Group #10695047 — subgroups

Subgroups — #10695047

Order 768.

510 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 195 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×#4680	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic
3	#69777	C₂	1	2	yes	maximal
4	#69771	C₂	1	2	yes	maximal
5	#69765	C₂	1	1	yes	maximal, characteristic
6	#9841	—	3	1	no	maximal, Sylow
7	C₂×#869	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#818	—	2	2	no	—
11	C₂×#817	—	2	2	no	—
12	C₂×#816	—	2	4	no	—
13	#4935	C₂²	1	1	yes	characteristic
14	#4935	C₂²	1	1	yes	characteristic
15	#4680	—	2	2	no	—
16	#4680	C₄	1	4	yes	complemented
17	C₃×#322	—	2	2	no	—
18	#4551	—	2	4	no	—
19	C₂×#498	—	3	1	no	—
20	#1990	—	3	2	no	—
21	#1956	—	3	2	no	—
22	#1845	—	3	1	no	—
23	C₂×#322	D₃	1	1	yes	characteristic
24	C₂²×Dic₆	—	2	2	no	—
25	C₂²×C₃×C₈	D₄	1	4	yes	—
26	C₂²×C₃×C₈	D₄	1	2	yes	—
27	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₃	—	2	2	no	—
29	C₂×C₄⋊Dic₃	—	2	2	no	—
30	C₂×C₄⋊Dic₃	—	2	2	no	—
31	C₂×C₄⋊Dic₃	—	2	2	no	—
32	C₂×C₄⋊Dic₃	—	2	1	no	—
33	C₂×C₄⋊Dic₃	—	2	1	no	—
34	#869	—	2	2	no	—
35	#869	—	2	2	no	—
36	#869	—	2	2	no	—
37	#869	C₂×C₄	1	4	yes	—
38	C₃×C₄×C₈	—	2	2	no	—
39	C₃×C₄×C₈	—	2	2	no	—
40	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
41	C₃×C₄×C₈	D₄	1	2	yes	—
42	#818	—	4	4	no	—
43	#817	—	2	8	no	—
44	#816	—	4	8	no	—
45	C₂×C₄⋊Q₈	—	3	2	no	—
46	#498	—	3	4	no	—
47	#498	—	6	2	no	—
48	C₂×#107	—	6	2	no	—
49	C₂×#106	—	6	2	no	—
50	C₂×#103	—	6	4	no	—
51	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
52	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	#384	—	3	1	no	—
54	#384	—	3	1	no	—
55	#340	—	6	4	no	—
56	#322	—	2	2	no	—
57	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
59	C₂²×Dic₃	—	4	2	no	—
60	C₂²×Dic₃	—	4	1	no	—
61	C₂²×Dic₃	—	4	1	no	—
62	C₂×Dic₆	—	2	4	no	—
63	C₂×Dic₆	—	4	4	no	—
64	C₂×Dic₆	—	4	4	no	—
65	C₂×Dic₆	—	4	2	no	—
66	C₂×C₃×C₈	—	2	4	no	—
67	C₂×C₃×C₈	—	2	4	no	—
68	C₂×C₃×C₈	—	2	4	no	—
69	C₂×C₃×C₈	C₂²⋊C₄	1	4	yes	—
70	C₂×C₃×C₈	SD₁₆	1	4	yes	—
71	C₂×C₃×C₈	Q₁₆	1	4	yes	—
72	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
73	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₃×C₄²	#41	1	2	yes	—
75	C₄⋊Dic₃	—	2	4	no	—
76	C₄⋊Dic₃	—	2	4	no	—
77	C₄⋊Dic₃	—	4	2	no	—
78	C₄⋊Dic₃	—	4	2	no	—
79	C₄⋊Dic₃	—	4	2	no	—
80	C₄⋊Dic₃	—	4	2	no	—
81	C₄⋊Dic₃	—	4	4	no	—
82	C₄⋊Dic₃	—	4	4	no	—
83	C₂²×Q₈	—	6	2	no	—
84	C₂²×C₈	D₁₂	1	4	yes	—
85	C₂²×C₈	D₁₂	1	2	yes	—
86	C₄⋊Q₈	—	3	4	no	—
87	C₄⋊Q₈	—	6	2	no	—
88	C₄⋊Q₈	—	6	2	no	—
89	C₄⋊Q₈	—	6	2	no	—
90	C₂×C₄⋊C₄	—	6	2	no	—
91	C₂×C₄⋊C₄	—	6	2	no	—
92	C₂×C₄⋊C₄	—	6	2	no	—
93	C₂×C₄⋊C₄	—	6	2	no	—
94	C₂×C₄⋊C₄	—	6	1	no	—
95	C₂×C₄⋊C₄	—	6	1	no	—
96	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
97	#107	—	12	4	no	—
98	#106	—	6	8	no	—
99	#103	—	12	8	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	C₄×D₃	1	2	yes	—
103	C₄×C₈	#67	1	2	yes	—
