Group #28507174 — subgroups

Subgroups — #28507174

Order 15104.

462 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 143 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	#21425852	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837218	C₂	1	4	yes	maximal
3	C₄×#14273663	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	C₂²	1	2	yes	—
6	C₂×#14273663	C₂²	1	2	yes	—
7	C₄×#11745664	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
9	#16837218	—	2	8	no	—
10	#16837167	—	2	1	no	—
11	C₂²×Q₁₆×C₅₉	—	2	1	no	—
12	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
14	C₅₉×#494	—	2	2	no	—
15	C₅₉×#494	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745664	—	2	2	no	—
18	#14273667	—	2	4	no	—
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	#14273663	—	2	2	no	—
21	#14273663	D₄	1	4	yes	—
22	#14273638	—	4	1	no	—
23	C₄×C₅₉⋊C₈	—	4	1	no	—
24	C₂²×Q₈×C₅₉	—	2	2	no	—
25	C₂×Q₁₆×C₅₉	—	2	4	no	—
26	C₂×Q₁₆×C₅₉	—	2	4	no	—
27	C₂×Q₁₆×C₅₉	—	2	2	no	—
28	C₂×Q₁₆×C₅₉	—	2	4	no	—
29	C₂×Q₁₆×C₅₉	—	4	4	no	—
30	C₂×Q₁₆×C₅₉	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₅₉	—	2	4	no	—
34	C₄⋊Q₈×C₅₉	—	4	1	no	—
35	C₄⋊Q₈×C₅₉	—	4	1	no	—
36	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
37	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₉	D₄	1	2	yes	—
40	C₂×Q₈×C₅₉	—	2	4	no	—
41	C₂×Q₈×C₅₉	—	4	2	no	—
42	C₂×Q₈×C₅₉	—	4	4	no	—
43	C₂×Q₈×C₅₉	—	4	4	no	—
44	C₂²×C₄×C₅₉	—	4	2	no	—
45	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
46	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	Q₁₆×C₅₉	—	2	8	no	—
48	Q₁₆×C₅₉	—	4	8	no	—
49	Q₁₆×C₅₉	—	4	2	no	—
50	Q₁₆×C₅₉	—	4	2	no	—
51	C₂×C₈×C₅₉	—	2	2	no	—
52	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	#41	1	2	yes	—
57	C₄⋊C₄×C₅₉	—	4	4	no	—
58	C₄⋊C₄×C₅₉	—	4	4	no	—
59	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₅₉	D₈	1	1	yes	characteristic
61	C₄²×C₅₉	D₈	1	1	yes	characteristic
62	#11745664	—	2	4	no	—
63	#11745664	—	4	2	no	—
64	C₂×C₅₉⋊C₈	—	4	2	no	—
65	C₅₉⋊C₁₆	—	4	2	no	—
66	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₅₉	—	4	4	no	—
68	Q₈×C₅₉	—	4	4	no	—
69	Q₈×C₅₉	—	8	4	no	—
70	C₂×C₄×C₅₉	—	2	4	no	—
71	C₂×C₄×C₅₉	—	4	4	no	—
72	C₂×C₄×C₅₉	—	8	4	no	—
73	C₂×C₄×C₅₉	#102	1	2	yes	—
74	C₂×C₄×C₅₉	#102	1	2	yes	—
75	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₉	#121	1	2	yes	—
77	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
79	C₈×C₅₉	—	2	2	no	—
80	C₈×C₅₉	C₄×D₄	1	2	yes	—
81	C₈×C₅₉	#124	1	1	yes	characteristic
82	C₈×C₅₉	#127	1	1	yes	characteristic
83	C₅₉⋊C₈	—	4	4	no	—
84	#13256	—	59	1	no	maximal, Sylow
85	C₂²×C₅₉	—	2	2	no	—
86	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
87	C₂²×C₅₉	#460	1	2	yes	—
88	C₄×C₅₉	—	2	2	no	—
89	C₄×C₅₉	—	8	4	no	—
90	C₄×C₅₉	#362	1	4	yes	complemented
91	C₄×C₅₉	#390	1	1	yes	characteristic
92	C₄×C₅₉	#418	1	1	yes	characteristic
