Group #28507173 — subgroups

Subgroups — #28507173

Order 15104.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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×#16837782	C₂	1	1	yes	maximal, characteristic
2	#21425852	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837217	C₂	1	4	yes	maximal
4	C₄×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₃₆	—	2	1	no	—
6	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
7	C₂×#14273727	—	2	1	no	—
8	C₂×#14273667	C₂²	1	2	yes	—
9	#16837782	—	2	2	no	—
10	#16837782	C₄	1	4	yes	—
11	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
12	#16837217	—	2	8	no	—
13	C₂×#108×C₅₉	C₂²	1	2	yes	—
14	C₄×M₁₆×C₅₉	—	2	1	no	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#341	—	2	1	no	—
17	C₂²×Dic₁₁₈	—	2	2	no	—
18	C₂×Dic₂₃₆	—	2	4	no	—
19	C₂×Dic₂₃₆	—	2	4	no	—
20	C₂×Dic₂₃₆	—	2	4	no	—
21	C₂×Dic₂₃₆	—	2	2	no	—
22	C₂×Dic₂₃₆	—	4	4	no	—
23	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
24	#14273727	—	2	4	no	—
25	#14273727	—	4	1	no	—
26	#14273727	—	4	1	no	—
27	C₂×C₄⋊Dic₅₉	—	4	2	no	—
28	#14273667	—	2	4	no	—
29	C₂×C₅₉⋊C₁₆	—	2	2	no	—
30	C₂×M₁₆×C₅₉	—	2	2	no	—
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
34	#108×C₅₉	—	2	2	no	—
35	#108×C₅₉	D₄	1	4	yes	—
36	#97×C₅₉	—	4	1	no	—
37	C₄×C₈×C₅₉	—	4	1	no	—
38	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₉	D₄	1	2	yes	—
40	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
41	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅₉	—	4	2	no	—
43	C₂×Dic₁₁₈	—	2	4	no	—
44	C₂×Dic₁₁₈	—	4	4	no	—
45	C₂×Dic₁₁₈	—	4	4	no	—
46	C₂×Dic₁₁₈	—	4	2	no	—
47	M₁₆×C₅₉	—	2	4	no	—
48	M₁₆×C₅₉	—	4	2	no	—
49	C₂×C₈×C₅₉	—	2	2	no	—
50	C₂×C₈×C₅₉	—	4	2	no	—
51	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	#41	1	2	yes	—
56	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₅₉	D₈	1	1	yes	characteristic
58	C₄²×C₅₉	D₈	1	1	yes	characteristic
59	C₄⋊Dic₅₉	—	4	4	no	—
60	C₄⋊Dic₅₉	—	4	4	no	—
61	Dic₂₃₆	—	2	8	no	—
62	Dic₂₃₆	—	4	8	no	—
63	Dic₂₃₆	—	4	2	no	—
64	Dic₂₃₆	—	4	2	no	—
65	C₅₉⋊C₁₆	—	4	2	no	—
66	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₉	—	2	4	no	—
68	C₂×C₄×C₅₉	#102	1	2	yes	—
69	C₂×C₄×C₅₉	#102	1	2	yes	—
70	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#121	1	2	yes	—
72	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×Dic₅₉	—	4	4	no	—
75	C₂×Dic₅₉	—	8	4	no	—
76	Dic₁₁₈	—	4	4	no	—
77	Dic₁₁₈	—	4	4	no	—
78	Dic₁₁₈	—	8	4	no	—
79	C₈×C₅₉	—	2	2	no	—
80	C₈×C₅₉	—	4	4	no	—
81	C₈×C₅₉	C₄×D₄	1	2	yes	—
82	C₈×C₅₉	#124	1	1	yes	characteristic
83	C₈×C₅₉	#127	1	1	yes	characteristic
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	#362	1	4	yes	complemented
90	C₄×C₅₉	#390	1	1	yes	characteristic
91	C₄×C₅₉	#418	1	1	yes	characteristic
92	C₄×C₅₉	#459	1	2	yes	—
93	Dic₅₉	—	8	4	no	—
