Group #28487396 — subgroups

Subgroups — #28487396

Order 15104.

290 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 119 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	#21424869	C₂	1	2	yes	maximal
2	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
3	#21422385	C₂	1	1	yes	maximal, characteristic
4	#21421991	C₂	1	2	yes	maximal
5	C₅₉×#1566	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	1	yes	characteristic
7	C₂×#14273662	—	2	2	no	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₄×#11745664	C₂²	1	1	yes	characteristic
10	#16837166	—	2	2	no	—
11	#16837155	C₄	1	2	yes	—
12	#16837155	C₄	1	2	yes	—
13	#16837124	—	2	1	no	—
14	#16837123	—	2	1	no	—
15	C₅₉×#423	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#334	—	2	1	no	—
18	C₅₉×#334	—	2	1	no	—
19	C₂×#11745664	D₄	1	2	yes	—
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	#14273662	—	4	4	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
24	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
25	C₂²⋊(C₅₉⋊C₈)	Q₈	1	2	yes	—
26	#14273638	—	2	1	no	—
27	C₄×C₅₉⋊C₈	—	2	1	no	—
28	C₄×C₅₉⋊C₈	—	2	1	no	—
29	C₂×M₁₆×C₅₉	—	2	2	no	—
30	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₉	—	2	2	no	—
34	C₄⋊C₈×C₅₉	—	2	2	no	—
35	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₅₉	—	4	2	no	—
43	M₁₆×C₅₉	—	4	2	no	—
44	C₂×C₈×C₅₉	—	2	2	no	—
45	C₂×C₈×C₅₉	—	4	1	no	—
46	C₂×C₈×C₅₉	—	4	2	no	—
47	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅₉	#41	1	1	yes	characteristic
54	C₄²×C₅₉	#41	1	1	yes	characteristic
55	#11745664	—	2	2	no	—
56	#11745664	—	2	2	no	—
57	C₂×C₅₉⋊C₈	—	2	4	no	—
58	C₂×C₅₉⋊C₈	—	2	2	no	—
59	C₂×C₅₉⋊C₈	—	2	2	no	—
60	C₂×C₅₉⋊C₈	—	4	2	no	—
61	C₂×C₅₉⋊C₈	—	4	2	no	—
62	C₂³×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	1	no	—
64	C₂×C₄×C₅₉	—	2	1	no	—
65	C₂×C₄×C₅₉	—	4	1	no	—
66	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
72	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₄×C₅₉	#137	1	2	yes	—
74	C₈×C₅₉	—	2	1	no	—
75	C₈×C₅₉	—	2	1	no	—
76	C₈×C₅₉	—	2	1	no	—
77	C₈×C₅₉	—	8	2	no	—
78	C₈×C₅₉	#108	1	2	yes	complemented
79	C₅₉⋊C₈	—	4	2	no	—
80	C₅₉⋊C₈	—	4	4	no	—
81	C₅₉⋊C₈	—	4	2	no	—
82	#9625	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	—	2	1	no	—
84	C₂²×C₅₉	—	2	1	no	—
85	C₂²×C₅₉	#389	1	1	yes	characteristic
86	C₂²×C₅₉	#501	1	2	yes	—
87	C₄×C₅₉	—	2	1	no	—
88	C₄×C₅₉	—	2	1	no	—
89	C₄×C₅₉	—	2	1	no	—
90	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
91	C₄×C₅₉	#430	1	1	yes	characteristic
92	C₄×C₅₉	#441	1	1	yes	characteristic
93	C₄×C₅₉	#442	1	1	yes	characteristic
94	C₄×C₅₉	#468	1	2	yes	—
95	#1950	—	59	2	no	—
96	#1566	D₅₉	1	1	yes	characteristic
97	C₂×#335	—	59	1	no	—
98	#1451	—	59	1	no	—
99	#1301	—	59	2	no	—
100	C₂×C₅₉	—	2	2	no	—
101	C₂×C₅₉	#1943	1	1	yes	characteristic
102	C₂×C₅₉	#1950	1	1	yes	characteristic
