Group #12018837 — subgroups

Subgroups — #12018837

Order 1280.

418 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 95 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×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic
2	#10602834	C₂	1	2	yes	maximal, complemented
3	C₂×#64783	C₂	1	2	yes	maximal, complemented
4	#10602194	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598299	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×D₈	C₂²	1	2	yes	—
7	C₂×C₅×D₁₆	—	2	2	no	—
8	C₂×C₅×D₁₆	—	2	4	no	—
9	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#3965	—	2	2	no	—
11	#65504	—	2	2	no	—
12	C₂×#3951	C₂²	1	2	yes	—
13	#65390	C₂²	1	1	yes	characteristic
14	C₂×#3944	C₂²	1	1	yes	characteristic
15	#64783	—	2	4	no	—
16	#64721	—	2	2	no	—
17	#13439	—	5	1	no	maximal, Sylow
18	C₂²×C₅×D₄	—	2	2	no	—
19	C₂×C₅×D₈	—	2	2	no	—
20	C₂×C₅×D₈	—	2	2	no	—
21	C₂×C₅×D₈	—	2	2	no	—
22	C₂×C₅×D₈	—	2	2	no	—
23	C₂×C₅×D₈	—	2	2	no	—
24	C₂×C₅×D₈	D₄	1	4	yes	—
25	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
26	#4072	D₄	1	1	yes	characteristic
27	C₂×#681	—	2	2	no	—
28	C₂×#681	D₄	1	1	yes	characteristic
29	C₅×D₁₆	—	2	8	no	—
30	C₂×C₅×C₁₆	—	2	2	no	—
31	C₂×C₅×C₁₆	D₄	1	2	yes	—
32	#3965	—	4	4	no	—
33	#3964	—	4	4	no	—
34	#3951	—	2	2	no	—
35	#3951	—	2	2	no	—
36	#3947	—	2	1	no	—
37	C₈×Dic₅	—	2	1	no	—
38	#3944	—	2	2	no	—
39	C₂×#672	—	2	1	no	—
40	C₂²×D₁₆	D₅	1	1	yes	characteristic
41	#2151	—	5	1	no	—
42	C₂×#361	—	5	2	no	—
43	#1978	—	5	2	no	—
44	#1381	—	5	1	no	—
45	C₂⁴×C₅	—	4	2	no	—
46	C₂×C₅×D₄	—	2	4	no	—
47	C₂×C₅×D₄	—	4	2	no	—
48	C₂×C₅×D₄	—	4	2	no	—
49	C₂×C₅×D₄	—	4	2	no	—
50	C₂×C₅×D₄	—	4	2	no	—
51	C₂×C₅×D₄	—	4	2	no	—
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₅×D₈	—	2	4	no	—
54	C₅×D₈	—	2	4	no	—
55	C₅×D₈	—	4	2	no	—
56	C₅×D₈	—	4	2	no	—
57	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
58	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₅×C₈	#41	1	1	yes	characteristic
60	C₂×C₅×C₈	#41	1	2	yes	—
61	C₂²⋊Dic₅	—	2	1	no	—
62	C₄⋊Dic₅	D₈	1	2	yes	—
63	C₄×Dic₅	—	2	1	no	—
64	#681	—	2	2	no	—
65	#681	—	4	2	no	—
66	#681	—	4	2	no	—
67	C₂×#168	—	2	1	no	—
68	C₂×#168	—	4	2	no	—
69	C₅×C₁₆	—	2	2	no	—
70	#672	—	4	1	no	—
71	C₂²×D₈	C₂×D₅	1	2	yes	—
72	C₂×D₁₆	—	2	4	no	—
73	C₂×D₁₆	—	2	2	no	—
74	C₂×M₃₂	—	5	1	no	—
75	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
76	C₂×#108	—	5	2	no	—
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	—	4	4	no	—
83	C₂³×C₅	—	8	2	no	—
84	C₂³×C₅	—	8	2	no	—
85	C₂³×C₅	—	16	4	no	—
86	C₂³×C₅	#135	1	1	yes	characteristic
87	C₅×D₄	—	4	4	no	—
88	C₅×D₄	—	4	4	no	—
89	C₅×D₄	—	8	2	no	—
90	C₅×D₄	—	8	2	no	—
91	C₂×C₄×C₅	#121	1	1	yes	characteristic
