Group #28483192 — subgroups

Subgroups — #28483192

Order 15104.

412 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 107 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	#21424925	C₂	1	1	yes	maximal, characteristic
2	#21422884	C₂	1	2	yes	maximal, complemented
3	#21422758	C₂	1	1	yes	maximal, characteristic
4	#21421928	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2979	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	#16837426	C₂²	1	1	yes	characteristic
8	C₂×#14273654	—	2	2	no	—
9	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837154	—	2	2	no	—
11	#16837152	—	2	2	no	—
12	#16837151	C₄	1	2	yes	complemented
13	#16837146	C₄	1	1	yes	characteristic, complemented
14	#16837146	C₄	1	1	yes	characteristic, complemented
15	C₅₉×#532	C₂²	1	1	yes	characteristic
16	C₅₉×#436	—	2	2	no	—
17	C₈×D₄×C₅₉	—	2	2	no	—
18	C₅₉×#431	C₂²	1	2	yes	—
19	C₅₉×#408	—	2	2	no	—
20	C₂×#11745664	—	2	1	no	—
21	C₂×C₄×Dic₅₉	—	2	1	no	—
22	C₂×C₄×Dic₅₉	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
25	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
26	#14273654	—	4	4	no	—
27	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
28	C₄×C₅₉⋊C₈	—	2	2	no	—
29	C₂×#41×C₅₉	D₄	1	2	yes	—
30	C₂×M₁₆×C₅₉	—	2	2	no	—
31	C₂²×C₈×C₅₉	—	2	2	no	—
32	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	#127×C₅₉	C₂×C₄	1	1	yes	characteristic
34	#124×C₅₉	Q₈	1	2	yes	—
35	#121×C₅₉	—	2	2	no	—
36	C₄×D₄×C₅₉	—	2	2	no	—
37	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
39	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
40	C₂²⋊C₈×C₅₉	—	2	4	no	—
41	C₂²⋊C₈×C₅₉	—	2	2	no	—
42	C₄×C₈×C₅₉	—	2	2	no	—
43	#41×C₅₉	—	4	2	no	—
44	#41×C₅₉	—	4	2	no	—
45	C₂×Q₈×C₅₉	C₄⋊C₄	1	2	yes	—
46	C₂×D₄×C₅₉	—	2	2	no	—
47	C₂×D₄×C₅₉	—	2	2	no	—
48	C₂×D₄×C₅₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₉	—	2	2	no	—
50	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	#41	1	2	yes	—
52	C₂²×Dic₅₉	—	4	1	no	—
53	M₁₆×C₅₉	—	8	2	no	—
54	C₂×C₈×C₅₉	—	2	2	no	—
55	C₂×C₈×C₅₉	—	2	2	no	—
56	C₂×C₈×C₅₉	—	8	2	no	—
57	C₄⋊C₄×C₅₉	—	2	2	no	—
58	C₂²⋊C₄×C₅₉	—	2	4	no	—
59	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₅₉	C₂×D₄	1	2	yes	—
61	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
62	C₄×Dic₅₉	—	2	2	no	—
63	C₄×Dic₅₉	—	2	4	no	—
64	C₄×Dic₅₉	—	4	2	no	—
65	C₄×Dic₅₉	—	4	1	no	—
66	#11745664	—	4	2	no	—
67	C₂×C₅₉⋊C₈	—	2	2	no	—
68	C₂×C₅₉⋊C₈	—	2	2	no	—
69	C₂×C₅₉⋊C₈	—	4	1	no	—
70	C₂³×C₅₉	—	2	2	no	—
71	C₂³×C₅₉	#123	1	1	yes	characteristic
72	Q₈×C₅₉	—	4	2	no	—
73	D₄×C₅₉	—	4	2	no	—
74	D₄×C₅₉	—	4	2	no	—
75	D₄×C₅₉	—	4	2	no	—
76	D₄×C₅₉	—	8	2	no	—
77	C₂×C₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	—	2	2	no	—
79	C₂×C₄×C₅₉	—	2	1	no	—
80	C₂×C₄×C₅₉	—	2	1	no	—
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	—	4	2	no	—
83	C₂×C₄×C₅₉	—	4	2	no	—
84	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
88	C₂×Dic₅₉	—	4	2	no	—
89	C₂×Dic₅₉	—	4	4	no	—
90	C₈×C₅₉	—	4	2	no	—
91	C₈×C₅₉	—	4	2	no	—
92	C₅₉⋊C₈	—	4	2	no	—
93	C₅₉⋊C₈	—	4	2	no	—
94	C₅₉⋊C₈	—	8	1	no	—
95	#8934	—	59	1	no	maximal, Sylow
96	C₂²×C₅₉	—	2	1	no	—
97	C₂²×C₅₉	—	2	2	no	—
98	C₂²×C₅₉	—	4	2	no	—
99	C₂²×C₅₉	—	4	4	no	—
100	C₂²×C₅₉	#385	1	1	yes	characteristic
