Group #28508078 — subgroups

Subgroups — #28508078

Order 15104.

436 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16837218	C₂	1	2	yes	maximal
2	#21425816	C₂	1	2	yes	maximal
3	#21424844	C₂	1	1	yes	maximal, characteristic
4	#21422178	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂×#14273663	C₂²	1	1	yes	characteristic
8	#16837440	C₂²	1	1	yes	characteristic
9	#16837428	—	2	1	no	—
10	#16837218	—	2	4	no	—
11	#16837206	—	2	4	no	—
12	#16837162	—	2	1	no	—
13	C₂²×Q₁₆×C₅₉	—	2	1	no	—
14	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
16	C₅₉×#494	—	2	4	no	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×#11745664	—	2	1	no	—
19	#14273686	—	2	1	no	—
20	#14273667	—	2	2	no	—
21	#14273667	—	2	2	no	—
22	C₂×C₅₉⋊C₁₆	—	2	2	no	—
23	#14273663	—	2	2	no	—
24	#14273661	D₄	1	2	yes	—
25	#14273659	—	2	1	no	—
26	Dic₅₉⋊C₈	—	4	2	no	—
27	C₂²×Q₈×C₅₉	—	2	2	no	—
28	C₂×Q₁₆×C₅₉	—	2	4	no	—
29	C₂×Q₁₆×C₅₉	—	2	2	no	—
30	C₂×Q₁₆×C₅₉	—	2	2	no	—
31	C₂×Q₁₆×C₅₉	—	2	2	no	—
32	C₂×Q₁₆×C₅₉	—	2	2	no	—
33	C₂×Q₁₆×C₅₉	—	2	2	no	—
34	C₂×Q₁₆×C₅₉	—	4	2	no	—
35	C₂×Q₁₆×C₅₉	—	4	2	no	—
36	C₂×Q₁₆×C₅₉	D₄	1	4	yes	—
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₅₉	—	2	4	no	—
40	C₄⋊Q₈×C₅₉	—	4	2	no	—
41	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
42	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₉	—	2	2	no	—
44	C₂×Q₈×C₅₉	—	2	4	no	—
45	C₂×Q₈×C₅₉	—	4	2	no	—
46	C₂×Q₈×C₅₉	—	4	2	no	—
47	C₂×Q₈×C₅₉	—	4	2	no	—
48	C₂×Q₈×C₅₉	—	4	2	no	—
49	C₂×Q₈×C₅₉	—	4	2	no	—
50	C₂²×C₄×C₅₉	—	2	1	no	—
51	C₂²×C₄×C₅₉	—	4	2	no	—
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₅₉	—	2	4	no	—
54	Q₁₆×C₅₉	—	2	4	no	—
55	Q₁₆×C₅₉	—	4	8	no	—
56	Q₁₆×C₅₉	—	4	2	no	—
57	Q₁₆×C₅₉	—	4	2	no	—
58	C₂×C₈×C₅₉	—	2	1	no	—
59	C₂×C₈×C₅₉	—	2	1	no	—
60	C₂×C₈×C₅₉	D₈	1	2	yes	—
61	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₉	#41	1	2	yes	—
64	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
65	C₄⋊C₄×C₅₉	—	4	8	no	—
66	C₄²×C₅₉	—	2	1	no	—
67	C₄²×C₅₉	SD₁₆	1	2	yes	—
68	C₂²⋊Dic₅₉	—	4	1	no	—
69	C₄⋊Dic₅₉	—	2	2	no	—
70	C₄×Dic₅₉	—	4	1	no	—
71	#11745664	—	4	1	no	—
72	#11745664	—	4	1	no	—
73	C₂×C₅₉⋊C₈	—	4	1	no	—
74	C₅₉⋊C₁₆	—	4	2	no	—
75	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Q₈×C₅₉	—	4	4	no	—
77	Q₈×C₅₉	—	4	4	no	—
78	Q₈×C₅₉	—	8	2	no	—
79	Q₈×C₅₉	—	8	2	no	—
80	C₂×C₄×C₅₉	—	2	4	no	—
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	—	4	4	no	—
83	C₂×C₄×C₅₉	—	8	2	no	—
84	C₂×C₄×C₅₉	—	8	2	no	—
85	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
86	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
87	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
88	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×Dic₅₉	—	4	2	no	—
92	C₈×C₅₉	—	2	2	no	—
93	C₈×C₅₉	#121	1	2	yes	—
94	C₈×C₅₉	#123	1	2	yes	—
95	C₅₉⋊C₈	—	8	1	no	—
96	#13415	—	59	1	no	maximal, Sylow
97	C₂²×C₅₉	—	2	2	no	—
98	C₂²×C₅₉	#450	1	1	yes	characteristic
99	C₂²×C₅₉	#459	1	1	yes	characteristic
100	C₂²×C₅₉	#490	1	1	yes	characteristic
101	C₄×C₅₉	—	2	4	no	—
