Group #12018853 — subgroups

Subgroups — #12018853

Order 1280.

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

normal subgroups only

This group has 378 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×Dic₄₀	C₂	1	1	yes	maximal, characteristic
2	#10602194	C₂	1	1	yes	maximal, characteristic
3	#10601503	C₂	1	2	yes	maximal
4	C₂×#64716	C₂	1	2	yes	maximal
5	C₅×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀	C₂²	1	2	yes	—
7	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₅×#108	C₂²	1	2	yes	—
9	C₂×#3956	—	2	2	no	—
10	#65409	—	2	2	no	—
11	#65390	C₂²	1	1	yes	characteristic
12	C₂×#3944	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₀	—	2	4	no	—
14	C₂×Dic₄₀	—	2	2	no	—
15	#64721	—	2	2	no	—
16	#64716	—	2	4	no	—
17	#13441	—	5	1	no	maximal, Sylow
18	C₂²×Dic₁₀	—	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×Dic₂₀	—	2	2	no	—
24	C₂×Dic₂₀	—	2	2	no	—
25	C₂×Dic₂₀	—	2	2	no	—
26	C₂×Dic₂₀	—	2	2	no	—
27	C₂×Dic₂₀	—	2	2	no	—
28	C₂×Dic₂₀	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	#3956	—	4	4	no	—
34	#3948	—	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	Dic₄₀	—	2	8	no	—
40	C₂²×Q₃₂	—	5	1	no	—
41	#2151	—	5	1	no	—
42	C₂×#362	—	5	2	no	—
43	#1979	—	5	2	no	—
44	#1381	D₅	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₅	—	4	2	no	—
47	C₂×Dic₁₀	—	2	4	no	—
48	C₂×Dic₁₀	—	4	2	no	—
49	C₂×Dic₁₀	—	4	2	no	—
50	C₂×Dic₁₀	—	4	2	no	—
51	C₂×Dic₁₀	—	4	2	no	—
52	C₂×Dic₁₀	—	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	Dic₂₀	—	2	4	no	—
66	Dic₂₀	—	2	4	no	—
67	Dic₂₀	—	4	2	no	—
68	Dic₂₀	—	4	2	no	—
69	C₅×C₁₆	—	2	2	no	—
70	#672	—	4	1	no	—
71	C₂²×Q₁₆	—	5	2	no	—
72	C₂×Q₃₂	—	10	4	no	—
73	C₂×Q₃₂	—	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	#419	—	10	2	no	—
78	C₂×#101	—	10	2	no	—
79	#405	—	5	1	no	—
80	C₄⋊C₁₆	—	10	2	no	—
81	#362	—	10	4	no	—
82	C₂³×C₅	#135	1	1	yes	characteristic
83	C₂×C₄×C₅	#121	1	2	yes	—
84	C₂×C₄×C₅	#121	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
87	C₂×Dic₅	—	2	2	no	—
88	C₂×Dic₅	—	4	4	no	—
89	C₂×Dic₅	—	8	2	no	—
90	C₂×Dic₅	—	8	2	no	—
91	Dic₁₀	—	4	4	no	—
92	Dic₁₀	—	4	4	no	—
93	Dic₁₀	—	8	2	no	—
94	Dic₁₀	—	8	2	no	—
95	C₅×C₈	—	8	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	C₂²×Q₈	—	10	2	no	—
101	C₂×Q₁₆	—	5	4	no	—
102	C₂×Q₁₆	—	10	2	no	—
103	C₂×Q₁₆	—	10	2	no	—
104	C₂×Q₁₆	—	10	2	no	—
