Group #28508521 — subgroups

Subgroups — #28508521

Order 15104.

302 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 93 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×#16837219	C₂	1	2	yes	maximal
2	#21424967	C₂	1	1	yes	maximal, characteristic
3	#21424271	C₂	1	1	yes	maximal, characteristic
4	#21422195	C₂	1	2	yes	maximal
5	C₅₉×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	C₂×#14273663	—	2	1	no	—
8	C₂×#14273663	C₂²	1	2	yes	—
9	C₂×#14273659	C₂²	1	1	yes	characteristic
10	#16837426	—	2	1	no	—
11	C₂×#14273652	—	2	1	no	—
12	#16837308	—	2	1	no	—
13	#16837219	—	2	4	no	—
14	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₁₈	—	2	1	no	—
17	C₂×Dic₂₃₆	—	2	2	no	—
18	C₂×Dic₂₃₆	—	2	2	no	—
19	C₂×Dic₂₃₆	—	2	1	no	—
20	C₂×Dic₂₃₆	D₄	1	2	yes	—
21	C₂×#11745664	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	—	2	1	no	—
23	#14273663	—	2	2	no	—
24	#14273663	—	2	2	no	—
25	#14273663	—	4	2	no	—
26	#14273659	—	2	2	no	—
27	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
28	#14273652	—	4	2	no	—
29	C₅₉⋊#95	—	4	2	no	—
30	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
33	M₃₂×C₅₉	—	2	2	no	—
34	M₃₂×C₅₉	—	2	2	no	—
35	C₂×C₁₆×C₅₉	—	2	2	no	—
36	#97×C₅₉	—	2	2	no	—
37	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
38	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅₉	—	4	1	no	—
40	C₂²×Dic₅₉	—	4	1	no	—
41	C₂×Dic₁₁₈	—	2	2	no	—
42	C₂×Dic₁₁₈	—	4	1	no	—
43	C₂×Dic₁₁₈	—	4	1	no	—
44	C₂×Dic₁₁₈	—	4	1	no	—
45	C₂×Dic₁₁₈	—	4	1	no	—
46	C₂×Dic₁₁₈	—	4	1	no	—
47	C₂×C₈×C₅₉	—	2	1	no	—
48	C₂×C₈×C₅₉	—	2	1	no	—
49	C₂×C₈×C₅₉	D₈	1	2	yes	—
50	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	#41	1	2	yes	—
52	C₂×C₈×C₅₉	#41	1	2	yes	—
53	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
54	C₄²×C₅₉	—	2	1	no	—
55	C₄²×C₅₉	—	2	1	no	—
56	C₄⋊Dic₅₉	—	4	1	no	—
57	C₄⋊Dic₅₉	—	4	1	no	—
58	#11745664	—	4	2	no	—
59	#11745664	—	4	2	no	—
60	C₂×C₅₉⋊C₈	—	4	2	no	—
61	Dic₂₃₆	—	2	4	no	—
62	Dic₂₃₆	—	4	2	no	—
63	C₁₆×C₅₉	—	4	2	no	—
64	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₉	—	2	2	no	—
66	C₂×C₄×C₅₉	—	4	2	no	—
67	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
68	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
73	C₂×Dic₅₉	—	4	2	no	—
74	C₂×Dic₅₉	—	8	1	no	—
75	C₂×Dic₅₉	—	8	1	no	—
76	C₂×Dic₅₉	—	8	2	no	—
77	C₂×Dic₅₉	—	8	1	no	—
78	Dic₁₁₈	—	4	2	no	—
79	Dic₁₁₈	—	4	2	no	—
80	Dic₁₁₈	—	8	1	no	—
81	Dic₁₁₈	—	8	1	no	—
82	C₈×C₅₉	—	2	2	no	—
83	C₈×C₅₉	#123	1	2	yes	—
84	C₈×C₅₉	#126	1	2	yes	—
85	C₅₉⋊C₈	—	8	2	no	—
86	#13488	—	59	1	no	maximal, Sylow
87	C₂²×C₅₉	—	2	2	no	—
88	C₂²×C₅₉	#461	1	1	yes	characteristic
89	C₂²×C₅₉	#481	1	1	yes	characteristic
90	C₂²×C₅₉	#486	1	1	yes	characteristic
91	C₄×C₅₉	—	4	2	no	—
92	C₄×C₅₉	#397	1	1	yes	characteristic
93	C₄×C₅₉	#459	1	1	yes	characteristic
94	C₄×C₅₉	#480	1	1	yes	characteristic
95	C₄×C₅₉	#489	1	1	yes	characteristic
96	Dic₅₉	—	8	2	no	—
97	Dic₅₉	—	16	1	no	—
98	C₂×#362	—	59	2	no	—
99	#1971	—	59	1	no	—
