Group #12003467 — subgroups

Subgroups — #12003467

Order 1280.

326 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 75 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	#10610800	C₂	1	1	yes	maximal, characteristic
2	C₅×#1838	C₂	1	1	yes	maximal, characteristic
3	#10601303	C₂	1	2	yes	maximal
4	#10600594	C₂	1	2	yes	maximal
5	#10598207	C₂	1	1	yes	maximal, characteristic
6	#65804	C₂²	1	2	yes	—
7	C₅×#423	C₂²	1	1	yes	characteristic
8	C₅×#423	C₂²	1	1	yes	characteristic
9	C₂×C₅×#97	C₂²	1	1	yes	characteristic
10	#65426	—	2	2	no	—
11	#65215	C₂²	1	2	yes	—
12	#65140	—	2	2	no	—
13	#65138	—	2	2	no	—
14	C₅×#341	—	2	2	no	—
15	#64702	—	2	2	no	—
16	#64701	—	2	2	no	—
17	#64685	—	2	4	no	—
18	#10529	—	5	1	no	maximal, Sylow
19	C₂×C₅×M₁₆	—	2	1	no	—
20	C₂×C₅×M₁₆	D₄	1	2	yes	—
21	C₂²×C₅×C₈	—	2	1	no	—
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	#4072	—	2	2	no	—
24	#4070	—	2	2	no	—
25	#4070	—	2	2	no	—
26	C₂×Dic₂₀	—	2	2	no	—
27	#4016	—	2	2	no	—
28	#4015	D₄	1	2	yes	—
29	#4015	D₄	1	2	yes	—
30	C₄×Dic₁₀	—	2	2	no	—
31	C₄×Dic₁₀	—	2	2	no	—
32	C₅×C₄⋊C₈	—	2	2	no	—
33	C₅×C₄⋊C₈	—	2	2	no	—
34	C₅×#97	—	2	1	no	—
35	C₅×#97	D₄	1	2	yes	—
36	C₂²⋊#168	—	2	2	no	—
37	#3949	—	2	2	no	—
38	#3948	—	2	4	no	—
39	C₄⋊#168	—	2	2	no	—
40	#3935	—	2	2	no	—
41	#3255	—	5	1	no	—
42	#1838	D₅	1	1	yes	characteristic
43	#1559	—	5	2	no	—
44	#1531	—	5	2	no	—
45	#1307	—	5	1	no	—
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	#41	1	2	yes	—
48	C₂×Dic₁₀	—	2	2	no	—
49	C₂×Dic₁₀	—	2	2	no	—
50	C₅×M₁₆	—	4	2	no	—
51	C₅×M₁₆	—	4	2	no	—
52	C₂×C₅×C₈	—	2	2	no	—
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	—	4	1	no	—
55	C₂×C₅×C₈	—	4	1	no	—
56	C₂×C₅×C₈	—	4	1	no	—
57	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅	C₂×D₄	1	2	yes	—
59	C₄²×C₅	#41	1	1	yes	characteristic
60	C₂²⋊Dic₅	—	4	2	no	—
61	C₄⋊Dic₅	—	2	2	no	—
62	C₄⋊Dic₅	—	2	2	no	—
63	C₄⋊Dic₅	—	2	2	no	—
64	C₄⋊Dic₅	—	2	2	no	—
65	#683	—	4	2	no	—
66	#683	—	4	2	no	—
67	C₄×Dic₅	—	4	2	no	—
68	C₂×#168	—	4	2	no	—
69	Dic₂₀	—	8	2	no	—
70	#533	—	5	2	no	—
71	#475	—	10	2	no	—
72	#470	—	10	2	no	—
73	#462	—	10	2	no	—
74	#432	—	5	2	no	—
75	#423	C₂×D₅	1	1	yes	characteristic
76	#423	C₂×D₅	1	1	yes	characteristic
77	C₂×#97	C₂×D₅	1	1	yes	characteristic
78	#341	—	2	2	no	—
79	#333	—	10	2	no	—
80	#332	—	10	2	no	—
81	#330	—	10	4	no	—
82	C₂³×C₅	#124	1	1	yes	characteristic
83	C₂×C₄×C₅	—	2	1	no	—
84	C₂×C₄×C₅	—	2	1	no	—
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	#120	1	1	yes	characteristic
87	C₂×C₄×C₅	#121	1	2	yes	—