105	C₂×Q₁₆	—	10	2	no	—
106	C₂×Q₁₆	—	10	2	no	—
107	C₂×M₁₆	—	2	2	no	—
108	C₂×M₁₆	—	5	1	no	—
109	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
110	#117	—	5	1	no	—
111	Q₃₂	—	10	8	no	—
112	M₃₂	—	10	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	10	1	no	—
115	C₂×C₁₆	D₂₀	1	2	yes	—
116	#108	—	2	2	no	—
117	#108	—	2	2	no	—
118	#103	—	20	4	no	—
119	#101	—	20	4	no	—
120	#97	—	10	1	no	—
121	C₄×C₈	—	10	1	no	—
122	C₂²×C₅	—	2	2	no	—
123	C₂²×C₅	#463	1	2	yes	—
124	C₂²×C₅	#466	1	1	yes	characteristic
125	C₄×C₅	#394	1	1	yes	characteristic
126	C₄×C₅	#459	1	2	yes	—
127	C₄×C₅	#469	1	1	yes	characteristic
128	Dic₅	—	4	2	no	—
129	Dic₅	—	8	4	no	—
130	C₂×Q₈	—	10	4	no	—
131	C₂×Q₈	—	20	2	no	—
132	C₂×Q₈	—	20	2	no	—
133	C₂×Q₈	—	20	2	no	—
134	C₂×Q₈	—	20	2	no	—
135	C₂×Q₈	—	20	2	no	—
136	C₂²×C₄	—	20	2	no	—
137	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
138	Q₁₆	—	10	4	no	—
139	Q₁₆	—	10	4	no	—
140	Q₁₆	—	20	2	no	—
141	Q₁₆	—	20	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	10	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	10	1	no	—
147	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
148	C₂×C₈	#709	1	2	yes	—
149	C₂×C₈	D₄×D₅	1	2	yes	—
150	C₂×C₈	#713	1	1	yes	characteristic
151	C₄⋊C₄	—	5	2	no	—
152	C₂²⋊C₄	—	10	1	no	—
153	C₄²	—	10	1	no	—
154	C₁₆	—	2	2	no	—
155	C₁₆	—	20	1	no	—
156	C₂×C₅	—	2	2	no	—
157	C₂×C₅	#2054	1	1	yes	characteristic
158	C₂×C₅	#2222	1	2	yes	—
159	C₂³	#4065	1	1	yes	characteristic
160	Q₈	—	20	4	no	—
161	Q₈	—	20	4	no	—
162	Q₈	—	40	2	no	—
163	Q₈	—	40	2	no	—
164	C₂×C₄	—	10	2	no	—
165	C₂×C₄	—	20	4	no	—
166	C₂×C₄	—	40	2	no	—
167	C₂×C₄	—	40	2	no	—
168	C₂×C₄	#4030	1	2	yes	—
169	C₂×C₄	#4041	1	1	yes	characteristic
170	C₂×C₄	D₅×D₈	1	1	yes	characteristic
171	C₂×C₄	#4060	1	2	yes	—
172	C₈	—	8	2	no	—
173	C₈	—	20	1	no	—
174	C₈	#4021	1	1	yes	characteristic
175	C₈	#4028	1	1	yes	characteristic
176	C₈	#4040	1	2	yes	—
177	C₅	#13441	1	1	yes	characteristic, complemented, Sylow
178	C₂²	—	2	2	no	—
179	C₂²	#65108	1	2	yes	—
180	C₂²	#65172	1	1	yes	characteristic
181	C₄	—	20	2	no	—
182	C₄	—	40	4	no	—
183	C₄	#64961	1	1	yes	characteristic
184	C₄	#65075	1	2	yes	—
185	C₄	#65154	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	#10599655	1	1	yes	characteristic
188	C₂	#10600155	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₀