94	C₂×#494	—	59	1	no	—
95	#2150	—	59	1	no	—
96	C₂×#362	—	59	4	no	—
97	C₄×#108	D₅₉	1	1	yes	characteristic
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	—	59	2	no	—
102	C₂²×Q₁₆	—	118	1	no	—
103	C₂×C₄⋊Q₈	—	118	1	no	—
104	C₂×M₃₂	—	59	2	no	—
105	#494	—	59	4	no	—
106	#494	—	118	2	no	—
107	C₂×#108	C₂×D₅₉	1	2	yes	—
108	C₄×M₁₆	—	2	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	—	2	1	no	—
113	C₅₉	#13256	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	118	2	no	—
115	C₂×Q₁₆	—	59	4	no	—
116	C₂×Q₁₆	—	118	4	no	—
117	C₂×Q₁₆	—	118	4	no	—
118	C₂×Q₁₆	—	118	2	no	—
119	C₂×Q₁₆	—	118	4	no	—
120	C₂×Q₁₆	—	236	4	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
124	C₄⋊Q₈	—	118	4	no	—
125	C₄⋊Q₈	—	236	1	no	—
126	C₄⋊Q₈	—	236	1	no	—
127	C₂×C₄⋊C₄	—	236	2	no	—
128	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
129	M₃₂	—	118	4	no	—
130	C₂×C₁₆	—	118	2	no	—
131	#108	—	2	2	no	—
132	#108	D₂₃₆	1	4	yes	—
133	#97	—	4	1	no	—
134	C₄×C₈	—	4	1	no	—
135	C₄×C₈	C₄×D₅₉	1	2	yes	—
136	C₄×C₈	D₂₃₆	1	2	yes	—
137	C₂×Q₈	—	118	4	no	—
138	C₂×Q₈	—	236	4	no	—
139	C₂×Q₈	—	236	4	no	—
140	C₂×Q₈	—	236	2	no	—
141	C₂²×C₄	—	236	2	no	—
142	C₂²×C₄	#11745671	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
144	Q₁₆	—	118	8	no	—
145	Q₁₆	—	236	8	no	—
146	Q₁₆	—	236	2	no	—
147	Q₁₆	—	236	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
154	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
155	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
156	C₂×C₈	#11745685	1	2	yes	—
157	C₄⋊C₄	—	236	4	no	—
158	C₄⋊C₄	—	236	4	no	—
159	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
160	C₄²	#11745669	1	1	yes	characteristic
161	C₄²	#11745669	1	1	yes	characteristic
162	C₁₆	—	236	2	no	—
163	C₂³	C₂×#11745671	1	1	yes	characteristic
164	Q₈	—	472	4	no	—
165	Q₈	—	236	4	no	—
166	Q₈	—	236	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	472	4	no	—
169	C₂×C₄	—	236	4	no	—
170	C₂×C₄	#14273647	1	2	yes	—
171	C₂×C₄	#14273647	1	2	yes	—
172	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#14273701	1	2	yes	—
174	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
175	C₂×C₄	#14273752	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	4	4	no	—
178	C₈	C₄×D₂₃₆	1	2	yes	—
179	C₈	#14273715	1	1	yes	characteristic
180	C₈	#14273724	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₂×#14273647	1	1	yes	characteristic
183	C₂²	#16837602	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	472	4	no	—
186	C₄	#16837217	1	4	yes	complemented
187	C₄	#16837304	1	1	yes	characteristic
188	C₄	#16837406	1	1	yes	characteristic
189	C₄	#16837597	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	#21425023	1	1	yes	characteristic
192	C₂	C₂×#16837217	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837782

