Group #12018689 — subgroups

Subgroups — #12018689

Order 1280.

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×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#64785	C₂	1	2	yes	maximal
3	#10602854	C₂	1	2	yes	maximal
4	#10600587	C₂	1	1	yes	maximal, characteristic
5	#10598149	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×Q₁₆	—	2	1	no	—
7	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₅×#494	—	2	4	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3951	C₂²	1	1	yes	characteristic
12	#65395	C₂²	1	1	yes	characteristic
13	#65391	—	2	1	no	—
14	C₂×#3944	C₂²	1	2	yes	—
15	#64785	—	2	4	no	—
16	#64784	—	2	4	no	—
17	#64743	—	2	1	no	—
18	#13415	—	5	1	no	maximal, Sylow
19	C₂²×C₅×Q₈	—	2	2	no	—
20	C₂×C₅×Q₁₆	—	2	4	no	—
21	C₂×C₅×Q₁₆	—	2	2	no	—
22	C₂×C₅×Q₁₆	—	2	2	no	—
23	C₂×C₅×Q₁₆	—	2	2	no	—
24	C₂×C₅×Q₁₆	—	2	2	no	—
25	C₂×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₅×Q₁₆	—	4	2	no	—
27	C₂×C₅×Q₁₆	—	4	2	no	—
28	C₂×C₅×Q₁₆	D₄	1	4	yes	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₅×C₄⋊Q₈	—	2	4	no	—
32	C₅×C₄⋊Q₈	—	4	2	no	—
33	C₂×C₅×C₄⋊C₄	—	4	2	no	—
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	#4072	—	2	1	no	—
36	C₂×#681	—	2	1	no	—
37	C₄×C₅×C₈	—	2	2	no	—
38	#3951	—	2	2	no	—
39	#3950	D₄	1	2	yes	—
40	#3949	—	2	1	no	—
41	#3946	—	4	2	no	—
42	#3944	—	2	2	no	—
43	#3944	—	2	2	no	—
44	C₂×#672	—	2	2	no	—
45	C₂×#494	D₅	1	1	yes	characteristic
46	C₂×#362	—	5	2	no	—
47	#2141	—	5	2	no	—
48	#1945	—	5	1	no	—
49	#1376	—	5	1	no	—
50	C₂×C₅×Q₈	—	2	4	no	—
51	C₂×C₅×Q₈	—	4	2	no	—
52	C₂×C₅×Q₈	—	4	2	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂×C₅×Q₈	—	4	2	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	—	4	2	no	—
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₅×Q₁₆	—	2	4	no	—
60	C₅×Q₁₆	—	2	4	no	—
61	C₅×Q₁₆	—	4	8	no	—
62	C₅×Q₁₆	—	4	2	no	—
63	C₅×Q₁₆	—	4	2	no	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	D₈	1	2	yes	—
67	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₅×C₈	#41	1	2	yes	—
70	C₂×C₅×C₈	#41	1	1	yes	characteristic
71	C₅×C₄⋊C₄	—	4	8	no	—
72	C₄²×C₅	—	2	1	no	—
73	C₄²×C₅	SD₁₆	1	2	yes	—
74	C₂²⋊Dic₅	—	4	1	no	—
75	C₄⋊Dic₅	—	2	2	no	—
76	C₄×Dic₅	—	4	1	no	—
77	#681	—	4	1	no	—
78	#681	—	4	1	no	—
79	C₂×#168	—	4	1	no	—
80	#672	—	4	2	no	—
81	C₂²×Q₁₆	—	2	1	no	—
82	C₂²×Q₁₆	C₂×D₅	1	2	yes	—
83	C₂×C₄⋊Q₈	—	2	1	no	—
84	C₂×M₃₂	—	5	2	no	—
85	#494	—	2	4	no	—
86	C₂×#108	—	5	1	no	—
87	#427	—	5	1	no	—
88	#424	—	10	1	no	—
89	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
90	#362	—	10	4	no	—
91	#358	—	10	4	no	—
92	#339	—	10	1	no	—
93	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₅×Q₈	—	4	4	no	—
95	C₅×Q₈	—	4	4	no	—
96	C₅×Q₈	—	8	2	no	—
97	C₅×Q₈	—	8	2	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	4	4	no	—
101	C₂×C₄×C₅	—	8	2	no	—
102	C₂×C₄×C₅	—	8	2	no	—
103	C₂×C₄×C₅	#120	1	1	yes	characteristic
104	C₂×C₄×C₅	#121	1	1	yes	characteristic