92	C₂×C₄×C₅	#121	1	2	yes	—
93	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×Dic₅	—	2	2	no	—
96	C₅×C₈	#120	1	1	yes	characteristic
97	C₅×C₈	#123	1	2	yes	—
98	C₅×C₈	#124	1	1	yes	characteristic
99	#168	—	4	1	no	—
100	#168	—	8	2	no	—
101	C₂²×D₄	—	2	2	no	—
102	C₂×D₈	—	2	2	no	—
103	C₂×D₈	—	2	2	no	—
104	C₂×D₈	—	2	2	no	—
105	C₂×D₈	—	2	2	no	—
106	C₂×D₈	—	2	2	no	—
107	C₂×D₈	C₂²⋊D₅	1	4	yes	—
108	C₂×M₁₆	—	5	1	no	—
109	C₂×M₁₆	—	10	2	no	—
110	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
111	#117	—	5	1	no	—
112	D₁₆	—	2	8	no	—
113	M₃₂	—	10	2	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	—	10	1	no	—
116	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
117	#108	—	10	2	no	—
118	#108	—	10	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	—	8	4	no	—
125	C₂²×C₅	—	16	4	no	—
126	C₂²×C₅	—	16	4	no	—
127	C₂²×C₅	—	16	4	no	—
128	C₂²×C₅	#463	1	2	yes	—
129	C₂²×C₅	#466	1	1	yes	characteristic
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	—	4	2	no	—
135	C₂×D₄	—	2	4	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂×D₄	—	4	2	no	—
138	C₂×D₄	—	4	2	no	—
139	C₂×D₄	—	4	2	no	—
140	C₂×D₄	—	4	2	no	—
141	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
142	D₈	—	2	4	no	—
143	D₈	—	2	4	no	—
144	D₈	—	4	2	no	—
145	D₈	—	4	2	no	—
146	M₁₆	—	10	2	no	—
147	M₁₆	—	20	2	no	—
148	M₁₆	—	20	2	no	—
149	C₂×C₈	—	10	1	no	—
150	C₂×C₈	—	20	2	no	—
151	C₂×C₈	#711	1	1	yes	characteristic
152	C₂×C₈	#711	1	2	yes	—
153	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
154	C₂×C₈	C₂×C₂²⋊D₅	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	—	16	4	no	—
162	C₂×C₅	#2054	1	1	yes	characteristic
163	C₂×C₅	#2221	1	2	yes	—
164	C₂³	—	4	4	no	—
165	C₂³	—	8	2	no	—
166	C₂³	—	8	2	no	—
167	C₂³	—	16	4	no	—
168	C₂³	#4058	1	1	yes	characteristic
169	D₄	—	4	4	no	—
170	D₄	—	4	4	no	—
171	D₄	—	8	2	no	—
172	D₄	—	8	2	no	—
173	C₂×C₄	—	10	2	no	—
174	C₂×C₄	D₅×D₈	1	1	yes	characteristic
175	C₂×C₄	#4057	1	2	yes	—
176	C₂×C₄	#4084	1	1	yes	characteristic
177	C₂×C₄	#4085	1	2	yes	—
178	C₈	—	20	1	no	—
179	C₈	—	40	2	no	—
180	C₈	#4081	1	2	yes	—
181	C₈	#4082	1	1	yes	characteristic
182	C₈	#4099	1	1	yes	characteristic
183	C₅	#13439	1	1	yes	characteristic, complemented, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	—	16	4	no	—
188	C₂²	—	16	4	no	—
189	C₂²	#65442	1	2	yes	—
190	C₂²	#65446	1	1	yes	characteristic
191	C₄	—	20	2	no	—
192	C₄	#65439	1	2	yes	—
193	C₄	#65448	1	1	yes	characteristic
194	C₄	#65511	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	16	4	no	—
197	C₂	#10602265	1	2	yes	—
198	C₂	#10602833	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅×D₁₆

