Group #28481681 — subgroups

Subgroups — #28481681

Order 15104.

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

normal subgroups only

This group has 388 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	#21422859	C₂	1	2	yes	maximal, complemented
2	#21422731	C₂	1	2	yes	maximal
3	#21422007	C₂	1	2	yes	maximal, complemented
4	C₅₉×#2931	C₂	1	1	yes	maximal, characteristic
5	#16837459	C₂²	1	4	yes	—
6	#16837155	—	2	2	no	—
7	#16837153	—	2	2	no	—
8	#16837142	—	2	4	no	—
9	C₂×#131×C₅₉	C₄	1	2	yes	—
10	C₅₉×#532	C₂²	1	1	yes	characteristic
11	C₅₉×#436	—	2	2	no	—
12	C₈×D₄×C₅₉	—	2	2	no	—
13	C₅₉×#423	C₂²	1	1	yes	characteristic
14	C₅₉×#408	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
17	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
18	#14273638	—	2	4	no	—
19	C₂×#41×C₅₉	C₂×C₄	1	2	yes	—
20	#131×C₅₉	—	4	4	no	—
21	C₂×M₁₆×C₅₉	—	2	2	no	—
22	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₉	—	2	2	no	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₄⋊Q₈×C₅₉	Q₈	1	1	yes	characteristic
27	#127×C₅₉	Q₈	1	1	yes	characteristic
28	#124×C₅₉	D₄	1	2	yes	—
29	#121×C₅₉	—	2	2	no	—
30	C₄×D₄×C₅₉	—	2	2	no	—
31	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₅₉	—	2	2	no	—
33	C₂²⋊C₈×C₅₉	—	2	4	no	—
34	C₄×C₈×C₅₉	D₄	1	2	yes	—
35	C₄×C₈×C₅₉	D₄	1	2	yes	—
36	#41×C₅₉	—	4	2	no	—
37	#41×C₅₉	—	4	2	no	—
38	C₂×Q₈×C₅₉	C₄⋊C₄	1	2	yes	—
39	C₂×D₄×C₅₉	—	2	2	no	—
40	C₂×D₄×C₅₉	—	2	2	no	—
41	C₂×D₄×C₅₉	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₉	—	2	2	no	—
43	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	#41	1	2	yes	—
45	M₁₆×C₅₉	—	4	4	no	—
46	M₁₆×C₅₉	—	4	2	no	—
47	C₂×C₈×C₅₉	—	2	2	no	—
48	C₂×C₈×C₅₉	—	2	2	no	—
49	C₂×C₈×C₅₉	—	4	1	no	—
50	C₂×C₈×C₅₉	—	4	4	no	—
51	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₄⋊C₄×C₅₉	—	2	2	no	—
55	C₂²⋊C₄×C₅₉	—	2	4	no	—
56	C₄²×C₅₉	C₂×D₄	1	2	yes	—
57	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₅₉⋊C₈	—	4	4	no	—
60	C₂³×C₅₉	—	2	2	no	—
61	C₂³×C₅₉	#117	1	1	yes	characteristic
62	Q₈×C₅₉	—	4	2	no	—
63	D₄×C₅₉	—	4	2	no	—
64	D₄×C₅₉	—	4	4	no	—
65	D₄×C₅₉	—	4	4	no	—
66	C₂×C₄×C₅₉	—	2	1	no	—
67	C₂×C₄×C₅₉	—	2	1	no	—
68	C₂×C₄×C₅₉	—	2	2	no	—
69	C₂×C₄×C₅₉	—	4	2	no	—
70	C₂×C₄×C₅₉	—	4	2	no	—
71	C₂×C₄×C₅₉	—	4	2	no	—
72	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#121	1	2	yes	—
75	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
76	C₈×C₅₉	—	2	1	no	—
77	C₈×C₅₉	—	2	2	no	—
78	C₈×C₅₉	—	4	2	no	—
79	C₈×C₅₉	#100	1	2	yes	—
80	C₅₉⋊C₈	—	8	4	no	—
81	#8630	—	59	1	no	maximal, Sylow
82	C₂²×C₅₉	—	2	1	no	—
83	C₂²×C₅₉	—	2	2	no	—
84	C₂²×C₅₉	—	4	2	no	—
85	C₂²×C₅₉	—	4	4	no	—
86	C₂²×C₅₉	#380	1	1	yes	characteristic
87	C₄×C₅₉	—	2	1	no	—
88	C₄×C₅₉	—	2	2	no	—
