Group #28506727 — subgroups

Subgroups — #28506727

Order 15104.

318 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16837230	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837177	C₂	1	1	yes	maximal, characteristic
3	#21424827	C₂	1	2	yes	maximal
4	#21422187	C₂	1	2	yes	maximal
5	C₂×C₅₉×#366	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273661	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
9	#16837298	—	2	2	no	—
10	#16837230	C₄	1	4	yes	complemented
11	#16837177	—	2	2	no	—
12	#16837164	—	2	2	no	—
13	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#366	—	2	2	no	—
16	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×Dic₅₉⋊C₄	—	4	2	no	—
19	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	—	2	1	no	—
21	C₂×C₅₉⋊C₁₆	—	2	1	no	—
22	C₂×C₅₉⋊C₁₆	Q₈	1	2	yes	—
23	#14273661	—	2	2	no	—
24	#14273661	C₂×C₄	1	4	yes	—
25	C₈×Dic₅₉	—	2	1	no	—
26	C₈×Dic₅₉	—	2	1	no	—
27	C₅₉⋊#95	—	4	2	no	—
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
33	#107×C₅₉	—	2	2	no	—
34	#107×C₅₉	—	2	2	no	—
35	C₂²×C₄×C₅₉	—	4	2	no	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₉	—	4	2	no	—
38	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
39	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
45	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₅₉	—	4	2	no	—
47	C₄⋊C₄×C₅₉	—	4	2	no	—
48	C₄⋊Dic₅₉	—	2	4	no	—
49	C₄⋊Dic₅₉	—	4	2	no	—
50	Dic₅₉⋊C₄	—	8	4	no	—
51	C₄×Dic₅₉	—	2	1	no	—
52	C₄×Dic₅₉	—	2	1	no	—
53	C₂×C₅₉⋊C₈	—	2	1	no	—
54	C₂×C₅₉⋊C₈	—	2	1	no	—
55	C₂×C₅₉⋊C₈	Q₁₆	1	2	yes	—
56	C₁₆×C₅₉	C₄⋊C₄	1	4	yes	complemented
57	C₅₉⋊C₁₆	—	2	2	no	—
58	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	8	2	no	—
60	C₂×C₄×C₅₉	—	8	2	no	—
61	C₂×C₄×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
68	C₂×Dic₅₉	—	2	1	no	—
69	C₂×Dic₅₉	—	2	1	no	—
70	C₂×Dic₅₉	—	2	2	no	—
71	C₂×Dic₅₉	—	2	2	no	—
72	C₂×Dic₅₉	—	4	4	no	—
73	C₂×Dic₅₉	—	8	2	no	—
74	C₂×Dic₅₉	—	8	2	no	—
75	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₈×C₅₉	#125	1	1	yes	characteristic
79	C₅₉⋊C₈	—	2	2	no	—
80	#13167	—	59	1	no	maximal, Sylow
81	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
82	C₂²×C₅₉	#466	1	1	yes	characteristic
83	C₂²×C₅₉	#500	1	1	yes	characteristic
84	C₂²×C₅₉	#509	1	2	yes	—
85	C₂²×C₅₉	#510	1	2	yes	—
86	C₄×C₅₉	—	16	2	no	—
87	C₄×C₅₉	#389	1	1	yes	characteristic
88	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
89	C₄×C₅₉	#467	1	1	yes	characteristic
90	C₄×C₅₉	#499	1	1	yes	characteristic
91	Dic₅₉	—	4	1	no	—
92	Dic₅₉	—	4	1	no	—
93	Dic₅₉	—	8	4	no	—
94	C₂×#367	—	59	1	no	—
95	C₂×#366	D₅₉	1	1	yes	characteristic
96	C₂×#346	—	59	1	no	—
97	#1942	—	59	2	no	—
98	#1380	—	59	2	no	—
99	C₂×C₅₉	#1942	1	1	yes	characteristic
100	C₂×C₅₉	#2175	1	2	yes	—
101	C₂×C₅₉	#2218	1	1	yes	characteristic
102	C₂×C₅₉	#2219	1	1	yes	characteristic
103	C₂×C₅₉	#2255	1	2	yes	—
104	C₂²×C₁₆	—	59	1	no	—
