Group #28488810 — subgroups

Subgroups — #28488810

Order 15104.

430 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 163 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	#21432459	C₂	1	1	yes	maximal, characteristic
2	#21424213	C₂	1	1	yes	maximal, characteristic
3	#21421980	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273719	—	2	1	no	—
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	#16837300	—	2	2	no	—
8	#16837300	C₄	1	4	yes	—
9	#16837272	C₂²	1	2	yes	—
10	#16837268	—	2	1	no	—
11	#16837157	—	2	2	no	—
12	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₉	—	2	1	no	—
14	C₄³×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#336	—	2	1	no	—
16	C₂²×C₅₉⋊C₈	—	2	2	no	—
17	#14273719	—	4	4	no	—
18	C₂×C₄⋊Dic₅₉	—	2	2	no	—
19	C₂×C₄⋊Dic₅₉	—	2	2	no	—
20	C₂×C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
21	C₂×Dic₅₉⋊C₄	—	2	4	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
23	C₅₉⋊#95	—	2	4	no	—
24	C₂²×C₈×C₅₉	—	2	2	no	—
25	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₉	D₄	1	2	yes	—
27	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₉	—	2	2	no	—
31	C₄⋊C₈×C₅₉	Q₈	1	4	yes	—
32	C₄×C₈×C₅₉	—	4	2	no	—
33	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₉	—	2	4	no	—
39	C₂×C₈×C₅₉	—	2	4	no	—
40	C₂×C₈×C₅₉	—	4	2	no	—
41	C₂×C₈×C₅₉	—	4	2	no	—
42	C₄²×C₅₉	—	2	4	no	—
43	C₄²×C₅₉	—	2	2	no	—
44	C₄²×C₅₉	—	4	2	no	—
45	C₄²×C₅₉	SD₁₆	1	2	yes	—
46	C₄²×C₅₉	Q₁₆	1	2	yes	—
47	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
48	C₄²×C₅₉	#41	1	2	yes	—
49	C₄⋊Dic₅₉	—	4	4	no	—
50	C₄⋊Dic₅₉	—	4	4	no	—
51	C₄⋊Dic₅₉	—	4	4	no	—
52	Dic₅₉⋊C₄	—	4	8	no	—
53	C₂×C₅₉⋊C₈	—	4	2	no	—
54	C₂×C₅₉⋊C₈	—	4	4	no	—
55	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	—	2	2	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	—	2	2	no	—
61	C₂×C₄×C₅₉	—	2	2	no	—
62	C₂×C₄×C₅₉	#103	1	4	yes	—
63	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#135	1	4	yes	—
70	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₅₉	—	4	4	no	—
72	C₂×Dic₅₉	—	4	4	no	—
73	C₂×Dic₅₉	—	4	4	no	—
74	C₈×C₅₉	—	4	4	no	—
75	C₅₉⋊C₈	—	8	2	no	—
76	#9897	—	59	1	no	maximal, Sylow
77	C₂²×C₅₉	#391	1	1	yes	characteristic
78	C₂²×C₅₉	#413	1	2	yes	—
79	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
80	C₂²×C₅₉	#416	1	2	yes	—
81	C₂²×C₅₉	#418	1	1	yes	characteristic
82	C₄×C₅₉	—	2	4	no	—
83	C₄×C₅₉	—	2	2	no	—
84	C₄×C₅₉	—	4	2	no	—
85	C₄×C₅₉	#476	1	2	yes	—
86	C₄×C₅₉	#477	1	2	yes	—
87	C₄×C₅₉	#488	1	2	yes	—
88	C₄×C₅₉	#499	1	2	yes	—
89	Dic₅₉	—	8	4	no	—
