Group #10713672 — subgroups

Subgroups — #10713672

Order 768.

462 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 143 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#4565	C₂	1	4	yes	maximal
3	C₄×#819	C₂	1	1	yes	maximal, characteristic
4	#69743	C₂	1	1	yes	maximal, characteristic
5	#13256	—	3	1	no	maximal, Sylow
6	C₂²×C₃×Q₁₆	—	2	1	no	—
7	C₂²×C₃×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
9	C₃×#494	—	2	2	no	—
10	C₃×#494	C₄	1	4	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#819	C₂²	1	2	yes	—
13	C₂×#812	C₂²	1	2	yes	—
14	C₄×#208	—	2	1	no	—
15	#4565	—	2	8	no	—
16	#4552	—	2	1	no	—
17	C₄⋊(C₃⋊C₁₆)	—	2	2	no	—
18	C₂×#494	D₃	1	1	yes	characteristic
19	#2150	—	3	1	no	—
20	C₂×#362	—	3	4	no	—
21	C₄×#108	—	3	1	no	—
22	C₂²×C₃×Q₈	—	2	2	no	—
23	C₂×C₃×Q₁₆	—	2	4	no	—
24	C₂×C₃×Q₁₆	—	2	4	no	—
25	C₂×C₃×Q₁₆	—	2	2	no	—
26	C₂×C₃×Q₁₆	—	2	4	no	—
27	C₂×C₃×Q₁₆	—	4	4	no	—
28	C₂×C₃×Q₁₆	C₂×C₄	1	4	yes	—
29	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
31	C₃×C₄⋊Q₈	—	2	4	no	—
32	C₃×C₄⋊Q₈	—	4	1	no	—
33	C₃×C₄⋊Q₈	—	4	1	no	—
34	C₂×C₃×C₄⋊C₄	—	4	2	no	—
35	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
36	C₂×#208	—	2	2	no	—
37	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₃×C₄×C₈	D₄	1	2	yes	—
39	#819	—	2	2	no	—
40	#819	D₄	1	4	yes	—
41	#812	—	2	4	no	—
42	C₂×C₃⋊C₁₆	—	2	2	no	—
43	#803	—	4	1	no	—
44	C₄×C₃⋊C₈	—	4	1	no	—
45	C₂²×Q₁₆	—	2	1	no	—
46	C₂²×Q₁₆	C₂×D₃	1	2	yes	—
47	C₂×C₄⋊Q₈	—	2	1	no	—
48	C₂×M₃₂	—	3	2	no	—
49	#494	—	2	2	no	—
50	#494	Dic₃	1	4	yes	—
51	C₂×#108	—	3	2	no	—
52	C₄×M₁₆	—	6	1	no	—
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	C₄⋊C₁₆	—	6	2	no	—
55	#362	—	6	8	no	—
56	#341	—	6	1	no	—
57	C₂×C₃×Q₈	—	2	4	no	—
58	C₂×C₃×Q₈	—	4	2	no	—
59	C₂×C₃×Q₈	—	4	4	no	—
60	C₂×C₃×Q₈	—	4	4	no	—
61	C₂²×C₃×C₄	—	4	2	no	—
62	C₂²×C₃×C₄	SD₁₆	1	2	yes	—
63	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
64	C₃×Q₁₆	—	2	8	no	—
65	C₃×Q₁₆	—	4	8	no	—
66	C₃×Q₁₆	—	4	2	no	—
67	C₃×Q₁₆	—	4	2	no	—
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
72	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₈	#41	1	2	yes	—
74	C₃×C₄⋊C₄	—	4	4	no	—
75	C₃×C₄⋊C₄	—	4	4	no	—
76	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
77	C₃×C₄²	D₈	1	1	yes	characteristic
78	C₃×C₄²	D₈	1	1	yes	characteristic
79	#208	—	2	4	no	—
80	#208	—	4	2	no	—
81	C₂×C₃⋊C₈	—	4	2	no	—
82	C₃⋊C₁₆	—	4	2	no	—
83	C₂²×Q₈	—	2	2	no	—
84	C₂×Q₁₆	—	2	4	no	—
85	C₂×Q₁₆	—	2	4	no	—
86	C₂×Q₁₆	—	2	4	no	—
87	C₂×Q₁₆	—	2	2	no	—
88	C₂×Q₁₆	—	4	4	no	—
89	C₂×Q₁₆	C₂×Dic₃	1	4	yes	—
90	C₂×M₁₆	—	6	2	no	—
91	C₂²×C₈	#67	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
93	C₄⋊Q₈	—	2	4	no	—
94	C₄⋊Q₈	—	4	1	no	—
95	C₄⋊Q₈	—	4	1	no	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄²	#67	1	1	yes	characteristic
98	M₃₂	—	6	4	no	—
99	C₂×C₁₆	—	6	2	no	—