105	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
106	C₂×#107	—	59	2	no	—
107	C₂×#107	C₂×D₅₉	1	2	yes	—
108	C₂×C₄×C₈	—	59	1	no	—
109	#389	—	118	2	no	—
110	#367	—	59	4	no	—
111	#366	—	2	2	no	—
112	#346	—	118	2	no	—
113	#340	—	118	2	no	—
114	C₅₉	#13167	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	59	1	no	—
116	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	118	2	no	—
119	C₂×C₄⋊C₄	—	236	2	no	—
120	C₂×C₄²	—	59	1	no	—
121	C₂×C₁₆	—	59	2	no	—
122	C₂×C₁₆	—	118	1	no	—
123	C₂×C₁₆	—	118	1	no	—
124	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
125	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
126	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	2	2	no	—
129	#107	—	59	4	no	—
130	#107	—	118	2	no	—
131	C₄×C₈	—	118	1	no	—
132	C₄×C₈	—	118	1	no	—
133	#95	—	236	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	59	1	no	—
136	C₂²×C₄	—	59	1	no	—
137	C₂²×C₄	—	236	2	no	—
138	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
139	C₂×C₈	—	59	2	no	—
140	C₂×C₈	—	118	1	no	—
141	C₂×C₈	—	118	1	no	—
142	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
144	C₂×C₈	#11745687	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
146	C₂×C₈	#11745689	1	1	yes	characteristic
147	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	118	4	no	—
151	C₄⋊C₄	—	472	4	no	—
152	C₄⋊C₄	—	236	2	no	—
153	C₄²	—	118	1	no	—
154	C₄²	—	118	1	no	—
155	C₁₆	—	118	2	no	—
156	C₁₆	Dic₅₉⋊C₄	1	4	yes	complemented
157	C₂³	D₈×D₅₉	1	1	yes	characteristic
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	118	1	no	—
162	C₂×C₄	—	118	1	no	—
163	C₂×C₄	—	118	2	no	—
164	C₂×C₄	—	118	2	no	—
165	C₂×C₄	—	472	2	no	—
166	C₂×C₄	—	472	2	no	—
167	C₂×C₄	—	236	4	no	—
168	C₂×C₄	#14273689	1	1	yes	characteristic
169	C₂×C₄	#14273698	1	1	yes	characteristic
170	C₂×C₄	#14273726	1	1	yes	characteristic
171	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
172	C₂×C₄	#14273746	1	1	yes	characteristic
173	C₂×C₄	#14273750	1	1	yes	characteristic
174	C₈	—	118	2	no	—
175	C₈	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
176	C₈	#14273692	1	1	yes	characteristic
177	C₈	#14273703	1	1	yes	characteristic
178	C₈	#14273718	1	1	yes	characteristic
179	C₂²	#16837484	1	1	yes	characteristic
180	C₂²	#16837635	1	1	yes	characteristic
181	C₂²	#16837806	1	1	yes	characteristic
182	C₂²	#16837836	1	2	yes	—
183	C₂²	#16837844	1	2	yes	—
184	C₄	—	16	2	no	—
185	C₄	—	472	4	no	—
186	C₄	—	236	1	no	—
187	C₄	—	236	1	no	—
188	C₄	#16837298	1	1	yes	characteristic
189	C₄	#16837496	1	1	yes	characteristic
190	C₄	#16837640	1	1	yes	characteristic
191	C₄	#16837803	1	1	yes	characteristic
192	C₂	#21424827	1	1	yes	characteristic
193	C₂	#21425959	1	2	yes	—
194	C₂	#21426196	1	1	yes	characteristic
195	C₂	#21426201	1	1	yes	characteristic
196	C₂	#21426380	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837230