90	#2859	—	59	1	no	—
91	#1831	—	59	1	no	—
92	C₄×C₄⋊C₈	D₅₉	1	1	yes	characteristic
93	#1296	—	59	4	no	—
94	C₂×C₅₉	#1542	1	2	yes	—
95	C₂×C₅₉	#1553	1	2	yes	—
96	C₂×C₅₉	#1987	1	1	yes	characteristic
97	C₂×C₅₉	#1991	1	1	yes	characteristic
98	C₂×C₅₉	#1998	1	1	yes	characteristic
99	C₂×#125	—	118	1	no	—
100	C₂×C₄⋊C₈	—	59	2	no	—
101	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
102	C₂×C₄×C₈	—	2	1	no	—
103	#389	—	59	4	no	—
104	#389	—	118	2	no	—
105	#384	—	59	2	no	—
106	#382	—	118	1	no	—
107	C₄³	C₂×D₅₉	1	1	yes	characteristic
108	#336	—	2	1	no	—
109	#336	—	118	2	no	—
110	C₅₉	#9897	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	118	2	no	—
113	#125	—	236	4	no	—
114	C₂×C₄⋊C₄	—	59	4	no	—
115	C₂×C₄⋊C₄	—	118	4	no	—
116	C₂×C₄⋊C₄	—	118	2	no	—
117	C₂×C₄⋊C₄	—	118	2	no	—
118	C₂×C₄²	C₄×D₅₉	1	2	yes	—
119	C₂×C₄²	D₂₃₆	1	2	yes	—
120	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	118	4	no	—
125	C₄⋊C₈	Dic₁₁₈	1	4	yes	—
126	C₄×C₈	—	4	2	no	—
127	#95	—	118	4	no	—
128	C₂²×C₄	—	118	4	no	—
129	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
130	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	236	2	no	—
138	C₂×C₈	—	236	4	no	—
139	C₄⋊C₄	—	236	4	no	—
140	C₄⋊C₄	—	236	8	no	—
141	C₄⋊C₄	—	236	4	no	—
142	C₄⋊C₄	—	236	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#11745670	1	2	yes	—
147	C₄²	#11745672	1	2	yes	—
148	C₄²	C₂×Dic₁₁₈	1	2	yes	—
149	C₄²	#11745685	1	2	yes	—
150	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	236	4	no	—
158	C₂×C₄	—	236	4	no	—
159	C₂×C₄	—	236	4	no	—
160	C₂×C₄	#14273650	1	4	yes	—
161	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
162	C₂×C₄	#14273706	1	2	yes	—
163	C₂×C₄	#14273715	1	1	yes	characteristic
164	C₂×C₄	#14273727	1	1	yes	characteristic
165	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
166	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
167	C₂×C₄	#14273749	1	4	yes	—
168	C₂×C₄	#14273752	1	2	yes	—
169	C₈	—	4	4	no	—
170	C₈	—	472	2	no	—
171	C₂²	#16837308	1	1	yes	characteristic
172	C₂²	#16837384	1	2	yes	—
173	C₂²	C₂×#14273650	1	1	yes	characteristic
174	C₂²	#16837399	1	2	yes	—
175	C₂²	#16837406	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	472	4	no	—
180	C₄	#16837692	1	2	yes	—
181	C₄	#16837697	1	2	yes	—
182	C₄	#16837757	1	2	yes	—
183	C₄	#16837804	1	2	yes	—
184	C₂	#21422754	1	2	yes	—
185	C₂	#21422799	1	2	yes	—
186	C₂	#21425044	1	1	yes	characteristic
187	C₂	#21425061	1	1	yes	characteristic
188	C₂	#21425089	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21432459

