Group #12018841 — subgroups

Subgroups — #12018841

Order 1280.

418 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₂²×D₈₀	C₂	1	1	yes	maximal, characteristic
2	#10602194	C₂	1	1	yes	maximal, characteristic, complemented
3	#10601502	C₂	1	2	yes	maximal, complemented
4	C₂×#64715	C₂	1	2	yes	maximal, complemented
5	C₅×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₀	C₂²	1	2	yes	—
7	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	#65420	—	2	2	no	—
10	C₂×#3955	—	2	2	no	—
11	#65390	C₂²	1	1	yes	characteristic
12	C₂×#3944	C₂²	1	1	yes	characteristic
13	C₂×D₈₀	—	2	4	no	—
14	C₂×D₈₀	—	2	2	no	—
15	#64721	—	2	2	no	—
16	#64715	—	2	4	no	—
17	#13439	—	5	1	no	maximal, Sylow
18	C₂²×D₂₀	—	2	2	no	—
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	#4072	D₄	1	1	yes	characteristic
22	C₂×#681	D₄	1	1	yes	characteristic
23	C₂×D₄₀	—	2	2	no	—
24	C₂×D₄₀	—	2	2	no	—
25	C₂×D₄₀	—	2	2	no	—
26	C₂×D₄₀	—	2	2	no	—
27	C₂×D₄₀	—	2	2	no	—
28	C₂×D₄₀	D₄	1	4	yes	—
29	C₂×C₅×C₁₆	—	2	2	no	—
30	C₂×C₅×C₁₆	D₄	1	2	yes	—
31	C₅×#108	—	2	2	no	—
32	C₅×#108	—	2	2	no	—
33	#3955	—	4	4	no	—
34	#3953	—	4	4	no	—
35	#3947	—	2	1	no	—
36	C₈×Dic₅	—	2	1	no	—
37	#3944	—	2	2	no	—
38	C₂×#672	—	2	1	no	—
39	D₈₀	—	2	8	no	—
40	C₂²×D₁₆	—	5	1	no	—
41	#2151	—	5	1	no	—
42	C₂×#361	—	5	2	no	—
43	#1978	—	5	2	no	—
44	#1381	D₅	1	1	yes	characteristic, complemented
45	C₂³×D₅	—	4	2	no	—
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂×D₂₀	—	2	4	no	—
48	C₂×D₂₀	—	4	2	no	—
49	C₂×D₂₀	—	4	2	no	—
50	C₂×D₂₀	—	4	2	no	—
51	C₂×D₂₀	—	4	2	no	—
52	C₂×D₂₀	—	4	2	no	—
53	C₅×M₁₆	—	4	2	no	—
54	C₅×M₁₆	—	4	2	no	—
55	C₂×C₅×C₈	—	4	2	no	—
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
58	C₂×C₅×C₈	#41	1	2	yes	—
59	C₂×C₅×C₈	#41	1	1	yes	characteristic
60	C₂²⋊Dic₅	—	2	1	no	—
61	C₄⋊Dic₅	D₈	1	2	yes	—
62	C₄×Dic₅	—	2	1	no	—
63	#681	—	2	2	no	—
64	C₂×#168	—	2	1	no	—
65	D₄₀	—	2	4	no	—
66	D₄₀	—	2	4	no	—
67	D₄₀	—	4	2	no	—
68	D₄₀	—	4	2	no	—
69	C₅×C₁₆	—	2	2	no	—
70	#672	—	4	1	no	—
71	C₂²×D₈	—	5	2	no	—
72	C₂×D₁₆	—	10	4	no	—
73	C₂×D₁₆	—	10	2	no	—
74	C₂×M₃₂	—	5	1	no	—
75	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
76	C₂×#108	C₂×D₅	1	2	yes	—
77	#418	—	10	2	no	—
78	C₂×#100	—	10	2	no	—
79	#405	—	5	1	no	—
80	C₄⋊C₁₆	—	10	2	no	—
81	#361	—	10	4	no	—
82	C₂³×C₅	#135	1	1	yes	characteristic
83	C₂²×D₅	—	4	4	no	—
84	C₂²×D₅	—	8	2	no	—
85	C₂²×D₅	—	8	2	no	—
86	C₂²×D₅	—	16	4	no	—
87	C₂×C₄×C₅	#121	1	2	yes	—
88	C₂×C₄×C₅	#121	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×Dic₅	—	2	2	no	—
92	D₂₀	—	4	4	no	—
93	D₂₀	—	4	4	no	—
94	D₂₀	—	8	2	no	—
95	D₂₀	—	8	2	no	—
96	C₅×C₈	—	8	2	no	—
97	C₅×C₈	#120	1	1	yes	characteristic
98	C₅×C₈	#123	1	2	yes	—
99	C₅×C₈	#124	1	1	yes	characteristic