89	C₄×C₅₉	—	2	2	no	—
90	C₄×C₅₉	—	4	2	no	—
91	C₄×C₅₉	C₂×#100	1	1	yes	characteristic
92	C₄×C₅₉	#413	1	1	yes	characteristic
93	#2931	D₅₉	1	1	yes	characteristic
94	#1568	—	59	2	no	—
95	#1536	—	59	2	no	—
96	#1307	—	59	2	no	—
97	C₂×C₅₉	—	4	1	no	—
98	C₂×C₅₉	—	8	2	no	—
99	C₂×C₅₉	#1800	1	1	yes	characteristic
100	C₂×C₅₉	#1814	1	2	yes	—
101	C₂×#131	Dic₅₉	1	2	yes	—
102	#532	C₂×D₅₉	1	1	yes	characteristic
103	#436	—	2	2	no	—
104	C₈×D₄	—	2	2	no	—
105	#432	—	59	4	no	—
106	#423	C₂×D₅₉	1	1	yes	characteristic
107	#408	Dic₅₉	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	#335	—	118	2	no	—
110	#334	—	118	2	no	—
111	#329	—	118	4	no	—
112	C₅₉	#8630	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	C₂×Dic₅₉	1	2	yes	—
114	#131	—	4	4	no	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
120	C₄⋊Q₈	Dic₁₁₈	1	1	yes	characteristic
121	#127	Dic₁₁₈	1	1	yes	characteristic
122	#124	D₂₃₆	1	2	yes	—
123	#121	—	2	2	no	—
124	C₄×D₄	—	2	2	no	—
125	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	118	4	no	—
128	C₂²⋊C₈	—	2	4	no	—
129	C₂²⋊C₈	—	118	4	no	—
130	#97	—	118	4	no	—
131	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
132	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
133	#41	—	4	2	no	—
134	#41	—	4	2	no	—
135	C₂×Q₈	C₄⋊Dic₅₉	1	2	yes	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	C₄⋊Dic₅₉	1	2	yes	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	#11745685	1	2	yes	—
141	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
142	M₁₆	—	4	4	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	4	no	—
148	C₂×C₈	—	236	4	no	—
149	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
150	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	4	no	—
154	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
155	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
156	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
157	C₂³	—	2	2	no	—
158	C₂³	#14273686	1	1	yes	characteristic
159	Q₈	—	4	2	no	—
160	D₄	—	4	2	no	—
161	D₄	—	4	4	no	—
162	D₄	—	4	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
171	C₂×C₄	#14273701	1	2	yes	—
172	C₂×C₄	#14273706	1	2	yes	—
173	C₈	—	2	1	no	—
174	C₈	—	2	2	no	—
175	C₈	—	4	2	no	—
176	C₈	—	472	4	no	—
177	C₈	#14273644	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	2	no	—
182	C₂²	#16837262	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	1	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂×#14273644	1	1	yes	characteristic
188	C₄	#16837386	1	1	yes	characteristic
189	C₂	—	4	1	no	—
190	C₂	—	8	2	no	—
191	C₂	#21424088	1	1	yes	characteristic
192	C₂	#21424150	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21422859

