Group #11987628 — subgroups

Subgroups — #11987628

Order 1280.

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

normal subgroups only

This group has 456 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	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10600185	C₂	1	2	yes	maximal
3	#10598133	C₂	1	4	yes	maximal
4	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₅×Q₈	C₄	1	2	yes	—
6	C₅×#391	C₂²	1	2	yes	—
7	C₅×#389	—	2	2	no	—
8	C₅×#384	C₄	1	1	yes	characteristic
9	C₄×C₅×C₄⋊C₄	—	2	2	no	—
10	C₄³×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#168	—	2	2	no	—
12	#64744	—	2	2	no	—
13	#64744	C₂²	1	4	yes	—
14	#7653	—	5	1	no	maximal, Sylow
15	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
16	C₅×C₄⋊Q₈	—	2	4	no	—
17	C₄×C₅×Q₈	—	4	4	no	—
18	C₂×C₅×C₄⋊C₄	—	2	2	no	—
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×C₄²×C₅	—	2	2	no	—
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	D₄	1	2	yes	—
26	C₂×C₄²×C₅	D₄	1	2	yes	—
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂²×#168	—	2	4	no	—
29	C₅×#95	—	2	4	no	—
30	C₄⋊#168	—	4	4	no	—
31	C₄×C₄⋊Q₈	D₅	1	1	yes	characteristic
32	#1831	—	5	2	no	—
33	#1272	—	5	4	no	—
34	C₂×C₅×Q₈	—	2	2	no	—
35	C₂×C₅×Q₈	—	4	2	no	—
36	C₂×C₅×Q₈	—	4	2	no	—
37	C₂×C₅×Q₈	M₁₆	1	4	yes	—
38	C₂²×C₄×C₅	—	2	2	no	—
39	C₂²×C₄×C₅	—	2	2	no	—
40	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
45	C₅×C₄⋊C₄	—	2	8	no	—
46	C₅×C₄⋊C₄	—	4	2	no	—
47	C₅×C₄⋊C₄	—	4	4	no	—
48	C₅×C₄⋊C₄	—	4	4	no	—
49	C₄²×C₅	—	2	2	no	—
50	C₄²×C₅	—	2	2	no	—
51	C₄²×C₅	—	4	2	no	—
52	C₄²×C₅	—	4	1	no	—
53	C₄²×C₅	—	4	4	no	—
54	C₄²×C₅	SD₁₆	1	4	yes	—
55	C₄²×C₅	Q₁₆	1	4	yes	—
56	C₂×#168	—	4	4	no	—
57	C₂×#168	—	4	4	no	—
58	C₂×#168	—	4	4	no	—
59	C₂×C₄⋊Q₈	Dic₅	1	1	yes	characteristic
60	C₂×C₄×Q₈	Dic₅	1	2	yes	—
61	C₂×C₄⋊C₈	—	10	2	no	—
62	#391	C₂×D₅	1	2	yes	—
63	#389	—	2	2	no	—
64	#384	Dic₅	1	1	yes	characteristic
65	C₄×C₄⋊C₄	—	2	2	no	—
66	C₄³	C₂×D₅	1	1	yes	characteristic
67	#336	—	5	4	no	—
68	#336	—	10	2	no	—
69	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₅×Q₈	—	4	4	no	—
71	C₅×Q₈	—	4	4	no	—
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	2	no	—
74	C₂×C₄×C₅	—	2	1	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	4	no	—
79	C₂×C₄×C₅	—	2	1	no	—
80	C₂×C₄×C₅	—	2	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	—	4	4	no	—
85	C₂×C₄×C₅	#100	1	2	yes	—
86	C₂×C₄×C₅	#103	1	4	yes	—
87	C₂×C₄×C₅	#103	1	4	yes	—
88	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
91	#168	—	8	4	no	—
92	C₂²×Q₈	C₂×Dic₅	1	2	yes	—
93	C₂²×C₈	—	10	4	no	—
94	C₄⋊Q₈	—	2	4	no	—
95	C₄×Q₈	—	4	4	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
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×C₄²	—	2	2	no	—
101	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
104	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
105	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
106	C₄⋊C₈	—	20	4	no	—
107	#95	—	2	4	no	—
108	C₂²×C₅	#407	1	1	yes	characteristic
109	C₂²×C₅	#410	1	1	yes	characteristic
110	C₂²×C₅	C₂×#100	1	1	yes	characteristic
111	C₂²×C₅	C₂×#103	1	2	yes	—
112	C₂²×C₅	#421	1	2	yes	—
113	C₄×C₅	—	2	2	no	—
114	C₄×C₅	—	2	4	no	—
115	C₄×C₅	—	2	2	no	—
116	C₄×C₅	—	2	2	no	—
117	C₄×C₅	—	4	1	no	—
118	C₄×C₅	—	4	4	no	—
119	C₄×C₅	—	8	2	no	—
120	C₄×C₅	#333	1	4	yes	—
121	C₂×Q₈	—	2	2	no	—
122	C₂×Q₈	—	4	2	no	—
123	C₂×Q₈	—	4	2	no	—
124	C₂×Q₈	#681	1	4	yes	—
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	—	20	4	no	—
133	C₂×C₈	—	20	4	no	—
134	C₂×C₈	—	20	4	no	—
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	4	2	no	—
142	C₄²	—	4	1	no	—
143	C₄²	—	4	4	no	—
144	C₄²	#688	1	4	yes	—
145	C₄²	#689	1	4	yes	—
146	C₂×C₅	#1466	1	1	yes	characteristic
147	C₂×C₅	#1487	1	2	yes	—
148	C₂×C₅	#1518	1	2	yes	—
149	C₂×C₅	#1536	1	1	yes	characteristic
150	C₂×C₅	C₂×#333	1	1	yes	characteristic
151	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
152	Q₈	—	4	4	no	—
153	Q₈	—	4	4	no	—
154	C₂×C₄	—	2	2	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	—	2	1	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	#3965	1	2	yes	—
168	C₂×C₄	#3967	1	4	yes	—
169	C₂×C₄	#3967	1	4	yes	—
170	C₂×C₄	C₂×#681	1	2	yes	—
171	C₂×C₄	C₂×#688	1	2	yes	—
172	C₂×C₄	C₂×#689	1	2	yes	—
173	C₈	—	40	4	no	—
174	C₅	#7653	1	1	yes	characteristic, complemented, Sylow
175	C₂²	#65496	1	1	yes	characteristic
176	C₂²	C₂×#3965	1	1	yes	characteristic
177	C₂²	C₂×#3967	1	2	yes	—
178	C₂²	#65525	1	2	yes	—
179	C₂²	#65529	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	4	no	—
186	C₄	—	8	2	no	—
187	C₄	#64766	1	4	yes	—
188	C₂	#10602396	1	1	yes	characteristic
189	C₂	#10602415	1	2	yes	—
190	C₂	C₂×#64766	1	1	yes	characteristic
191	C₂	#10602592	1	1	yes	characteristic
192	C₂	#10602631	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅×C₄⋊Q₈

