Group #11999428 — subgroups

Subgroups — #11999428

Order 1280.

430 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 159 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	#10605038	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	#10598193	C₂	1	4	yes	maximal
5	C₂×#4016	—	2	1	no	—
6	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄³×C₅	C₂²	1	1	yes	characteristic
8	#65225	—	2	1	no	—
9	#65224	C₂²	1	2	yes	—
10	#65217	—	2	2	no	—
11	#65217	C₄	1	4	yes	—
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	C₂×C₄×#168	—	2	1	no	—
14	C₅×#336	—	2	2	no	—
15	#64744	—	2	1	no	—
16	#9899	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	—	2	2	no	—
18	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₅	D₄	1	2	yes	—
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₅	—	2	2	no	—
24	C₂×C₄⋊Dic₅	—	2	2	no	—
25	C₂×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
26	C₂×#683	—	2	4	no	—
27	C₂²×#168	—	2	2	no	—
28	#4016	—	4	2	no	—
29	#4016	—	4	2	no	—
30	C₅×C₄⋊C₈	—	2	4	no	—
31	#3963	—	2	4	no	—
32	C₄⋊#168	—	2	2	no	—
33	C₄⋊#168	Q₈	1	4	yes	—
34	C₄×#168	—	4	2	no	—
35	#2859	—	5	1	no	—
36	#1831	D₅	1	1	yes	characteristic
37	C₄×C₄⋊C₈	—	5	1	no	—
38	#1296	—	5	4	no	—
39	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅	—	2	4	no	—
45	C₂×C₅×C₈	—	4	2	no	—
46	C₂×C₅×C₈	—	4	4	no	—
47	C₄²×C₅	—	2	4	no	—
48	C₄²×C₅	—	2	2	no	—
49	C₄²×C₅	—	4	2	no	—
50	C₄²×C₅	D₈	1	2	yes	—
51	C₄²×C₅	SD₁₆	1	2	yes	—
52	C₄²×C₅	C₂×Q₈	1	2	yes	—
53	C₄²×C₅	#41	1	2	yes	—
54	C₄⋊Dic₅	—	4	2	no	—
55	C₄⋊Dic₅	—	4	2	no	—
56	C₄⋊Dic₅	—	4	4	no	—
57	C₄⋊Dic₅	—	4	4	no	—
58	#683	—	4	8	no	—
59	C₂×#168	—	2	4	no	—
60	C₂×#168	—	4	2	no	—
61	C₂×#168	—	4	2	no	—
62	C₂×#125	—	10	1	no	—
63	C₂×C₄⋊C₈	—	5	2	no	—
64	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
65	C₂×C₄×C₈	—	10	1	no	—
66	#389	—	5	4	no	—
67	#389	—	10	2	no	—
68	#384	—	5	2	no	—
69	#382	—	10	1	no	—
70	C₄³	C₂×D₅	1	1	yes	characteristic
71	#336	—	2	2	no	—
72	#336	—	10	1	no	—
73	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅	—	2	2	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	2	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	#102	1	4	yes	—
81	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
82	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₅	#124	1	1	yes	characteristic
84	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅	#135	1	4	yes	—
88	C₂×C₄×C₅	#137	1	2	yes	—
89	C₂×Dic₅	—	4	4	no	—
90	C₂×Dic₅	—	4	4	no	—
91	C₂×Dic₅	—	4	4	no	—
92	C₅×C₈	—	8	2	no	—
93	#168	—	4	4	no	—
94	C₂²×C₈	—	2	2	no	—
95	C₂²×C₈	—	10	2	no	—
96	#125	—	20	2	no	—
97	#125	—	20	2	no	—