100	#108	—	3	4	no	—
101	#108	—	6	2	no	—
102	#97	—	12	1	no	—
103	C₄×C₈	—	12	1	no	—
104	C₄×C₈	C₂×Dic₃	1	2	yes	—
105	C₄×C₈	#67	1	2	yes	—
106	C₂³×C₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₃×Q₈	—	4	4	no	—
108	C₃×Q₈	—	4	4	no	—
109	C₃×Q₈	—	8	4	no	—
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	—	4	4	no	—
112	C₂×C₃×C₄	—	8	4	no	—
113	C₂×C₃×C₄	#102	1	2	yes	—
114	C₂×C₃×C₄	#102	1	2	yes	—
115	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₃×C₄	#121	1	2	yes	—
117	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
118	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
119	C₃×C₈	—	2	2	no	—
120	C₃×C₈	C₄×D₄	1	2	yes	—
121	C₃×C₈	#124	1	1	yes	characteristic
122	C₃×C₈	#127	1	1	yes	characteristic
123	C₃⋊C₈	—	4	4	no	—
124	C₂×Q₈	—	2	4	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	4	4	no	—
127	C₂×Q₈	—	4	4	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	#214	1	2	yes	—
130	C₂²×C₄	C₂×#67	1	1	yes	characteristic
131	Q₁₆	—	2	8	no	—
132	Q₁₆	—	4	8	no	—
133	Q₁₆	—	4	2	no	—
134	Q₁₆	—	4	2	no	—
135	M₁₆	—	6	4	no	—
136	M₁₆	—	12	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	12	2	no	—
139	C₂×C₈	#217	1	2	yes	—
140	C₂×C₈	D₃×D₄	1	2	yes	—
141	C₂×C₈	#237	1	2	yes	—
142	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
143	C₂×C₈	C₂×#67	1	1	yes	characteristic
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄²	#213	1	1	yes	characteristic
147	C₄²	#213	1	1	yes	characteristic
148	C₄²	#217	1	2	yes	—
149	C₁₆	—	12	2	no	—
150	C₂²×C₃	—	2	2	no	—
151	C₂²×C₃	C₂×#102	1	1	yes	characteristic
152	C₂²×C₃	#460	1	2	yes	—
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	8	4	no	—
155	C₃×C₄	#362	1	4	yes	complemented
156	C₃×C₄	#390	1	1	yes	characteristic
157	C₃×C₄	#418	1	1	yes	characteristic
158	C₃×C₄	#459	1	2	yes	—
159	C₂³	C₂×#214	1	1	yes	characteristic
160	Q₈	—	4	4	no	—
161	Q₈	—	4	4	no	—
162	Q₈	—	8	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	8	4	no	—
166	C₂×C₄	#832	1	2	yes	—
167	C₂×C₄	#832	1	2	yes	—
168	C₂×C₄	C₂×#213	1	1	yes	characteristic
169	C₂×C₄	#932	1	2	yes	—
170	C₂×C₄	#938	1	2	yes	—
171	C₂×C₄	C₂×#217	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	12	4	no	—
174	C₈	D₄×Dic₃	1	2	yes	—
175	C₈	#936	1	1	yes	characteristic
176	C₈	#940	1	1	yes	characteristic
177	C₂×C₃	—	2	2	no	—
178	C₂×C₃	#1982	1	1	yes	characteristic
179	C₂×C₃	C₂×#362	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	#5075	1	2	yes	—
182	C₂²	C₂×#832	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	#4565	1	4	yes	complemented
186	C₄	#5072	1	2	yes	—
187	C₄	#5215	1	1	yes	characteristic
188	C₄	#5221	1	1	yes	characteristic
189	C₃	#13256	1	1	yes	characteristic, complemented, Sylow
190	C₂	—	2	2	no	—
191	C₂	#71761	1	1	yes	characteristic
192	C₂	C₂×#4565	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#494