C₂

yes

maximal, complemented

#21422731

C₂

yes

maximal

#21422007

C₂

yes

maximal, complemented

C₅₉×#2931

C₂

yes

maximal, characteristic

#16837459

C₂²

yes

—

#16837155

—

#16837153

—

#16837142

—

C₂×#131×C₅₉

C₄

yes

—

C₅₉×#532

C₂²

yes

characteristic

C₅₉×#436

—

C₈×D₄×C₅₉

—

C₅₉×#423

C₂²

yes

characteristic

C₅₉×#408

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

C₂²⋊(C₅₉⋊C₈)

—

#14273638

—

C₂×#41×C₅₉

C₂×C₄

yes

—

#131×C₅₉

—

C₂×M₁₆×C₅₉

—

C₂×M₁₆×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₅₉

Q₈

yes

characteristic

#127×C₅₉

Q₈

yes

characteristic

#124×C₅₉

D₄

yes

—

#121×C₅₉

—

C₄×D₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

C₄⋊C₄

yes

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₄⋊C₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

#117

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#100

yes

—

C₅₉⋊C₈

—

#8630

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#380

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1800

yes

characteristic

100

C₂×C₅₉

#1814

yes

—

101

C₂×#131

Dic₅₉

yes

—

102

#532

C₂×D₅₉

yes

characteristic

103

#436

—

104

C₈×D₄

—

105

#432

—

106

#423

C₂×D₅₉

yes

characteristic

107

#408

Dic₅₉

yes

—

108

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

109

#335

—

118

—

110

#334

—

118

—

111

#329

—

118

—

112

C₅₉

#8630

yes

characteristic, complemented, Sylow

113

C₂×#41

C₂×Dic₅₉

yes

—

114

#131

—

115

C₂×M₁₆

—

116

C₂×M₁₆

C₂×Dic₅₉

yes

—

117

C₂²×C₈

—

118

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

119

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

120

C₄⋊Q₈

Dic₁₁₈

yes

characteristic

121

#127

Dic₁₁₈

yes

characteristic

122

#124

D₂₃₆

yes

—

123

#121

—

124

C₄×D₄

—

125

C₂×C₄²

C₂²×D₅₉

yes

characteristic

126

C₄⋊C₈

—

127

C₄⋊C₈

—

118

—

128

C₂²⋊C₈

—

129

C₂²⋊C₈

—

118

—

130

#97

—

118

—

131

C₄×C₈

C₂²⋊D₅₉

yes

—

132

C₄×C₈

C₂²⋊D₅₉

yes

—

133

#41

—

134

#41

—

135

C₂×Q₈

C₄⋊Dic₅₉

yes

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

C₄⋊Dic₅₉

yes

—

139

C₂²×C₄

—

140

C₂²×C₄

#11745685

yes

—

141

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

236

—

149

C₂×C₈

C₂²⋊Dic₅₉

yes

—

150

C₂×C₈

C₂²⋊Dic₅₉

yes

characteristic

151

C₂×C₈

C₂²⋊Dic₅₉

yes

characteristic

152

C₄⋊C₄

—

153

C₂²⋊C₄

—

154

C₄²

C₂×Dic₁₁₈

yes

characteristic

155

C₄²

C₂×D₂₃₆

yes

characteristic

156

C₄²

C₂×C₂²⋊D₅₉

yes

—

157

C₂³

—

158

C₂³

#14273686

yes

characteristic

159

Q₈

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

170

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

171

C₂×C₄

#14273701

yes

—

172

C₂×C₄

#14273706

yes

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

472

—

177

C₈

#14273644

yes

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

#16837262

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

C₂×#14273644

yes

characteristic

188

C₄

#16837386

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#21424088

yes

characteristic

192

C₂

#21424150

yes

—

	subgroup	G/H
center	C₂²	#16837262
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2931	C₂
socle	C₂²×C₅₉	#380
prime radical	C₅₉×#532	C₂²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#2931	D₅₉
hypocenter	C₅₉	#8630
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

subgroup

G/H

center

C₂²

#16837262

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#2931

C₂

socle

C₂²×C₅₉

#380

prime radical

C₅₉×#532

C₂²

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

N	H ≅ G/N	type
1. #21422859	C₂	semidirect
3. #21422007	C₂	semidirect
112. C₅₉	#8630	semidirect

H ≅ G/N

type

1. #21422859

C₂

semidirect

3. #21422007

C₂

semidirect

112. C₅₉

#8630

semidirect

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

subgroup	index
#21422859	2
#21422731	2
#21422007	2
C₅₉×#2931	2
#8630	59

subgroup

index

C₅₉×#2931

Subgroups — #28481681

Conjugacy classes of subgroups

Complements

Maximal subgroups