Group #11998382 — subgroups

Subgroups — #11998382

Order 1280.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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	C₅×#2868	C₂	1	1	yes	maximal, characteristic
2	C₂×#64766	C₂	1	4	yes	maximal
3	#10600185	C₂	1	1	yes	maximal, characteristic
4	#10600184	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×C₄⋊Q₈	C₄	1	4	yes	—
6	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₅×#384	—	2	1	no	—
8	C₅×#384	—	2	2	no	—
9	C₅×#384	—	2	1	no	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#168	C₂²	1	2	yes	—
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	C₂×#3935	—	2	1	no	—
14	#64766	—	2	8	no	—
15	#64744	—	2	1	no	—
16	#64744	—	2	2	no	—
17	#9702	—	5	1	no	maximal, Sylow
18	C₂²×C₅×Q₈	C₂×C₄	1	4	yes	—
19	C₅×C₄⋊Q₈	—	2	4	no	—
20	C₅×C₄⋊Q₈	—	2	4	no	—
21	C₅×C₄⋊Q₈	—	4	8	no	—
22	C₂×C₅×C₄⋊C₄	—	2	4	no	—
23	C₂×C₅×C₄⋊C₄	—	2	4	no	—
24	C₂×C₅×C₄⋊C₄	—	2	2	no	—
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₂²×#168	—	2	2	no	—
32	C₂²×#168	—	2	2	no	—
33	C₄⋊#168	—	2	2	no	—
34	C₄⋊#168	—	2	4	no	—
35	C₄⋊#168	D₄	1	4	yes	—
36	#3935	—	4	2	no	—
37	#2868	D₅	1	1	yes	characteristic
38	#1831	—	5	1	no	—
39	#1767	—	5	1	no	—
40	C₂×#333	—	5	4	no	—
41	C₂×C₅×Q₈	—	2	8	no	—
42	C₂×C₅×Q₈	—	4	8	no	—
43	C₂²×C₄×C₅	—	2	4	no	—
44	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₅×C₄⋊C₄	—	4	8	no	—
50	C₅×C₄⋊C₄	—	4	8	no	—
51	C₅×C₄⋊C₄	—	4	4	no	—
52	C₅×C₄⋊C₄	—	4	2	no	—
53	C₅×C₄⋊C₄	—	4	2	no	—
54	C₄²×C₅	—	2	4	no	—
55	C₄²×C₅	—	2	2	no	—
56	C₄²×C₅	—	2	4	no	—
57	C₄²×C₅	SD₁₆	1	2	yes	—
58	C₄²×C₅	Q₁₆	1	2	yes	—
59	C₄²×C₅	C₂×D₄	1	2	yes	—
60	C₄²×C₅	#41	1	2	yes	—
61	C₂×#168	—	2	4	no	—
62	C₂×#168	—	4	2	no	—
63	C₂×#168	—	4	2	no	—
64	C₂×#168	—	4	2	no	—
65	C₂×#168	—	4	4	no	—
66	C₂×C₄⋊Q₈	Dic₅	1	4	yes	—
67	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
68	C₂×C₄⋊C₈	—	5	2	no	—
69	C₂×C₄⋊C₈	—	5	2	no	—
70	C₂×#97	—	10	1	no	—
71	#384	—	2	1	no	—
72	#384	—	2	2	no	—
73	#384	—	2	1	no	—
74	C₄³	C₂×D₅	1	1	yes	characteristic
75	#336	—	10	2	no	—
76	#336	—	10	1	no	—
77	#333	—	10	8	no	—
78	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₅×Q₈	—	8	8	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	—	2	2	no	—
85	C₂×C₄×C₅	—	4	4	no	—
86	C₂×C₄×C₅	—	4	8	no	—
87	C₂×C₄×C₅	#100	1	4	yes	—
88	C₂×C₄×C₅	#103	1	4	yes	—
89	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₅	#121	1	2	yes	—
91	C₂×C₄×C₅	#124	1	1	yes	characteristic
92	C₂×C₄×C₅	#127	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅	#137	1	2	yes	—
96	C₂×C₄×C₅	#137	1	4	yes	—
97	#168	—	4	4	no	—
98	#168	—	8	2	no	—
99	C₂²×Q₈	C₂×Dic₅	1	4	yes	—
100	C₂²×C₈	—	10	2	no	—
101	C₂²×C₈	—	10	2	no	—
102	C₄⋊Q₈	—	2	4	no	—
103	C₄⋊Q₈	—	2	4	no	—