104	C₂³×C₃	#127	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
107	C₂×C₃×C₄	#121	1	2	yes	—
108	C₂×C₃×C₄	#124	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
110	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
111	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
112	C₂×C₃×C₄	#137	1	4	yes	—
113	C₂×Dic₃	—	4	4	no	—
114	C₂×Dic₃	—	8	2	no	—
115	C₂×Dic₃	—	8	2	no	—
116	C₂×Dic₃	—	8	1	no	—
117	C₂×Dic₃	—	8	1	no	—
118	C₂×Dic₃	—	8	2	no	—
119	C₂×Dic₃	—	8	2	no	—
120	Dic₆	—	4	8	no	—
121	Dic₆	—	8	4	no	—
122	C₃×C₈	—	4	4	no	—
123	C₃×C₈	#103	1	8	yes	complemented
124	C₂×Q₈	—	6	4	no	—
125	C₂×Q₈	—	12	2	no	—
126	C₂×Q₈	—	12	4	no	—
127	C₂×Q₈	—	12	4	no	—
128	C₂²×C₄	—	12	2	no	—
129	C₂²×C₄	—	12	1	no	—
130	C₂²×C₄	—	12	1	no	—
131	C₂²×C₄	C₂×D₁₂	1	2	yes	—
132	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	#204	1	4	yes	—
137	C₂×C₈	Dic₁₂	1	4	yes	—
138	C₂×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
139	C₄⋊C₄	—	6	4	no	—
140	C₄⋊C₄	—	6	4	no	—
141	C₄⋊C₄	—	12	2	no	—
142	C₄⋊C₄	—	12	2	no	—
143	C₄⋊C₄	—	12	2	no	—
144	C₄⋊C₄	—	12	2	no	—
145	C₄⋊C₄	—	12	4	no	—
146	C₄⋊C₄	—	12	4	no	—
147	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
148	C₄²	#235	1	2	yes	—
149	C₄²	C₂×#67	1	1	yes	characteristic
150	C₂²×C₃	#390	1	1	yes	characteristic
151	C₂²×C₃	#492	1	1	yes	characteristic
152	C₂²×C₃	#494	1	1	yes	characteristic
153	C₂²×C₃	#496	1	2	yes	—
154	C₂²×C₃	#497	1	2	yes	—
155	C₃×C₄	C₂×#103	1	2	yes	—
156	C₃×C₄	#440	1	4	yes	—
157	C₃×C₄	#468	1	4	yes	—
158	C₃×C₄	#487	1	2	yes	—
159	Dic₃	—	8	4	no	—
160	Dic₃	—	16	1	no	—
161	Dic₃	—	16	1	no	—
162	C₂³	#874	1	1	yes	characteristic
163	Q₈	—	12	8	no	—
164	Q₈	—	24	4	no	—
165	C₂×C₄	—	12	4	no	—
166	C₂×C₄	—	24	2	no	—
167	C₂×C₄	—	24	2	no	—
168	C₂×C₄	—	24	1	no	—
169	C₂×C₄	—	24	1	no	—
170	C₂×C₄	—	24	2	no	—
171	C₂×C₄	—	24	2	no	—
172	C₂×C₄	C₄×D₁₂	1	2	yes	—
173	C₂×C₄	#875	1	1	yes	characteristic
174	C₂×C₄	C₂×#204	1	2	yes	—
175	C₂×C₄	C₂×Dic₁₂	1	2	yes	—
176	C₂×C₄	#908	1	4	yes	—
177	C₂×C₄	#909	1	4	yes	—
178	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
179	C₂×C₄	#930	1	2	yes	—
180	C₈	—	4	4	no	—
181	C₈	#816	1	8	yes	complemented
182	C₂×C₃	#1715	1	1	yes	characteristic
183	C₂×C₃	#1716	1	1	yes	characteristic
184	C₂×C₃	#1723	1	2	yes	—
185	C₂×C₃	#1956	1	1	yes	characteristic
186	C₂×C₃	#1973	1	2	yes	—
187	C₂²	#4693	1	1	yes	characteristic
188	C₂²	#4699	1	1	yes	characteristic
189	C₂²	#4712	1	2	yes	—
190	C₂²	#4715	1	2	yes	—
191	C₂²	#4939	1	1	yes	characteristic
192	C₄	—	24	4	no	—
193	C₄	—	48	1	no	—
194	C₄	—	48	1	no	—
195	C₄	#4681	1	2	yes	—
196	C₄	#4710	1	4	yes	—
197	C₄	C₂×#816	1	2	yes	—
198	C₄	#5134	1	4	yes	—
199	C₃	#9841	1	1	yes	characteristic, complemented, Sylow
200	C₂	#68091	1	1	yes	characteristic
201	C₂	#68095	1	1	yes	characteristic
202	C₂	#68099	1	2	yes	—
203	C₂	#69771	1	1	yes	characteristic
204	C₂	#69826	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4680

