Group #11990226 — subgroups

Subgroups — #11990226

Order 1280.

444 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₂×#64763	C₂	1	2	yes	maximal
3	#10600182	C₂	1	2	yes	maximal
4	#10598121	C₂	1	2	yes	maximal
5	C₂×C₅×#125	C₂²	1	1	yes	characteristic
6	C₅×#389	C₄	1	2	yes	—
7	C₅×#387	—	2	2	no	—
8	C₅×#382	C₂²	1	1	yes	characteristic
9	C₄×C₅×C₄⋊C₄	—	2	2	no	—
10	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
11	C₄³×C₅	C₂²	1	1	yes	characteristic
12	C₂×#3935	C₂²	1	4	yes	—
13	#64763	C₄	1	8	yes	complemented
14	#64744	—	2	2	no	—
15	#64744	—	2	2	no	—
16	#8225	—	5	1	no	maximal, Sylow
17	C₅×#125	—	2	2	no	—
18	C₅×#125	C₂×C₄	1	4	yes	—
19	C₂×C₅×C₄⋊C₄	—	2	2	no	—
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
22	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	—	2	2	no	—
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	D₄	1	2	yes	—
27	C₂×C₄²×C₅	D₄	1	2	yes	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂²×#168	—	2	4	no	—
30	C₅×#95	—	2	2	no	—
31	C₅×#95	—	2	2	no	—
32	C₅×#95	—	2	2	no	—
33	#3935	—	2	4	no	—
34	#3935	C₂×C₄	1	8	yes	—
35	C₄×#125	D₅	1	1	yes	characteristic
36	#1745	—	5	2	no	—
37	C₂×#330	—	5	2	no	—
38	#1295	—	5	2	no	—
39	C₂²×C₄×C₅	—	2	2	no	—
40	C₂²×C₄×C₅	—	2	2	no	—
41	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
46	C₅×C₄⋊C₄	—	2	2	no	—
47	C₅×C₄⋊C₄	—	2	8	no	—
48	C₅×C₄⋊C₄	—	4	2	no	—
49	C₅×C₄⋊C₄	—	4	4	no	—
50	C₅×C₄⋊C₄	C₄²	1	4	yes	—
51	C₄²×C₅	—	2	2	no	—
52	C₄²×C₅	—	4	1	no	—
53	C₄²×C₅	—	4	2	no	—
54	C₄²×C₅	—	4	2	no	—
55	C₄²×C₅	—	4	4	no	—
56	C₄²×C₅	C₂²×C₄	1	2	yes	—
57	C₄²×C₅	#41	1	2	yes	—
58	C₂×#168	—	2	8	no	—
59	C₂×#168	—	4	4	no	—
60	C₂×#168	—	4	4	no	—
61	C₂×#125	C₂×D₅	1	1	yes	characteristic
62	C₂×#97	—	5	4	no	—
63	#389	Dic₅	1	2	yes	—
64	#387	—	2	2	no	—
65	#382	C₂×D₅	1	1	yes	characteristic
66	C₄×C₄⋊C₄	—	2	2	no	—
67	C₄×C₄⋊C₄	Dic₅	1	2	yes	—
68	C₄³	C₂×D₅	1	1	yes	characteristic
69	#336	—	10	2	no	—
70	#336	—	10	2	no	—
71	#330	—	5	8	no	—
72	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅	—	2	1	no	—
74	C₂×C₄×C₅	—	2	1	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	4	no	—
79	C₂×C₄×C₅	—	2	1	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	4	2	no	—
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	—	4	2	no	—
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	—	4	2	no	—
89	C₂×C₄×C₅	#101	1	2	yes	—
90	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₅	#117	1	1	yes	characteristic
92	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
93	#168	—	4	8	no	—
94	C₂²×C₈	—	10	4	no	—
95	#125	—	2	2	no	—
96	#125	C₄×D₅	1	4	yes	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	—	2	2	no	—
99	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
100	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
101	C₂×C₄²	—	2	2	no	—
102	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
103	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
105	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
106	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
107	#97	—	5	8	no	—
108	#97	—	10	4	no	—
109	#95	—	2	2	no	—
110	#95	—	2	2	no	—
111	#95	—	2	2	no	—
112	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₅	C₂×#101	1	1	yes	characteristic
114	C₂²×C₅	#413	1	1	yes	characteristic
115	C₂²×C₅	#421	1	2	yes	—
116	C₂²×C₅	#421	1	2	yes	—
117	C₄×C₅	—	2	1	no	—
118	C₄×C₅	—	2	1	no	—
119	C₄×C₅	—	2	4	no	—
120	C₄×C₅	—	2	2	no	—
121	C₄×C₅	—	4	1	no	—
122	C₄×C₅	—	4	4	no	—
123	C₄×C₅	—	4	2	no	—
124	C₄×C₅	—	8	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
128	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
129	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
130	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
132	C₂×C₈	—	10	8	no	—
133	C₂×C₈	—	20	4	no	—
134	C₂×C₈	—	20	4	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	8	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	C₄×Dic₅	1	4	yes	—
140	C₄²	—	2	2	no	—
141	C₄²	—	4	1	no	—
142	C₄²	—	4	2	no	—
143	C₄²	—	4	2	no	—
144	C₄²	—	4	4	no	—
145	C₄²	C₂×C₄×D₅	1	2	yes	—
146	C₄²	#709	1	2	yes	—
147	C₂×C₅	#1534	1	1	yes	characteristic
148	C₂×C₅	#1535	1	2	yes	—
149	C₂×C₅	#1537	1	1	yes	characteristic
150	C₂×C₅	C₄×#101	1	1	yes	characteristic
151	C₂×C₅	#1759	1	2	yes	—
152	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	#3968	1	2	yes	—
170	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
171	C₂×C₄	#4072	1	1	yes	characteristic
172	C₂×C₄	C₄×C₂²⋊D₅	1	4	yes	—
173	C₈	—	20	8	no	—
174	C₅	#8225	1	1	yes	characteristic, complemented, Sylow
175	C₂²	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
176	C₂²	C₂×#3968	1	1	yes	characteristic
177	C₂²	#65525	1	2	yes	—
178	C₂²	#65525	1	2	yes	—
179	C₂²	#65526	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	4	no	—
187	C₄	—	8	2	no	—
188	C₂	C₄×#3968	1	1	yes	characteristic
189	C₂	#10602436	1	2	yes	—
190	C₂	#10602529	1	1	yes	characteristic
191	C₂	#10602542	1	2	yes	—
192	C₂	#10602616	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅×#125

