Group #11998380 — subgroups

Subgroups — #11998380

Order 1280.

538 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 167 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	#10605042	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂×#64701	C₂	1	4	yes	maximal
4	#10600184	C₂	1	1	yes	maximal, characteristic
5	C₂×#4015	C₄	1	4	yes	—
6	C₂×#4015	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄³×C₅	C₂²	1	1	yes	characteristic
9	#65224	—	2	1	no	—
10	#65224	—	2	1	no	—
11	#65224	—	2	2	no	—
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	C₂×#3935	—	2	1	no	—
14	C₅×#336	—	2	2	no	—
15	#64744	—	2	1	no	—
16	#64701	—	2	8	no	—
17	#9702	—	5	1	no	maximal, Sylow
18	C₂²×Dic₁₀	C₂×C₄	1	4	yes	—
19	C₂²×C₅×C₈	—	2	2	no	—
20	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	D₄	1	2	yes	—
24	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₅	—	2	4	no	—
26	C₂×C₄⋊Dic₅	—	2	4	no	—
27	C₂×C₄⋊Dic₅	—	2	2	no	—
28	C₂×C₄⋊Dic₅	—	2	2	no	—
29	C₂²×#168	—	2	2	no	—
30	#4015	—	2	4	no	—
31	#4015	—	2	4	no	—
32	#4015	—	4	8	no	—
33	C₅×C₄⋊C₈	—	2	4	no	—
34	C₄⋊#168	—	2	2	no	—
35	C₄⋊#168	D₄	1	4	yes	—
36	#3935	—	4	2	no	—
37	#2868	—	5	1	no	—
38	#1831	D₅	1	1	yes	characteristic
39	#1767	—	5	1	no	—
40	C₂×#333	—	5	4	no	—
41	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₅	—	2	4	no	—
47	C₂×Dic₁₀	—	2	8	no	—
48	C₂×Dic₁₀	—	4	8	no	—
49	C₂×C₅×C₈	—	4	2	no	—
50	C₂×C₅×C₈	—	4	4	no	—
51	C₄²×C₅	—	2	4	no	—
52	C₄²×C₅	—	2	4	no	—
53	C₄²×C₅	—	2	2	no	—
54	C₄²×C₅	SD₁₆	1	2	yes	—
55	C₄²×C₅	Q₁₆	1	2	yes	—
56	C₄²×C₅	C₂×D₄	1	2	yes	—
57	C₄²×C₅	#41	1	2	yes	—
58	C₄⋊Dic₅	—	4	8	no	—
59	C₄⋊Dic₅	—	4	8	no	—
60	C₄⋊Dic₅	—	4	4	no	—
61	C₄⋊Dic₅	—	4	2	no	—
62	C₄⋊Dic₅	—	4	2	no	—
63	C₂×#168	—	2	4	no	—
64	C₂×#168	—	4	2	no	—
65	C₂×#168	—	4	2	no	—
66	C₂×C₄⋊Q₈	—	5	4	no	—
67	C₂×C₄⋊Q₈	—	5	2	no	—
68	C₂×C₄⋊C₈	—	5	2	no	—
69	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
70	C₂×#97	—	10	1	no	—
71	#384	—	10	1	no	—
72	#384	—	10	1	no	—
73	#384	—	10	2	no	—
74	C₄³	C₂×D₅	1	1	yes	characteristic
75	#336	—	2	2	no	—
76	#336	—	10	1	no	—
77	#333	—	10	8	no	—
78	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅	—	2	2	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	#100	1	4	yes	—
85	C₂×C₄×C₅	#103	1	4	yes	—
86	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₅	#121	1	2	yes	—
88	C₂×C₄×C₅	#124	1	1	yes	characteristic
89	C₂×C₄×C₅	#127	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅	#137	1	4	yes	—
93	C₂×C₄×C₅	#137	1	2	yes	—
94	C₂×Dic₅	—	4	4	no	—
95	C₂×Dic₅	—	4	8	no	—
96	Dic₁₀	—	8	8	no	—
97	C₅×C₈	—	8	2	no	—
98	#168	—	4	4	no	—
99	C₂²×Q₈	—	5	4	no	—
100	C₂²×C₈	—	2	2	no	—
101	C₂²×C₈	—	10	2	no	—
102	C₄⋊Q₈	—	10	4	no	—
103	C₄⋊Q₈	—	10	4	no	—
104	C₄⋊Q₈	—	20	8	no	—
105	C₂×C₄⋊C₄	—	10	4	no	—
106	C₂×C₄⋊C₄	—	10	4	no	—
107	C₂×C₄⋊C₄	—	10	2	no	—
108	C₂×C₄⋊C₄	—	10	2	no	—
109	C₂×C₄²	C₄×D₅	1	2	yes	—
110	C₂×C₄²	D₂₀	1	1	yes	characteristic
111	C₂×C₄²	D₂₀	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
113	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
114	C₄⋊C₈	—	2	4	no	—
115	C₄⋊C₈	—	5	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	#461	1	2	yes	—
127	C₄×C₅	#462	1	2	yes	—
128	C₄×C₅	#491	1	2	yes	—
129	C₄×C₅	#497	1	2	yes	—
130	Dic₅	—	8	4	no	—
131	C₂×Q₈	—	10	8	no	—
132	C₂×Q₈	—	20	8	no	—
133	C₂²×C₄	—	10	4	no	—
134	C₂²×C₄	#685	1	2	yes	—
135	C₂²×C₄	#685	1	2	yes	—
136	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	10	4	no	—
142	C₂×C₈	—	20	2	no	—
143	C₂×C₈	—	20	2	no	—
144	C₄⋊C₄	—	20	8	no	—
145	C₄⋊C₄	—	20	8	no	—
146	C₄⋊C₄	—	20	4	no	—
147	C₄⋊C₄	—	20	2	no	—
148	C₄⋊C₄	—	20	2	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	2	no	—
152	C₄²	#677	1	2	yes	—
153	C₄²	Dic₂₀	1	2	yes	—
154	C₄²	D₄×D₅	1	2	yes	—
155	C₄²	#713	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×#685	1	1	yes	characteristic
162	Q₈	—	40	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	—	20	8	no	—
169	C₂×C₄	—	20	4	no	—
170	C₂×C₄	#3948	1	4	yes	—
171	C₂×C₄	#3955	1	4	yes	—
172	C₂×C₄	#4039	1	2	yes	—
173	C₂×C₄	#4041	1	2	yes	—
174	C₂×C₄	C₂×#677	1	1	yes	characteristic
175	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
176	C₂×C₄	#4055	1	2	yes	—
177	C₂×C₄	#4086	1	1	yes	characteristic
178	C₂×C₄	#4093	1	1	yes	characteristic
179	C₂×C₄	#4097	1	4	yes	—
180	C₈	—	8	2	no	—
181	C₈	—	20	4	no	—
182	C₅	#9702	1	1	yes	characteristic, complemented, Sylow
183	C₂²	#65277	1	1	yes	characteristic
184	C₂²	#65293	1	2	yes	—
185	C₂²	C₂×#3948	1	1	yes	characteristic
186	C₂²	#65409	1	1	yes	characteristic
187	C₂²	C₂×#3955	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	40	4	no	—
192	C₄	#65130	1	2	yes	—
193	C₄	#65138	1	2	yes	—
194	C₄	#65349	1	2	yes	—
195	C₄	#65358	1	2	yes	—
196	C₂	#10600345	1	1	yes	characteristic
197	C₂	#10601249	1	1	yes	characteristic
198	C₂	#10601314	1	1	yes	characteristic
199	C₂	#10601572	1	2	yes	—
200	C₂	#10601617	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10605042