C₂

yes

maximal, characteristic

C₂×#16837177

C₂

yes

maximal, characteristic

#21424827

C₂

yes

maximal

#21422187

C₂

yes

maximal

C₂×C₅₉×#366

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14273661

C₂²

yes

—

C₂×C₈×Dic₅₉

C₂²

yes

characteristic

#16837298

—

#16837230

C₄

yes

complemented

#16837177

—

#16837164

—

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×#107×C₅₉

C₂²

yes

—

C₅₉×#366

—

C₂²×C₅₉⋊C₈

D₄

yes

characteristic

C₂×C₄⋊Dic₅₉

—

C₂×Dic₅₉⋊C₄

—

C₂×C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

Q₈

yes

—

#14273661

—

#14273661

C₂×C₄

yes

—

C₈×Dic₅₉

—

C₈×Dic₅₉

—

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

D₄

yes

—

C₂×C₁₆×C₅₉

Q₈

yes

—

#107×C₅₉

—

#107×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

D₈

yes

characteristic

C₂²×Dic₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

C₂×C₈×C₅₉

yes

C₂×C₈×C₅₉

yes

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

Q₁₆

yes

—

C₁₆×C₅₉

C₄⋊C₄

yes

complemented

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₄×Q₈

yes

characteristic

C₈×C₅₉

C₂²⋊Q₈

yes

characteristic

C₈×C₅₉

#125

yes

characteristic

C₅₉⋊C₈

—

#13167

—

maximal, Sylow

C₂²×C₅₉

C₄×D₈

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

—

C₂²×C₅₉

#510

yes

—

C₄×C₅₉

—

C₄×C₅₉

#389

yes

characteristic

C₄×C₅₉

C₄×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

C₂×#367

—

C₂×#366

D₅₉

yes

—

—

—

C₂×C₅₉

#1942

yes

characteristic

100

C₂×C₅₉

#2175

yes

—

101

C₂×C₅₉

#2218

yes

characteristic

102

C₂×C₅₉

#2219

yes

characteristic

103

C₂×C₅₉

#2255

yes

—

104

C₂²×C₁₆

—

105

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

106

C₂×#107

—

107

C₂×#107

C₂×D₅₉

yes

—

108

C₂×C₄×C₈

—

109

#389

—

118

—

110

#367

—

111

#366

—

112

#346

—

118

—

113

#340

—

118

—

114

C₅₉

#13167

yes

characteristic, complemented, Sylow

115

C₂²×C₈

—

116

C₂²×C₈

C₂²×D₅₉

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

118

—

119

C₂×C₄⋊C₄

—

236

—

120

C₂×C₄²

—

121

C₂×C₁₆

—

122

C₂×C₁₆

—

118

—

123

C₂×C₁₆

—

118

—

124

C₂×C₁₆

Dic₁₁₈

yes

—

125

C₂×C₁₆

C₄×D₅₉

yes

—

126

C₂×C₁₆

C₂²⋊D₅₉

yes

—

127

#107

—

128

#107

—

129

#107

—

130

#107

—

118

—

131

C₄×C₈

—

118

—

132

C₄×C₈

—

118

—

133

#95

—

236

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

236

—

138

C₂²×C₄

D₄×D₅₉

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

118

—

141

C₂×C₈

—

118

—

142

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

143

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

144

C₂×C₈

#11745687

yes

characteristic

145

C₂×C₈

Q₈×D₅₉

yes

characteristic

146

C₂×C₈

#11745689

yes

characteristic

147

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

118

—

151

C₄⋊C₄

—

472

—

152

C₄⋊C₄

—

236

—

153

C₄²

—

118

—

154

C₄²

—

118

—

155

C₁₆

—

118

—

156

C₁₆

Dic₅₉⋊C₄

yes

complemented

157

C₂³

D₈×D₅₉

yes

characteristic

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

118

—

162

C₂×C₄

—

118

—

163

C₂×C₄

—

118

—

164

C₂×C₄

—

118

—

165

C₂×C₄

—

472

—

166

C₂×C₄

—

472

—

167

C₂×C₄

—

236

—

168

C₂×C₄

#14273689

yes

characteristic

169

C₂×C₄

#14273698

yes

characteristic

170

C₂×C₄

#14273726

yes

characteristic

171

C₂×C₄

Q₁₆×D₅₉

yes

characteristic

172

C₂×C₄

#14273746

yes

characteristic

173

C₂×C₄

#14273750

yes

characteristic

174

C₈

—

118

—

175

C₈

C₂×Dic₅₉⋊C₄

yes

characteristic

176

C₈

#14273692

yes

characteristic

177

C₈

#14273703

yes

characteristic

178

C₈

#14273718

yes

characteristic

179

C₂²

#16837484

yes

characteristic

180

C₂²

#16837635

yes

characteristic

181

C₂²

#16837806

yes

characteristic

182

C₂²

#16837836

yes

—

183

C₂²

#16837844

yes

—

184

C₄

—

185

C₄

—

472

—

186

C₄

—

236

—

187

C₄

—

236

—

188

C₄

#16837298

yes

characteristic

189

C₄

#16837496

yes

characteristic

190

C₄

#16837640

yes

characteristic

191

C₄

#16837803

yes

characteristic

192

C₂

#21424827

yes

characteristic

193

C₂

#21425959

yes

—

194

C₂

#21426196

yes

characteristic

195

C₂

#21426201

yes

characteristic

196

C₂

#21426380

yes

—

	subgroup	G/H
center	C₂³	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#366	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#366	D₅₉
hypocenter	C₅₉	#13167
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₅₉

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#366

C₂

socle, prime radical

C₂³×C₅₉

C₂×D₈

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
10. #16837230	C₄	semidirect
56. C₁₆×C₅₉	C₄⋊C₄	semidirect
114. C₅₉	#13167	semidirect
156. C₁₆	Dic₅₉⋊C₄	semidirect

H ≅ G/N

type

10. #16837230

C₄

semidirect

56. C₁₆×C₅₉

C₄⋊C₄

semidirect

114. C₅₉

#13167

semidirect

156. C₁₆

Dic₅₉⋊C₄

semidirect

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

subgroup	index
C₂×#16837230	2
C₂×#16837177	2
#21424827	2
#21422187	2
C₂×C₅₉×#366	2
#13167	59

subgroup

index

C₂×C₅₉×#366

Subgroups — #28506727

Conjugacy classes of subgroups

Complements

Maximal subgroups