C₂

yes

maximal, characteristic

#10602834

C₂

yes

maximal, complemented

C₂×#64783

C₂

yes

maximal, complemented

#10602194

C₂

yes

maximal, characteristic, complemented

#10598299

C₂

yes

maximal, characteristic, complemented

C₂²×C₅×D₈

C₂²

yes

—

C₂×C₅×D₁₆

—

C₂×C₅×D₁₆

—

C₂²×C₅×C₁₆

C₂²

yes

—

—

C₂²

yes

—

#65390

C₂²

yes

characteristic

C₂×#3944

C₂²

yes

—

—

—

C₂²×C₅×D₄

—

C₂×C₅×D₈

—

C₂×C₅×D₈

—

C₂×C₅×D₈

—

C₂×C₅×D₈

—

C₂×C₅×D₈

—

C₂×C₅×D₈

D₄

yes

—

C₂²×C₅×C₈

C₂³

yes

characteristic

#4072

D₄

yes

characteristic

C₂×#681

—

C₂×#681

D₄

yes

characteristic

C₅×D₁₆

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

D₄

yes

—

—

—

—

—

—

C₈×Dic₅

—

#3944

—

C₂×#672

—

C₂²×D₁₆

D₅

yes

—

—

—

—

C₂⁴×C₅

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×D₈

—

C₅×D₈

—

C₅×D₈

—

C₅×D₈

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

C₂×D₄

yes

C₂×C₅×C₈

yes

C₂×C₅×C₈

yes

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

D₈

yes

—

C₄×Dic₅

—

—

—

—

—

—

C₅×C₁₆

—

#672

—

C₂²×D₈

C₂×D₅

yes

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₅

yes

—

—

—

—

C₄⋊C₁₆

—

#361

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

#135

yes

characteristic

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×Dic₅

—

C₅×C₈

#120

yes

characteristic

C₅×C₈

#123

yes

—

C₅×C₈

#124

yes

characteristic

#168

—

100

#168

—

101

C₂²×D₄

—

102

C₂×D₈

—

103

C₂×D₈

—

104

C₂×D₈

—

105

C₂×D₈

—

106

C₂×D₈

—

107

C₂×D₈

C₂²⋊D₅

yes

—

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

C₂²⋊D₅

yes

—

117

#108

—

118

#108

—

119

#102

—

120

#100

—

121

#97

—

122

C₄×C₈

—

123

C₂²×C₅

—

124

C₂²×C₅

—

125

C₂²×C₅

—

126

C₂²×C₅

—

127

C₂²×C₅

—

128

C₂²×C₅

#463

yes

—

129

C₂²×C₅

#466

yes

characteristic

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

#711

yes

characteristic

152

C₂×C₈

#711

yes

—

153

C₂×C₈

C₂×C₂²⋊D₅

yes

—

154

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₁₆

—

159

C₁₆

—

160

C₂×C₅

—

161

C₂×C₅

—

162

C₂×C₅

#2054

yes

characteristic

163

C₂×C₅

#2221

yes

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

#4058

yes

characteristic

169

D₄

—

170

D₄

—

171

D₄

—

172

D₄

—

173

C₂×C₄

—

174

C₂×C₄

D₅×D₈

yes

characteristic

175

C₂×C₄

#4057

yes

—

176

C₂×C₄

#4084

yes

characteristic

177

C₂×C₄

#4085

yes

—

178

C₈

—

179

C₈

—

180

C₈

#4081

yes

—

181

C₈

#4082

yes

characteristic

182

C₈

#4099

yes

characteristic

183

C₅

#13439

yes

characteristic, complemented, Sylow

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

#65442

yes

—

190

C₂²

#65446

yes

characteristic

191

C₄

—

192

C₄

#65439

yes

—

193

C₄

#65448

yes

characteristic

194

C₄

#65511

yes

characteristic

195

C₂

—

196

C₂

—

197

C₂

#10602265

yes

—

198

C₂

#10602833

yes

characteristic

	subgroup	G/H
center	C₂²	#65446
commutator, cosocle	C₂²×C₅×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₅
Fitting, prime radical	C₂²×C₅×D₁₆	C₂
socle	C₂²×C₅	#466
hypercenter	C₂²×D₁₆	D₅
hypocenter	C₅	#13439
LCS	G ▷ C₂²×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

subgroup

G/H

center

C₂²

#65446

commutator, cosocle

C₂²×C₅×C₈

C₂³

Frattini

C₂²×C₈

C₂²×D₅

Fitting, prime radical

C₂²×C₅×D₁₆

C₂

socle

C₂²×C₅

#466

hypercenter

C₂²×D₁₆

D₅

hypocenter

C₅

#13439

LCS

G ▷ C₂²×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

N	H ≅ G/N	type
2. #10602834	C₂	semidirect
3. C₂×#64783	C₂	semidirect
4. #10602194	C₂	semidirect
5. #10598299	C₂	semidirect
183. C₅	#13439	semidirect

H ≅ G/N

type

2. #10602834

C₂

semidirect

3. C₂×#64783

C₂

semidirect

4. #10602194

C₂

semidirect

5. #10598299

C₂

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×C₅×D₁₆	2
#10602834	2
C₂×#64783	2
#10602194	2
#10598299	2
#13439	5

subgroup

index

C₂²×C₅×D₁₆

Subgroups — #12018837

Conjugacy classes of subgroups

Complements

Maximal subgroups