98	C₂×C₄⋊C₄	—	5	4	no	—
99	C₂×C₄⋊C₄	—	10	4	no	—
100	C₂×C₄⋊C₄	—	10	2	no	—
101	C₂×C₄⋊C₄	—	10	2	no	—
102	C₂×C₄²	C₄×D₅	1	2	yes	—
103	C₂×C₄²	D₂₀	1	1	yes	characteristic
104	C₂×C₄²	D₂₀	1	1	yes	characteristic
105	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
106	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
107	C₄⋊C₈	—	2	4	no	—
108	C₄⋊C₈	—	5	4	no	—
109	C₄⋊C₈	—	10	2	no	—
110	C₄×C₈	—	20	2	no	—
111	#95	—	10	4	no	—
112	C₂²×C₅	#391	1	1	yes	characteristic
113	C₂²×C₅	#413	1	2	yes	—
114	C₂²×C₅	C₂×#102	1	1	yes	characteristic
115	C₂²×C₅	#416	1	2	yes	—
116	C₂²×C₅	#419	1	1	yes	characteristic
117	C₄×C₅	—	2	4	no	—
118	C₄×C₅	—	2	2	no	—
119	C₄×C₅	—	4	2	no	—
120	C₄×C₅	#474	1	2	yes	—
121	C₄×C₅	#475	1	2	yes	—
122	C₄×C₅	#488	1	2	yes	—
123	C₄×C₅	#499	1	2	yes	—
124	Dic₅	—	8	4	no	—
125	C₂²×C₄	—	10	4	no	—
126	C₂²×C₄	#685	1	2	yes	—
127	C₂²×C₄	#685	1	2	yes	—
128	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
129	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	10	4	no	—
134	C₂×C₈	—	20	2	no	—
135	C₂×C₈	—	20	2	no	—
136	C₄⋊C₄	—	20	4	no	—
137	C₄⋊C₄	—	20	4	no	—
138	C₄⋊C₄	—	20	8	no	—
139	C₄⋊C₄	—	20	2	no	—
140	C₄⋊C₄	—	20	2	no	—
141	C₄²	—	2	4	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	4	2	no	—
144	C₄²	#677	1	2	yes	—
145	C₄²	D₄₀	1	2	yes	—
146	C₄²	#711	1	2	yes	—
147	C₄²	Q₈×D₅	1	2	yes	—
148	C₂×C₅	#1540	1	2	yes	—
149	C₂×C₅	#1554	1	2	yes	—
150	C₂×C₅	#1988	1	1	yes	characteristic
151	C₂×C₅	#1991	1	1	yes	characteristic
152	C₂×C₅	#1998	1	1	yes	characteristic
153	C₂³	C₂×#685	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	20	4	no	—
161	C₂×C₄	—	20	4	no	—
162	C₂×C₄	—	20	4	no	—
163	C₂×C₄	#3953	1	4	yes	—
164	C₂×C₄	#4033	1	2	yes	—
165	C₂×C₄	#4043	1	2	yes	—
166	C₂×C₄	C₂×#677	1	1	yes	characteristic
167	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
168	C₂×C₄	#4055	1	2	yes	—
169	C₂×C₄	#4082	1	1	yes	characteristic
170	C₂×C₄	#4090	1	1	yes	characteristic
171	C₂×C₄	#4096	1	4	yes	—
172	C₈	—	8	2	no	—
173	C₈	—	20	4	no	—
174	C₅	#9899	1	1	yes	characteristic, complemented, Sylow
175	C₂²	#65263	1	1	yes	characteristic
176	C₂²	#65291	1	2	yes	—
177	C₂²	C₂×#3953	1	1	yes	characteristic
178	C₂²	#65409	1	1	yes	characteristic
179	C₂²	#65421	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	40	4	no	—
184	C₄	#65135	1	2	yes	—
185	C₄	#65140	1	2	yes	—
186	C₄	#65345	1	2	yes	—
187	C₄	#65353	1	2	yes	—
188	C₂	#10600347	1	1	yes	characteristic
189	C₂	#10601253	1	1	yes	characteristic
190	C₂	#10601318	1	1	yes	characteristic
191	C₂	#10601565	1	2	yes	—
192	C₂	#10601614	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10605038

