Group #28488630 — subgroups

Subgroups — #28488630

Order 15104.

530 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 151 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	#21434524	C₂	1	1	yes	maximal, characteristic
2	#21422546	C₂	1	4	yes	maximal, complemented
3	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14273749	—	2	2	no	—
6	#16837952	C₂²	1	2	yes	—
7	#16837786	—	2	2	no	—
8	#16837782	C₄	1	1	yes	characteristic
9	#16837779	C₄	1	1	yes	characteristic
10	#16837775	C₄	1	2	yes	—
11	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
12	#16837129	—	2	8	no	—
13	C₈×C₅₉⋊C₈	—	2	2	no	—
14	C₂×#108×C₅₉	—	2	2	no	—
15	C₅₉×#423	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745685	—	2	2	no	—
18	#14273749	—	4	4	no	—
19	#14273749	—	4	4	no	—
20	C₂×Dic₂₃₆	—	2	2	no	—
21	C₂×#11745660	—	2	4	no	—
22	C₂×D₄₇₂	—	2	2	no	—
23	C₂²×C₅₉⋊C₈	—	2	2	no	—
24	#14273727	C₂×C₄	1	2	yes	—
25	#14273724	C₂×C₄	1	2	yes	—
26	#14273715	—	2	2	no	—
27	#14273701	—	2	4	no	—
28	C₄×D₂₃₆	—	2	4	no	—
29	C₄×C₅₉⋊C₈	—	2	2	no	—
30	C₄×C₅₉⋊C₈	D₄	1	4	yes	—
31	C₂×M₁₆×C₅₉	—	2	2	no	—
32	C₂²×C₈×C₅₉	D₄	1	2	yes	—
33	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	#108×C₅₉	—	4	4	no	—
35	C₄⋊C₈×C₅₉	—	2	4	no	—
36	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	D₄	1	2	yes	—
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	#41	1	2	yes	—
41	C₂×C₂²⋊D₅₉	—	4	2	no	—
42	#11745685	—	4	4	no	—
43	#11745685	—	4	4	no	—
44	C₂×D₂₃₆	—	2	2	no	—
45	C₂×D₂₃₆	—	2	4	no	—
46	C₂×C₄×D₅₉	—	4	2	no	—
47	C₂×Dic₁₁₈	—	2	2	no	—
48	M₁₆×C₅₉	—	4	4	no	—
49	C₂×C₈×C₅₉	—	4	2	no	—
50	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₉	D₈	1	4	yes	—
53	C₂×C₈×C₅₉	SD₁₆	1	4	yes	—
54	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₉	C₂×D₄	1	2	yes	—
57	(C₂×D₅₉)⋊C₄	—	4	4	no	—
58	C₄⋊Dic₅₉	—	2	4	no	—
59	C₂×C₅₉⋊C₈	—	2	8	no	—
60	C₂×C₅₉⋊C₈	—	2	4	no	—
61	Dic₂₃₆	—	4	4	no	—
62	D₄₇₂	—	4	4	no	—
63	#11745660	—	4	8	no	—
64	C₂³×C₅₉	#124	1	1	yes	characteristic
65	C₂²×D₅₉	—	4	2	no	—
66	C₂×C₄×C₅₉	—	2	4	no	—
67	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
69	C₂×C₄×C₅₉	#121	1	2	yes	—
70	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
73	C₂²⋊D₅₉	—	8	4	no	—
74	C₂×Dic₅₉	—	4	2	no	—
75	D₂₃₆	—	4	4	no	—
76	D₂₃₆	—	4	8	no	—
77	C₄×D₅₉	—	8	2	no	—
78	C₄×D₅₉	—	8	2	no	—
79	Dic₁₁₈	—	4	4	no	—
80	C₈×C₅₉	—	8	2	no	—
81	C₈×C₅₉	#102	1	4	yes	—
82	C₈×C₅₉	#102	1	4	yes	—
83	C₅₉⋊C₈	—	2	8	no	—
84	#9856	—	59	1	no	maximal, Sylow
85	C₂²×C₅₉	—	2	2	no	—
86	C₂²×C₅₉	#390	1	1	yes	characteristic
87	C₂²×C₅₉	#486	1	2	yes	—
88	C₂×D₅₉	—	8	2	no	—
89	C₂×D₅₉	—	8	4	no	—
90	C₄×C₅₉	—	2	2	no	—
91	C₄×C₅₉	C₂×#102	1	2	yes	—
92	C₄×C₅₉	#443	1	4	yes	—
93	C₄×C₅₉	#492	1	1	yes	characteristic
94	C₄×C₅₉	#493	1	1	yes	characteristic
95	Dic₅₉	—	8	2	no	—
96	#3147	—	59	1	no	—
97	#1848	D₅₉	1	1	yes	characteristic, complemented
98	#1492	—	59	4	no	—
99	C₂×C₈²	—	59	1	no	—
100	C₂×C₅₉	—	2	2	no	—
101	C₂×C₅₉	#1957	1	1	yes	characteristic
102	C₂×C₅₉	#1974	1	2	yes	—
103	D₅₉	—	16	2	no	—