88	C₂×C₄×C₅	#121	1	1	yes	characteristic
89	C₂×C₄×C₅	#123	1	2	yes	—
90	C₂×Dic₅	—	4	2	no	—
91	C₂×Dic₅	—	4	2	no	—
92	Dic₁₀	—	4	2	no	—
93	Dic₁₀	—	4	2	no	—
94	Dic₁₀	—	8	2	no	—
95	C₅×C₈	—	4	2	no	—
96	C₅×C₈	—	4	2	no	—
97	C₅×C₈	—	8	1	no	—
98	#168	—	8	2	no	—
99	C₂×Q₁₆	—	10	2	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	C₂²⋊D₅	1	2	yes	—
102	C₂²×C₈	—	2	1	no	—
103	C₄⋊Q₈	—	5	2	no	—
104	C₄⋊Q₈	—	5	2	no	—
105	#125	—	10	2	no	—
106	C₂²⋊Q₈	—	10	2	no	—
107	C₂²⋊Q₈	—	10	2	no	—
108	C₄×Q₈	—	10	2	no	—
109	C₄×Q₈	—	10	2	no	—
110	#117	—	10	2	no	—
111	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
112	#106	—	10	2	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	10	2	no	—
116	#103	—	10	4	no	—
117	C₂²⋊C₈	—	10	2	no	—
118	#97	—	2	1	no	—
119	#97	—	10	2	no	—
120	#97	D₂₀	1	2	yes	—
121	C₂²×C₅	—	2	2	no	—
122	C₂²×C₅	—	2	1	no	—
123	C₂²×C₅	#394	1	1	yes	characteristic
124	C₄×C₅	—	2	2	no	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	2	1	no	—
127	C₄×C₅	#454	1	1	yes	characteristic
128	C₄×C₅	#455	1	1	yes	characteristic
129	Dic₅	—	8	2	no	—
130	Dic₅	—	8	2	no	—
131	C₂×Q₈	—	10	2	no	—
132	C₂×Q₈	—	10	2	no	—
133	C₂²×C₄	#713	1	2	yes	—
134	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
135	Q₁₆	—	40	2	no	—
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₂×C₈	—	20	2	no	—
144	C₄⋊C₄	—	10	2	no	—
145	C₄⋊C₄	—	10	2	no	—
146	C₄⋊C₄	—	10	2	no	—
147	C₄⋊C₄	—	10	2	no	—
148	C₄⋊C₄	—	20	2	no	—
149	C₄⋊C₄	—	20	2	no	—
150	C₂²⋊C₄	—	20	2	no	—
151	C₄²	—	20	2	no	—
152	C₄²	C₂×D₂₀	1	1	yes	characteristic
153	C₄²	#709	1	1	yes	characteristic
154	C₄²	D₄×D₅	1	2	yes	—
155	C₂×C₅	—	4	1	no	—
156	C₂×C₅	#2040	1	1	yes	characteristic
157	C₂×C₅	#2049	1	2	yes	—
158	C₂³	#4093	1	1	yes	characteristic
159	Q₈	—	20	2	no	—
160	Q₈	—	20	2	no	—
161	Q₈	—	40	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	20	2	no	—
166	C₂×C₄	—	20	2	no	—
167	C₂×C₄	#4040	1	2	yes	—
168	C₂×C₄	#4041	1	2	yes	—
169	C₂×C₄	#4076	1	1	yes	characteristic
170	C₂×C₄	#4083	1	1	yes	characteristic
171	C₈	—	4	2	no	—
172	C₈	—	4	2	no	—
173	C₈	—	8	1	no	—
174	C₈	—	40	2	no	—
175	C₅	#10529	1	1	yes	characteristic, complemented, Sylow
176	C₂²	—	2	2	no	—
177	C₂²	—	2	1	no	—
178	C₂²	#65280	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	1	no	—
182	C₄	—	40	2	no	—
183	C₄	—	40	2	no	—
184	C₄	#65350	1	1	yes	characteristic
185	C₄	#65351	1	1	yes	characteristic
186	C₂	—	4	1	no	—
187	C₂	#10600356	1	1	yes	characteristic
188	C₂	#10601530	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10610800

