Group #11987564 — subgroups

Subgroups — #11987564

Order 1280.

436 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₄×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	#10600184	C₂	1	2	yes	maximal
3	#10598129	C₂	1	4	yes	maximal
4	C₂×C₅×#125	C₄	1	1	yes	characteristic
5	C₅×#389	C₂²	1	2	yes	—
6	C₅×#387	—	2	2	no	—
7	C₅×#382	C₄	1	1	yes	characteristic
8	C₄×C₅×C₄⋊C₄	—	2	2	no	—
9	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#168	—	2	2	no	—
12	C₂×#3935	—	2	2	no	—
13	#64744	C₂²	1	4	yes	—
14	#7636	—	5	1	no	maximal, Sylow
15	C₅×#125	—	2	4	no	—
16	C₂×C₅×C₄⋊C₄	—	2	2	no	—
17	C₂×C₅×C₄⋊C₄	—	2	2	no	—
18	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
19	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅	—	2	2	no	—
21	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	D₄	1	2	yes	—
24	C₂×C₄²×C₅	D₄	1	2	yes	—
25	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
26	C₂²×#168	—	2	4	no	—
27	C₅×#95	—	2	2	no	—
28	C₅×#95	—	2	2	no	—
29	C₅×#95	—	2	2	no	—
30	C₄⋊#168	—	2	8	no	—
31	#3935	—	4	4	no	—
32	C₄×#125	D₅	1	1	yes	characteristic
33	#1767	—	5	2	no	—
34	#1271	—	5	4	no	—
35	C₂²×C₄×C₅	—	2	2	no	—
36	C₂²×C₄×C₅	—	2	2	no	—
37	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
42	C₅×C₄⋊C₄	—	2	8	no	—
43	C₅×C₄⋊C₄	—	2	2	no	—
44	C₅×C₄⋊C₄	—	4	2	no	—
45	C₅×C₄⋊C₄	—	4	2	no	—
46	C₅×C₄⋊C₄	—	4	2	no	—
47	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
48	C₄²×C₅	—	2	2	no	—
49	C₄²×C₅	—	2	4	no	—
50	C₄²×C₅	—	2	2	no	—
51	C₄²×C₅	—	2	2	no	—
52	C₄²×C₅	—	2	2	no	—
53	C₄²×C₅	—	4	1	no	—
54	C₄²×C₅	—	4	4	no	—
55	C₂×#168	—	2	8	no	—
56	C₂×#168	—	4	4	no	—
57	C₂×#168	—	4	4	no	—
58	C₂×#125	Dic₅	1	1	yes	characteristic
59	C₂×C₄⋊C₈	—	10	2	no	—
60	C₂×#97	—	10	2	no	—
61	#389	C₂×D₅	1	2	yes	—
62	#387	—	2	2	no	—
63	#382	Dic₅	1	1	yes	characteristic
64	C₄×C₄⋊C₄	—	2	2	no	—
65	C₄×C₄⋊C₄	Dic₅	1	2	yes	—
66	C₄³	C₂×D₅	1	1	yes	characteristic
67	#336	—	5	4	no	—
68	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	—	2	1	no	—
71	C₂×C₄×C₅	—	2	1	no	—
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	4	no	—
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	4	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	—	4	2	no	—
84	C₂×C₄×C₅	#101	1	2	yes	—
85	C₂×C₄×C₅	#104	1	4	yes	—
86	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×C₄×C₅	#137	1	2	yes	—
89	#168	—	4	8	no	—
90	C₂²×C₈	—	10	4	no	—
91	#125	—	2	4	no	—
92	C₂×C₄⋊C₄	—	2	2	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
95	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
96	C₂×C₄²	—	2	2	no	—
97	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
98	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
100	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
101	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
102	C₄⋊C₈	—	10	8	no	—
103	#97	—	20	4	no	—
104	#95	—	2	2	no	—
105	#95	—	2	2	no	—
106	#95	—	2	2	no	—
107	C₂²×C₅	#407	1	1	yes	characteristic
108	C₂²×C₅	#410	1	1	yes	characteristic
109	C₂²×C₅	C₂×#101	1	1	yes	characteristic
110	C₂²×C₅	#417	1	2	yes	—
111	C₂²×C₅	C₂×#104	1	2	yes	—
112	C₄×C₅	—	2	2	no	—
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	—	2	2	no	—
115	C₄×C₅	—	4	1	no	—
116	C₄×C₅	—	4	2	no	—
117	C₄×C₅	—	4	4	no	—
118	C₄×C₅	—	8	2	no	—
119	C₄×C₅	#330	1	4	yes	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
123	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
124	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
125	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
127	C₂×C₈	—	10	8	no	—
128	C₂×C₈	—	20	4	no	—
129	C₂×C₈	—	20	4	no	—
130	C₄⋊C₄	—	2	8	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	#681	1	4	yes	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	4	1	no	—
142	C₄²	—	4	4	no	—
143	C₂×C₅	#1467	1	1	yes	characteristic
144	C₂×C₅	#1489	1	2	yes	—
145	C₂×C₅	#1509	1	2	yes	—
146	C₂×C₅	C₂×#330	1	1	yes	characteristic
147	C₂×C₅	#1561	1	1	yes	characteristic
148	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	#3968	1	2	yes	—
165	C₂×C₄	#3969	1	4	yes	—
166	C₂×C₄	C₂×#681	1	2	yes	—
167	C₂×C₄	#4095	1	2	yes	—
168	C₂×C₄	#4097	1	2	yes	—
169	C₈	—	20	8	no	—
170	C₅	#7636	1	1	yes	characteristic, complemented, Sylow
171	C₂²	#65496	1	1	yes	characteristic
172	C₂²	C₂×#3968	1	1	yes	characteristic
173	C₂²	#65521	1	2	yes	—
174	C₂²	C₂×#3969	1	2	yes	—
175	C₂²	#65529	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	4	1	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	4	no	—
182	C₄	—	8	2	no	—
183	C₄	#64763	1	4	yes	—
184	C₂	#10602418	1	1	yes	characteristic
185	C₂	#10602434	1	2	yes	—
186	C₂	C₂×#64763	1	1	yes	characteristic
187	C₂	#10602548	1	2	yes	—
188	C₂	#10602627	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅×#125