C₂

yes

maximal, characteristic

C₂×C₃×#322

C₂

yes

maximal, characteristic

#69777

C₂

yes

maximal

#69771

C₂

yes

maximal

#69765

C₂

yes

maximal, characteristic

#9841

—

maximal, Sylow

C₂×#869

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

#4935

C₂²

yes

characteristic

#4680

—

#4680

C₄

yes

—

—

—

—

—

—

D₃

yes

characteristic

C₂²×Dic₆

—

C₂²×C₃×C₈

D₄

yes

—

C₂²×C₃×C₈

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

—

—

—

C₂×C₄

yes

—

C₃×C₄×C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#818

—

#817

—

#816

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₃

yes

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

SD₁₆

yes

—

C₂×C₃×C₈

Q₁₆

yes

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

#41

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×Q₈

—

C₂²×C₈

D₁₂

yes

—

C₂²×C₈

D₁₂

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₃

yes

—

—

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

C₄×D₃

yes

—

103

C₄×C₈

#67

yes

—

104

C₂³×C₃

#127

yes

characteristic

105

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

106

C₂×C₃×C₄

C₄×D₄

yes

—

107

C₂×C₃×C₄

#121

yes

—

108

C₂×C₃×C₄

#124

yes

characteristic

109

C₂×C₃×C₄

C₂×SD₁₆

yes

—

110

C₂×C₃×C₄

C₂×Q₁₆

yes

—

111

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

112

C₂×C₃×C₄

#137

yes

—

113

C₂×Dic₃

—

114

C₂×Dic₃

—

115

C₂×Dic₃

—

116

C₂×Dic₃

—

117

C₂×Dic₃

—

118

C₂×Dic₃

—

119

C₂×Dic₃

—

120

Dic₆

—

121

Dic₆

—

122

C₃×C₈

—

123

C₃×C₈

#103

yes

complemented

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂×D₁₂

yes

—

132

C₂²×C₄

C₂×D₁₂

yes

characteristic

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

#204

yes

—

137

C₂×C₈

Dic₁₂

yes

—

138

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄²

C₂×C₄×D₃

yes

characteristic

148

C₄²

#235

yes

—

149

C₄²

C₂×#67

yes

characteristic

150

C₂²×C₃

#390

yes

characteristic

151

C₂²×C₃

#492

yes

characteristic

152

C₂²×C₃

#494

yes

characteristic

153

C₂²×C₃

#496

yes

—

154

C₂²×C₃

#497

yes

—

155

C₃×C₄

C₂×#103

yes

—

156

C₃×C₄

#440

yes

—

157

C₃×C₄

#468

yes

—

158

C₃×C₄

#487

yes

—

159

Dic₃

—

160

Dic₃

—

161

Dic₃

—

162

C₂³

#874

yes

characteristic

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

C₄×D₁₂

yes

—

173

C₂×C₄

#875

yes

characteristic

174

C₂×C₄

C₂×#204

yes

—

175

C₂×C₄

C₂×Dic₁₂

yes

—

176

C₂×C₄

#908

yes

—

177

C₂×C₄

#909

yes

—

178

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

179

C₂×C₄

#930

yes

—

180

C₈

—

181

C₈

#816

yes

complemented

182

C₂×C₃

#1715

yes

characteristic

183

C₂×C₃

#1716

yes

characteristic

184

C₂×C₃

#1723

yes

—

185

C₂×C₃

#1956

yes

characteristic

186

C₂×C₃

#1973

yes

—

187

C₂²

#4693

yes

characteristic

188

C₂²

#4699

yes

characteristic

189

C₂²

#4712

yes

—

190

C₂²

#4715

yes

—

191

C₂²

#4939

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

#4681

yes

—

196

C₄

#4710

yes

—

197

C₄

C₂×#816

yes

—

198

C₄

#5134

yes

—

199

C₃

#9841

yes

characteristic, complemented, Sylow

200

C₂

#68091

yes

characteristic

201

C₂

#68095

yes

characteristic

202

C₂

#68099

yes

—

203

C₂

#69771

yes

characteristic

204

C₂

#69826

yes

—

	subgroup	G/H
center	C₂³	#874
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#322	C₂
socle, prime radical	C₂³×C₃	#127
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#322	D₃
hypocenter	C₃	#9841
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#874

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₂×C₃×#322

C₂

socle, prime radical

C₂³×C₃

#127

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
16. #4680	C₄	semidirect
123. C₃×C₈	#103	semidirect
181. C₈	#816	semidirect
199. C₃	#9841	semidirect

H ≅ G/N

type

16. #4680

C₄

semidirect

123. C₃×C₈

#103

semidirect

181. C₈

#816

semidirect

199. C₃

#9841

semidirect

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

subgroup	index
C₂×#4680	2
C₂×C₃×#322	2
#69777	2
#69771	2
#69765	2
#9841	3

subgroup

index

C₂×C₃×#322

Subgroups — #10695047

Conjugacy classes of subgroups

Complements

Maximal subgroups