C₂

yes

maximal, characteristic

C₂×#4565

C₂

yes

maximal

C₄×#819

C₂

yes

maximal, characteristic

#69743

C₂

yes

maximal, characteristic

#13256

—

maximal, Sylow

C₂²×C₃×Q₁₆

—

C₂²×C₃×Q₁₆

C₂²

yes

—

C₂×C₃×C₄⋊Q₈

—

C₃×#494

—

C₃×#494

C₄

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₂×#819

C₂²

yes

—

C₂×#812

C₂²

yes

—

C₄×#208

—

#4565

—

#4552

—

C₄⋊(C₃⋊C₁₆)

—

C₂×#494

D₃

yes

—

—

—

C₂²×C₃×Q₈

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

C₂×C₄

yes

—

C₂²×C₃×C₈

D₄

yes

characteristic

C₂²×C₃×C₈

C₂³

yes

characteristic

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×#208

—

C₃×C₄×C₈

C₂×C₄

yes

—

C₃×C₄×C₈

D₄

yes

—

#819

—

#819

D₄

yes

—

#812

—

C₂×C₃⋊C₁₆

—

#803

—

C₄×C₃⋊C₈

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂×D₃

yes

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

#494

—

#494

Dic₃

yes

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

C₄⋊C₁₆

—

#362

—

#341

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

SD₁₆

yes

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₃×Q₁₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²×C₄

yes

characteristic

C₂×C₃×C₈

C₂×D₄

yes

—

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

#41

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²⋊C₄

yes

—

C₃×C₄²

D₈

yes

characteristic

C₃×C₄²

D₈

yes

characteristic

#208

—

#208

—

C₂×C₃⋊C₈

—

C₃⋊C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×Dic₃

yes

—

C₂×M₁₆

—

C₂²×C₈

#67

yes

characteristic

C₂²×C₈

C₂²×D₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

#67

yes

characteristic

M₃₂

—

C₂×C₁₆

—

100

#108

—

101

#108

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

C₂×Dic₃

yes

—

105

C₄×C₈

#67

yes

—

106

C₂³×C₃

C₂×SD₁₆

yes

characteristic

107

C₃×Q₈

—

108

C₃×Q₈

—

109

C₃×Q₈

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

#102

yes

—

114

C₂×C₃×C₄

#102

yes

—

115

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

116

C₂×C₃×C₄

#121

yes

—

117

C₂×C₃×C₄

C₂×D₈

yes

characteristic

118

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

119

C₃×C₈

—

120

C₃×C₈

C₄×D₄

yes

—

121

C₃×C₈

#124

yes

characteristic

122

C₃×C₈

#127

yes

characteristic

123

C₃⋊C₈

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂²×C₄

—

129

C₂²×C₄

#214

yes

—

130

C₂²×C₄

C₂×#67

yes

characteristic

131

Q₁₆

—

132

Q₁₆

—

133

Q₁₆

—

134

Q₁₆

—

135

M₁₆

—

136

M₁₆

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

#217

yes

—

140

C₂×C₈

D₃×D₄

yes

—

141

C₂×C₈

#237

yes

—

142

C₂×C₈

C₂²×Dic₃

yes

characteristic

143

C₂×C₈

C₂×#67

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄²

#213

yes

characteristic

147

C₄²

#213

yes

characteristic

148

C₄²

#217

yes

—

149

C₁₆

—

150

C₂²×C₃

—

151

C₂²×C₃

C₂×#102

yes

characteristic

152

C₂²×C₃

#460

yes

—

153

C₃×C₄

—

154

C₃×C₄

—

155

C₃×C₄

#362

yes

complemented

156

C₃×C₄

#390

yes

characteristic

157

C₃×C₄

#418

yes

characteristic

158

C₃×C₄

#459

yes

—

159

C₂³

C₂×#214

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

#832

yes

—

167

C₂×C₄

#832

yes

—

168

C₂×C₄

C₂×#213

yes

characteristic

169

C₂×C₄

#932

yes

—

170

C₂×C₄

#938

yes

—

171

C₂×C₄

C₂×#217

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

D₄×Dic₃

yes

—

175

C₈

#936

yes

characteristic

176

C₈

#940

yes

characteristic

177

C₂×C₃

—

178

C₂×C₃

#1982

yes

characteristic

179

C₂×C₃

C₂×#362

yes

—

180

C₂²

—

181

C₂²

#5075

yes

—

182

C₂²

C₂×#832

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

#4565

yes

complemented

186

C₄

#5072

yes

—

187

C₄

#5215

yes

characteristic

188

C₄

#5221

yes

characteristic

189

C₃

#13256

yes

characteristic, complemented, Sylow

190

C₂

—

191

C₂

#71761

yes

characteristic

192

C₂

C₂×#4565

yes

—

	subgroup	G/H
center	C₂²	C₂×#832
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#494	C₂
socle	C₂²×C₃	C₂×#102
prime radical	C₂³×C₃	C₂×SD₁₆
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#494	D₃
hypocenter	C₃	#13256
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂²

C₂×#832

commutator

C₂×C₃×C₈

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₃

Fitting

C₂×C₃×#494

C₂

socle

C₂²×C₃

C₂×#102

prime radical

C₂³×C₃

C₂×SD₁₆

cosocle

C₂²×C₃×C₈

C₂³

D₃

C₃

G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

N	H ≅ G/N	type
155. C₃×C₄	#362	semidirect
185. C₄	#4565	semidirect
189. C₃	#13256	semidirect

H ≅ G/N

type

155. C₃×C₄

#362

semidirect

185. C₄

#4565

semidirect

189. C₃

#13256

semidirect

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

subgroup	index
C₂×C₃×#494	2
C₂×#4565	2
C₄×#819	2
#69743	2
#13256	3

subgroup

index

C₂×C₃×#494

Subgroups — #10713672

Conjugacy classes of subgroups

Complements

Maximal subgroups