C₂

yes

maximal, characteristic

#10602194

C₂

yes

maximal, characteristic

#10601503

C₂

yes

maximal

C₂×#64716

C₂

yes

maximal

C₅×#1381

C₂

yes

maximal, characteristic

C₂²×Dic₂₀

C₂²

yes

—

C₂²×C₅×C₁₆

C₂²

yes

characteristic

C₂×C₅×#108

C₂²

yes

—

C₂×#3956

—

#65409

—

#65390

C₂²

yes

characteristic

C₂×#3944

C₂²

yes

characteristic

C₂×Dic₄₀

—

C₂×Dic₄₀

—

—

—

—

C₂²×Dic₁₀

—

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

C₂³

yes

characteristic

#4072

D₄

yes

characteristic

C₂×#681

D₄

yes

characteristic

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

D₄

yes

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

D₄

yes

—

—

—

—

—

—

C₈×Dic₅

—

#3944

—

C₂×#672

—

Dic₄₀

—

C₂²×Q₃₂

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

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

—

Dic₂₀

—

Dic₂₀

—

Dic₂₀

—

Dic₂₀

—

C₅×C₁₆

—

#672

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₅

yes

characteristic

C₂×#108

C₂×D₅

yes

—

#419

—

C₂×#101

—

#405

—

C₄⋊C₁₆

—

#362

—

C₂³×C₅

#135

yes

characteristic

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

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

#120

yes

characteristic

C₅×C₈

#123

yes

—

C₅×C₈

#124

yes

characteristic

#168

—

100

C₂²×Q₈

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂×Q₁₆

—

105

C₂×Q₁₆

—

106

C₂×Q₁₆

—

107

C₂×M₁₆

—

108

C₂×M₁₆

—

109

C₂²×C₈

C₂²×D₅

yes

characteristic

110

#117

—

111

Q₃₂

—

112

M₃₂

—

113

C₂×C₁₆

—

114

C₂×C₁₆

—

115

C₂×C₁₆

D₂₀

yes

—

116

#108

—

117

#108

—

118

#103

—

119

#101

—

120

#97

—

121

C₄×C₈

—

122

C₂²×C₅

—

123

C₂²×C₅

#463

yes

—

124

C₂²×C₅

#466

yes

characteristic

125

C₄×C₅

#394

yes

characteristic

126

C₄×C₅

#459

yes

—

127

C₄×C₅

#469

yes

characteristic

128

Dic₅

—

129

Dic₅

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

D₄×D₅

yes

characteristic

138

Q₁₆

—

139

Q₁₆

—

140

Q₁₆

—

141

Q₁₆

—

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

C₂×D₂₀

yes

characteristic

148

C₂×C₈

#709

yes

—

149

C₂×C₈

D₄×D₅

yes

—

150

C₂×C₈

#713

yes

characteristic

151

C₄⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₁₆

—

155

C₁₆

—

156

C₂×C₅

—

157

C₂×C₅

#2054

yes

characteristic

158

C₂×C₅

#2222

yes

—

159

C₂³

#4065

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

#4030

yes

—

169

C₂×C₄

#4041

yes

characteristic

170

C₂×C₄

D₅×D₈

yes

characteristic

171

C₂×C₄

#4060

yes

—

172

C₈

—

173

C₈

—

174

C₈

#4021

yes

characteristic

175

C₈

#4028

yes

characteristic

176

C₈

#4040

yes

—

177

C₅

#13441

yes

characteristic, complemented, Sylow

178

C₂²

—

179

C₂²

#65108

yes

—

180

C₂²

#65172

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

#64961

yes

characteristic

184

C₄

#65075

yes

—

185

C₄

#65154

yes

characteristic

186

C₂

—

187

C₂

#10599655

yes

characteristic

188

C₂

#10600155

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	C₂³×C₅	#135
hypercenter	#1381	D₅
hypocenter	C₅	#13441
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

C₂³×C₅

D₅

C₅

G ▷ C₂²×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
177. C₅	#13441	semidirect

H ≅ G/N

type

177. C₅

#13441

semidirect

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

subgroup	index
C₂²×Dic₄₀	2
#10602194	2
#10601503	2
C₂×#64716	2
C₅×#1381	2
#13441	5

subgroup

index

C₂²×Dic₄₀

Subgroups — #12018853

Conjugacy classes of subgroups

Complements

Maximal subgroups