101	C₄×C₅₉	—	2	1	no	—
102	C₄×C₅₉	—	2	2	no	—
103	C₄×C₅₉	—	4	2	no	—
104	C₄×C₅₉	#443	1	2	yes	—
105	C₄×C₅₉	#444	1	2	yes	—
106	C₄×C₅₉	#453	1	1	yes	characteristic
107	C₄×C₅₉	#455	1	1	yes	characteristic
108	Dic₅₉	—	4	4	no	—
109	#2979	D₅₉	1	1	yes	characteristic
110	#1962	—	59	1	no	—
111	#1574	—	59	2	no	—
112	#1543	—	59	1	no	—
113	#1276	—	59	2	no	—
114	C₂×C₅₉	—	4	1	no	—
115	C₂×C₅₉	—	8	2	no	—
116	C₂×C₅₉	#1856	1	1	yes	characteristic
117	C₂×C₅₉	#1866	1	2	yes	—
118	#532	C₂×D₅₉	1	1	yes	characteristic
119	#436	—	2	2	no	—
120	C₈×D₄	—	2	2	no	—
121	#431	—	59	2	no	—
122	#431	C₂×D₅₉	1	2	yes	—
123	#423	—	59	1	no	—
124	C₂×#104	—	118	2	no	—
125	#408	—	2	2	no	—
126	C₄³	—	59	1	no	—
127	#335	—	118	2	no	—
128	#334	—	118	2	no	—
129	#333	—	59	2	no	—
130	#331	—	59	1	no	—
131	#331	—	59	1	no	—
132	C₅₉	#8934	1	1	yes	characteristic, complemented, Sylow
133	C₂×#41	C₂²⋊D₅₉	1	2	yes	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	—	118	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₄⋊Q₈	C₄×D₅₉	1	1	yes	characteristic
138	#127	C₄×D₅₉	1	1	yes	characteristic
139	#124	Dic₁₁₈	1	2	yes	—
140	#121	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₂×C₄²	—	118	2	no	—
143	C₂×C₄²	—	118	1	no	—
144	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
145	C₄⋊C₈	—	59	2	no	—
146	C₄⋊C₈	—	59	2	no	—
147	C₄⋊C₈	—	118	2	no	—
148	C₄⋊C₈	C₂²⋊D₅₉	1	2	yes	—
149	C₄⋊C₈	C₂²⋊D₅₉	1	2	yes	—
150	#104	—	236	4	no	—
151	C₂²⋊C₈	—	2	4	no	—
152	C₂²⋊C₈	—	2	2	no	—
153	C₂²⋊C₈	—	118	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	118	2	no	—
156	#41	—	4	2	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	Dic₅₉⋊C₄	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	Dic₅₉⋊C₄	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	236	1	no	—
164	C₂²×C₄	#11745685	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
166	M₁₆	—	8	2	no	—
167	M₁₆	—	236	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	8	2	no	—
171	C₂×C₈	—	118	2	no	—
172	C₂×C₈	—	118	2	no	—
173	C₂×C₈	—	236	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	118	2	no	—
177	C₄²	—	118	4	no	—
178	C₄²	—	236	2	no	—
179	C₄²	—	236	1	no	—
180	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
181	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
183	C₂³	—	2	2	no	—
184	C₂³	#14273710	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	D₄	—	8	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	236	2	no	—
198	C₂×C₄	—	236	4	no	—
199	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
200	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
201	C₂×C₄	#14273696	1	1	yes	characteristic
202	C₂×C₄	#14273706	1	2	yes	—
203	C₈	—	4	2	no	—
204	C₈	—	4	2	no	—
205	C₈	—	472	1	no	—
206	C₈	—	236	2	no	—
207	C₈	—	236	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	#16837275	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	2	no	—
216	C₄	—	236	4	no	—
217	C₄	#16837517	1	2	yes	—
218	C₄	#16837522	1	2	yes	—
219	C₄	#16837570	1	1	yes	characteristic
220	C₄	#16837580	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	—	8	2	no	—
223	C₂	#21424306	1	1	yes	characteristic
224	C₂	#21424371	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21424925