100	#1849	—	59	1	no	—
101	#1383	—	59	2	no	—
102	#1377	D₅₉	1	1	yes	characteristic
103	C₂×C₅₉	—	2	1	no	—
104	C₂×C₅₉	—	2	1	no	—
105	C₂×C₅₉	#2071	1	1	yes	characteristic
106	C₂×C₅₉	#2239	1	2	yes	—
107	C₂²×Q₁₆	—	59	1	no	—
108	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
109	C₂×#108	—	59	2	no	—
110	C₂×#108	—	118	1	no	—
111	C₂×#106	—	59	1	no	—
112	#423	—	118	1	no	—
113	C₂×#103	—	118	1	no	—
114	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
115	#391	—	118	1	no	—
116	#362	—	118	4	no	—
117	C₅₉	#13488	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	—	118	1	no	—
119	C₂×Q₁₆	—	59	2	no	—
120	C₂×Q₁₆	—	118	2	no	—
121	C₂×Q₁₆	—	118	2	no	—
122	C₂×Q₁₆	—	118	1	no	—
123	C₂×M₁₆	—	118	2	no	—
124	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	118	1	no	—
127	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	#108	—	118	2	no	—
132	#108	—	118	2	no	—
133	#108	—	236	2	no	—
134	#106	—	118	2	no	—
135	C₄⋊C₈	—	236	2	no	—
136	#103	—	236	2	no	—
137	#97	—	2	2	no	—
138	#95	—	236	2	no	—
139	C₂×Q₈	—	118	2	no	—
140	C₂×Q₈	—	236	1	no	—
141	C₂×Q₈	—	236	1	no	—
142	C₂×Q₈	—	236	1	no	—
143	C₂×Q₈	—	236	1	no	—
144	C₂×Q₈	—	236	1	no	—
145	C₂²×C₄	—	236	1	no	—
146	C₂²×C₄	—	236	1	no	—
147	C₂²×C₄	#11745660	1	2	yes	—
148	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
149	Q₁₆	—	118	4	no	—
150	Q₁₆	—	236	2	no	—
151	M₁₆	—	236	2	no	—
152	M₁₆	—	236	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	236	2	no	—
156	C₂×C₈	D₄₇₂	1	2	yes	—
157	C₂×C₈	#11745685	1	2	yes	—
158	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
159	C₂×C₈	#11745687	1	2	yes	—
160	C₂×C₈	#11745689	1	1	yes	characteristic
161	C₄⋊C₄	—	236	1	no	—
162	C₄⋊C₄	—	236	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₁₆	—	4	2	no	—
166	C₂³	C₂×#11745660	1	1	yes	characteristic
167	Q₈	—	472	1	no	—
168	Q₈	—	472	1	no	—
169	Q₈	—	236	2	no	—
170	Q₈	—	236	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	472	2	no	—
174	C₂×C₄	—	472	1	no	—
175	C₂×C₄	—	472	1	no	—
176	C₂×C₄	—	472	1	no	—
177	C₂×C₄	—	236	2	no	—
178	C₂×C₄	#14273702	1	1	yes	characteristic
179	C₂×C₄	#14273711	1	1	yes	characteristic
180	C₂×C₄	#14273715	1	1	yes	characteristic
181	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
182	C₂×C₄	#14273755	1	1	yes	characteristic
183	C₂×C₄	#14273760	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	472	2	no	—
186	C₈	#14273709	1	2	yes	—
187	C₈	#14273722	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	#16837608	1	1	yes	characteristic
190	C₂²	#16837721	1	1	yes	characteristic
191	C₂²	#16837749	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	944	1	no	—
194	C₄	—	472	2	no	—
195	C₄	#16837329	1	1	yes	characteristic
196	C₄	#16837598	1	1	yes	characteristic
197	C₄	#16837716	1	1	yes	characteristic
198	C₄	#16837762	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#21425501	1	1	yes	characteristic
202	C₂	#21426316	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837219

