Group #28505048 — subgroups

Subgroups — #28505048

Order 15104.

342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 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	#21425851	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#21422178	C₂	1	1	yes	maximal, characteristic
4	#21422177	C₂	1	2	yes	maximal
5	C₅₉×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₉	C₄	1	1	yes	characteristic
13	Dic₅₉⋊C₁₆	—	2	4	no	—
14	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₉	—	2	2	no	—
19	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
21	#14273667	—	2	2	no	—
22	#14273667	—	2	2	no	—
23	C₂×C₅₉⋊C₁₆	—	2	2	no	—
24	C₄×C₅₉⋊C₈	—	2	2	no	—
25	C₈×Dic₅₉	—	2	2	no	—
26	C₈×Dic₅₉	—	2	2	no	—
27	C₈×Dic₅₉	C₈	1	2	yes	—
28	C₈×Dic₅₉	C₈	1	2	yes	—
29	C₈×Dic₅₉	D₄	1	2	yes	—
30	C₈×Dic₅₉	Q₈	1	2	yes	—
31	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₅₉	—	2	2	no	—
35	M₃₂×C₅₉	—	2	2	no	—
36	C₂×C₁₆×C₅₉	—	2	2	no	—
37	C₄×C₈×C₅₉	—	2	2	no	—
38	C₂²×C₄×C₅₉	—	2	1	no	—
39	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅₉	—	2	1	no	—
41	C₂²×Dic₅₉	M₁₆	1	2	yes	—
42	C₂×C₈×C₅₉	—	2	1	no	—
43	C₂×C₈×C₅₉	—	2	1	no	—
44	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₉	#41	1	2	yes	—
48	C₂×C₈×C₅₉	#41	1	2	yes	—
49	C₄²×C₅₉	—	2	1	no	—
50	C₄²×C₅₉	M₁₆	1	2	yes	—
51	C₄×Dic₅₉	—	2	8	no	—
52	C₄×Dic₅₉	—	2	1	no	—
53	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
54	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
55	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
56	C₄×Dic₅₉	M₁₆	1	2	yes	—
57	C₂×C₅₉⋊C₈	—	2	1	no	—
58	C₂×C₅₉⋊C₈	—	2	1	no	—
59	C₂×C₅₉⋊C₈	—	2	1	no	—
60	C₂×C₅₉⋊C₈	—	2	1	no	—
61	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
63	C₁₆×C₅₉	—	4	2	no	—
64	C₅₉⋊C₁₆	—	4	2	no	—
65	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₉	—	2	4	no	—
67	C₂×C₄×C₅₉	—	2	2	no	—
68	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₅₉	—	2	4	no	—
75	C₂×Dic₅₉	—	2	2	no	—
76	C₂×Dic₅₉	—	2	4	no	—
77	C₂×Dic₅₉	C₄⋊C₈	1	2	yes	—
78	C₂×Dic₅₉	C₄⋊C₈	1	2	yes	—
79	C₈×C₅₉	—	2	2	no	—
80	C₈×C₅₉	#124	1	2	yes	—
81	C₈×C₅₉	#125	1	2	yes	—
82	C₅₉⋊C₈	—	2	2	no	—
83	C₅₉⋊C₈	—	2	2	no	—
84	#12883	—	59	1	no	maximal, Sylow
85	C₂²×C₅₉	—	2	2	no	—
86	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₅₉	#432	1	1	yes	characteristic
88	C₂²×C₅₉	#432	1	1	yes	characteristic
89	C₄×C₅₉	—	2	4	no	—
90	C₄×C₅₉	#382	1	1	yes	characteristic
91	C₄×C₅₉	#423	1	1	yes	characteristic
92	C₄×C₅₉	#431	1	1	yes	characteristic
93	C₄×C₅₉	#431	1	1	yes	characteristic
94	Dic₅₉	—	2	4	no	—
95	Dic₅₉	—	2	2	no	—
96	Dic₅₉	#364	1	4	yes	—
97	#2150	—	59	2	no	—
98	C₄²×C₈	—	59	1	no	—
99	#1376	—	59	2	no	—
100	#1376	—	59	1	no	—
101	#1376	D₅₉	1	1	yes	characteristic
102	C₂×C₅₉	—	2	1	no	—
103	C₂×C₅₉	—	2	1	no	—