100	#168	—	4	1	no	—
101	C₂²×D₄	—	10	2	no	—
102	C₂×D₈	—	5	4	no	—
103	C₂×D₈	—	10	2	no	—
104	C₂×D₈	—	10	2	no	—
105	C₂×D₈	—	10	2	no	—
106	C₂×D₈	—	10	2	no	—
107	C₂×D₈	—	10	2	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	5	1	no	—
110	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
111	#117	—	5	1	no	—
112	D₁₆	—	10	8	no	—
113	M₃₂	—	10	2	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	—	10	1	no	—
116	C₂×C₁₆	D₂₀	1	2	yes	—
117	#108	—	2	2	no	—
118	#108	—	2	2	no	—
119	#102	—	20	4	no	—
120	#100	—	20	4	no	—
121	#97	—	10	1	no	—
122	C₄×C₈	—	10	1	no	—
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	#463	1	2	yes	—
125	C₂²×C₅	#466	1	1	yes	characteristic
126	C₂×D₅	—	8	4	no	—
127	C₂×D₅	—	16	4	no	—
128	C₂×D₅	—	16	4	no	—
129	C₂×D₅	—	16	4	no	—
130	C₄×C₅	#394	1	1	yes	characteristic
131	C₄×C₅	#459	1	2	yes	—
132	C₄×C₅	#469	1	1	yes	characteristic
133	Dic₅	—	4	2	no	—
134	C₂⁴	—	20	2	no	—
135	C₂×D₄	—	10	4	no	—
136	C₂×D₄	—	20	2	no	—
137	C₂×D₄	—	20	2	no	—
138	C₂×D₄	—	20	2	no	—
139	C₂×D₄	—	20	2	no	—
140	C₂×D₄	—	20	2	no	—
141	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
142	D₈	—	10	4	no	—
143	D₈	—	10	4	no	—
144	D₈	—	20	2	no	—
145	D₈	—	20	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	10	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	10	1	no	—
151	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
152	C₂×C₈	#709	1	2	yes	—
153	C₂×C₈	D₄×D₅	1	2	yes	—
154	C₂×C₈	#713	1	1	yes	characteristic
155	C₄⋊C₄	—	5	2	no	—
156	C₂²⋊C₄	—	10	1	no	—
157	C₄²	—	10	1	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	20	1	no	—
160	C₂×C₅	—	2	2	no	—
161	C₂×C₅	#2054	1	1	yes	characteristic
162	C₂×C₅	#2221	1	2	yes	—
163	D₅	—	16	4	no	—
164	C₂³	—	20	4	no	—
165	C₂³	—	40	2	no	—
166	C₂³	—	40	2	no	—
167	C₂³	—	80	4	no	—
168	C₂³	#4065	1	1	yes	characteristic
169	D₄	—	20	4	no	—
170	D₄	—	20	4	no	—
171	D₄	—	40	2	no	—
172	D₄	—	40	2	no	—
173	C₂×C₄	—	10	2	no	—
174	C₂×C₄	#4030	1	2	yes	—
175	C₂×C₄	#4041	1	1	yes	characteristic
176	C₂×C₄	D₅×D₈	1	1	yes	characteristic
177	C₂×C₄	#4060	1	2	yes	—
178	C₈	—	8	2	no	—
179	C₈	—	20	1	no	—
180	C₈	#4021	1	1	yes	characteristic
181	C₈	#4028	1	1	yes	characteristic
182	C₈	#4040	1	2	yes	—
183	C₅	#13439	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	—	40	4	no	—
186	C₂²	—	80	4	no	—
187	C₂²	—	80	4	no	—
188	C₂²	—	80	4	no	—
189	C₂²	#65108	1	2	yes	—
190	C₂²	#65172	1	1	yes	characteristic
191	C₄	—	20	2	no	—
192	C₄	#64961	1	1	yes	characteristic
193	C₄	#65075	1	2	yes	—
194	C₄	#65154	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	80	4	no	—
197	C₂	#10599655	1	1	yes	characteristic
198	C₂	#10600154	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₈₀