C₂

yes

maximal

#21424967

C₂

yes

maximal, characteristic

#21424271

C₂

yes

maximal, characteristic

#21422195

C₂

yes

maximal

C₅₉×#1377

C₂

yes

maximal, characteristic

C₂²×Dic₂₃₆

C₂²

yes

characteristic

C₂×#14273663

—

C₂×#14273663

C₂²

yes

—

C₂×#14273659

C₂²

yes

—

—

—

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×#97×C₅₉

C₂²

yes

characteristic

C₂²×Dic₁₁₈

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

D₄

yes

—

C₂×#11745664

—

C₂×C₄⋊Dic₅₉

—

—

—

—

—

C₄⋊(C₅₉⋊C₈)

—

#14273652

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

#97×C₅₉

—

C₂²×C₄×C₅₉

SD₁₆

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₂³×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

C₂×D₈

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#123

yes

—

C₈×C₅₉

#126

yes

—

C₅₉⋊C₈

—

#13488

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

C₂×#362

—

#1971

—

100

#1849

—

101

#1383

—

102

#1377

D₅₉

yes

characteristic

103

C₂×C₅₉

—

104

C₂×C₅₉

—

105

C₂×C₅₉

#2071

yes

characteristic

106

C₂×C₅₉

#2239

yes

—

107

C₂²×Q₁₆

—

108

C₂×M₃₂

C₂×D₅₉

yes

—

109

C₂×#108

—

110

C₂×#108

—

118

—

111

C₂×#106

—

112

#423

—

118

—

113

C₂×#103

—

118

—

114

C₂×#97

C₂×D₅₉

yes

characteristic

115

#391

—

118

—

116

#362

—

118

—

117

C₅₉

#13488

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

—

118

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

118

—

121

C₂×Q₁₆

—

118

—

122

C₂×Q₁₆

—

118

—

123

C₂×M₁₆

—

118

—

124

C₂²×C₈

D₂₃₆

yes

characteristic

125

C₂²×C₈

C₂²×D₅₉

yes

characteristic

126

C₂×C₄⋊C₄

—

118

—

127

C₂×C₄²

D₂₃₆

yes

characteristic

128

M₃₂

—

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

118

—

132

#108

—

118

—

133

#108

—

236

—

134

#106

—

118

—

135

C₄⋊C₈

—

236

—

136

#103

—

236

—

137

#97

—

138

#95

—

236

—

139

C₂×Q₈

—

118

—

140

C₂×Q₈

—

236

—

141

C₂×Q₈

—

236

—

142

C₂×Q₈

—

236

—

143

C₂×Q₈

—

236

—

144

C₂×Q₈

—

236

—

145

C₂²×C₄

—

236

—

146

C₂²×C₄

—

236

—

147

C₂²×C₄

#11745660

yes

—

148

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

149

Q₁₆

—

118

—

150

Q₁₆

—

236

—

151

M₁₆

—

236

—

152

M₁₆

—

236

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

236

—

156

C₂×C₈

D₄₇₂

yes

—

157

C₂×C₈

#11745685

yes

—

158

C₂×C₈

D₄×D₅₉

yes

characteristic

159

C₂×C₈

#11745687

yes

—

160

C₂×C₈

#11745689

yes

characteristic

161

C₄⋊C₄

—

236

—

162

C₄⋊C₄

—

236

—

163

C₄²

—

164

C₄²

—

165

C₁₆

—

166

C₂³

C₂×#11745660

yes

characteristic

167

Q₈

—

472

—

168

Q₈

—

472

—

169

Q₈

—

236

—

170

Q₈

—

236

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

472

—

174

C₂×C₄

—

472

—

175

C₂×C₄

—

472

—

176

C₂×C₄

—

472

—

177

C₂×C₄

—

236

—

178

C₂×C₄

#14273702

yes

characteristic

179

C₂×C₄

#14273711

yes

characteristic

180

C₂×C₄

#14273715

yes

characteristic

181

C₂×C₄

C₂×D₄₇₂

yes

characteristic

182

C₂×C₄

#14273755

yes

characteristic

183

C₂×C₄

#14273760

yes

characteristic

184

C₈

—

185

C₈

—

472

—

186

C₈

#14273709

yes

—

187

C₈

#14273722

yes

—

188

C₂²

—

189

C₂²

#16837608

yes

characteristic

190

C₂²

#16837721

yes

characteristic

191

C₂²

#16837749

yes

characteristic

192

C₄

—

193

C₄

—

944

—

194

C₄

—

472

—

195

C₄

#16837329

yes

characteristic

196

C₄

#16837598

yes

characteristic

197

C₄

#16837716

yes

characteristic

198

C₄

#16837762

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#21425501

yes

characteristic

202

C₂

#21426316

yes

—

	subgroup	G/H
center	C₂²	#16837721
commutator, cosocle	C₂²×C₈×C₅₉	C₂³
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1377	C₂
socle	C₂²×C₅₉	#481
prime radical	C₂³×C₅₉	C₂×SD₁₆
hypercenter	#1377	D₅₉
hypocenter	C₅₉	#13488
LCS	G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

subgroup

G/H

center

C₂²

#16837721

commutator, cosocle

C₂²×C₈×C₅₉

C₂³

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1377

C₂

socle

C₂²×C₅₉

#481

prime radical

C₂³×C₅₉

C₂×SD₁₆

D₅₉

C₅₉

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

N	H ≅ G/N	type
117. C₅₉	#13488	semidirect

H ≅ G/N

type

117. C₅₉

#13488

semidirect

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

subgroup	index
C₂×#16837219	2
#21424967	2
#21424271	2
#21422195	2
C₅₉×#1377	2
#13488	59

subgroup

index

C₅₉×#1377

Subgroups — #28508521

Conjugacy classes of subgroups

Complements

Maximal subgroups