104	C₄⋊Q₈	—	4	8	no	—
105	C₂×C₄⋊C₄	—	2	4	no	—
106	C₂×C₄⋊C₄	—	2	4	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄²	C₂×Dic₅	1	2	yes	—
110	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
111	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
112	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
113	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
114	C₄⋊C₈	—	5	4	no	—
115	C₄⋊C₈	—	10	4	no	—
116	C₄⋊C₈	—	10	2	no	—
117	#97	—	20	2	no	—
118	C₂²×C₅	#390	1	1	yes	characteristic
119	C₂²×C₅	C₂×#100	1	2	yes	—
120	C₂²×C₅	C₂×#103	1	1	yes	characteristic
121	C₂²×C₅	#419	1	2	yes	—
122	C₂²×C₅	#419	1	1	yes	characteristic
123	C₄×C₅	—	2	4	no	—
124	C₄×C₅	—	2	4	no	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	8	4	no	—
127	C₄×C₅	#461	1	2	yes	—
128	C₄×C₅	#462	1	2	yes	—
129	C₄×C₅	#491	1	2	yes	—
130	C₄×C₅	#497	1	2	yes	—
131	C₂×Q₈	—	2	8	no	—
132	C₂×Q₈	—	4	8	no	—
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
135	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
136	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
139	C₂×C₈	—	10	4	no	—
140	C₂×C₈	—	20	2	no	—
141	C₂×C₈	—	20	2	no	—
142	C₂×C₈	—	20	2	no	—
143	C₂×C₈	—	20	4	no	—
144	C₄⋊C₄	—	4	8	no	—
145	C₄⋊C₄	—	4	8	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	2	no	—
152	C₄²	#688	1	2	yes	—
153	C₄²	#689	1	2	yes	—
154	C₄²	D₄×D₅	1	2	yes	—
155	C₄²	#711	1	2	yes	—
156	C₂×C₅	#1548	1	2	yes	—
157	C₂×C₅	#1558	1	2	yes	—
158	C₂×C₅	#1960	1	1	yes	characteristic
159	C₂×C₅	#1965	1	1	yes	characteristic
160	C₂×C₅	#1981	1	1	yes	characteristic
161	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
162	Q₈	—	8	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	8	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	#3965	1	4	yes	—
171	C₂×C₄	#3967	1	4	yes	—
172	C₂×C₄	D₄×Dic₅	1	2	yes	—
173	C₂×C₄	#4082	1	1	yes	characteristic
174	C₂×C₄	#4084	1	2	yes	—
175	C₂×C₄	#4086	1	1	yes	characteristic
176	C₂×C₄	C₂×#688	1	1	yes	characteristic
177	C₂×C₄	#4088	1	4	yes	—
178	C₂×C₄	#4088	1	2	yes	—
179	C₂×C₄	C₂×#689	1	1	yes	characteristic
180	C₈	—	20	4	no	—
181	C₈	—	40	2	no	—
182	C₅	#9702	1	1	yes	characteristic, complemented, Sylow
183	C₂²	C₂×#3965	1	2	yes	—
184	C₂²	#65510	1	1	yes	characteristic
185	C₂²	C₂×#3967	1	1	yes	characteristic
186	C₂²	#65514	1	2	yes	—
187	C₂²	#65514	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	8	4	no	—
192	C₄	#65371	1	2	yes	—
193	C₄	#65372	1	2	yes	—
194	C₄	#65375	1	2	yes	—
195	C₄	#65379	1	2	yes	—
196	C₂	#10602559	1	1	yes	characteristic
197	C₂	#10602567	1	2	yes	—
198	C₂	#10602570	1	2	yes	—
199	C₂	#10602571	1	1	yes	characteristic
200	C₂	#10602579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2868

