Group #28744190 — subgroups

Subgroups — #28744190

Order 15104.

482 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 187 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	#21434608	C₂	1	2	yes	maximal
2	#21426381	C₂	1	4	yes	maximal, complemented
3	#21426380	C₂	1	4	yes	maximal, complemented
4	#21425880	C₂	1	1	yes	maximal, characteristic
5	#21425878	C₂	1	2	yes	maximal
6	C₂×#16837177	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#2158	C₂	1	1	yes	maximal, characteristic
8	#16837955	—	2	2	no	—
9	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	#16837806	C₂²	1	4	yes	—
11	#16837803	C₂²	1	4	yes	—
12	#16837802	—	2	2	no	—
13	#16837799	—	2	2	no	—
14	#16837440	C₂²	1	2	yes	—
15	#16837438	—	2	2	no	—
16	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
17	#16837230	C₂²	1	2	yes	—
18	#16837229	C₂²	1	4	yes	—
19	#16837228	C₂²	1	2	yes	—
20	#16837227	C₂²	1	4	yes	—
21	#16837177	C₂²	1	2	yes	—
22	#16837177	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#427	C₂²	1	2	yes	—
25	C₅₉×#367	C₂²	1	2	yes	—
26	C₅₉×#366	C₂²	1	2	yes	—
27	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
28	#14273726	—	2	4	no	—
29	#14273718	—	2	4	no	—
30	#14273704	—	4	4	no	—
31	C₄×Dic₁₁₈	—	4	4	no	—
32	#14273686	—	2	2	no	—
33	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
34	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	complemented
35	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
36	#14273661	C₂³	1	4	yes	—
37	#14273660	—	2	4	no	—
38	#14273659	—	2	2	no	—
39	#14273658	—	2	4	no	—
40	C₈×Dic₅₉	D₄	1	2	yes	—
41	C₈×Dic₅₉	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	#117×C₅₉	—	2	2	no	—
44	C₂×C₁₆×C₅₉	Q₈	1	4	yes	complemented
45	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
47	#107×C₅₉	C₂³	1	4	yes	—
48	#106×C₅₉	—	2	2	no	—
49	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅₉	D₈	1	2	yes	—
51	C₂×Dic₁₁₈	—	4	4	no	—
52	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₅₉	—	2	4	no	—
57	C₂²⋊C₄×C₅₉	—	4	2	no	—
58	C₄²×C₅₉	—	4	2	no	—
59	C₂²⋊Dic₅₉	—	4	2	no	—
60	C₄⋊Dic₅₉	—	2	4	no	—
61	C₄⋊Dic₅₉	—	4	4	no	—
62	Dic₅₉⋊C₄	—	4	4	no	—
63	Dic₅₉⋊C₄	—	4	4	no	—
64	C₄×Dic₅₉	—	4	2	no	—
65	C₄×Dic₅₉	D₈	1	2	yes	—
66	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
67	C₂×C₅₉⋊C₈	Q₁₆	1	4	yes	—
68	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
69	C₁₆×C₅₉	C₂×Q₈	1	2	yes	—
70	C₁₆×C₅₉	C₂×Q₈	1	2	yes	—
71	C₅₉⋊C₁₆	C₂×Q₈	1	4	yes	—
72	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅₉	—	4	4	no	—
74	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₅₉	—	4	2	no	—
79	C₂×Dic₅₉	—	4	4	no	—
80	C₂×Dic₅₉	C₂×D₈	1	4	yes	—
81	Dic₁₁₈	—	8	8	no	—
82	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
83	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
85	C₅₉⋊C₈	C₂×Q₁₆	1	4	yes	—
86	#33728	—	59	1	no	maximal, Sylow
87	C₂²×C₅₉	—	2	2	no	—
88	C₂²×C₅₉	#500	1	2	yes	—
89	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
90	C₄×C₅₉	—	8	4	no	—
91	C₄×C₅₉	#500	1	2	yes	—
92	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
94	Dic₅₉	—	2	4	no	—
95	Dic₅₉	—	8	4	no	—
96	#3160	—	59	2	no	—
97	#2255	—	59	4	no	—
98	#2255	—	59	4	no	—
99	#2158	—	59	1	no	—
100	#2158	—	59	2	no	—
101	#2158	D₅₉	1	1	yes	characteristic
102	C₂×#346	—	59	1	no	—
103	C₂×C₅₉	—	2	2	no	—
104	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
105	C₂×C₅₉	#3414	1	2	yes	—
106	#533	—	118	2	no	—
107	C₂²×C₁₆	—	59	1	no	—
108	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
109	#500	—	59	4	no	—
110	#499	—	59	4	no	—
111	#499	—	118	2	no	—
112	#498	—	118	2	no	—
113	#427	—	59	2	no	—
114	#427	—	118	2	no	—
115	#427	C₂×D₅₉	1	2	yes	—
116	C₂×C₄×C₈	—	59	1	no	—
117	#367	—	59	2	no	—
118	#367	—	59	4	no	—
119	#367	C₂×D₅₉	1	2	yes	—
120	#366	—	59	2	no	—
121	#366	—	59	4	no	—
122	#366	C₂×D₅₉	1	2	yes	—
123	#346	—	59	2	no	—
124	#346	—	59	2	no	—
125	C₅₉	#33728	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	59	1	no	—
127	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
128	C₄⋊Q₈	—	118	4	no	—
129	#125	—	118	4	no	—
130	C₂²⋊Q₈	—	236	4	no	—
131	C₄×Q₈	—	236	4	no	—
132	#117	—	2	2	no	—
133	#117	—	118	2	no	—
134	C₂×C₄²	—	59	1	no	—
135	C₂×C₁₆	—	59	4	no	—
136	C₂×C₁₆	—	59	2	no	—
137	C₂×C₁₆	Dic₁₁₈	1	4	yes	complemented
138	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
139	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
140	#107	—	59	4	no	—
141	#107	—	118	4	no	—
142	#107	C₂²×D₅₉	1	4	yes	—
143	#106	—	2	2	no	—
144	#106	—	118	4	no	—
145	#106	—	118	2	no	—
146	C₄×C₈	—	59	2	no	—
147	C₄×C₈	—	59	2	no	—
148	C₂×Q₈	—	236	4	no	—
149	C₂²×C₄	—	59	2	no	—
150	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
151	C₂×C₈	—	59	4	no	—
152	C₂×C₈	—	59	2	no	—
153	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
154	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₅₉	1	2	yes	—
156	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	118	4	no	—
159	C₄⋊C₄	—	236	4	no	—
160	C₄⋊C₄	—	236	4	no	—
161	C₄⋊C₄	—	236	4	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	236	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	59	2	no	—
166	C₄²	—	59	2	no	—
167	C₄²	—	236	2	no	—
168	C₁₆	—	59	4	no	—
169	C₁₆	C₂×Dic₁₁₈	1	2	yes	—
170	C₁₆	C₂×Dic₁₁₈	1	2	yes	—
171	C₂³	D₈×D₅₉	1	1	yes	characteristic
172	Q₈	—	472	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	59	4	no	—
175	C₂×C₄	—	236	2	no	—
176	C₂×C₄	—	236	4	no	—
177	C₂×C₄	#14273726	1	2	yes	—
178	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₅₉	1	2	yes	—
180	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
181	C₈	—	59	4	no	—
182	C₈	#14273726	1	2	yes	—
183	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
184	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
185	C₂²	—	2	2	no	—
186	C₂²	#16837806	1	2	yes	—
187	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	118	4	no	—
190	C₄	—	472	4	no	—
191	C₄	#16837806	1	2	yes	—
192	C₄	C₂×#14273726	1	1	yes	characteristic
193	C₄	C₂×Q₁₆×D₅₉	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₂×#16837806	1	1	yes	characteristic
196	C₂	#21437229	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21434608

