#14273724

yes

characteristic

202

C₄

C₂²×D₄₇₂

yes

characteristic

203

C₂

—

204

C₂

—

944

—

205

C₂

C₂×#16837779

yes

characteristic

206

C₂

#21437232

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	#33716
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₄₇₂

commutator, cosocle

C₂×C₈×C₅₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₉

Fitting

C₂×C₅₉×#346

C₂

socle

C₂²×C₅₉

C₂²×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
1. C₂×#16837834	C₂	semidirect
2. #21434524	C₂	semidirect
3. #21426367	C₂	semidirect
4. #21426362	C₂	semidirect
5. C₂×C₅₉×#346	C₂	semidirect
41. C₂×C₁₆×C₅₉	D₄	semidirect
104. C₂×#346	D₅₉	semidirect
124. C₅₉	#33716	semidirect
145. C₂×C₁₆	D₂₃₆	semidirect

H ≅ G/N

type

1. C₂×#16837834

C₂

semidirect

2. #21434524

C₂

semidirect

3. #21426367

C₂

semidirect

4. #21426362

C₂

semidirect

5. C₂×C₅₉×#346

C₂

semidirect

41. C₂×C₁₆×C₅₉

D₄

semidirect

104. C₂×#346

D₅₉

semidirect

124. C₅₉

#33716

semidirect

145. C₂×C₁₆

D₂₃₆

semidirect

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

subgroup	index
C₂×#16837834	2
#21434524	2
#21426367	2
#21426362	2
C₂×C₅₉×#346	2
#33716	59

subgroup

index

C₂×C₅₉×#346

Subgroups — #28744113

Conjugacy classes of subgroups

Complements

Maximal subgroups