C₂

yes

maximal, characteristic

C₂×#64766

C₂

yes

maximal

#10600185

C₂

yes

maximal, characteristic

#10600184

C₂

yes

maximal, characteristic

C₂×C₅×C₄⋊Q₈

C₄

yes

—

C₂×C₅×C₄⋊Q₈

C₂²

yes

—

C₅×#384

—

C₅×#384

—

C₅×#384

—

C₄³×C₅

C₂²

yes

characteristic

C₂×C₄⋊#168

C₂²

yes

—

C₂×C₄⋊#168

C₂²

yes

—

—

—

—

—

—

C₂²×C₅×Q₈

C₂×C₄

yes

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂×C₄

yes

—

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂²×#168

—

C₂²×#168

—

C₄⋊#168

—

C₄⋊#168

—

C₄⋊#168

D₄

yes

—

#3935

—

#2868

D₅

yes

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

SD₁₆

yes

—

C₄²×C₅

Q₁₆

yes

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

#41

yes

—

—

—

—

—

—

C₂×C₄⋊Q₈

Dic₅

yes

—

C₂×C₄⋊Q₈

C₂×D₅

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#384

—

#384

—

#384

—

C₄³

C₂×D₅

yes

—

—

—

C₂³×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#100

yes

—

C₂×C₄×C₅

#103

yes

—

C₂×C₄×C₅

C₄×D₄

yes

—

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅

#137

yes

—

C₂×C₄×C₅

#137

yes

—

#168

—

#168

—

C₂²×Q₈

C₂×Dic₅

yes

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

C₂×Dic₅

yes

—

110

C₂×C₄²

C₂²⋊D₅

yes

characteristic

111

C₂×C₄²

C₂²⋊D₅

yes

—

112

C₂×C₄²

C₂²⋊D₅

yes

characteristic

113

C₂×C₄²

C₂²×D₅

yes

characteristic

114

C₄⋊C₈

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

#97

—

118

C₂²×C₅

#390

yes

characteristic

119

C₂²×C₅

C₂×#100

yes

—

120

C₂²×C₅

C₂×#103

yes

characteristic

121

C₂²×C₅

#419

yes

—

122

C₂²×C₅

#419

yes

characteristic

123

C₄×C₅

—

124

C₄×C₅

—

125

C₄×C₅

—

126

C₄×C₅

—

127

C₄×C₅

#461

yes

—

128

C₄×C₅

#462

yes

—

129

C₄×C₅

#491

yes

—

130

C₄×C₅

#497

yes

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂²×C₄

—

134

C₂²×C₄

C₂²⋊Dic₅

yes

—

135

C₂²×C₄

C₂²⋊Dic₅

yes

—

136

C₂²×C₄

C₂²×Dic₅

yes

characteristic

137

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

138

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

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

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

#688

yes

—

153

C₄²

#689

yes

—

154

C₄²

D₄×D₅

yes

—

155

C₄²

#711

yes

—

156

C₂×C₅

#1548

yes

—

157

C₂×C₅

#1558

yes

—

158

C₂×C₅

#1960

yes

characteristic

159

C₂×C₅

#1965

yes

characteristic

160

C₂×C₅

#1981

yes

characteristic

161

C₂³

C₂×C₂²⋊Dic₅

yes

characteristic

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#3965

yes

—

171

C₂×C₄

#3967

yes

—

172

C₂×C₄

D₄×Dic₅

yes

—

173

C₂×C₄

#4082

yes

characteristic

174

C₂×C₄

#4084

yes

—

175

C₂×C₄

#4086

yes

characteristic

176

C₂×C₄

C₂×#688

yes

characteristic

177

C₂×C₄

#4088

yes

—

178

C₂×C₄

#4088

yes

—

179

C₂×C₄

C₂×#689

yes

characteristic

180

C₈

—

181

C₈

—

182

C₅

#9702

yes

characteristic, complemented, Sylow

183

C₂²

C₂×#3965

yes

—

184

C₂²

#65510

yes

characteristic

185

C₂²

C₂×#3967

yes

characteristic

186

C₂²

#65514

yes

—

187

C₂²

#65514

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#65371

yes

—

193

C₄

#65372

yes

—

194

C₄

#65375

yes

—

195

C₄

#65379

yes

—

196

C₂

#10602559

yes

characteristic

197

C₂

#10602567

yes

—

198

C₂

#10602570

yes

—

199

C₂

#10602571

yes

characteristic

200

C₂

#10602579

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2868	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2868	D₅
hypocenter	C₅	#9702
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₅×#2868

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 ◁ #2868

N	H ≅ G/N	type
182. C₅	#9702	semidirect

H ≅ G/N

type

182. C₅

#9702

semidirect

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

subgroup	index
C₅×#2868	2
C₂×#64766	2
#10600185	2
#10600184	2
#9702	5

subgroup

index

Subgroups — #11998382

Conjugacy classes of subgroups

Complements

Maximal subgroups