C₂

yes

maximal, characteristic

#10602194

C₂

yes

maximal, characteristic, complemented

#10601502

C₂

yes

maximal, complemented

C₂×#64715

C₂

yes

maximal, complemented

C₅×#1381

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₀

C₂²

yes

—

C₂²×C₅×C₁₆

C₂²

yes

characteristic

C₂×C₅×#108

C₂²

yes

—

#65420

—

C₂×#3955

—

#65390

C₂²

yes

characteristic

C₂×#3944

C₂²

yes

characteristic

C₂×D₈₀

—

C₂×D₈₀

—

—

—

—

C₂²×D₂₀

—

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

C₂³

yes

characteristic

#4072

D₄

yes

characteristic

C₂×#681

D₄

yes

characteristic

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

D₄

yes

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

D₄

yes

—

—

—

—

—

—

C₈×Dic₅

—

#3944

—

C₂×#672

—

D₈₀

—

C₂²×D₁₆

—

—

—

—

D₅

yes

characteristic, complemented

C₂³×D₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

#41

yes

—

C₂×C₅×C₈

#41

yes

characteristic

C₂²⋊Dic₅

—

C₄⋊Dic₅

D₈

yes

—

C₄×Dic₅

—

#681

—

C₂×#168

—

D₄₀

—

D₄₀

—

D₄₀

—

D₄₀

—

C₅×C₁₆

—

#672

—

C₂²×D₈

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₅

yes

characteristic

C₂×#108

C₂×D₅

yes

—

#418

—

C₂×#100

—

#405

—

C₄⋊C₁₆

—

#361

—

C₂³×C₅

#135

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₅×C₈

—

C₅×C₈

#120

yes

characteristic

C₅×C₈

#123

yes

—

C₅×C₈

#124

yes

characteristic

100

#168

—

101

C₂²×D₄

—

102

C₂×D₈

—

103

C₂×D₈

—

104

C₂×D₈

—

105

C₂×D₈

—

106

C₂×D₈

—

107

C₂×D₈

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

110

C₂²×C₈

C₂²×D₅

yes

characteristic

111

#117

—

112

D₁₆

—

113

M₃₂

—

114

C₂×C₁₆

—

115

C₂×C₁₆

—

116

C₂×C₁₆

D₂₀

yes

—

117

#108

—

118

#108

—

119

#102

—

120

#100

—

121

#97

—

122

C₄×C₈

—

123

C₂²×C₅

—

124

C₂²×C₅

#463

yes

—

125

C₂²×C₅

#466

yes

characteristic

126

C₂×D₅

—

127

C₂×D₅

—

128

C₂×D₅

—

129

C₂×D₅

—

130

C₄×C₅

#394

yes

characteristic

131

C₄×C₅

#459

yes

—

132

C₄×C₅

#469

yes

characteristic

133

Dic₅

—

134

C₂⁴

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂²×C₄

D₄×D₅

yes

characteristic

142

D₈

—

143

D₈

—

144

D₈

—

145

D₈

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

C₂×D₂₀

yes

characteristic

152

C₂×C₈

#709

yes

—

153

C₂×C₈

D₄×D₅

yes

—

154

C₂×C₈

#713

yes

characteristic

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₁₆

—

159

C₁₆

—

160

C₂×C₅

—

161

C₂×C₅

#2054

yes

characteristic

162

C₂×C₅

#2221

yes

—

163

D₅

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

#4065

yes

characteristic

169

D₄

—

170

D₄

—

171

D₄

—

172

D₄

—

173

C₂×C₄

—

174

C₂×C₄

#4030

yes

—

175

C₂×C₄

#4041

yes

characteristic

176

C₂×C₄

D₅×D₈

yes

characteristic

177

C₂×C₄

#4060

yes

—

178

C₈

—

179

C₈

—

180

C₈

#4021

yes

characteristic

181

C₈

#4028

yes

characteristic

182

C₈

#4040

yes

—

183

C₅

#13439

yes

characteristic, complemented, Sylow

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

#65108

yes

—

190

C₂²

#65172

yes

characteristic

191

C₄

—

192

C₄

#64961

yes

characteristic

193

C₄

#65075

yes

—

194

C₄

#65154

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#10599655

yes

characteristic

198

C₂

#10600154

yes

—

	subgroup	G/H
center	C₂²	#65172
commutator, cosocle	C₂²×C₅×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1381	C₂
socle	C₂²×C₅	#466
prime radical	C₂²×D₈₀	C₂
hypercenter	#1381	D₅
hypocenter	C₅	#13439
LCS	G ▷ C₂²×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#65172

commutator, cosocle

C₂²×C₅×C₈

C₂³

Frattini

C₂²×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂²×D₈₀

C₂

D₅

C₅

G ▷ C₂²×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
2. #10602194	C₂	semidirect
3. #10601502	C₂	semidirect
4. C₂×#64715	C₂	semidirect
5. C₅×#1381	C₂	semidirect
44. #1381	D₅	semidirect
183. C₅	#13439	semidirect

H ≅ G/N

type

2. #10602194

C₂

semidirect

3. #10601502

C₂

semidirect

4. C₂×#64715

C₂

semidirect

5. C₅×#1381

C₂

semidirect

44. #1381

D₅

semidirect

183. C₅

#13439

semidirect

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

subgroup	index
C₂²×D₈₀	2
#10602194	2
#10601502	2
C₂×#64715	2
C₅×#1381	2
#13439	5

subgroup

index

C₂²×D₈₀

Subgroups — #12018841

Conjugacy classes of subgroups

Complements

Maximal subgroups