C₂

yes

maximal, characteristic

#21422884

C₂

yes

maximal, complemented

#21422758

C₂

yes

maximal, characteristic

#21421928

C₂

yes

maximal, complemented

C₅₉×#2979

C₂

yes

maximal, characteristic

#16837455

C₂²

yes

—

#16837426

C₂²

yes

characteristic

C₂×#14273654

—

C₄²×Dic₅₉

C₂²

yes

—

—

C₄

yes

complemented

#16837146

C₄

yes

characteristic, complemented

#16837146

C₄

yes

characteristic, complemented

C₅₉×#532

C₂²

yes

characteristic

C₅₉×#436

—

C₈×D₄×C₅₉

—

C₅₉×#431

C₂²

yes

—

C₅₉×#408

—

C₂×#11745664

—

C₂×C₄×Dic₅₉

—

C₂×C₄×Dic₅₉

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂×C₄

yes

—

C₄⋊(C₅₉⋊C₈)

C₂×C₄

yes

—

#14273654

—

C₂²⋊(C₅₉⋊C₈)

—

C₄×C₅₉⋊C₈

—

C₂×#41×C₅₉

D₄

yes

—

C₂×M₁₆×C₅₉

—

C₂²×C₈×C₅₉

—

C₄⋊Q₈×C₅₉

C₂×C₄

yes

characteristic

#127×C₅₉

C₂×C₄

yes

characteristic

#124×C₅₉

Q₈

yes

—

#121×C₅₉

—

C₄×D₄×C₅₉

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

D₄

yes

—

C₄⋊C₈×C₅₉

D₄

yes

—

C₂²⋊C₈×C₅₉

—

C₂²⋊C₈×C₅₉

—

C₄×C₈×C₅₉

—

#41×C₅₉

—

#41×C₅₉

—

C₂×Q₈×C₅₉

C₄⋊C₄

yes

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

C₄⋊C₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₉

#41

yes

—

C₂²×Dic₅₉

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×Q₈

yes

characteristic

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

#123

yes

characteristic

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×D₄

yes

—

C₂×C₄×C₅₉

#120

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#8934

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

100

C₂²×C₅₉

#385

yes

characteristic

101

C₄×C₅₉

—

102

C₄×C₅₉

—

103

C₄×C₅₉

—

104

C₄×C₅₉

#443

yes

—

105

C₄×C₅₉

#444

yes

—

106

C₄×C₅₉

#453

yes

characteristic

107

C₄×C₅₉

#455

yes

characteristic

108

Dic₅₉

—

109

#2979

D₅₉

yes

characteristic

110

#1962

—

111

#1574

—

112

#1543

—

113

#1276

—

114

C₂×C₅₉

—

115

C₂×C₅₉

—

116

C₂×C₅₉

#1856

yes

characteristic

117

C₂×C₅₉

#1866

yes

—

118

#532

C₂×D₅₉

yes

characteristic

119

#436

—

120

C₈×D₄

—

121

#431

—

122

#431

C₂×D₅₉

yes

—

123

#423

—

124

C₂×#104

—

118

—

125

#408

—

126

C₄³

—

127

#335

—

118

—

128

#334

—

118

—

129

#333

—

130

#331

—

131

#331

—

132

C₅₉

#8934

yes

characteristic, complemented, Sylow

133

C₂×#41

C₂²⋊D₅₉

yes

—

134

C₂×M₁₆

—

135

C₂×M₁₆

—

118

—

136

C₂²×C₈

—

137

C₄⋊Q₈

C₄×D₅₉

yes

characteristic

138

#127

C₄×D₅₉

yes

characteristic

139

#124

Dic₁₁₈

yes

—

140

#121

—

141

C₄×D₄

—

142

C₂×C₄²

—

118

—

143

C₂×C₄²

—

118

—

144

C₂×C₄²

C₂²×D₅₉

yes

characteristic

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

118

—