C₂

yes

maximal, characteristic

#21424213

C₂

yes

maximal, characteristic

#21421980

C₂

yes

maximal

C₄×C₄⋊C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#14273719

—

C₂×C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#16837300

—

#16837300

C₄

yes

—

#16837272

C₂²

yes

—

#16837268

—

#16837157

—

C₂×C₄⋊C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

—

C₄³×C₅₉

C₂²

yes

characteristic

C₅₉×#336

—

C₂²×C₅₉⋊C₈

—

#14273719

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

—

C₂×C₄⋊Dic₅₉

C₂×C₄

yes

—

C₂×Dic₅₉⋊C₄

—

C₄⋊(C₅₉⋊C₈)

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

Q₈

yes

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

SD₁₆

yes

—

C₄²×C₅₉

Q₁₆

yes

—

C₄²×C₅₉

C₂×Q₈

yes

—

C₄²×C₅₉

#41

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#103

yes

—

C₂×C₄×C₅₉

C₄×Q₈

yes

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

#124

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

C₂×#103

yes

characteristic

C₂²×C₅₉

#416

yes

—

C₂²×C₅₉

#418

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#476

yes

—

C₄×C₅₉

#477

yes

—

C₄×C₅₉

#488

yes

—

C₄×C₅₉

#499

yes

—

Dic₅₉

—

#2859

—

#1831

—

C₄×C₄⋊C₈

D₅₉

yes

characteristic

#1296

—

C₂×C₅₉

#1542

yes

—

C₂×C₅₉

#1553

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

118

—

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

C₂×D₅₉

yes

—

102

C₂×C₄×C₈

—

103

#389

—

104

#389

—

118

—

105

#384

—

106

#382

—

118

—

107

C₄³

C₂×D₅₉

yes

characteristic

108

#336

—

109

#336

—

118

—

110

C₅₉

#9897

yes

characteristic, complemented, Sylow

111

C₂²×C₈

—

112

C₂²×C₈

—

118

—

113

#125

—

236

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

118

—

116

C₂×C₄⋊C₄

—

118

—

117

C₂×C₄⋊C₄

—

118

—

118

C₂×C₄²

C₄×D₅₉

yes

—

119

C₂×C₄²

D₂₃₆

yes

—

120

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

121

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

122

C₂×C₄²

C₂²×D₅₉

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

118

—

125

C₄⋊C₈

Dic₁₁₈

yes

—

126

C₄×C₈

—

127

#95

—

118

—

128

C₂²×C₄

—

118

—

129

C₂²×C₄

(C₂×D₅₉)⋊C₄

yes

—

130

C₂²×C₄

(C₂×D₅₉)⋊C₄

yes

—

131

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

132

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

133

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

236

—

138

C₂×C₈

—

236

—

139

C₄⋊C₄

—

236

—

140

C₄⋊C₄

—

236

—

141

C₄⋊C₄

—

236

—

142

C₄⋊C₄

—

236

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#11745670

yes

—

147

C₄²

#11745672

yes

—

148

C₄²

C₂×Dic₁₁₈

yes

—

149

C₄²

#11745685

yes

—

150

C₂³

C₂×(C₂×D₅₉)⋊C₄

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

236

—

158

C₂×C₄

—

236

—

159

C₂×C₄

—

236

—

160

C₂×C₄

#14273650

yes

—

161

C₂×C₄

C₄×Dic₁₁₈

yes

—

162

C₂×C₄

#14273706

yes

—

163

C₂×C₄

#14273715

yes

characteristic

164

C₂×C₄

#14273727

yes

characteristic

165

C₂×C₄

C₂×#11745670

yes

characteristic

166

C₂×C₄

C₂×#11745672

yes

characteristic

167

C₂×C₄

#14273749

yes

—

168

C₂×C₄

#14273752

yes

—

169

C₈

—

170

C₈

—

472

—

171

C₂²

#16837308

yes

characteristic

172

C₂²

#16837384

yes

—

173

C₂²

C₂×#14273650

yes

characteristic

174

C₂²

#16837399

yes

—

175

C₂²

#16837406

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

472

—

180

C₄

#16837692

yes

—

181

C₄

#16837697

yes

—

182

C₄

#16837757

yes

—

183

C₄

#16837804

yes

—

184

C₂

#21422754

yes

—

185

C₂

#21422799

yes

—

186

C₂

#21425044

yes

characteristic

187

C₂

#21425061

yes

characteristic

188

C₂

#21425089

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₅₉)⋊C₄
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×C₄⋊C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#9897
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×(C₂×D₅₉)⋊C₄

commutator

C₂²×C₄×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₄×C₄⋊C₈×C₅₉

C₂

socle, prime radical

C₂³×C₅₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅₉

C₂³

hypercenter

C₄×C₄⋊C₈

D₅₉

hypocenter

C₅₉

#9897

LCS

G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

N	H ≅ G/N	type
110. C₅₉	#9897	semidirect

H ≅ G/N

type

110. C₅₉

#9897

semidirect

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

subgroup	index
#21432459	2
#21424213	2
#21421980	2
C₄×C₄⋊C₈×C₅₉	2
#9897	59

subgroup

index

C₄×C₄⋊C₈×C₅₉

Subgroups — #28488810

Conjugacy classes of subgroups

Complements

Maximal subgroups