102	C₄×C₅₉	—	8	4	no	—
103	C₄×C₅₉	#394	1	1	yes	characteristic
104	C₄×C₅₉	#447	1	1	yes	characteristic
105	C₄×C₅₉	#460	1	1	yes	characteristic
106	C₄×C₅₉	#486	1	1	yes	characteristic
107	Dic₅₉	—	8	2	no	—
108	C₂×#494	D₅₉	1	1	yes	characteristic
109	C₂×#362	—	59	2	no	—
110	#2141	—	59	2	no	—
111	#1945	—	59	1	no	—
112	#1376	—	59	1	no	—
113	C₂×C₅₉	—	2	1	no	—
114	C₂×C₅₉	—	2	1	no	—
115	C₂×C₅₉	#2041	1	1	yes	characteristic
116	C₂×C₅₉	#2214	1	2	yes	—
117	C₂²×Q₁₆	—	2	1	no	—
118	C₂²×Q₁₆	C₂×D₅₉	1	2	yes	—
119	C₂×C₄⋊Q₈	—	2	1	no	—
120	C₂×M₃₂	—	59	2	no	—
121	#494	—	2	4	no	—
122	C₂×#108	—	59	1	no	—
123	#427	—	59	1	no	—
124	#424	—	118	1	no	—
125	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
126	#362	—	118	4	no	—
127	#358	—	118	4	no	—
128	#339	—	118	1	no	—
129	C₅₉	#13415	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₈	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×Q₁₆	—	4	2	no	—
139	C₂×Q₁₆	C₂²⋊D₅₉	1	4	yes	—
140	C₂×M₁₆	—	118	1	no	—
141	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	—	4	2	no	—
145	#117	—	118	1	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
148	M₃₂	—	118	2	no	—
149	M₃₂	—	118	2	no	—
150	C₂×C₁₆	—	118	2	no	—
151	#108	—	118	2	no	—
152	#107	—	59	2	no	—
153	#106	—	118	1	no	—
154	C₄⋊C₈	—	236	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
165	Q₁₆	—	2	4	no	—
166	Q₁₆	—	2	4	no	—
167	Q₁₆	—	4	8	no	—
168	Q₁₆	—	4	2	no	—
169	Q₁₆	—	4	2	no	—
170	M₁₆	—	236	1	no	—
171	M₁₆	—	236	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	236	1	no	—
175	C₂×C₈	#11745669	1	2	yes	—
176	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
177	C₂×C₈	#11745687	1	2	yes	—
178	C₂×C₈	#11745687	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	118	2	no	—
182	C₂²⋊C₄	—	236	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	236	1	no	—
185	C₄²	#11745670	1	2	yes	—
186	C₁₆	—	236	2	no	—
187	C₂³	#14273753	1	1	yes	characteristic
188	Q₈	—	4	4	no	—
189	Q₈	—	4	4	no	—
190	Q₈	—	8	2	no	—
191	Q₈	—	8	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	236	2	no	—
198	C₂×C₄	#14273696	1	1	yes	characteristic
199	C₂×C₄	#14273698	1	1	yes	characteristic
200	C₂×C₄	#14273714	1	1	yes	characteristic
201	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
202	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
203	C₂×C₄	#14273753	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	472	1	no	—
206	C₈	#14273699	1	2	yes	—
207	C₈	#14273708	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	#16837553	1	1	yes	characteristic
210	C₂²	#16837594	1	1	yes	characteristic
211	C₂²	#16837766	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	8	4	no	—
214	C₄	—	472	2	no	—
215	C₄	#16837316	1	1	yes	characteristic
216	C₄	#16837536	1	1	yes	characteristic
217	C₄	#16837599	1	1	yes	characteristic
218	C₄	#16837748	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#21425327	1	1	yes	characteristic
222	C₂	#21426174	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837218