C₂

yes

maximal, characteristic

#21425852

C₂

yes

maximal, characteristic

C₂×#16837217

C₂

yes

maximal

C₄×#108×C₅₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₃₆

—

C₂²×Dic₂₃₆

C₂²

yes

—

C₂×#14273727

—

C₂×#14273667

C₂²

yes

—

#16837782

—

#16837782

C₄

yes

—

C₄⋊(C₅₉⋊C₁₆)

—

#16837217

—

C₂×#108×C₅₉

C₂²

yes

—

C₄×M₁₆×C₅₉

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#341

—

C₂²×Dic₁₁₈

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

C₂×C₄

yes

—

#14273727

—

#14273727

—

#14273727

—

C₂×C₄⋊Dic₅₉

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

#108×C₅₉

—

#108×C₅₉

D₄

yes

—

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

SD₁₆

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

M₁₆×C₅₉

—

C₂×C₈×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

—

C₄²×C₅₉

C₂²⋊C₄

yes

—

C₄²×C₅₉

D₈

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×SD₁₆

yes

characteristic

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

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₄×D₄

yes

—

C₈×C₅₉

yes

C₈×C₅₉

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×#102

yes

characteristic

C₂²×C₅₉

#460

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

#1982

yes

characteristic

100

C₂×C₅₉

C₂×#362

yes

—

101

C₂²×Q₁₆

—

102

C₂²×Q₁₆

—

118

—

103

C₂×C₄⋊Q₈

—

118

—

104

C₂×M₃₂

—

105

#494

—

106

#494

—

118

—

107

C₂×#108

C₂×D₅₉

yes

—

108

C₄×M₁₆

—

109

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

110

C₄⋊C₁₆

—

118

—

111

#362

—

118

—

112

#341

—

113

C₅₉

#13256

yes

characteristic, complemented, Sylow

114

C₂²×Q₈

—

118

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

118

—

117

C₂×Q₁₆

—

118

—

118

C₂×Q₁₆

—

118

—

119

C₂×Q₁₆

—

118

—

120

C₂×Q₁₆

—

236

—

121

C₂×M₁₆

—

122

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

123

C₂²×C₈

C₂²×D₅₉

yes

characteristic

124

C₄⋊Q₈

—

118

—

125

C₄⋊Q₈

—

236

—

126

C₄⋊Q₈

—

236

—

127

C₂×C₄⋊C₄

—

236

—

128

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

129

M₃₂

—

118

—

130

C₂×C₁₆

—

118

—

131

#108

—

132

#108

D₂₃₆

yes

—

133

#97

—

134

C₄×C₈

—

135

C₄×C₈

C₄×D₅₉

yes

—

136

C₄×C₈

D₂₃₆

yes

—

137

C₂×Q₈

—

118

—

138

C₂×Q₈

—

236

—

139

C₂×Q₈

—

236

—

140

C₂×Q₈

—

236

—

141

C₂²×C₄

—

236

—

142

C₂²×C₄

#11745671

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

144

Q₁₆

—

118

—

145

Q₁₆

—

236

—

146

Q₁₆

—

236

—

147

Q₁₆

—

236

—

148

M₁₆

—

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

153

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

154

C₂×C₈

C₂×D₂₃₆

yes

—

155

C₂×C₈

C₂×D₂₃₆

yes

characteristic

156

C₂×C₈

#11745685

yes

—

157

C₄⋊C₄

—

236

—

158

C₄⋊C₄

—

236

—

159

C₄²

(C₂×D₅₉)⋊C₄

yes

—

160

C₄²

#11745669

yes

characteristic

161

C₄²

#11745669

yes

characteristic

162

C₁₆

—

236

—

163

C₂³

C₂×#11745671

yes

characteristic

164

Q₈

—

472

—

165

Q₈

—

236

—

166

Q₈

—

236

—

167

C₂×C₄

—

168

C₂×C₄

—

472

—

169

C₂×C₄

—

236

—

170

C₂×C₄

#14273647

yes

—

171

C₂×C₄

#14273647

yes

—

172

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

173

C₂×C₄

#14273701

yes

—

174

C₂×C₄

C₂×#11745669

yes

characteristic

175

C₂×C₄

#14273752

yes

—

176

C₈

—

177

C₈

—

178

C₈

C₄×D₂₃₆

yes

—

179

C₈

#14273715

yes

characteristic

180

C₈

#14273724

yes

characteristic

181

C₂²

—

182

C₂²

C₂×#14273647

yes

characteristic

183

C₂²

#16837602

yes

—

184

C₄

—

185

C₄

—

472

—

186

C₄

#16837217

yes

complemented

187

C₄

#16837304

yes

characteristic

188

C₄

#16837406

yes

characteristic

189

C₄

#16837597

yes

—

190

C₂

—

191

C₂

#21425023

yes

characteristic

192

C₂

C₂×#16837217

yes

—

	subgroup	G/H
center	C₂²	C₂×#14273647
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×#108×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×#108	D₅₉
hypocenter	C₅₉	#13256
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂²

C₂×#14273647

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₄×#108×C₅₉

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 ◁ C₄×#108

N	H ≅ G/N	type
89. C₄×C₅₉	#362	semidirect
113. C₅₉	#13256	semidirect
186. C₄	#16837217	semidirect

H ≅ G/N

type

89. C₄×C₅₉

#362

semidirect

113. C₅₉

#13256

semidirect

186. C₄

#16837217

semidirect

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

subgroup	index
C₂×#16837782	2
#21425852	2
C₂×#16837217	2
C₄×#108×C₅₉	2
#13256	59

subgroup

index

C₄×#108×C₅₉

Subgroups — #28507173

Conjugacy classes of subgroups

Complements

Maximal subgroups