104	C₂×#135	—	118	2	no	—
105	#532	—	59	2	no	—
106	#495	—	118	2	no	—
107	#494	—	59	1	no	—
108	#493	—	59	1	no	—
109	#492	—	59	2	no	—
110	C₂×#108	—	2	2	no	—
111	#423	C₂×D₅₉	1	2	yes	—
112	C₂×C₄×C₈	—	59	2	no	—
113	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
114	D₄⋊C₈	—	118	8	no	—
115	C₈²	—	118	2	no	—
116	C₅₉	#9856	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	—	118	2	no	—
118	#135	—	236	4	no	—
119	#135	—	236	4	no	—
120	C₂×Q₁₆	—	118	2	no	—
121	C₂×SD₁₆	—	118	4	no	—
122	C₂×D₈	—	118	2	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂²×C₈	—	118	2	no	—
125	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
126	C₄⋊Q₈	—	59	2	no	—
127	#127	—	59	2	no	—
128	#124	—	118	2	no	—
129	#121	—	118	4	no	—
130	C₄×D₄	—	118	4	no	—
131	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
132	#108	—	4	4	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄×C₈	—	59	4	no	—
135	C₄×C₈	—	118	2	no	—
136	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
137	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
138	C₄×C₈	D₂₃₆	1	2	yes	—
139	#41	—	236	4	no	—
140	#41	—	236	4	no	—
141	C₂×Q₈	—	118	2	no	—
142	C₂×D₄	—	118	2	no	—
143	C₂×D₄	—	118	4	no	—
144	C₂×D₄	—	236	2	no	—
145	C₂²×C₄	—	236	2	no	—
146	C₂²×C₄	#11745689	1	2	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
148	Q₁₆	—	236	4	no	—
149	SD₁₆	—	236	8	no	—
150	D₈	—	236	4	no	—
151	M₁₆	—	4	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	118	8	no	—
154	C₂×C₈	—	118	4	no	—
155	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
156	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
157	C₂×C₈	#11745669	1	4	yes	—
158	C₂×C₈	#11745671	1	4	yes	—
159	C₄⋊C₄	—	118	4	no	—
160	C₂²⋊C₄	—	236	4	no	—
161	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
162	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
163	C₄²	D₄×D₅₉	1	2	yes	—
164	C₂³	—	236	2	no	—
165	C₂³	#14273716	1	1	yes	characteristic
166	Q₈	—	236	4	no	—
167	D₄	—	472	4	no	—
168	D₄	—	236	4	no	—
169	D₄	—	236	8	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	472	2	no	—
172	C₂×C₄	—	472	2	no	—
173	C₂×C₄	—	236	2	no	—
174	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
175	C₂×C₄	#14273689	1	2	yes	—
176	C₂×C₄	#14273702	1	2	yes	—
177	C₂×C₄	#14273723	1	1	yes	characteristic
178	C₂×C₄	C₂×#11745669	1	2	yes	—
179	C₂×C₄	C₂×#11745671	1	2	yes	—
180	C₈	—	8	2	no	—
181	C₈	—	118	8	no	—
182	C₈	#14273647	1	4	yes	—
183	C₈	#14273647	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	472	2	no	—
186	C₂²	—	472	4	no	—
187	C₂²	#16837303	1	1	yes	characteristic
188	C₂²	#16837750	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	472	2	no	—
191	C₄	C₂×#14273647	1	2	yes	—
192	C₄	#16837518	1	4	yes	—
193	C₄	#16837776	1	1	yes	characteristic
194	C₄	#16837778	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	944	2	no	—
197	C₂	#21424901	1	1	yes	characteristic
198	C₂	#21424982	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21434524