103	C₂×C₅₉	#1953	1	1	yes	characteristic
104	#431	—	59	1	no	—
105	C₂×#108	—	118	2	no	—
106	#423	C₂×D₅₉	1	2	yes	—
107	C₂×C₄⋊C₈	—	59	2	no	—
108	C₄×M₁₆	—	59	1	no	—
109	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
110	#341	—	118	2	no	—
111	#335	—	59	2	no	—
112	#335	—	59	2	no	—
113	#334	—	2	1	no	—
114	#334	—	2	1	no	—
115	#322	—	118	1	no	—
116	#322	—	118	1	no	—
117	C₅₉	#9625	1	1	yes	characteristic, complemented, Sylow
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	59	2	no	—
120	C₂²×C₈	—	118	2	no	—
121	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
122	C₂²×C₈	Dic₁₁₈	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
124	#108	—	236	4	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	59	4	no	—
128	C₄⋊C₈	—	118	2	no	—
129	C₄⋊C₈	—	118	1	no	—
130	C₂²⋊C₈	—	59	2	no	—
131	#97	—	118	1	no	—
132	C₄×C₈	—	118	1	no	—
133	C₄×C₈	—	118	1	no	—
134	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
135	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
139	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
141	M₁₆	—	4	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	118	2	no	—
144	M₁₆	—	118	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	118	4	no	—
149	C₂×C₈	—	118	2	no	—
150	C₂×C₈	—	118	2	no	—
151	C₂×C₈	—	236	2	no	—
152	C₂×C₈	—	236	2	no	—
153	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
154	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
155	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
156	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
157	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
158	C₄²	#11745687	1	1	yes	characteristic
159	C₄²	#11745689	1	1	yes	characteristic
160	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
161	C₂³	#14273726	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
166	C₂×C₄	#14273689	1	1	yes	characteristic
167	C₂×C₄	#14273692	1	1	yes	characteristic
168	C₂×C₄	#14273698	1	1	yes	characteristic
169	C₂×C₄	#14273703	1	1	yes	characteristic
170	C₂×C₄	#14273718	1	1	yes	characteristic
171	C₂×C₄	#14273754	1	2	yes	—
172	C₂×C₄	#14273756	1	2	yes	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	8	2	no	—
177	C₈	—	236	2	no	—
178	C₈	—	236	2	no	—
179	C₈	—	236	4	no	—
180	C₈	#14273662	1	2	yes	complemented
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	#16837298	1	1	yes	characteristic
184	C₂²	#16837809	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	C₂×#14273662	1	1	yes	characteristic
189	C₄	#16837450	1	1	yes	characteristic
190	C₄	#16837508	1	1	yes	characteristic
191	C₄	#16837513	1	1	yes	characteristic
192	C₄	#16837645	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#21424832	1	1	yes	characteristic
195	C₂	#21424869	1	1	yes	characteristic
196	C₂	#21424884	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424869

