Group #12004075 — subgroups

Subgroups — #12004075

Order 1280.

292 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 93 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	#10601569	C₂	1	2	yes	maximal
2	#10600234	C₂	1	1	yes	maximal, characteristic
3	#10600232	C₂	1	1	yes	maximal, characteristic
4	C₅×#1274	C₂	1	1	yes	maximal, characteristic
5	#10598236	C₂	1	2	yes	maximal
6	#65804	C₂²	1	1	yes	characteristic
7	C₅×#431	C₂²	1	2	yes	—
8	C₅×#376	C₂²	1	1	yes	characteristic
9	#65213	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#168	—	2	1	no	—
11	C₂×C₄⋊#168	—	2	1	no	—
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	#64743	—	2	1	no	—
14	#64743	—	2	1	no	—
15	#64702	—	2	2	no	—
16	#64701	—	2	2	no	—
17	#10628	—	5	1	no	maximal, Sylow
18	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
21	#4072	—	2	1	no	—
22	#4070	—	2	2	no	—
23	C₂×#681	—	2	1	no	—
24	C₂²×#168	—	2	2	no	—
25	#4016	—	2	1	no	—
26	#4015	D₄	1	2	yes	—
27	C₄×Dic₁₀	—	2	2	no	—
28	C₅×C₄⋊C₈	—	2	2	no	—
29	C₅×C₂²⋊C₈	—	2	2	no	—
30	C₄×C₅×C₈	—	2	2	no	—
31	C₅×#95	—	2	2	no	—
32	C₄⋊#168	—	2	2	no	—
33	C₄⋊#168	—	2	2	no	—
34	C₄⋊#168	—	2	2	no	—
35	C₄⋊#168	—	4	2	no	—
36	C₄⋊#168	—	4	2	no	—
37	#1995	—	5	1	no	—
38	#1843	—	5	1	no	—
39	#1542	—	5	2	no	—
40	#1301	—	5	2	no	—
41	#1274	D₅	1	1	yes	characteristic
42	C₂²×C₄×C₅	—	2	1	no	—
43	C₂²×C₄×C₅	—	2	1	no	—
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	#41	1	2	yes	—
46	C₂×Dic₁₀	—	2	2	no	—
47	C₂×C₅×C₈	—	4	2	no	—
48	C₄²×C₅	—	2	1	no	—
49	C₄²×C₅	—	2	1	no	—
50	C₄²×C₅	SD₁₆	1	2	yes	—
51	C₄²×C₅	Q₁₆	1	2	yes	—
52	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅	#41	1	2	yes	—
54	C₄²×C₅	#41	1	1	yes	characteristic
55	C₂²⋊Dic₅	—	4	1	no	—
56	C₄⋊Dic₅	—	2	2	no	—
57	C₄⋊Dic₅	—	2	2	no	—
58	#683	—	4	2	no	—
59	C₄×Dic₅	—	4	1	no	—
60	#681	—	4	1	no	—
61	#681	—	4	1	no	—
62	C₂×#168	—	4	2	no	—
63	C₂×#168	—	4	1	no	—
64	C₂×#168	—	4	2	no	—
65	C₂×#168	—	4	2	no	—
66	#533	—	5	1	no	—
67	#431	C₂×D₅	1	2	yes	—
68	#423	—	5	1	no	—
69	C₂×C₄⋊C₈	—	5	2	no	—
70	C₂×C₄⋊C₈	—	10	1	no	—
71	C₂×C₄⋊C₈	—	10	1	no	—
72	#376	C₂×D₅	1	1	yes	characteristic
73	#339	—	10	1	no	—
74	#339	—	10	1	no	—
75	#333	—	10	2	no	—
76	#332	—	10	2	no	—
77	C₂³×C₅	#123	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	4	2	no	—
81	C₂×C₄×C₅	—	4	2	no	—
82	C₂×C₄×C₅	—	4	2	no	—
83	C₂×C₄×C₅	#121	1	1	yes	characteristic
84	C₂×C₄×C₅	#123	1	2	yes	—
85	C₂×C₄×C₅	#124	1	1	yes	characteristic
86	C₂×C₄×C₅	#126	1	2	yes	—
87	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×C₄×C₅	#137	1	1	yes	characteristic
91	C₂×Dic₅	—	4	2	no	—
92	Dic₁₀	—	8	2	no	—
93	C₅×C₈	—	8	2	no	—