C₂

yes

maximal, characteristic

C₂×#64763

C₂

yes

maximal

#10600182

C₂

yes

maximal

#10598121

C₂

yes

maximal

C₂×C₅×#125

C₂²

yes

characteristic

C₅×#389

C₄

yes

—

C₅×#387

—

C₅×#382

C₂²

yes

characteristic

C₄×C₅×C₄⋊C₄

—

C₄×C₅×C₄⋊C₄

C₄

yes

—

C₄³×C₅

C₂²

yes

characteristic

C₂×#3935

C₂²

yes

—

#64763

C₄

yes

—

—

—

—

C₂×C₄

yes

—

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

—

#3935

—

#3935

C₂×C₄

yes

—

C₄×#125

D₅

yes

—

—

—

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

C₄²

yes

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂²×C₄

yes

—

C₄²×C₅

#41

yes

—

—

—

—

C₂×D₅

yes

characteristic

C₂×#97

—

#389

Dic₅

yes

—

#387

—

#382

C₂×D₅

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₅

yes

—

C₄³

C₂×D₅

yes

—

—

—

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

—

C₂×C₄×C₅

#101

yes

—

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

#117

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

—

#168

—

C₂²×C₈

—

#125

—

#125

C₄×D₅

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₅

yes

—

100

C₂×C₄⋊C₄

C₂×Dic₅

yes

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×Dic₅

yes

characteristic

103

C₂×C₄²

C₂×Dic₅

yes

characteristic

104

C₂×C₄²

C₂²⋊D₅

yes

—

105

C₂×C₄²

C₂²⋊D₅

yes

—

106

C₂×C₄²

C₂²×D₅

yes

characteristic

107

#97

—

108

#97

—

109

#95

—

110

#95

—

111

#95

—

112

C₂²×C₅

C₄×C₂²⋊C₄

yes

characteristic

113

C₂²×C₅

C₂×#101

yes

characteristic

114

C₂²×C₅

#413

yes

characteristic

115

C₂²×C₅

#421

yes

—

116

C₂²×C₅

#421

yes

—

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

—

120

C₄×C₅

—

121

C₄×C₅

—

122

C₄×C₅

—

123

C₄×C₅

—

124

C₄×C₅

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂²⋊Dic₅

yes

characteristic

128

C₂²×C₄

C₂²⋊Dic₅

yes

—

129

C₂²×C₄

C₂²⋊Dic₅

yes

characteristic

130

C₂²×C₄

C₂²×Dic₅

yes

characteristic

131

C₂²×C₄

C₂×C₂²⋊D₅

yes

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

C₄×Dic₅

yes

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

C₂×C₄×D₅

yes

—

146

C₄²

#709

yes

—

147

C₂×C₅

#1534

yes

characteristic

148

C₂×C₅

#1535

yes

—

149

C₂×C₅

#1537

yes

characteristic

150

C₂×C₅

C₄×#101

yes

characteristic

151

C₂×C₅

#1759

yes

—

152

C₂³

C₂×C₂²⋊Dic₅

yes

characteristic

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

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

#3968

yes

—

170

C₂×C₄

C₂×C₄×Dic₅

yes

characteristic

171

C₂×C₄

#4072

yes

characteristic

172

C₂×C₄

C₄×C₂²⋊D₅

yes

—

173

C₈

—

174

C₅

#8225

yes

characteristic, complemented, Sylow

175

C₂²

C₄×C₂²⋊Dic₅

yes

characteristic

176

C₂²

C₂×#3968

yes

characteristic

177

C₂²

#65525

yes

—

178

C₂²

#65525

yes

—

179

C₂²

#65526

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₂

C₄×#3968

yes

characteristic

189

C₂

#10602436

yes

—

190

C₂

#10602529

yes

characteristic

191

C₂

#10602542

yes

—

192

C₂

#10602616

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
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	#8225
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅

commutator

C₂×C₄×C₅

C₂×C₄²

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 ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
13. #64763	C₄	semidirect
174. C₅	#8225	semidirect

H ≅ G/N

type

13. #64763

C₄

semidirect

174. C₅

#8225

semidirect

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

subgroup	index
C₄×C₅×#125	2
C₂×#64763	2
#10600182	2
#10598121	2
#8225	5

subgroup

index

C₄×C₅×#125

Subgroups — #11990226

Conjugacy classes of subgroups

Complements

Maximal subgroups