Group #10694860 — subgroups

Subgroups — #10694860

Order 768.

434 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₂×C₃×#500	C₂	1	1	yes	maximal, characteristic
2	#71789	C₂	1	2	yes	maximal
3	#71773	C₂	1	2	yes	maximal
4	#69730	C₂	1	1	yes	maximal, characteristic
5	#69725	C₂	1	1	yes	maximal, characteristic
6	#9808	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃×#500	—	2	1	no	—
9	C₃×#500	—	2	1	no	—
10	C₃×#500	—	2	2	no	—
11	C₂×C₃×#107	C₄	1	4	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×#836	—	2	4	no	—
14	#4925	C₂²	1	1	yes	characteristic
15	#4924	C₂²	1	2	yes	—
16	C₄×#208	C₂²	1	1	yes	characteristic
17	#4547	—	2	4	no	—
18	#4457	—	2	2	no	—
19	#4455	—	2	1	no	—
20	#4454	—	2	1	no	—
21	C₂×#500	D₃	1	1	yes	characteristic
22	#1966	—	3	2	no	—
23	#1566	—	3	1	no	—
24	#1546	—	3	2	no	—
25	#1451	—	3	1	no	—
26	C₂²×C₃×Q₈	—	2	2	no	—
27	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
28	C₃×C₄⋊Q₈	—	2	2	no	—
29	C₃×C₄⋊Q₈	—	2	2	no	—
30	C₃×C₄⋊Q₈	—	2	2	no	—
31	C₃×C₄⋊Q₈	—	2	2	no	—
32	C₂×C₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
35	C₂×#208	—	2	2	no	—
36	C₂×#208	D₄	1	2	yes	—
37	C₃×#107	—	2	4	no	—
38	C₃×#107	—	2	4	no	—
39	C₃×C₄×C₈	D₄	1	1	yes	characteristic
40	C₃×C₄×C₈	D₄	1	1	yes	characteristic
41	C₃×C₄×C₈	D₄	1	2	yes	—
42	#836	—	4	8	no	—
43	#830	—	2	1	no	—
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₃⋊C₈)	—	2	2	no	—
46	C₄⋊(C₃⋊C₈)	D₄	1	2	yes	—
47	#803	—	2	1	no	—
48	C₄×C₃⋊C₈	—	2	1	no	—
49	C₄×C₃⋊C₈	—	2	1	no	—
50	C₂×C₄⋊Q₈	C₂×D₃	1	2	yes	—
51	#500	—	2	1	no	—
52	#500	—	2	1	no	—
53	#500	—	2	2	no	—
54	#431	—	3	1	no	—
55	C₂×#107	Dic₃	1	4	yes	—
56	#423	—	3	2	no	—
57	C₂×#101	—	6	4	no	—
58	C₄×M₁₆	—	3	1	no	—
59	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	#334	—	6	2	no	—
61	#332	—	6	4	no	—
62	#322	—	6	1	no	—
63	#322	—	6	1	no	—
64	C₂×C₃×Q₈	—	4	2	no	—
65	C₂×C₃×Q₈	—	4	2	no	—
66	C₂×C₃×Q₈	—	4	4	no	—
67	C₂×C₃×Q₈	—	4	4	no	—
68	C₂²×C₃×C₄	—	2	4	no	—
69	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	#41	1	1	yes	characteristic
72	C₂×C₃×C₈	—	2	4	no	—
73	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
75	C₃×C₄⋊C₄	—	2	4	no	—
76	C₃×C₄⋊C₄	—	2	4	no	—
77	C₃×C₄⋊C₄	—	4	2	no	—
78	C₃×C₄⋊C₄	—	4	2	no	—
79	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₃×C₄²	#41	1	1	yes	characteristic
83	#208	—	2	4	no	—
84	#208	—	4	4	no	—
85	C₂×C₃⋊C₈	—	2	2	no	—
86	C₂×C₃⋊C₈	—	2	2	no	—
87	C₂×C₃⋊C₈	—	4	2	no	—
88	C₂²×Q₈	—	2	2	no	—
89	C₂×M₁₆	—	3	2	no	—
90	C₂×M₁₆	—	6	2	no	—
91	C₂²×C₈	C₂×Dic₃	1	2	yes	—
92	C₄⋊Q₈	—	2	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	—	2	2	no	—
95	C₄⋊Q₈	—	2	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
98	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
99	#107	—	2	4	no	—
100	#107	—	2	4	no	—
101	C₄⋊C₈	—	3	2	no	—
102	C₄⋊C₈	—	6	2	no	—
103	C₄⋊C₈	—	6	2	no	—
104	#101	—	12	8	no	—
