Group #10713326 — subgroups

Subgroups — #10713326

Order 768.

310 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₃×#1844	C₂	1	1	yes	maximal, characteristic
2	C₂×#4487	C₂	1	2	yes	maximal
3	#69779	C₂	1	1	yes	maximal, characteristic
4	C₂×#4463	C₂	1	1	yes	maximal, characteristic
5	#67721	C₂	1	2	yes	maximal
6	#13186	—	3	1	no	maximal, Sylow
7	C₂×C₃×#106	—	2	1	no	—
8	C₂×C₃×#106	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×#384	—	2	1	no	—
11	C₂×#819	C₂²	1	1	yes	characteristic
12	#5106	C₂²	1	1	yes	characteristic
13	#5101	—	2	1	no	—
14	C₂×#812	C₂²	1	2	yes	—
15	#4523	—	2	1	no	—
16	#4487	—	2	4	no	—
17	#4463	C₄	1	4	yes	complemented
18	C₂×#365	—	3	2	no	—
19	C₂×#364	—	3	1	no	—
20	#1844	D₃	1	1	yes	characteristic
21	#1772	—	3	1	no	—
22	#1386	—	3	2	no	—
23	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₃×C₄⋊C₄	—	4	2	no	—
27	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
28	#926	—	2	1	no	—
29	C₂×#208	—	2	1	no	—
30	C₃×#106	—	2	2	no	—
31	C₃×#106	—	2	2	no	—
32	C₃×#106	—	4	2	no	—
33	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄×C₈	Q₈	1	2	yes	—
35	#819	—	2	1	no	—
36	#819	D₄	1	2	yes	—
37	#818	Q₈	1	2	yes	—
38	#817	—	2	1	no	—
39	#815	—	4	1	no	—
40	C₈×Dic₃	—	4	1	no	—
41	#812	—	2	2	no	—
42	#812	C₂×C₄	1	4	yes	—
43	C₂×C₃⋊C₁₆	—	2	2	no	—
44	C₂×M₃₂	—	3	2	no	—
45	C₂×#108	—	3	1	no	—
46	#427	—	3	1	no	—
47	C₂×#106	—	2	1	no	—
48	C₂×#106	C₂×D₃	1	2	yes	—
49	#405	—	6	1	no	—
50	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
51	#384	—	2	1	no	—
52	#365	—	6	4	no	—
53	#364	—	3	4	no	—
54	#339	—	6	1	no	—
55	C₂²×C₃×C₄	—	2	1	no	—
56	C₂²×C₃×C₄	—	4	2	no	—
57	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
59	C₂×C₃×C₈	SD₁₆	1	2	yes	—
60	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
61	C₂×C₃×C₈	SD₁₆	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃×C₈	#41	1	1	yes	characteristic
67	C₂×C₃×C₈	#41	1	1	yes	characteristic
68	C₃×C₄⋊C₄	—	4	2	no	—
69	C₃×C₄⋊C₄	—	4	2	no	—
70	C₃×C₄⋊C₄	—	8	4	no	—
71	C₃×C₄²	—	2	1	no	—
72	C₃×C₄²	C₄⋊C₄	1	2	yes	—
73	#217	—	4	1	no	—
74	C₄⋊Dic₃	—	2	2	no	—
75	C₄×Dic₃	—	4	1	no	—
76	#208	—	2	2	no	—
77	#208	—	4	1	no	—
78	C₂×C₃⋊C₈	—	4	1	no	—
79	C₃⋊C₁₆	—	2	4	no	—
80	C₂×M₁₆	—	6	1	no	—
81	C₂²×C₈	#67	1	1	yes	characteristic
82	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
83	#117	—	6	1	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄⋊C₄	—	4	2	no	—
86	C₂×C₄²	#67	1	1	yes	characteristic
87	M₃₂	—	3	4	no	—
88	M₃₂	—	6	2	no	—
89	C₂×C₁₆	—	6	2	no	—
90	#108	—	3	2	no	—
91	#108	—	6	1	no	—
92	#107	—	3	2	no	—
93	#106	—	2	2	no	—
94	#106	—	2	2	no	—
95	#106	—	4	2	no	—
96	#106	—	6	1	no	—
97	#97	—	12	1	no	—
98	C₄×C₈	—	12	1	no	—
99	C₄×C₈	Dic₆	1	2	yes	—
100	C₄×C₈	C₄×D₃	1	2	yes	—
101	C₂³×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₃×C₄	—	2	4	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	8	4	no	—
105	C₂×C₃×C₄	—	8	2	no	—
106	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₃×C₄	#121	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₃×C₄	#137	1	1	yes	characteristic
112	C₂×Dic₃	—	4	2	no	—
113	C₃×C₈	#106	1	2	yes	—
114	C₃×C₈	#106	1	2	yes	—
115	C₃×C₈	C₄×D₄	1	1	yes	characteristic
116	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
117	C₃×C₈	#125	1	1	yes	characteristic
118	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
119	C₃⋊C₈	—	4	2	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	C₂×#67	1	1	yes	characteristic
123	M₁₆	—	6	2	no	—
124	M₁₆	—	12	1	no	—
125	C₂×C₈	—	12	1	no	—
126	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
127	C₂×C₈	#214	1	2	yes	—
128	C₂×C₈	#215	1	1	yes	characteristic
129	C₂×C₈	#215	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
132	C₂×C₈	D₃×D₄	1	1	yes	characteristic
133	C₂×C₈	#237	1	1	yes	characteristic
134	C₂×C₈	D₃×Q₈	1	1	yes	characteristic
135	C₂×C₈	#239	1	1	yes	characteristic
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	6	2	no	—
139	C₄⋊C₄	—	8	4	no	—
140	C₂²⋊C₄	—	12	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	12	1	no	—
143	C₄²	Dic₃⋊C₄	1	2	yes	—
144	C₁₆	—	6	4	no	—
145	C₂²×C₃	—	2	2	no	—
146	C₂²×C₃	#428	1	1	yes	characteristic
147	C₂²×C₃	#460	1	1	yes	characteristic
148	C₂²×C₃	#476	1	1	yes	characteristic
149	C₃×C₄	—	2	4	no	—
150	C₃×C₄	—	16	2	no	—
151	C₃×C₄	#389	1	1	yes	characteristic
152	C₃×C₄	C₂×#106	1	1	yes	characteristic
153	C₃×C₄	#461	1	1	yes	characteristic
154	C₃×C₄	#475	1	1	yes	characteristic
155	Dic₃	—	8	2	no	—
156	C₂³	#932	1	1	yes	characteristic
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	8	4	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	12	2	no	—
162	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
163	C₂×C₄	C₂×#214	1	1	yes	characteristic
164	C₂×C₄	#938	1	1	yes	characteristic
165	C₂×C₄	C₂×#215	1	1	yes	characteristic
166	C₂×C₄	#942	1	1	yes	characteristic
167	C₂×C₄	#946	1	1	yes	characteristic
168	C₈	—	12	2	no	—
169	C₈	#808	1	2	yes	—
170	C₈	#808	1	2	yes	—
171	C₈	#887	1	1	yes	characteristic
172	C₈	#888	1	1	yes	characteristic
173	C₈	#890	1	1	yes	characteristic
174	C₈	#893	1	1	yes	characteristic
175	C₂×C₃	—	2	1	no	—
176	C₂×C₃	—	2	1	no	—
177	C₂×C₃	#1948	1	1	yes	characteristic
178	C₂×C₃	#2230	1	2	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	#4964	1	1	yes	characteristic
181	C₂²	#5075	1	1	yes	characteristic
182	C₂²	#5084	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	16	2	no	—
185	C₄	—	24	2	no	—
186	C₄	C₂×#808	1	1	yes	characteristic
187	C₄	#4972	1	1	yes	characteristic
188	C₄	#5083	1	1	yes	characteristic
189	C₄	#5094	1	1	yes	characteristic
190	C₃	#13186	1	1	yes	characteristic, complemented, Sylow
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#69568	1	1	yes	characteristic
194	C₂	#71350	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1844