C₂

yes

maximal, characteristic

C₅×#1831

C₂

yes

maximal, characteristic

C₄×C₄⋊#168

C₂

yes

maximal, characteristic

#10598193

C₂

yes

maximal

C₂×#4016

—

C₂×C₅×C₄⋊C₈

C₂²

yes

—

C₄³×C₅

C₂²

yes

characteristic

#65225

—

#65224

C₂²

yes

—

#65217

—

#65217

C₄

yes

—

C₂×C₄⋊#168

C₂²

yes

—

C₂×C₄×#168

—

—

—

—

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

yes

—

C₂×#683

—

C₂²×#168

—

#4016

—

#4016

—

C₅×C₄⋊C₈

—

#3963

—

C₄⋊#168

—

C₄⋊#168

Q₈

yes

—

C₄×#168

—

#2859

—

#1831

D₅

yes

characteristic

C₄×C₄⋊C₈

—

#1296

—

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×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

D₈

yes

—

C₄²×C₅

SD₁₆

yes

—

C₄²×C₅

C₂×Q₈

yes

—

C₄²×C₅

#41

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₅

yes

—

C₂×C₄×C₈

—

—

—

—

—

C₄³

C₂×D₅

yes

characteristic

#336

—

#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

#102

yes

—

C₂×C₄×C₅

C₄×Q₈

yes

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

#124

yes

characteristic

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅

#135

yes

—

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

#168

—

C₂²×C₈

—

C₂²×C₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄²

C₄×D₅

yes

—

103

C₂×C₄²

D₂₀

yes

characteristic

104

C₂×C₄²

D₂₀

yes

characteristic

105

C₂×C₄²

C₂²⋊D₅

yes

—

106

C₂×C₄²

C₂²×D₅

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄×C₈

—

111

#95

—

112

C₂²×C₅

#391

yes

characteristic

113

C₂²×C₅

#413

yes

—

114

C₂²×C₅

C₂×#102

yes

characteristic

115

C₂²×C₅

#416

yes

—

116

C₂²×C₅

#419

yes

characteristic

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

—

120

C₄×C₅

#474

yes

—

121

C₄×C₅

#475

yes

—

122

C₄×C₅

#488

yes

—

123

C₄×C₅

#499

yes

—

124

Dic₅

—

125

C₂²×C₄

—

126

C₂²×C₄

#685

yes

—

127

C₂²×C₄

#685

yes

—

128

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

129

C₂²×C₄

C₂×D₂₀

yes

characteristic

130

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

#677

yes

—

145

C₄²

D₄₀

yes

—

146

C₄²

#711

yes

—

147

C₄²

Q₈×D₅

yes

—

148

C₂×C₅

#1540

yes

—

149

C₂×C₅

#1554

yes

—

150

C₂×C₅

#1988

yes

characteristic

151

C₂×C₅

#1991

yes

characteristic

152

C₂×C₅

#1998

yes

characteristic

153

C₂³

C₂×#685

yes

characteristic

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

#3953

yes

—

164

C₂×C₄

#4033

yes

—

165

C₂×C₄

#4043

yes

—

166

C₂×C₄

C₂×#677

yes

characteristic

167

C₂×C₄

C₂×D₄₀

yes

characteristic

168

C₂×C₄

#4055

yes

—

169

C₂×C₄

#4082

yes

characteristic

170

C₂×C₄

#4090

yes

characteristic

171

C₂×C₄

#4096

yes

—

172

C₈

—

173

C₈

—

174

C₅

#9899

yes

characteristic, complemented, Sylow

175

C₂²

#65263

yes

characteristic

176

C₂²

#65291

yes

—

177

C₂²

C₂×#3953

yes

characteristic

178

C₂²

#65409

yes

characteristic

179

C₂²

#65421

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#65135

yes

—

185

C₄

#65140

yes

—

186

C₄

#65345

yes

—

187

C₄

#65353

yes

—

188

C₂

#10600347

yes

characteristic

189

C₂

#10601253

yes

characteristic

190

C₂

#10601318

yes

characteristic

191

C₂

#10601565

yes

—

192

C₂

#10601614

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	#9899
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
174. C₅	#9899	semidirect

H ≅ G/N

type

174. C₅

#9899

semidirect

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

subgroup	index
#10605038	2
C₅×#1831	2
C₄×C₄⋊#168	2
#10598193	2
#9899	5

subgroup

index

C₄×C₄⋊#168

Subgroups — #11999428

Conjugacy classes of subgroups

Complements

Maximal subgroups