148

C₄⋊C₈

C₂²⋊D₅₉

yes

—

149

C₄⋊C₈

C₂²⋊D₅₉

yes

—

150

#104

—

236

—

151

C₂²⋊C₈

—

152

C₂²⋊C₈

—

153

C₂²⋊C₈

—

118

—

154

C₄×C₈

—

155

C₄×C₈

—

118

—

156

#41

—

157

#41

—

158

C₂×Q₈

Dic₅₉⋊C₄

yes

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

Dic₅₉⋊C₄

yes

—

162

C₂²×C₄

—

163

C₂²×C₄

—

236

—

164

C₂²×C₄

#11745685

yes

—

165

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

166

M₁₆

—

167

M₁₆

—

236

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

118

—

172

C₂×C₈

—

118

—

173

C₂×C₈

—

236

—

174

C₄⋊C₄

—

175

C₂²⋊C₄

—

176

C₄²

—

118

—

177

C₄²

—

118

—

178

C₄²

—

236

—

179

C₄²

—

236

—

180

C₄²

C₂×Dic₁₁₈

yes

characteristic

181

C₄²

C₂×C₄×D₅₉

yes

characteristic

182

C₄²

C₂×C₂²⋊D₅₉

yes

—

183

C₂³

—

184

C₂³

#14273710

yes

characteristic

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

236

—

198

C₂×C₄

—

236

—

199

C₂×C₄

C₂×Dic₅₉⋊C₄

yes

characteristic

200

C₂×C₄

C₄×C₂²⋊D₅₉

yes

—

201

C₂×C₄

#14273696

yes

characteristic

202

C₂×C₄

#14273706

yes

—

203

C₈

—

204

C₈

—

205

C₈

—

472

—

206

C₈

—

236

—

207

C₈

—

236

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#16837275

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

236

—

217

C₄

#16837517

yes

—

218

C₄

#16837522

yes

—

219

C₄

#16837570

yes

characteristic

220

C₄

#16837580

yes

characteristic

221

C₂

—

222

C₂

—

223

C₂

#21424306

yes

characteristic

224

C₂

#21424371

yes

—

	subgroup	G/H
center	C₂×C₄	#14273696
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2979	C₂
socle	C₂²×C₅₉	#385
prime radical	C₅₉×#532	C₂²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#2979	D₅₉
hypocenter	C₅₉	#8934
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2979

subgroup

G/H

center

C₂×C₄

#14273696

commutator

C₄²×C₅₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₉

Fitting

C₅₉×#2979

C₂

socle

C₂²×C₅₉

#385

prime radical

C₅₉×#532

C₂²

cosocle

C₂×C₄²×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2979

N	H ≅ G/N	type
2. #21422884	C₂	semidirect
4. #21421928	C₂	semidirect
12. #16837151	C₄	semidirect
13. #16837146	C₄	semidirect
14. #16837146	C₄	semidirect
132. C₅₉	#8934	semidirect

H ≅ G/N

type

2. #21422884

C₂

semidirect

4. #21421928

C₂

semidirect

12. #16837151

C₄

semidirect

13. #16837146

C₄

semidirect

14. #16837146

C₄

semidirect

132. C₅₉

#8934

semidirect

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

subgroup	index
#21424925	2
#21422884	2
#21422758	2
#21421928	2
C₅₉×#2979	2
#8934	59

subgroup

index

C₅₉×#2979

Subgroups — #28483192

Conjugacy classes of subgroups

Complements

Maximal subgroups