105	C₂×C₄×C₅	#124	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×Dic₅	—	4	2	no	—
110	C₅×C₈	—	2	2	no	—
111	C₅×C₈	#121	1	2	yes	—
112	C₅×C₈	#123	1	2	yes	—
113	#168	—	8	1	no	—
114	C₂²×Q₈	—	2	2	no	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	—	2	2	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	4	2	no	—
122	C₂×Q₁₆	—	4	2	no	—
123	C₂×Q₁₆	C₂²⋊D₅	1	4	yes	—
124	C₂×M₁₆	—	10	1	no	—
125	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	4	no	—
128	C₄⋊Q₈	—	4	2	no	—
129	#117	—	10	1	no	—
130	C₂×C₄⋊C₄	—	4	2	no	—
131	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
132	M₃₂	—	10	2	no	—
133	M₃₂	—	10	2	no	—
134	C₂×C₁₆	—	10	2	no	—
135	#108	—	10	2	no	—
136	#107	—	5	2	no	—
137	#106	—	10	1	no	—
138	C₄⋊C₈	—	20	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₂²×C₅	—	2	2	no	—
141	C₂²×C₅	#450	1	1	yes	characteristic
142	C₂²×C₅	#459	1	1	yes	characteristic
143	C₂²×C₅	#490	1	1	yes	characteristic
144	C₄×C₅	—	2	4	no	—
145	C₄×C₅	—	8	4	no	—
146	C₄×C₅	#394	1	1	yes	characteristic
147	C₄×C₅	#447	1	1	yes	characteristic
148	C₄×C₅	#460	1	1	yes	characteristic
149	C₄×C₅	#486	1	1	yes	characteristic
150	Dic₅	—	8	2	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
160	Q₁₆	—	2	4	no	—
161	Q₁₆	—	2	4	no	—
162	Q₁₆	—	4	8	no	—
163	Q₁₆	—	4	2	no	—
164	Q₁₆	—	4	2	no	—
165	M₁₆	—	20	1	no	—
166	M₁₆	—	20	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	20	1	no	—
170	C₂×C₈	#686	1	2	yes	—
171	C₂×C₈	D₄×D₅	1	1	yes	characteristic
172	C₂×C₈	#711	1	1	yes	characteristic
173	C₂×C₈	#711	1	2	yes	—
174	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
175	C₄⋊C₄	—	4	8	no	—
176	C₄⋊C₄	—	10	2	no	—
177	C₂²⋊C₄	—	20	1	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	20	1	no	—
180	C₄²	#687	1	2	yes	—
181	C₁₆	—	20	2	no	—
182	C₂×C₅	—	2	1	no	—
183	C₂×C₅	—	2	1	no	—
184	C₂×C₅	#2041	1	1	yes	characteristic
185	C₂×C₅	#2214	1	2	yes	—
186	C₂³	#4078	1	1	yes	characteristic
187	Q₈	—	4	4	no	—
188	Q₈	—	4	4	no	—
189	Q₈	—	8	2	no	—
190	Q₈	—	8	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	20	2	no	—
197	C₂×C₄	C₂×#686	1	1	yes	characteristic
198	C₂×C₄	#4078	1	1	yes	characteristic
199	C₂×C₄	C₂×#687	1	1	yes	characteristic
200	C₂×C₄	#4082	1	1	yes	characteristic
201	C₂×C₄	#4084	1	1	yes	characteristic
202	C₂×C₄	#4099	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	40	1	no	—
205	C₈	#4081	1	2	yes	—
206	C₈	#4085	1	2	yes	—
207	C₅	#13415	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	2	2	no	—
209	C₂²	#65360	1	1	yes	characteristic
210	C₂²	#65361	1	1	yes	characteristic
211	C₂²	#65507	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	8	4	no	—
214	C₄	—	40	2	no	—
215	C₄	#65359	1	1	yes	characteristic
216	C₄	#65364	1	1	yes	characteristic
217	C₄	#65506	1	1	yes	characteristic
218	C₄	#65511	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#10602110	1	2	yes	—
222	C₂	#10602555	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#494