C₂

yes

maximal, characteristic

#10600185

C₂

yes

maximal

#10598133

C₂

yes

maximal

C₂×C₅×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄×C₅×Q₈

C₄

yes

—

C₅×#391

C₂²

yes

—

C₅×#389

—

C₅×#384

C₄

yes

characteristic

C₄×C₅×C₄⋊C₄

—

C₄³×C₅

C₂²

yes

characteristic

C₂×C₄⋊#168

—

#64744

—

#64744

C₂²

yes

—

#7653

—

maximal, Sylow

C₂²×C₅×Q₈

C₂×C₄

yes

—

C₅×C₄⋊Q₈

—

C₄×C₅×Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

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

—

C₄⋊#168

—

C₄×C₄⋊Q₈

D₅

yes

characteristic

#1831

—

#1272

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

M₁₆

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

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

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

SD₁₆

yes

—

C₄²×C₅

Q₁₆

yes

—

C₂×#168

—

C₂×#168

—

C₂×#168

—

C₂×C₄⋊Q₈

Dic₅

yes

characteristic

C₂×C₄×Q₈

Dic₅

yes

—

C₂×C₄⋊C₈

—

#391

C₂×D₅

yes

—

#389

—

#384

Dic₅

yes

characteristic

C₄×C₄⋊C₄

—

C₄³

C₂×D₅

yes

characteristic

#336

—

#336

—

C₂³×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₅×Q₈

—

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

—

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

#103

yes

—

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₂×C₄×C₅

C₂×Q₁₆

yes

—

#168

—

C₂²×Q₈

C₂×Dic₅

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₅

yes

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×Dic₅

yes

characteristic

102

C₂×C₄²

C₂×Dic₅

yes

characteristic

103

C₂×C₄²

C₂²⋊D₅

yes

—

104

C₂×C₄²

C₂²⋊D₅

yes

—

105

C₂×C₄²

C₂²×D₅

yes

characteristic

106

C₄⋊C₈

—

107

#95

—

108

C₂²×C₅

#407

yes

characteristic

109

C₂²×C₅

#410

yes

characteristic

110

C₂²×C₅

C₂×#100

yes

characteristic

111

C₂²×C₅

C₂×#103

yes

—

112

C₂²×C₅

#421

yes

—

113

C₄×C₅

—

114

C₄×C₅

—

115

C₄×C₅

—

116

C₄×C₅

—

117

C₄×C₅

—

118

C₄×C₅

—

119

C₄×C₅

—

120

C₄×C₅

#333

yes

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

#681

yes

—

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

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

#688

yes

—

145

C₄²

#689

yes

—

146

C₂×C₅

#1466

yes

characteristic

147

C₂×C₅

#1487

yes

—

148

C₂×C₅

#1518

yes

—

149

C₂×C₅

#1536

yes

characteristic

150

C₂×C₅

C₂×#333

yes

characteristic

151

C₂³

C₂×C₂²⋊Dic₅

yes

characteristic

152

Q₈

—

153

Q₈

—

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

#3965

yes

—

168

C₂×C₄

#3967

yes

—

169

C₂×C₄

#3967

yes

—

170

C₂×C₄

C₂×#681

yes

—

171

C₂×C₄

C₂×#688

yes

—

172

C₂×C₄

C₂×#689

yes

—

173

C₈

—

174

C₅

#7653

yes

characteristic, complemented, Sylow

175

C₂²

#65496

yes

characteristic

176

C₂²

C₂×#3965

yes

characteristic

177

C₂²

C₂×#3967

yes

—

178

C₂²

#65525

yes

—

179

C₂²

#65529

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

#64766

yes

—

188

C₂

#10602396

yes

characteristic

189

C₂

#10602415

yes

—

190

C₂

C₂×#64766

yes

characteristic

191

C₂

#10602592

yes

characteristic

192

C₂

#10602631

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅
hypocenter	C₅	#7653
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₅

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₄×C₅×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₅

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅

C₂³

hypercenter

C₄×C₄⋊Q₈

D₅

hypocenter

C₅

#7653

LCS

G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

N	H ≅ G/N	type
174. C₅	#7653	semidirect

H ≅ G/N

type

174. C₅

#7653

semidirect

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
#10600185	2
#10598133	2
#7653	5

subgroup

index

C₄×C₅×C₄⋊Q₈

#10600185

#10598133

#7653

Subgroups — #11987628

Conjugacy classes of subgroups

Complements

Maximal subgroups