C₂

yes

maximal

#21426381

C₂

yes

maximal, complemented

#21426380

C₂

yes

maximal, complemented

#21425880

C₂

yes

maximal, characteristic

#21425878

C₂

yes

maximal

C₂×#16837177

C₂

yes

maximal, characteristic

C₅₉×#2158

C₂

yes

maximal, characteristic

#16837955

—

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

#16837806

C₂²

yes

—

#16837803

C₂²

yes

—

#16837802

—

#16837799

—

#16837440

C₂²

yes

—

#16837438

—

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

#16837230

C₂²

yes

—

#16837229

C₂²

yes

—

#16837228

C₂²

yes

—

#16837227

C₂²

yes

—

#16837177

C₂²

yes

—

#16837177

C₂²

yes

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#427

C₂²

yes

—

C₅₉×#367

C₂²

yes

—

C₅₉×#366

C₂²

yes

—

C₂²×C₅₉⋊C₈

D₄

yes

—

—

—

C₄×Dic₁₁₈

—

#14273686

—

C₂×C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

Q₈

yes

complemented

C₂×C₅₉⋊C₁₆

C₂³

yes

—

#14273661

C₂³

yes

—

#14273660

—

#14273659

—

#14273658

—

C₈×Dic₅₉

D₄

yes

—

C₈×Dic₅₉

C₂³

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

—

C₂×C₁₆×C₅₉

Q₈

yes

complemented

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

#107×C₅₉

C₂³

yes

—

#106×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

D₈

yes

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₈

yes

—

C₄×Dic₅₉

C₂×D₄

yes

—

C₂×C₅₉⋊C₈

Q₁₆

yes

—

C₂×C₅₉⋊C₈

C₂×D₄

yes

—

C₁₆×C₅₉

C₂×Q₈

yes

—

C₁₆×C₅₉

C₂×Q₈

yes

—

C₅₉⋊C₁₆

C₂×Q₈

yes

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₂×C₄×C₅₉

C₂²×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₂×D₈

yes

—

Dic₁₁₈

—

C₈×C₅₉

C₄⋊Q₈

yes

—

C₈×C₅₉

C₂²×Q₈

yes

characteristic

C₈×C₅₉

C₂²×Q₈

yes

characteristic

C₅₉⋊C₈

C₂×Q₁₆

yes

—

#33728

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#500

yes

—

C₂²×C₅₉

C₂²×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#500

yes

—

C₄×C₅₉

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²×Q₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