C₂

yes

maximal, characteristic

C₂×#64785

C₂

yes

maximal

#10602854

C₂

yes

maximal

#10600587

C₂

yes

maximal, characteristic

#10598149

C₂

yes

maximal, characteristic

C₂²×C₅×Q₁₆

—

C₂²×C₅×Q₁₆

C₂²

yes

—

C₂×C₅×C₄⋊Q₈

—

C₅×#494

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3951

C₂²

yes

characteristic

#65395

C₂²

yes

characteristic

#65391

—

C₂×#3944

C₂²

yes

—

—

—

—

—

C₂²×C₅×Q₈

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

D₄

yes

—

C₂²×C₅×C₈

D₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

D₄

yes

characteristic

#4072

—

C₂×#681

—

C₄×C₅×C₈

—

#3951

—

#3950

D₄

yes

—

—

—

—

—

—

D₅

yes

—

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×Q₁₆

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

#41

yes

—

C₂×C₅×C₈

#41

yes

characteristic

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

SD₁₆

yes

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

—

—

—

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂×D₅

yes

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

—

—

—

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

C₂³×C₅

AGL(1,ℤ/8)

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

100

C₂×C₄×C₅

—

101

C₂×C₄×C₅

—

102

C₂×C₄×C₅

—

103

C₂×C₄×C₅

#120

yes

characteristic

104

C₂×C₄×C₅

#121

yes

characteristic

105

C₂×C₄×C₅

#124

yes

characteristic

106

C₂×C₄×C₅

C₂×D₈

yes

characteristic

107

C₂×C₄×C₅

C₂×SD₁₆

yes

characteristic

108

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

characteristic

109

C₂×Dic₅

—

110

C₅×C₈

—

111

C₅×C₈

#121

yes

—

112

C₅×C₈

#123

yes

—

113

#168

—

114

C₂²×Q₈

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

123

C₂×Q₁₆

C₂²⋊D₅

yes

—

124

C₂×M₁₆

—

125

C₂²×C₈

C₂²⋊D₅

yes

characteristic

126

C₂²×C₈

C₂²×D₅

yes

characteristic

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

#117

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄²

C₂²⋊D₅

yes

characteristic

132

M₃₂

—

133

M₃₂

—

134

C₂×C₁₆

—

135

#108

—

136

#107

—

137

#106

—

138

C₄⋊C₈

—

139

C₄×C₈

—

140

C₂²×C₅

—

141

C₂²×C₅

#450

yes

characteristic

142

C₂²×C₅

#459

yes

characteristic

143

C₂²×C₅

#490

yes

characteristic

144

C₄×C₅

—

145

C₄×C₅

—

146

C₄×C₅

#394

yes

characteristic

147

C₄×C₅

#447

yes

characteristic

148

C₄×C₅

#460

yes

characteristic

149

C₄×C₅

#486

yes

characteristic

150

Dic₅

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

160

Q₁₆

—

161

Q₁₆

—

162

Q₁₆

—

163

Q₁₆

—

164

Q₁₆

—

165

M₁₆

—

166

M₁₆

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

#686

yes

—

171

C₂×C₈

D₄×D₅

yes

characteristic

172

C₂×C₈

#711

yes

characteristic

173

C₂×C₈

#711

yes

—

174

C₂×C₈

C₂×C₂²⋊D₅

yes

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₂²⋊C₄

—

178

C₄²

—

179

C₄²

—

180

C₄²

#687

yes

—

181

C₁₆

—

182

C₂×C₅

—

183

C₂×C₅

—

184

C₂×C₅

#2041

yes

characteristic

185

C₂×C₅

#2214

yes

—

186

C₂³

#4078

yes

characteristic

187

Q₈

—

188

Q₈

—

189

Q₈

—

190

Q₈

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

C₂×#686

yes

characteristic

198

C₂×C₄

#4078

yes

characteristic

199

C₂×C₄

C₂×#687

yes

characteristic

200

C₂×C₄

#4082

yes

characteristic

201

C₂×C₄

#4084

yes

characteristic

202

C₂×C₄

#4099

yes

characteristic

203

C₈

—

204

C₈

—

205

C₈

#4081

yes

—

206

C₈

#4085

yes

—

207

C₅

#13415

yes

characteristic, complemented, Sylow

208

C₂²

—

209

C₂²

#65360

yes

characteristic

210

C₂²

#65361

yes

characteristic

211

C₂²

#65507

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

#65359

yes

characteristic

216

C₄

#65364

yes

characteristic

217

C₄

#65506

yes

characteristic

218

C₄

#65511

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#10602110

yes

—

222

C₂

#10602555

yes

characteristic

	subgroup	G/H
center	C₂²	#65361
commutator, cosocle	C₂²×C₅×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#494	C₂
socle	C₂²×C₅	#459
prime radical	C₂³×C₅	AGL(1,ℤ/8)
hypercenter	C₂×#494	D₅
hypocenter	C₅	#13415
LCS	G ▷ C₂²×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂²

#65361

commutator, cosocle

C₂²×C₅×C₈

C₂³

Frattini

C₂²×C₈

C₂²×D₅

Fitting

C₂×C₅×#494

C₂

C₂²×C₅

C₂³×C₅

D₅

C₅

G ▷ C₂²×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

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

N	H ≅ G/N	type
207. C₅	#13415	semidirect

H ≅ G/N

type

207. C₅

#13415

semidirect

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

subgroup	index
C₂×C₅×#494	2
C₂×#64785	2
#10602854	2
#10600587	2
#10598149	2
#13415	5

subgroup

index

C₂×C₅×#494

Subgroups — #12018689

Conjugacy classes of subgroups

Complements

Maximal subgroups