C₂

yes

maximal

#21425816

C₂

yes

maximal

#21424844

C₂

yes

maximal, characteristic

#21422178

C₂

yes

maximal, characteristic

C₂×C₅₉×#494

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

C₂×#14273663

C₂²

yes

characteristic

#16837440

C₂²

yes

—

—

—

—

C₂²×Q₁₆×C₅₉

—

C₂²×Q₁₆×C₅₉

C₂²

yes

—

C₂×C₄⋊Q₈×C₅₉

—

C₅₉×#494

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

—

—

—

C₂×C₅₉⋊C₁₆

—

#14273663

—

#14273661

D₄

yes

—

#14273659

—

Dic₅₉⋊C₈

—

C₂²×Q₈×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊Q₈×C₅₉

—

C₄⋊Q₈×C₅₉

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₄²×C₅₉

SD₁₆

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

yes

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#121

yes

—

C₈×C₅₉

#123

yes

—

C₅₉⋊C₈

—

#13415

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

100

C₂²×C₅₉

#490

yes

characteristic

101

C₄×C₅₉

—

102

C₄×C₅₉

—

103

C₄×C₅₉

#394

yes

characteristic

104

C₄×C₅₉

#447

yes

characteristic

105

C₄×C₅₉

#460

yes

characteristic

106

C₄×C₅₉

#486

yes

characteristic

107

Dic₅₉

—

108

C₂×#494

D₅₉

yes

characteristic

109

C₂×#362

—

110

#2141

—

111

#1945

—

112

#1376

—

113

C₂×C₅₉

—

114

C₂×C₅₉

—

115

C₂×C₅₉

#2041

yes

characteristic

116

C₂×C₅₉

#2214

yes

—

117

C₂²×Q₁₆

—

118

C₂²×Q₁₆

C₂×D₅₉

yes

—

119

C₂×C₄⋊Q₈

—

120

C₂×M₃₂

—

121

#494

—

122

C₂×#108

—

123

#427

—

124

#424

—

118

—

125

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

126

#362

—

118

—

127

#358

—

118

—

128

#339

—

118

—

129

C₅₉

#13415

yes

characteristic, complemented, Sylow

130

C₂²×Q₈

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂×Q₁₆

—

135

C₂×Q₁₆

—

136

C₂×Q₁₆

—

137

C₂×Q₁₆

—

138

C₂×Q₁₆

—

139

C₂×Q₁₆

C₂²⋊D₅₉

yes

—

140

C₂×M₁₆

—

118

—

141

C₂²×C₈

C₂²⋊D₅₉

yes

characteristic

142

C₂²×C₈

C₂²×D₅₉

yes

characteristic

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

#117

—

118

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄²

C₂²⋊D₅₉

yes

characteristic

148

M₃₂

—

118

—

149

M₃₂

—

118

—

150

C₂×C₁₆

—

118

—

151

#108

—

118

—

152

#107

—

153

#106

—

118

—

154

C₄⋊C₈

—

236

—

155

C₄×C₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

165

Q₁₆

—

166

Q₁₆

—

167

Q₁₆

—

168

Q₁₆

—

169

Q₁₆

—

170

M₁₆

—

236

—

171

M₁₆

—

236

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

236

—

175

C₂×C₈

#11745669

yes

—

176

C₂×C₈

D₄×D₅₉

yes

characteristic

177

C₂×C₈

#11745687

yes

—

178

C₂×C₈

#11745687

yes

characteristic

179

C₂×C₈

C₂×C₂²⋊D₅₉

yes

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

118

—

182

C₂²⋊C₄

—

236

—

183

C₄²

—

184

C₄²

—

236

—

185

C₄²

#11745670

yes

—

186

C₁₆

—

236

—

187

C₂³

#14273753

yes

characteristic

188

Q₈

—

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

236

—

198

C₂×C₄

#14273696

yes

characteristic

199

C₂×C₄

#14273698

yes

characteristic

200

C₂×C₄

#14273714

yes

characteristic

201

C₂×C₄

C₂×#11745669

yes

characteristic

202

C₂×C₄

C₂×#11745670

yes

characteristic

203

C₂×C₄

#14273753

yes

characteristic

204

C₈

—

205

C₈

—

472

—

206

C₈

#14273699

yes

—

207

C₈

#14273708

yes

—

208

C₂²

—

209

C₂²

#16837553

yes

characteristic

210

C₂²

#16837594

yes

characteristic

211

C₂²

#16837766

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

472

—

215

C₄

#16837316

yes

characteristic

216

C₄

#16837536

yes

characteristic

217

C₄

#16837599

yes

characteristic

218

C₄

#16837748

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#21425327

yes

characteristic

222

C₂

#21426174

yes

—

	subgroup	G/H
center	C₂²	#16837594
commutator, cosocle	C₂²×C₈×C₅₉	C₂³
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#494	C₂
socle	C₂²×C₅₉	#459
prime radical	C₂³×C₅₉	AGL(1,ℤ/8)
hypercenter	C₂×#494	D₅₉
hypocenter	C₅₉	#13415
LCS	G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂²

#16837594

commutator, cosocle

C₂²×C₈×C₅₉

C₂³

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#494

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
129. C₅₉	#13415	semidirect

H ≅ G/N

type

129. C₅₉

#13415

semidirect

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

subgroup	index
C₂×#16837218	2
#21425816	2
#21424844	2
#21422178	2
C₂×C₅₉×#494	2
#13415	59

subgroup

index

C₂×C₅₉×#494

Subgroups — #28508078

Conjugacy classes of subgroups

Complements

Maximal subgroups