105	C₂²⋊C₈	—	6	1	no	—
106	#97	—	6	1	no	—
107	C₄×C₈	—	6	1	no	—
108	C₄×C₈	—	6	1	no	—
109	C₄×C₈	#67	1	1	yes	characteristic
110	C₄×C₈	#67	1	1	yes	characteristic
111	C₄×C₈	#67	1	2	yes	—
112	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
113	C₃×Q₈	—	8	4	no	—
114	C₃×Q₈	—	8	2	no	—
115	C₃×Q₈	—	8	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	4	4	no	—
119	C₂×C₃×C₄	—	4	8	no	—
120	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
122	C₂×C₃×C₄	#121	1	1	yes	characteristic
123	C₂×C₃×C₄	#121	1	1	yes	characteristic
124	C₂×C₃×C₄	#124	1	1	yes	characteristic
125	C₂×C₃×C₄	#127	1	1	yes	characteristic
126	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
127	C₃×C₈	—	2	2	no	—
128	C₃×C₈	#101	1	4	yes	complemented
129	C₃⋊C₈	—	4	2	no	—
130	C₃⋊C₈	—	4	2	no	—
131	C₃⋊C₈	—	8	2	no	—
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	D₃×D₄	1	1	yes	characteristic
138	C₂²×C₄	#237	1	1	yes	characteristic
139	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
140	M₁₆	—	6	4	no	—
141	M₁₆	—	12	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	6	2	no	—
144	C₂×C₈	—	6	2	no	—
145	C₂×C₈	—	12	2	no	—
146	C₂×C₈	#217	1	2	yes	—
147	C₂×C₈	#217	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄²	D₃×D₄	1	1	yes	characteristic
153	C₄²	#237	1	1	yes	characteristic
154	C₄²	C₂×#67	1	1	yes	characteristic
155	C₄²	C₂×#67	1	1	yes	characteristic
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	#390	1	1	yes	characteristic
159	C₂²×C₃	#442	1	2	yes	—
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	—	8	4	no	—
163	C₃×C₄	C₂×#101	1	2	yes	—
164	C₃×C₄	#469	1	2	yes	—
165	C₃×C₄	#473	1	2	yes	—
166	C₃×C₄	#496	1	2	yes	—
167	C₂³	D₄×Dic₃	1	1	yes	characteristic
168	Q₈	—	8	2	no	—
169	Q₈	—	8	2	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	8	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	#911	1	4	yes	—
176	C₂×C₄	D₄×Dic₃	1	1	yes	characteristic
177	C₂×C₄	#936	1	1	yes	characteristic
178	C₂×C₄	#938	1	1	yes	characteristic
179	C₂×C₄	#938	1	1	yes	characteristic
180	C₂×C₄	#940	1	1	yes	characteristic
181	C₂×C₄	C₂×#217	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	12	2	no	—
184	C₈	—	12	2	no	—
185	C₈	—	24	2	no	—
186	C₈	#836	1	4	yes	complemented
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	#1966	1	1	yes	characteristic
189	C₂×C₃	#1972	1	1	yes	characteristic
190	C₂×C₃	#1979	1	1	yes	characteristic
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#5152	1	2	yes	—
194	C₂²	#5221	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	#5154	1	2	yes	—
199	C₄	#5159	1	2	yes	—
200	C₄	#5192	1	2	yes	—
201	C₄	C₂×#836	1	2	yes	—
202	C₃	#9808	1	1	yes	characteristic, complemented, Sylow
203	C₂	—	2	2	no	—
204	C₂	#71789	1	1	yes	characteristic
205	C₂	#72039	1	1	yes	characteristic
206	C₂	#72069	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#500