C₂

yes

maximal, characteristic

C₅×#1838

C₂

yes

maximal, characteristic

#10601303

C₂

yes

maximal

#10600594

C₂

yes

maximal

#10598207

C₂

yes

maximal, characteristic

#65804

C₂²

yes

—

C₅×#423

C₂²

yes

characteristic

C₅×#423

C₂²

yes

characteristic

C₂×C₅×#97

C₂²

yes

characteristic

#65426

—

#65215

C₂²

yes

—

—

—

—

—

—

—

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

D₄

yes

—

C₂²×C₅×C₈

—

C₂×C₄²×C₅

C₂³

yes

—

—

—

C₂×Dic₂₀

—

#4016

—

#4015

D₄

yes

—

#4015

D₄

yes

—

C₄×Dic₁₀

—

C₄×Dic₁₀

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×#97

—

C₅×#97

D₄

yes

—

C₂²⋊#168

—

—

—

—

—

—

D₅

yes

—

—

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

#41

yes

characteristic

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

#683

—

C₄×Dic₅

—

C₂×#168

—

Dic₂₀

—

—

—

—

—

—

C₂×D₅

yes

characteristic

#423

C₂×D₅

yes

characteristic

C₂×#97

C₂×D₅

yes

—

—

—

—

C₂³×C₅

#124

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#120

yes

characteristic

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

#168

—

C₂×Q₁₆

—

100

C₂×M₁₆

—

101

C₂×M₁₆

C₂²⋊D₅

yes

—

102

C₂²×C₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

107

C₂²⋊Q₈

—

108

C₄×Q₈

—

109

C₄×Q₈

—

110

#117

—

111

C₂×C₄²

C₂²×D₅

yes

characteristic

112

#106

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

#103

—

117

C₂²⋊C₈

—

118

#97

—

119

#97

—

120

#97

D₂₀

yes

—

121

C₂²×C₅

—

122

C₂²×C₅

—

123

C₂²×C₅

#394

yes

characteristic

124

C₄×C₅

—

125

C₄×C₅

—

126

C₄×C₅

—

127

C₄×C₅

#454

yes

characteristic

128

C₄×C₅

#455

yes

characteristic

129

Dic₅

—

130

Dic₅

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂²×C₄

#713

yes

—

134

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

135

Q₁₆

—

136

M₁₆

—

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₂²⋊C₄

—

151

C₄²

—

152

C₄²

C₂×D₂₀

yes

characteristic

153

C₄²

#709

yes

characteristic

154

C₄²

D₄×D₅

yes

—

155

C₂×C₅

—

156

C₂×C₅

#2040

yes

characteristic

157

C₂×C₅

#2049

yes

—

158

C₂³

#4093

yes

characteristic

159

Q₈

—

160

Q₈

—

161

Q₈

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

#4040

yes

—

168

C₂×C₄

#4041

yes

—

169

C₂×C₄

#4076

yes

characteristic

170

C₂×C₄

#4083

yes

characteristic

171

C₈

—

172

C₈

—

173

C₈

—

174

C₈

—

175

C₅

#10529

yes

characteristic, complemented, Sylow

176

C₂²

—

177

C₂²

—

178

C₂²

#65280

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#65350

yes

characteristic

185

C₄

#65351

yes

characteristic

186

C₂

—

187

C₂

#10600356

yes

characteristic

188

C₂

#10601530

yes

—

	subgroup	G/H
center	C₂²	#65280
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1838	C₂
socle	C₂²×C₅	#394
prime radical	C₂³×C₅	#124
hypercenter	#1838	D₅
hypocenter	C₅	#10529
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

subgroup

G/H

center

C₂²

#65280

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

N	H ≅ G/N	type
175. C₅	#10529	semidirect

H ≅ G/N

type

175. C₅

#10529

semidirect

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

subgroup	index
#10610800	2
C₅×#1838	2
#10601303	2
#10600594	2
#10598207	2
#10529	5

subgroup

index

Subgroups — #12003467

Conjugacy classes of subgroups

Complements

Maximal subgroups