93	C₄×C₅₉	#459	1	2	yes	—
94	C₂×#494	D₅₉	1	1	yes	characteristic
95	#2150	—	59	1	no	—
96	C₂×#362	—	59	4	no	—
97	C₄×#108	—	59	1	no	—
98	C₂×C₅₉	—	2	2	no	—
99	C₂×C₅₉	#1982	1	1	yes	characteristic
100	C₂×C₅₉	C₂×#362	1	2	yes	—
101	C₂²×Q₁₆	—	2	1	no	—
102	C₂²×Q₁₆	C₂×D₅₉	1	2	yes	—
103	C₂×C₄⋊Q₈	—	2	1	no	—
104	C₂×M₃₂	—	59	2	no	—
105	#494	—	2	2	no	—
106	#494	Dic₅₉	1	4	yes	—
107	C₂×#108	—	59	2	no	—
108	C₄×M₁₆	—	118	1	no	—
109	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
110	C₄⋊C₁₆	—	118	2	no	—
111	#362	—	118	8	no	—
112	#341	—	118	1	no	—
113	C₅₉	#13256	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	2	2	no	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	4	no	—
117	C₂×Q₁₆	—	2	4	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	4	4	no	—
120	C₂×Q₁₆	C₂×Dic₅₉	1	4	yes	—
121	C₂×M₁₆	—	118	2	no	—
122	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	1	no	—
126	C₄⋊Q₈	—	4	1	no	—
127	C₂×C₄⋊C₄	—	4	2	no	—
128	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
129	M₃₂	—	118	4	no	—
130	C₂×C₁₆	—	118	2	no	—
131	#108	—	59	4	no	—
132	#108	—	118	2	no	—
133	#97	—	236	1	no	—
134	C₄×C₈	—	236	1	no	—
135	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
136	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	4	2	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	#11745670	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	4	8	no	—
146	Q₁₆	—	4	2	no	—
147	Q₁₆	—	4	2	no	—
148	M₁₆	—	118	4	no	—
149	M₁₆	—	236	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	236	2	no	—
152	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
153	C₂×C₈	D₄×D₅₉	1	2	yes	—
154	C₂×C₈	#11745687	1	2	yes	—
155	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄²	#11745669	1	1	yes	characteristic
160	C₄²	#11745669	1	1	yes	characteristic
161	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
162	C₁₆	—	236	2	no	—
163	C₂³	C₂×#11745670	1	1	yes	characteristic
164	Q₈	—	4	4	no	—
165	Q₈	—	4	4	no	—
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	#14273648	1	2	yes	—
171	C₂×C₄	#14273648	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
173	C₂×C₄	#14273698	1	2	yes	—
174	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
175	C₂×C₄	#14273753	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	236	4	no	—
178	C₈	D₄×Dic₅₉	1	2	yes	—
179	C₈	#14273714	1	1	yes	characteristic
180	C₈	#14273723	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₂×#14273648	1	1	yes	characteristic
183	C₂²	#16837599	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	#16837218	1	4	yes	complemented
187	C₄	#16837305	1	1	yes	characteristic
188	C₄	#16837407	1	1	yes	characteristic
189	C₄	#16837594	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	#21425024	1	1	yes	characteristic
192	C₂	C₂×#16837218	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425852