104	C₂×C₅₉	#1831	1	1	yes	characteristic
105	C₂×C₅₉	C₂×#364	1	2	yes	—
106	C₂×M₃₂	—	59	2	no	—
107	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
108	C₂×C₄×C₈	—	59	1	no	—
109	C₂×C₄×C₈	—	59	1	no	—
110	C₂×C₄×C₈	—	59	1	no	—
111	C₂×C₄×C₈	—	59	1	no	—
112	C₂×C₄×C₈	—	59	1	no	—
113	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
114	C₄³	—	59	1	no	—
115	C₄⋊C₁₆	—	118	4	no	—
116	C₅₉	#12883	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	59	1	no	—
118	C₂²×C₈	—	59	1	no	—
119	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
121	C₂×C₄²	—	59	1	no	—
122	C₂×C₄²	—	59	1	no	—
123	C₂×C₄²	—	118	2	no	—
124	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	118	2	no	—
128	M₃₂	—	118	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	118	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	59	2	no	—
133	C₄×C₈	—	59	2	no	—
134	C₄×C₈	—	59	2	no	—
135	C₄×C₈	—	59	2	no	—
136	C₄×C₈	—	118	2	no	—
137	C₄×C₈	—	118	2	no	—
138	C₄×C₈	—	118	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	59	2	no	—
141	C₂²×C₄	—	118	1	no	—
142	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	59	2	no	—
146	C₂×C₈	—	59	2	no	—
147	C₂×C₈	—	118	1	no	—
148	C₂×C₈	—	118	1	no	—
149	C₂×C₈	—	118	1	no	—
150	C₂×C₈	—	118	1	no	—
151	C₂×C₈	C₈×D₅₉	1	2	yes	—
152	C₂×C₈	#11745685	1	2	yes	—
153	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
154	C₂×C₈	#11745687	1	2	yes	—
155	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
156	C₄²	—	2	1	no	—
157	C₄²	—	59	2	no	—
158	C₄²	—	59	1	no	—
159	C₄²	—	59	1	no	—
160	C₄²	—	59	2	no	—
161	C₄²	—	118	8	no	—
162	C₄²	—	118	1	no	—
163	C₄²	#11745659	1	2	yes	—
164	C₁₆	—	4	2	no	—
165	C₁₆	—	236	2	no	—
166	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	59	2	no	—
170	C₂×C₄	—	59	2	no	—
171	C₂×C₄	—	118	4	no	—
172	C₂×C₄	—	118	4	no	—
173	C₂×C₄	—	118	2	no	—
174	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
175	C₂×C₄	#14273683	1	1	yes	characteristic
176	C₂×C₄	#14273684	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
178	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
179	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	118	2	no	—
182	C₈	—	118	2	no	—
183	C₈	#14273713	1	2	yes	—
184	C₈	#14273717	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₄⋊C₈×D₅₉	1	1	yes	characteristic
187	C₂²	#16837456	1	1	yes	characteristic
188	C₂²	#16837457	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	59	4	no	—
191	C₄	—	118	4	no	—
192	C₄	—	118	2	no	—
193	C₄	#16837266	1	1	yes	characteristic
194	C₄	#16837424	1	1	yes	characteristic
195	C₄	#16837452	1	1	yes	characteristic
196	C₄	#16837453	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#21424211	1	1	yes	characteristic
200	C₂	#364×D₅₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425851