C₂

yes

maximal, characteristic

C₂×#4487

C₂

yes

maximal

#69779

C₂

yes

maximal, characteristic

C₂×#4463

C₂

yes

maximal, characteristic

#67721

C₂

yes

maximal

#13186

—

maximal, Sylow

C₂×C₃×#106

—

C₂×C₃×#106

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₃×#384

—

C₂×#819

C₂²

yes

characteristic

#5106

C₂²

yes

characteristic

#5101

—

C₂×#812

C₂²

yes

—

#4523

—

#4487

—

#4463

C₄

yes

—

—

D₃

yes

characteristic

#1772

—

#1386

—

C₂²×C₃×C₈

D₄

yes

characteristic

C₂²×C₃×C₈

C₂³

yes

characteristic

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

D₄

yes

—

—

—

—

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

Q₈

yes

—

#819

—

#819

D₄

yes

—

#818

Q₈

yes

—

#817

—

#815

—

C₈×Dic₃

—

#812

—

#812

C₂×C₄

yes

—

C₂×C₃⋊C₁₆

—

C₂×M₃₂

—

—

—

—

C₂×D₃

yes

—

#405

—

C₂×C₄×C₈

C₂×D₃

yes

—

—

—

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₈

C₄⋊C₄

yes

—

C₂×C₃×C₈

SD₁₆

yes

—

C₂×C₃×C₈

SD₁₆

yes

characteristic

C₂×C₃×C₈

SD₁₆

yes

characteristic

C₂×C₃×C₈

C₂²×C₄

yes

characteristic

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

C₂×Q₈

yes

characteristic

C₂×C₃×C₈

C₂×Q₈

yes

C₂×C₃×C₈

yes

C₂×C₃×C₈

yes

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₄⋊C₄

yes

—

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₃⋊C₁₆

—

C₂×M₁₆

—

C₂²×C₈

#67

yes

characteristic

C₂²×C₈

C₂²×D₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

#67

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

Dic₆

yes

—

100

C₄×C₈

C₄×D₃

yes

—

101

C₂³×C₃

AGL(1,ℤ/8)