94	#168	—	8	2	no	—
95	#168	—	8	1	no	—
96	C₂×M₁₆	—	10	1	no	—
97	C₂²×C₈	—	10	2	no	—
98	C₄⋊Q₈	—	5	2	no	—
99	#125	—	10	1	no	—
100	C₂²⋊Q₈	—	10	2	no	—
101	C₄×Q₈	—	10	2	no	—
102	#117	—	10	1	no	—
103	C₂×C₄²	D₂₀	1	1	yes	characteristic
104	C₂×C₄²	D₂₀	1	1	yes	characteristic
105	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	10	2	no	—
108	C₄⋊C₈	—	10	2	no	—
109	C₄⋊C₈	—	10	2	no	—
110	C₄⋊C₈	—	20	2	no	—
111	C₄⋊C₈	—	20	2	no	—
112	C₂²⋊C₈	—	2	2	no	—
113	C₄×C₈	—	2	2	no	—
114	#95	—	2	2	no	—
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	#397	1	1	yes	characteristic
117	C₂²×C₅	#483	1	1	yes	characteristic
118	C₂²×C₅	#484	1	1	yes	characteristic
119	C₄×C₅	—	2	2	no	—
120	C₄×C₅	—	2	2	no	—
121	C₄×C₅	—	4	2	no	—
122	C₄×C₅	—	4	2	no	—
123	C₄×C₅	#460	1	1	yes	characteristic
124	C₄×C₅	#462	1	1	yes	characteristic
125	C₄×C₅	#487	1	1	yes	characteristic
126	C₄×C₅	#489	1	1	yes	characteristic
127	Dic₅	—	8	2	no	—
128	C₂×Q₈	—	10	2	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
132	C₂²×C₄	#711	1	2	yes	—
133	M₁₆	—	20	1	no	—
134	M₁₆	—	20	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	20	1	no	—
137	C₂×C₈	—	20	2	no	—
138	C₂×C₈	—	20	2	no	—
139	C₂×C₈	—	20	2	no	—
140	C₄⋊C₄	—	10	2	no	—
141	C₄⋊C₄	—	10	2	no	—
142	C₄⋊C₄	—	20	2	no	—
143	C₂²⋊C₄	—	20	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	1	no	—
146	C₄²	—	20	1	no	—
147	C₄²	#677	1	2	yes	—
148	C₄²	Dic₂₀	1	2	yes	—
149	C₄²	#709	1	2	yes	—
150	C₄²	D₄×D₅	1	1	yes	characteristic
151	C₄²	#713	1	1	yes	characteristic
152	C₂×C₅	—	2	1	no	—
153	C₂×C₅	—	2	1	no	—
154	C₂×C₅	#2072	1	1	yes	characteristic
155	C₂×C₅	#2080	1	2	yes	—
156	C₂³	#4032	1	1	yes	characteristic
157	Q₈	—	40	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	20	2	no	—
164	C₂×C₄	#4021	1	1	yes	characteristic
165	C₂×C₄	#4029	1	2	yes	—
166	C₂×C₄	#4041	1	1	yes	characteristic
167	C₂×C₄	#4044	1	2	yes	—
168	C₂×C₄	C₂×#677	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
170	C₂×C₄	#4054	1	1	yes	characteristic
171	C₂×C₄	#4055	1	1	yes	characteristic
172	C₈	—	8	2	no	—
173	C₈	—	40	2	no	—
174	C₈	—	40	1	no	—
175	C₅	#10628	1	1	yes	characteristic, complemented, Sylow
176	C₂²	—	2	2	no	—
177	C₂²	#64963	1	1	yes	characteristic
178	C₂²	#65010	1	1	yes	characteristic
179	C₂²	#65011	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	40	2	no	—
185	C₄	#64970	1	1	yes	characteristic
186	C₄	#65003	1	1	yes	characteristic
187	C₄	#65137	1	1	yes	characteristic
188	C₄	#65138	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#10599429	1	1	yes	characteristic
192	C₂	#10599493	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10601569