C₂

yes

maximal

C₂×#16837155

C₂

yes

maximal, characteristic

#21422385

C₂

yes

maximal, characteristic

#21421991

C₂

yes

maximal

C₅₉×#1566

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

characteristic

C₂×#14273662

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₄×#11745664

C₂²

yes

characteristic

#16837166

—

#16837155

C₄

yes

—

#16837155

C₄

yes

—

#16837124

—

#16837123

—

C₅₉×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#334

—

C₅₉×#334

—

C₂×#11745664

D₄

yes

—

C₂²×C₅₉⋊C₈

—

#14273662

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₅₉⋊C₈)

Q₈

yes

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

Q₈

yes

characteristic

C₂²×C₈×C₅₉

Q₈

yes

characteristic

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₉

C₂×Q₈

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

C₄²×C₅₉

yes

C₄²×C₅₉

yes

—

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₄×Q₈

yes

characteristic

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

—

C₂×C₄×C₅₉

#137

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#108

yes

complemented

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#9625

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#389

yes

characteristic

C₂²×C₅₉

#501

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

#1950

—

#1566

D₅₉

yes

—

—

—

100

C₂×C₅₉

—

101

C₂×C₅₉

#1943

yes

characteristic

102

C₂×C₅₉

#1950

yes

characteristic

103

C₂×C₅₉

#1953

yes

characteristic

104

#431

—

105

C₂×#108

—

118

—

106

#423

C₂×D₅₉

yes

—

107

C₂×C₄⋊C₈

—

108

C₄×M₁₆

—

109

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

110

#341

—

118

—

111

#335

—

112

#335

—

113

#334

—

114

#334

—

115

#322

—

118

—

116

#322

—

118

—

117

C₅₉

#9625

yes

characteristic, complemented, Sylow

118

C₂×M₁₆

—

119

C₂×M₁₆

—

120

C₂²×C₈

—

118

—

121

C₂²×C₈

Dic₁₁₈

yes

characteristic

122

C₂²×C₈

Dic₁₁₈

yes

characteristic

123

C₂×C₄²

C₂²×D₅₉

yes

characteristic

124

#108

—

236

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

118

—

129

C₄⋊C₈

—

118

—

130

C₂²⋊C₈

—

131

#97

—

118

—

132

C₄×C₈

—

118

—

133

C₄×C₈

—

118

—

134

C₄×C₈

C₄×D₅₉

yes

characteristic

135

C₄×C₈

C₄×D₅₉

yes

characteristic

136

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

137

C₄×C₈

C₂²⋊D₅₉

yes

characteristic

138

C₂²×C₄

C₂×Dic₁₁₈

yes

characteristic

139

C₂²×C₄

D₄×D₅₉

yes

characteristic

140

C₂²×C₄

Q₈×D₅₉

yes

characteristic

141

M₁₆

—

142

M₁₆

—

143

M₁₆

—

118

—

144

M₁₆

—

118

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

118

—

149

C₂×C₈

—

118

—

150

C₂×C₈

—

118

—

151

C₂×C₈

—

236

—

152

C₂×C₈

—

236

—

153

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

154

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

155

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

156

C₂×C₈

Dic₅₉⋊C₄

yes

characteristic

157

C₄²

C₂×C₄×D₅₉

yes

characteristic

158

C₄²

#11745687

yes

characteristic

159

C₄²

#11745689

yes

characteristic

160

C₄²

C₂×C₂²⋊D₅₉

yes

characteristic

161

C₂³

#14273726

yes

characteristic

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

166

C₂×C₄

#14273689

yes

characteristic

167

C₂×C₄

#14273692

yes

characteristic

168

C₂×C₄

#14273698

yes

characteristic

169

C₂×C₄

#14273703

yes

characteristic

170

C₂×C₄

#14273718

yes

characteristic

171

C₂×C₄

#14273754

yes

—

172

C₂×C₄

#14273756

yes

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

236

—

178

C₈

—

236

—

179

C₈

—

236

—

180

C₈

#14273662

yes

complemented

181

C₂²

—

182

C₂²

—

183

C₂²

#16837298

yes

characteristic

184

C₂²

#16837809

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

C₂×#14273662

yes

characteristic

189

C₄

#16837450

yes

characteristic

190

C₄

#16837508

yes

characteristic

191

C₄

#16837513

yes

characteristic

192

C₄

#16837645

yes

—

193

C₂

—

194

C₂

#21424832

yes

characteristic

195

C₂

#21424869

yes

characteristic

196

C₂

#21424884

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14273698
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1566	C₂
socle	C₂²×C₅₉	#389
prime radical	C₂³×C₅₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1566	D₅₉
hypocenter	C₅₉	#9625
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#14273698

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1566

C₂

socle

C₂²×C₅₉

#389

prime radical

C₂³×C₅₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
78. C₈×C₅₉	#108	semidirect
117. C₅₉	#9625	semidirect
180. C₈	#14273662	semidirect

H ≅ G/N

type

78. C₈×C₅₉

#108

semidirect

117. C₅₉

#9625

semidirect

180. C₈

#14273662

semidirect

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

subgroup	index
#21424869	2
C₂×#16837155	2
#21422385	2
#21421991	2
C₅₉×#1566	2
#9625	59

subgroup

index

C₅₉×#1566

Subgroups — #28487396

Conjugacy classes of subgroups

Complements

Maximal subgroups