yes

characteristic

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

C₂×C₄⋊C₄

yes

characteristic

107

C₂×C₃×C₄

#121

yes

characteristic

108

C₂×C₃×C₄

C₂²⋊Q₈

yes

characteristic

109

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

110

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

111

C₂×C₃×C₄

#137

yes

characteristic

112

C₂×Dic₃

—

113

C₃×C₈

#106

yes

—

114

C₃×C₈

#106

yes

—

115

C₃×C₈

C₄×D₄

yes

characteristic

116

C₃×C₈

C₄×Q₈

yes

characteristic

117

C₃×C₈

#125

yes

characteristic

118

C₃×C₈

C₄⋊Q₈

yes

characteristic

119

C₃⋊C₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂×#67

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

Dic₃⋊C₄

yes

—

127

C₂×C₈

#214

yes

—

128

C₂×C₈

#215

yes

characteristic

129

C₂×C₈

#215

yes

characteristic

130

C₂×C₈

C₂×Dic₆

yes

characteristic

131

C₂×C₈

C₂×C₄×D₃

yes

characteristic

132

C₂×C₈

D₃×D₄

yes

characteristic

133

C₂×C₈

#237

yes

characteristic

134

C₂×C₈

D₃×Q₈

yes

characteristic

135

C₂×C₈

#239

yes

characteristic

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₂²⋊C₄

—

141

C₄²

—

142

C₄²

—

143

C₄²

Dic₃⋊C₄

yes

—

144

C₁₆

—

145

C₂²×C₃

—

146

C₂²×C₃

#428

yes

characteristic

147

C₂²×C₃

#460

yes

characteristic

148

C₂²×C₃

#476

yes

characteristic

149

C₃×C₄

—

150

C₃×C₄

—

151

C₃×C₄

#389

yes

characteristic

152

C₃×C₄

C₂×#106

yes

characteristic

153

C₃×C₄

#461

yes

characteristic

154

C₃×C₄

#475

yes

characteristic

155

Dic₃

—

156

C₂³

#932

yes

characteristic

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

C₂×Dic₃⋊C₄

yes

characteristic

163

C₂×C₄

C₂×#214

yes

characteristic

164

C₂×C₄

#938

yes

characteristic

165

C₂×C₄

C₂×#215

yes

characteristic

166

C₂×C₄

#942

yes

characteristic

167

C₂×C₄

#946

yes

characteristic

168

C₈

—

169

C₈

#808

yes

—

170

C₈

#808

yes

—

171

C₈

#887

yes

characteristic

172

C₈

#888

yes

characteristic

173

C₈

#890

yes

characteristic

174

C₈

#893

yes

characteristic

175

C₂×C₃

—

176

C₂×C₃

—

177

C₂×C₃

#1948

yes

characteristic

178

C₂×C₃

#2230

yes

—

179

C₂²

—

180

C₂²

#4964

yes

characteristic

181

C₂²

#5075

yes

characteristic

182

C₂²

#5084

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

C₂×#808

yes

characteristic

187

C₄

#4972

yes

characteristic

188

C₄

#5083

yes

characteristic

189

C₄

#5094

yes

characteristic

190

C₃

#13186

yes

characteristic, complemented, Sylow

191

C₂

—

192

C₂

—

193

C₂

#69568

yes

characteristic

194

C₂

#71350

yes

—

	subgroup	G/H
center	C₂²	#5084
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1844	C₂
socle	C₂²×C₃	#476
prime radical	C₂³×C₃	AGL(1,ℤ/8)
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1844	D₃
hypocenter	C₃	#13186
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

subgroup

G/H

center

C₂²

#5084

commutator

C₂×C₃×C₈

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

C₂²×C₃×C₈

C₂³

D₃

C₃

G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1844

N	H ≅ G/N	type
17. #4463	C₄	semidirect
190. C₃	#13186	semidirect

H ≅ G/N

type

17. #4463

C₄

semidirect

190. C₃

#13186

semidirect

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

subgroup	index
C₃×#1844	2
C₂×#4487	2
#69779	2
C₂×#4463	2
#67721	2
#13186	3

subgroup

index

Subgroups — #10713326

Conjugacy classes of subgroups

Complements

Maximal subgroups