C₂

yes

maximal, characteristic

C₅×#1831

C₂

yes

maximal, characteristic

C₂×#64701

C₂

yes

maximal

#10600184

C₂

yes

maximal, characteristic

C₂×#4015

C₄

yes

—

C₂×#4015

C₂²

yes

—

C₂×C₅×C₄⋊C₈

C₂²

yes

—

C₄³×C₅

C₂²

yes

—

—

—

C₂×C₄⋊#168

C₂²

yes

—

—

—

—

—

—

C₂²×Dic₁₀

C₂×C₄

yes

—

C₂²×C₅×C₈

—

C₂×C₄²×C₅

C₂×C₄

yes

—

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂²×#168

—

#4015

—

#4015

—

#4015

—

C₅×C₄⋊C₈

—

C₄⋊#168

—

C₄⋊#168

D₄

yes

—

#3935

—

#2868

—

#1831

D₅

yes

characteristic

#1767

—

C₂×#333

—

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

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

SD₁₆

yes

—

C₄²×C₅

Q₁₆

yes

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

#41

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

—

C₂×#168

—

C₂×#168

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₅

yes

—

C₂×#97

—

#384

—

#384

—

#384

—

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

#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

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₂²×Q₈

—

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

C₄×D₅

yes

—

110

C₂×C₄²

D₂₀

yes

characteristic

111

C₂×C₄²

D₂₀

yes

characteristic

112

C₂×C₄²

C₂²⋊D₅

yes

—

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

#461

yes

—

127

C₄×C₅

#462

yes

—

128

C₄×C₅

#491

yes

—

129

C₄×C₅

#497

yes

—

130

Dic₅

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂²×C₄

—

134

C₂²×C₄

#685

yes

—

135

C₂²×C₄

#685

yes

—

136

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

137

C₂²×C₄

C₂×D₂₀

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

#677

yes

—

153

C₄²

Dic₂₀

yes

—

154

C₄²

D₄×D₅

yes

—

155

C₄²

#713

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×#685

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

#3948

yes

—

171

C₂×C₄

#3955

yes

—

172

C₂×C₄

#4039

yes

—

173

C₂×C₄

#4041

yes

—

174

C₂×C₄

C₂×#677

yes

characteristic

175

C₂×C₄

C₂×Dic₂₀

yes

characteristic

176

C₂×C₄

#4055

yes

—

177

C₂×C₄

#4086

yes

characteristic

178

C₂×C₄

#4093

yes

characteristic

179

C₂×C₄

#4097

yes

—

180

C₈

—

181

C₈

—

182

C₅

#9702

yes

characteristic, complemented, Sylow

183

C₂²

#65277

yes

characteristic

184

C₂²

#65293

yes

—

185

C₂²

C₂×#3948

yes

characteristic

186

C₂²

#65409

yes

characteristic

187

C₂²

C₂×#3955

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#65130

yes

—

193

C₄

#65138

yes

—

194

C₄

#65349

yes

—

195

C₄

#65358

yes

—

196

C₂

#10600345

yes

characteristic

197

C₂

#10601249

yes

characteristic

198

C₂

#10601314

yes

characteristic

199

C₂

#10601572

yes

—

200

C₂

#10601617

yes

—

	subgroup	G/H
center	C₂³	C₂×#685
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1831	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1831	D₅
hypocenter	C₅	#9702
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×#685

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₅×#1831

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 ◁ #1831

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
#10605042	2
C₅×#1831	2
C₂×#64701	2
#10600184	2
#9702	5

subgroup

index

Subgroups — #11998380

Conjugacy classes of subgroups

Complements

Maximal subgroups