C₂

yes

maximal, characteristic

#10600184

C₂

yes

maximal

#10598129

C₂

yes

maximal

C₂×C₅×#125

C₄

yes

characteristic

C₅×#389

C₂²

yes

—

C₅×#387

—

C₅×#382

C₄

yes

characteristic

C₄×C₅×C₄⋊C₄

—

C₄×C₅×C₄⋊C₄

C₄

yes

—

C₄³×C₅

C₂²

yes

characteristic

C₂×C₄⋊#168

—

C₂×#3935

—

#64744

C₂²

yes

—

#7636

—

maximal, Sylow

C₅×#125

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₅×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₅

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂²×#168

—

C₅×#95

—

C₅×#95

—

C₅×#95

—

C₄⋊#168

—

#3935

—

C₄×#125

D₅

yes

characteristic

#1767

—

#1271

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

M₁₆

yes

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

—

—

—

Dic₅

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#97

—

#389

C₂×D₅

yes

—

#387

—

#382

Dic₅

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₅

yes

—

C₄³

C₂×D₅

yes

characteristic

#336

—

C₂³×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#101

yes

—

C₂×C₄×C₅

#104

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅

#137

yes

—

#168

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₅

yes

—

C₂×C₄⋊C₄

C₂×Dic₅

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₅

yes

characteristic

C₂×C₄²

C₂×Dic₅

yes

characteristic

C₂×C₄²

C₂²⋊D₅

yes

—

100

C₂×C₄²

C₂²⋊D₅

yes

—

101

C₂×C₄²

C₂²×D₅

yes

characteristic

102

C₄⋊C₈

—

103

#97

—

104

#95

—

105

#95

—

106

#95

—

107

C₂²×C₅

#407

yes

characteristic

108

C₂²×C₅

#410

yes

characteristic

109

C₂²×C₅

C₂×#101

yes

characteristic

110

C₂²×C₅

#417

yes

—

111

C₂²×C₅

C₂×#104

yes

—

112

C₄×C₅

—

113

C₄×C₅

—

114

C₄×C₅

—

115

C₄×C₅

—

116

C₄×C₅

—

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

#330

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂²⋊Dic₅

yes

characteristic

123

C₂²×C₄

C₂²⋊Dic₅

yes

—

124

C₂²×C₄

C₂²⋊Dic₅

yes

characteristic

125

C₂²×C₄

C₂²×Dic₅

yes

characteristic

126

C₂²×C₄

C₂×C₂²⋊D₅

yes

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

#681

yes

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₂×C₅

#1467

yes

characteristic

144

C₂×C₅

#1489

yes

—

145

C₂×C₅

#1509

yes

—

146

C₂×C₅

C₂×#330

yes

characteristic

147

C₂×C₅

#1561

yes

characteristic

148

C₂³

C₂×C₂²⋊Dic₅

yes

characteristic

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#3968

yes

—

165

C₂×C₄

#3969

yes

—

166

C₂×C₄

C₂×#681

yes

—

167

C₂×C₄

#4095

yes

—

168

C₂×C₄

#4097

yes

—

169

C₈

—

170

C₅

#7636

yes

characteristic, complemented, Sylow

171

C₂²

#65496

yes

characteristic

172

C₂²

C₂×#3968

yes

characteristic

173

C₂²

#65521

yes

—

174

C₂²

C₂×#3969

yes

—

175

C₂²

#65529

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

#64763

yes

—

184

C₂

#10602418

yes

characteristic

185

C₂

#10602434

yes

—

186

C₂

C₂×#64763

yes

characteristic

187

C₂

#10602548

yes

—

188

C₂

#10602627

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×#125	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×#125	D₅
hypocenter	C₅	#7636
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₄×C₅×#125

C₂

socle, prime radical

C₂³×C₅

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
170. C₅	#7636	semidirect

H ≅ G/N

type

170. C₅

#7636

semidirect

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

subgroup	index
C₄×C₅×#125	2
#10600184	2
#10598129	2
#7636	5

subgroup

index

C₄×C₅×#125

#10600184

#10598129

#7636

Subgroups — #11987564

Conjugacy classes of subgroups

Complements

Maximal subgroups