C₂

yes

maximal, characteristic

#21422546

C₂

yes

maximal, complemented

C₂×C₈×C₅₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₅₉×#1848

C₂

yes

maximal, characteristic, complemented

C₂×#14273749

—

#16837952

C₂²

yes

—

#16837786

—

#16837782

C₄

yes

characteristic

#16837779

C₄

yes

characteristic

#16837775

C₄

yes

—

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

#16837129

—

C₈×C₅₉⋊C₈

—

C₂×#108×C₅₉

—

C₅₉×#423

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

—

—

C₂×Dic₂₃₆

—

C₂×#11745660

—

C₂×D₄₇₂

—

C₂²×C₅₉⋊C₈

—

#14273727

C₂×C₄

yes

—

#14273724

C₂×C₄

yes

—

#14273715

—

#14273701

—

C₄×D₂₃₆

—

C₄×C₅₉⋊C₈

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

#108×C₅₉

—

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂×C₂²⋊D₅₉

—

#11745685

—

#11745685

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×C₄×D₅₉

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

C₂²⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

D₄₇₂

—

#11745660

—

C₂³×C₅₉

#124

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#121

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×SD₁₆

yes

—

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

#102

yes

—

C₈×C₅₉

#102

yes

—

C₅₉⋊C₈

—

#9856

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#390

yes

characteristic

C₂²×C₅₉

#486

yes

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×#102

yes

—

C₄×C₅₉

#443

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#3147

—

#1848

D₅₉

yes

characteristic, complemented

#1492

—

C₂×C₈²

—

100

C₂×C₅₉

—

101

C₂×C₅₉

#1957

yes

characteristic

102

C₂×C₅₉

#1974

yes

—

103

D₅₉

—

104

C₂×#135

—

118

—

105

#532

—

106

#495

—

118

—

107

#494

—

108

#493

—

109

#492

—

110

C₂×#108

—

111

#423

C₂×D₅₉

yes

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

114

D₄⋊C₈

—

118

—

115

C₈²

—

118

—

116

C₅₉

#9856

yes

characteristic, complemented, Sylow

117

C₂×#41

—

118

—

118

#135

—

236

—

119

#135

—

236

—

120

C₂×Q₁₆

—

118

—

121

C₂×SD₁₆

—

118

—

122

C₂×D₈

—

118

—

123

C₂×M₁₆

—

124

C₂²×C₈

—

118

—

125

C₂²×C₈

C₂²⋊D₅₉

yes

—

126

C₄⋊Q₈

—

127

#127

—

128

#124

—

118

—

129

#121

—

118

—

130

C₄×D₄

—

118

—

131

C₂×C₄²

C₂²×D₅₉

yes

characteristic

132

#108

—

133

C₄⋊C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

118

—

136

C₄×C₈

C₄×D₅₉

yes

characteristic

137

C₄×C₈

C₄×D₅₉

yes

characteristic

138

C₄×C₈

D₂₃₆

yes

—

139

#41

—

236

—

140

#41

—

236

—

141

C₂×Q₈

—

118

—

142

C₂×D₄

—

118

—

143

C₂×D₄

—

118

—

144

C₂×D₄

—

236

—

145

C₂²×C₄

—

236

—

146

C₂²×C₄

#11745689

yes

—

147

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

148

Q₁₆

—

236

—

149

SD₁₆

—

236

—

150

D₈

—

236

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

118

—

154

C₂×C₈

—

118

—

155

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

156

C₂×C₈

(C₂×D₅₉)⋊C₄

yes

—

157

C₂×C₈

#11745669

yes

—

158

C₂×C₈

#11745671

yes

—

159

C₄⋊C₄

—

118

—

160

C₂²⋊C₄

—

236

—

161

C₄²

C₂×C₄×D₅₉

yes

characteristic

162

C₄²

C₂×D₂₃₆

yes

characteristic

163

C₄²

D₄×D₅₉

yes

—

164

C₂³

—

236

—

165

C₂³

#14273716

yes

characteristic

166

Q₈

—

236

—

167

D₄

—

472

—

168

D₄

—

236

—

169

D₄

—

236

—

170

C₂×C₄

—

171

C₂×C₄

—

472

—

172

C₂×C₄

—

472

—

173

C₂×C₄

—

236

—

174

C₂×C₄

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

175

C₂×C₄

#14273689

yes

—

176

C₂×C₄

#14273702

yes

—

177

C₂×C₄

#14273723

yes

characteristic

178

C₂×C₄

C₂×#11745669

yes

—

179

C₂×C₄

C₂×#11745671

yes

—

180

C₈

—

181

C₈

—

118

—

182

C₈

#14273647

yes

—

183

C₈

#14273647

yes

—

184

C₂²

—

185

C₂²

—

472

—

186

C₂²

—

472

—

187

C₂²

#16837303

yes

characteristic

188

C₂²

#16837750

yes

—

189

C₄

—

190

C₄

—

472

—

191

C₄

C₂×#14273647

yes

—

192

C₄

#16837518

yes

—

193

C₄

#16837776

yes

characteristic

194

C₄

#16837778

yes

characteristic

195

C₂

—

196

C₂

—

944

—

197

C₂

#21424901

yes

characteristic

198

C₂

#21424982

yes

—

	subgroup	G/H
center	C₂×C₄	#14273723
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1848	C₂
socle	C₂²×C₅₉	#390
prime radical	#21434524	C₂
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1848	D₅₉
hypocenter	C₅₉	#9856
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#14273723

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#1848

C₂

C₂²×C₅₉

C₂

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
2. #21422546	C₂	semidirect
3. C₂×C₈×C₅₉⋊C₈	C₂	semidirect
4. C₅₉×#1848	C₂	semidirect
97. #1848	D₅₉	semidirect
116. C₅₉	#9856	semidirect

H ≅ G/N

type

2. #21422546

C₂

semidirect

3. C₂×C₈×C₅₉⋊C₈

C₂

semidirect

4. C₅₉×#1848

C₂

semidirect

97. #1848

D₅₉

semidirect

116. C₅₉

#9856

semidirect

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

subgroup	index
#21434524	2
#21422546	2
C₂×C₈×C₅₉⋊C₈	2
C₅₉×#1848	2
#9856	59

subgroup

index

#21434524

#21422546

C₂×C₈×C₅₉⋊C₈

C₅₉×#1848

#9856

Subgroups — #28488630

Conjugacy classes of subgroups

Complements

Maximal subgroups