C₂

yes

maximal, characteristic

#71789

C₂

yes

maximal

#71773

C₂

yes

maximal

#69730

C₂

yes

maximal, characteristic

#69725

C₂

yes

maximal, characteristic

#9808

—

maximal, Sylow

C₂×C₃×C₄⋊Q₈

C₂²

yes

—

C₃×#500

—

C₃×#500

—

C₃×#500

—

C₂×C₃×#107

C₄

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#836

—

#4925

C₂²

yes

characteristic

#4924

C₂²

yes

—

C₄×#208

C₂²

yes

—

—

—

—

D₃

yes

—

—

—

—

C₂²×C₃×Q₈

—

C₂²×C₃×C₈

C₂×C₄

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×#208

—

C₂×#208

D₄

yes

—

C₃×#107

—

C₃×#107

—

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

—

#836

—

#830

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

D₄

yes

—

#803

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂×C₄⋊Q₈

C₂×D₃

yes

—

—

—

—

—

Dic₃

yes

—

#423

—

C₂×#101

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

#41

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

C₃×C₄²

yes

—

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×Dic₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₃

yes

—

C₂×C₄²

C₂²×D₃

yes

characteristic

#107

—

100

#107

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

#101

—

105

C₂²⋊C₈

—

106

#97

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

#67

yes

characteristic

110

C₄×C₈

#67

yes

characteristic

111

C₄×C₈

#67

yes

—

112

C₂³×C₃

C₄×D₄

yes

characteristic

113

C₃×Q₈

—

114

C₃×Q₈

—

115

C₃×Q₈

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

—

118

C₂×C₃×C₄

—

119

C₂×C₃×C₄

—

120

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

121

C₂×C₃×C₄

C₄×D₄

yes

characteristic

122

C₂×C₃×C₄

#121

yes

characteristic

123

C₂×C₃×C₄

#121

yes

characteristic

124

C₂×C₃×C₄

#124

yes

characteristic

125

C₂×C₃×C₄

#127

yes

characteristic

126

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

127

C₃×C₈

—

128

C₃×C₈

#101

yes

complemented

129

C₃⋊C₈

—

130

C₃⋊C₈

—

131

C₃⋊C₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

D₃×D₄

yes

characteristic

138

C₂²×C₄

#237

yes

characteristic

139

C₂²×C₄

C₂²×Dic₃

yes

characteristic

140

M₁₆

—

141

M₁₆

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

#217

yes

—

147

C₂×C₈

#217

yes

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

D₃×D₄

yes

characteristic

153

C₄²

#237

yes

characteristic

154

C₄²

C₂×#67

yes

characteristic

155

C₄²

C₂×#67

yes

characteristic

156

C₂²×C₃

—

157

C₂²×C₃

—

158

C₂²×C₃

#390

yes

characteristic

159

C₂²×C₃

#442

yes

—

160

C₃×C₄

—

161

C₃×C₄

—

162

C₃×C₄

—

163

C₃×C₄

C₂×#101

yes

—

164

C₃×C₄

#469

yes

—

165

C₃×C₄

#473

yes

—

166

C₃×C₄

#496

yes

—

167

C₂³

D₄×Dic₃

yes

characteristic

168

Q₈

—

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#911

yes

—

176

C₂×C₄

D₄×Dic₃

yes

characteristic

177

C₂×C₄

#936

yes

characteristic

178

C₂×C₄

#938

yes

characteristic

179

C₂×C₄

#938

yes

characteristic

180

C₂×C₄

#940

yes

characteristic

181

C₂×C₄

C₂×#217

yes

characteristic

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

#836

yes

complemented

187

C₂×C₃

—

188

C₂×C₃

#1966

yes

characteristic

189

C₂×C₃

#1972

yes

characteristic

190

C₂×C₃

#1979

yes

characteristic

191

C₂²

—

192

C₂²

—

193

C₂²

#5152

yes

—

194

C₂²

#5221

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

#5154

yes

—

199

C₄

#5159

yes

—

200

C₄

#5192

yes

—

201

C₄

C₂×#836

yes

—

202

C₃

#9808

yes

characteristic, complemented, Sylow

203

C₂

—

204

C₂

#71789

yes

characteristic

205

C₂

#72039

yes

characteristic

206

C₂

#72069

yes

characteristic

	subgroup	G/H
center	C₂²	#5221
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#500	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	C₄×D₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#500	D₃
hypocenter	C₃	#9808
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#5221

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₂×C₃×#500

C₂

socle

C₂²×C₃

#390

prime radical

C₂³×C₃

C₄×D₄

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
128. C₃×C₈	#101	semidirect
186. C₈	#836	semidirect
202. C₃	#9808	semidirect

H ≅ G/N

type

128. C₃×C₈

#101

semidirect

186. C₈

#836

semidirect

202. C₃

#9808

semidirect

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

subgroup	index
C₂×C₃×#500	2
#71789	2
#71773	2
#69730	2
#69725	2
#9808	3

subgroup

index

C₂×C₃×#500

Subgroups — #10694860

Conjugacy classes of subgroups

Complements

Maximal subgroups