C₂

yes

maximal, characteristic

C₂×#16837218

C₂

yes

maximal

C₄×#14273663

C₂

yes

maximal, characteristic

C₂×C₅₉×#494

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₂×#14273663

C₂²

yes

—

C₄×#11745664

—

C₄⋊(C₅₉⋊C₁₆)

—

#16837218

—

#16837167

—

C₂²×Q₁₆×C₅₉

—

C₂²×Q₁₆×C₅₉

C₂²

yes

—

C₂×C₄⋊Q₈×C₅₉

—

C₅₉×#494

—

C₅₉×#494

C₄

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×#11745664

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

#14273663

—

#14273663

D₄

yes

—

#14273638

—

C₄×C₅₉⋊C₈

—

C₂²×Q₈×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

SD₁₆

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×SD₁₆

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

#102

yes

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₄×D₄

yes

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₅₉⋊C₈

—

#13256

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#102

yes

characteristic

C₂²×C₅₉

#460

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×#494

D₅₉

yes

—

—

—

C₂×C₅₉

—

C₂×C₅₉

#1982

yes

characteristic

100

C₂×C₅₉

C₂×#362

yes

—

101

C₂²×Q₁₆

—

102

C₂²×Q₁₆

C₂×D₅₉

yes

—

103

C₂×C₄⋊Q₈

—

104

C₂×M₃₂

—

105

#494

—

106

#494

Dic₅₉

yes

—

107

C₂×#108

—

108

C₄×M₁₆

—

118

—

109

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

110

C₄⋊C₁₆

—

118

—

111

#362

—

118

—

112

#341

—

118

—

113

C₅₉

#13256

yes

characteristic, complemented, Sylow

114

C₂²×Q₈

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

C₂×Dic₅₉

yes

—

121

C₂×M₁₆

—

118

—

122

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

123

C₂²×C₈

C₂²×D₅₉

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

129

M₃₂

—

118

—

130

C₂×C₁₆

—

118

—

131

#108

—

132

#108

—

118

—

133

#97

—

236

—

134

C₄×C₈

—

236

—

135

C₄×C₈

C₂×Dic₅₉

yes

—

136

C₄×C₈

C₂²⋊D₅₉

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂²×C₄

—

142

C₂²×C₄

#11745670

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

144

Q₁₆

—

145

Q₁₆

—

146

Q₁₆

—

147

Q₁₆

—

148

M₁₆

—

118

—

149

M₁₆

—

236

—

150

C₂×C₈

—

151

C₂×C₈

—

236

—

152

C₂×C₈

C₂²⋊Dic₅₉

yes

—

153

C₂×C₈

D₄×D₅₉

yes

—

154

C₂×C₈

#11745687

yes

—

155

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

156

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄²

#11745669

yes

characteristic

160

C₄²

#11745669

yes

characteristic

161

C₄²

C₂²⋊Dic₅₉

yes

—

162

C₁₆

—

236

—

163

C₂³

C₂×#11745670

yes

characteristic

164

Q₈

—

165

Q₈

—

166

Q₈

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#14273648

yes

—

171

C₂×C₄

#14273648

yes

—

172

C₂×C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

173

C₂×C₄

#14273698

yes

—

174

C₂×C₄

C₂×#11745669

yes

characteristic

175

C₂×C₄

#14273753

yes

—

176

C₈

—

177

C₈

—

236

—

178

C₈

D₄×Dic₅₉

yes

—

179

C₈

#14273714

yes

characteristic

180

C₈

#14273723

yes

characteristic

181

C₂²

—

182

C₂²

C₂×#14273648

yes

characteristic

183

C₂²

#16837599

yes

—

184

C₄

—

185

C₄

—

186

C₄

#16837218

yes

complemented

187

C₄

#16837305

yes

characteristic

188

C₄

#16837407

yes

characteristic

189

C₄

#16837594

yes

—

190

C₂

—

191

C₂

#21425024

yes

characteristic

192

C₂

C₂×#16837218

yes

—

	subgroup	G/H
center	C₂²	C₂×#14273648
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#494	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#494	D₅₉
hypocenter	C₅₉	#13256
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂²

C₂×#14273648

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#494

C₂

socle

C₂²×C₅₉

C₂×#102

prime radical

C₂³×C₅₉

C₂×SD₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
90. C₄×C₅₉	#362	semidirect
113. C₅₉	#13256	semidirect
186. C₄	#16837218	semidirect

H ≅ G/N

type

90. C₄×C₅₉

#362

semidirect

113. C₅₉

#13256

semidirect

186. C₄

#16837218

semidirect

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

subgroup	index
#21425852	2
C₂×#16837218	2
C₄×#14273663	2
C₂×C₅₉×#494	2
#13256	59

subgroup

index

C₂×C₅₉×#494

Subgroups — #28507174

Conjugacy classes of subgroups

Complements

Maximal subgroups