C₂

yes

maximal

#10600234

C₂

yes

maximal, characteristic

#10600232

C₂

yes

maximal, characteristic

C₅×#1274

C₂

yes

maximal, characteristic

#10598236

C₂

yes

maximal

#65804

C₂²

yes

characteristic

C₅×#431

C₂²

yes

—

C₅×#376

C₂²

yes

characteristic

#65213

C₂²

yes

characteristic

C₂×C₄⋊#168

—

C₂×C₄⋊#168

—

C₂×C₄⋊#168

C₂²

yes

—

—

—

—

—

—

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄²×C₅

C₂³

yes

—

—

—

C₂²×#168

—

#4016

—

#4015

D₄

yes

—

C₄×Dic₁₀

—

C₅×C₄⋊C₈

—

C₅×C₂²⋊C₈

—

C₄×C₅×C₈

—

C₅×#95

—

—

—

—

—

—

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

SD₁₆

yes

—

C₄²×C₅

Q₁₆

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

#41

yes

—

C₄²×C₅

#41

yes

characteristic

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

—

—

—

—

—

—

—

C₂×D₅

yes

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

#376

C₂×D₅

yes

—

—

—

—

C₂³×C₅

#123

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

#124

yes

characteristic

C₂×C₄×C₅

#126

yes

—

C₂×C₄×C₅

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅

C₂×Q₁₆

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

—

103

C₂×C₄²

D₂₀

yes

characteristic

104

C₂×C₄²

D₂₀

yes

characteristic

105

C₂×C₄²

C₂²×D₅

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₂²⋊C₈

—

113

C₄×C₈

—

114

#95

—

115

C₂²×C₅

—

116

C₂²×C₅

#397

yes

characteristic

117

C₂²×C₅

#483

yes

characteristic

118

C₂²×C₅

#484

yes

characteristic

119

C₄×C₅

—

120

C₄×C₅

—

121

C₄×C₅

—

122

C₄×C₅

—

123

C₄×C₅

#460

yes

characteristic

124

C₄×C₅

#462

yes

characteristic

125

C₄×C₅

#487

yes

characteristic

126

C₄×C₅

#489

yes

characteristic

127

Dic₅

—

128

C₂×Q₈

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂×D₂₀

yes

characteristic

132

C₂²×C₄

#711

yes

—

133

M₁₆

—

134

M₁₆

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

#677

yes

—

148

C₄²

Dic₂₀

yes

—

149

C₄²

#709

yes

—

150

C₄²

D₄×D₅

yes

characteristic

151

C₄²

#713

yes

characteristic

152

C₂×C₅

—

153

C₂×C₅

—

154

C₂×C₅

#2072

yes

characteristic

155

C₂×C₅

#2080

yes

—

156

C₂³

#4032

yes

characteristic

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#4021

yes

characteristic

165

C₂×C₄

#4029

yes

—

166

C₂×C₄

#4041

yes

characteristic

167

C₂×C₄

#4044

yes

—

168

C₂×C₄

C₂×#677

yes

characteristic

169

C₂×C₄

C₂×Dic₂₀

yes

characteristic

170

C₂×C₄

#4054

yes

characteristic

171

C₂×C₄

#4055

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

175

C₅

#10628

yes

characteristic, complemented, Sylow

176

C₂²

—

177

C₂²

#64963

yes

characteristic

178

C₂²

#65010

yes

characteristic

179

C₂²

#65011

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

#64970

yes

characteristic

186

C₄

#65003

yes

characteristic

187

C₄

#65137

yes

characteristic

188

C₄

#65138

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#10599429

yes

characteristic

192

C₂

#10599493

yes

—

	subgroup	G/H
center	C₂×C₄	#4021
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1274	C₂
socle	C₂²×C₅	#397
prime radical	C₂³×C₅	#123
hypercenter	#1274	D₅
hypocenter	C₅	#10628
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#4021

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 ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
175. C₅	#10628	semidirect

H ≅ G/N

type

175. C₅

#10628

semidirect

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

subgroup	index
#10601569	2
#10600234	2
#10600232	2
C₅×#1274	2
#10598236	2
#10628	5

subgroup

index

Subgroups — #12004075

Conjugacy classes of subgroups

Complements

Maximal subgroups