C₂

yes

maximal

C₄×C₈×Dic₅₉

C₂

yes

maximal, characteristic

#21422178

C₂

yes

maximal, characteristic

#21422177

C₂

yes

maximal

C₅₉×#1376

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₅₉

C₄

yes

characteristic

C₂×C₈×Dic₅₉

C₄

yes

characteristic

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

C₄²×Dic₅₉

C₄

yes

characteristic

Dic₅₉⋊C₁₆

—

C₂×M₃₂×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₅₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₉

C₂×C₄

yes

characteristic

#14273667

—

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₄×C₅₉⋊C₈

—

C₈×Dic₅₉

—

C₈×Dic₅₉

—

C₈×Dic₅₉

C₈

yes

—

C₈×Dic₅₉

C₈

yes

—

C₈×Dic₅₉

D₄

yes

—

C₈×Dic₅₉

Q₈

yes

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

M₃₂×C₅₉

—

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

M₁₆

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

M₁₆

yes

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

C₄⋊C₄

yes

—

C₄×Dic₅₉

C₂×C₈

yes

characteristic

C₄×Dic₅₉

C₂×C₈

yes

characteristic

C₄×Dic₅₉

M₁₆

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₈

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₄⋊C₈

yes

—

C₂×Dic₅₉

C₄⋊C₈

yes

—

C₈×C₅₉

—

C₈×C₅₉

#124

yes

—

C₈×C₅₉

#125

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#12883

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

#364

yes

—

#2150

—

C₄²×C₈

—

#1376

—

100

#1376

—

101

#1376

D₅₉

yes

characteristic

102

C₂×C₅₉

—

103

C₂×C₅₉

—

104

C₂×C₅₉

#1831

yes

characteristic

105

C₂×C₅₉

C₂×#364

yes

—

106

C₂×M₃₂

—

107

C₂×M₃₂

C₂×D₅₉

yes

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

—

111

C₂×C₄×C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

114

C₄³

—

115

C₄⋊C₁₆

—

118

—

116

C₅₉

#12883

yes

characteristic, complemented, Sylow

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₂²×C₈

C₄×D₅₉

yes

characteristic

120

C₂²×C₈

C₂²×D₅₉

yes

characteristic

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

—

118

—

124

C₂×C₄²

C₄×D₅₉

yes

characteristic

125

M₃₂

—

126

M₃₂

—

127

M₃₂

—

118

—

128

M₃₂

—

118

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

118

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

118

—

137

C₄×C₈

—

118

—

138

C₄×C₈

—

118

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

118

—

142

C₂²×C₄

C₂×C₄×D₅₉

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

118

—

148

C₂×C₈

—

118

—

149

C₂×C₈

—

118

—

150

C₂×C₈

—

118

—

151

C₂×C₈

C₈×D₅₉

yes

—

152

C₂×C₈

#11745685

yes

—

153

C₂×C₈

D₄×D₅₉

yes

characteristic

154

C₂×C₈

#11745687

yes

—

155

C₂×C₈

Q₈×D₅₉

yes

characteristic

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

118

—

162

C₄²

—

118

—

163

C₄²

#11745659

yes

—

164

C₁₆

—

165

C₁₆

—

236

—

166

C₂³

M₁₆×D₅₉

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

118

—

172

C₂×C₄

—

118

—

173

C₂×C₄

—

118

—

174

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

175

C₂×C₄

#14273683

yes

characteristic

176

C₂×C₄

#14273684

yes

characteristic

177

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

178

C₂×C₄

M₁₆×D₅₉

yes

characteristic

179

C₂×C₄

C₂×#11745659

yes

characteristic

180

C₈

—

181

C₈

—

118

—

182

C₈

—

118

—

183

C₈

#14273713

yes

—

184

C₈

#14273717

yes

—

185

C₂²

—

186

C₂²

C₄⋊C₈×D₅₉

yes

characteristic

187

C₂²

#16837456

yes

characteristic

188

C₂²

#16837457

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

118

—

192

C₄

—

118

—

193

C₄

#16837266

yes

characteristic

194

C₄

#16837424

yes

characteristic

195

C₄

#16837452

yes

characteristic

196

C₄

#16837453

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#21424211

yes

characteristic

200

C₂

#364×D₅₉

yes

—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1376	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1376	D₅₉
hypocenter	C₅₉	#12883
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₈

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1376

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₈

prime radical

C₂³×C₅₉

C₂×M₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

N	H ≅ G/N	type
116. C₅₉	#12883	semidirect

H ≅ G/N

type

116. C₅₉

#12883

semidirect

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

subgroup	index
#21425851	2
C₄×C₈×Dic₅₉	2
#21422178	2
#21422177	2
C₅₉×#1376	2
#12883	59

subgroup

index

C₄×C₈×Dic₅₉

C₅₉×#1376

Subgroups — #28505048

Conjugacy classes of subgroups

Complements

Maximal subgroups