—

—

—

100

#2158

—

101

#2158

D₅₉

yes

characteristic

102

C₂×#346

—

103

C₂×C₅₉

—

104

C₂×C₅₉

C₂×#500

yes

characteristic

105

C₂×C₅₉

#3414

yes

—

106

#533

—

118

—

107

C₂²×C₁₆

—

108

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

109

#500

—

110

#499

—

111

#499

—

118

—

112

#498

—

118

—

113

#427

—

114

#427

—

118

—

115

#427

C₂×D₅₉

yes

—

116

C₂×C₄×C₈

—

117

#367

—

118

#367

—

119

#367

C₂×D₅₉

yes

—

120

#366

—

121

#366

—

122

#366

C₂×D₅₉

yes

—

123

#346

—

124

#346

—

125

C₅₉

#33728

yes

characteristic, complemented, Sylow

126

C₂²×C₈

—

127

C₂²×C₈

C₂²×D₅₉

yes

characteristic

128

C₄⋊Q₈

—

118

—

129

#125

—

118

—

130

C₂²⋊Q₈

—

236

—

131

C₄×Q₈

—

236

—

132

#117

—

133

#117

—

118

—

134

C₂×C₄²

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

Dic₁₁₈

yes

complemented

138

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

139

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

140

#107

—

141

#107

—

118

—

142

#107

C₂²×D₅₉

yes

—

143

#106

—

144

#106

—

118

—

145

#106

—

118

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₂×Q₈

—

236

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₅₉

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂×Dic₁₁₈

yes

—

154

C₂×C₈

D₄×D₅₉

yes

characteristic

155

C₂×C₈

Q₈×D₅₉

yes

—

156

C₂×C₈

C₂³×D₅₉

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

118

—

159

C₄⋊C₄

—

236

—

160

C₄⋊C₄

—

236

—

161

C₄⋊C₄

—

236

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

236

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

236

—

168

C₁₆

—

169

C₁₆

C₂×Dic₁₁₈

yes

—

170

C₁₆

C₂×Dic₁₁₈

yes

—

171

C₂³

D₈×D₅₉

yes

characteristic

172

Q₈

—

472

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

236

—

176

C₂×C₄

—

236

—

177

C₂×C₄

#14273726

yes

—

178

C₂×C₄

D₈×D₅₉

yes

characteristic

179

C₂×C₄

Q₁₆×D₅₉

yes

—

180

C₂×C₄

C₂×D₄×D₅₉

yes

characteristic

181

C₈

—

182

C₈

#14273726

yes

—

183

C₈

C₂×Q₈×D₅₉

yes

characteristic

184

C₈

C₂²×Dic₁₁₈

yes

characteristic

185

C₂²

—

186

C₂²

#16837806

yes

—

187

C₂²

C₂×D₈×D₅₉

yes

characteristic

188

C₄

—

189

C₄

—

118

—

190

C₄

—

472

—

191

C₄

#16837806

yes

—

192

C₄

C₂×#14273726

yes

characteristic

193

C₄

C₂×Q₁₆×D₅₉

yes

characteristic

194

C₂

—

195

C₂

C₂×#16837806

yes

characteristic

196

C₂

#21437229

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2158	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	#2158	D₅₉
hypocenter	C₅₉	#33728
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂×C₄

D₈×D₅₉

commutator, cosocle

C₂×C₈×C₅₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₉

Fitting

C₅₉×#2158

C₂

socle

C₂²×C₅₉

C₂²×D₈

prime radical

C₂³×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
2. #21426381	C₂	semidirect
3. #21426380	C₂	semidirect
34. C₂×C₅₉⋊C₁₆	Q₈	semidirect
44. C₂×C₁₆×C₅₉	Q₈	semidirect
125. C₅₉	#33728	semidirect
137. C₂×C₁₆	Dic₁₁₈	semidirect

H ≅ G/N

type

2. #21426381

C₂

semidirect

3. #21426380

C₂

semidirect

34. C₂×C₅₉⋊C₁₆

Q₈

semidirect

44. C₂×C₁₆×C₅₉

Q₈

semidirect

125. C₅₉

#33728

semidirect

137. C₂×C₁₆

Dic₁₁₈

semidirect

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

subgroup	index
#21434608	2
#21426381	2
#21426380	2
#21425880	2
#21425878	2
C₂×#16837177	2
C₅₉×#2158	2
#33728	59

subgroup

index

C₅₉×#2158

Subgroups — #28744190

Conjugacy classes of subgroups

Complements

Maximal subgroups