Group #10695317 — subgroups

Subgroups — #10695317

Order 768.

430 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 167 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×#2859	C₂	1	1	yes	maximal, characteristic
2	#69392	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#67634	C₂	1	4	yes	maximal
5	#9899	—	3	1	no	maximal, Sylow
6	C₂×C₃×#125	—	2	1	no	—
7	C₃×#389	—	2	2	no	—
8	C₃×#389	C₄	1	4	yes	—
9	C₃×#384	C₂²	1	2	yes	—
10	C₃×#382	—	2	1	no	—
11	C₃×C₄³	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₃⋊C₈	—	2	1	no	—
15	#4524	—	2	1	no	—
16	#4524	—	2	2	no	—
17	#2859	D₃	1	1	yes	characteristic
18	#1831	—	3	1	no	—
19	C₄×C₄⋊C₈	—	3	1	no	—
20	#1296	—	3	4	no	—
21	C₃×#125	—	4	2	no	—
22	C₃×#125	—	4	2	no	—
23	C₂×C₃×C₄⋊C₄	—	2	4	no	—
24	C₂×C₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
27	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	D₄	1	2	yes	—
30	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
32	C₂²×C₃⋊C₈	—	2	2	no	—
33	C₂²×C₃⋊C₈	—	2	2	no	—
34	C₃×#95	—	2	4	no	—
35	C₄⋊(C₃⋊C₈)	—	2	2	no	—
36	C₄⋊(C₃⋊C₈)	—	2	4	no	—
37	C₄⋊(C₃⋊C₈)	Q₈	1	4	yes	—
38	C₄×C₃⋊C₈	—	4	2	no	—
39	C₂×#125	—	2	1	no	—
40	C₂×C₄⋊C₈	—	3	2	no	—
41	C₂×C₄⋊C₈	—	3	2	no	—
42	C₂×C₄×C₈	—	6	1	no	—
43	#389	—	2	2	no	—
44	#389	Dic₃	1	4	yes	—
45	#384	C₂×D₃	1	2	yes	—
46	#382	—	2	1	no	—
47	C₄³	C₂×D₃	1	1	yes	characteristic
48	#336	—	6	2	no	—
49	#336	—	6	1	no	—
50	C₂²×C₃×C₄	—	2	4	no	—
51	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
56	C₃×C₄⋊C₄	—	4	4	no	—
57	C₃×C₄⋊C₄	—	4	8	no	—
58	C₃×C₄⋊C₄	—	4	2	no	—
59	C₃×C₄⋊C₄	—	4	2	no	—
60	C₃×C₄⋊C₄	—	4	4	no	—
61	C₃×C₄²	—	2	4	no	—
62	C₃×C₄²	—	2	2	no	—
63	C₃×C₄²	—	4	2	no	—
64	C₃×C₄²	D₈	1	2	yes	—
65	C₃×C₄²	SD₁₆	1	2	yes	—
66	C₃×C₄²	C₂×Q₈	1	2	yes	—
67	C₃×C₄²	#41	1	2	yes	—
68	C₂×C₃⋊C₈	—	2	4	no	—
69	C₂×C₃⋊C₈	—	4	2	no	—
70	C₂×C₃⋊C₈	—	4	2	no	—
71	C₂×C₃⋊C₈	—	4	2	no	—
72	C₂×C₃⋊C₈	—	4	4	no	—
73	C₂²×C₈	—	6	2	no	—
74	C₂²×C₈	—	6	2	no	—
75	#125	—	4	2	no	—
76	#125	—	4	2	no	—
77	C₂×C₄⋊C₄	—	2	4	no	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
81	C₂×C₄²	C₂×Dic₃	1	2	yes	—
82	C₂×C₄²	#67	1	1	yes	characteristic
83	C₂×C₄²	#67	1	2	yes	—
84	C₂×C₄²	#67	1	1	yes	characteristic
85	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
86	C₄⋊C₈	—	3	4	no	—
87	C₄⋊C₈	—	6	2	no	—
88	C₄⋊C₈	—	6	4	no	—
89	C₄×C₈	—	12	2	no	—
90	#95	—	2	4	no	—
91	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	4	4	no	—
99	C₂×C₃×C₄	—	4	4	no	—
100	C₂×C₃×C₄	—	4	4	no	—
101	C₂×C₃×C₄	#102	1	4	yes	—
102	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
103	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₃×C₄	#124	1	1	yes	characteristic
105	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₃×C₄	#135	1	4	yes	—
109	C₂×C₃×C₄	#137	1	2	yes	—
110	C₃⋊C₈	—	4	4	no	—
111	C₃⋊C₈	—	8	2	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	#217	1	2	yes	—
114	C₂²×C₄	#217	1	2	yes	—
115	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
116	C₂²×C₄	C₂×#67	1	1	yes	characteristic
117	C₂²×C₄	C₂×#67	1	1	yes	characteristic
118	C₂×C₈	—	6	4	no	—
119	C₂×C₈	—	12	2	no	—
120	C₂×C₈	—	12	2	no	—
121	C₂×C₈	—	12	2	no	—
122	C₂×C₈	—	12	4	no	—
123	C₄⋊C₄	—	4	4	no	—
124	C₄⋊C₄	—	4	8	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	4	no	—
130	C₄²	—	4	2	no	—
131	C₄²	#213	1	2	yes	—
132	C₄²	#214	1	2	yes	—
133	C₄²	D₃×Q₈	1	2	yes	—
134	C₄²	#239	1	2	yes	—
135	C₂²×C₃	#391	1	1	yes	characteristic
136	C₂²×C₃	#413	1	2	yes	—
137	C₂²×C₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₃	#416	1	2	yes	—
139	C₂²×C₃	#419	1	1	yes	characteristic
140	C₃×C₄	—	2	4	no	—
141	C₃×C₄	—	2	2	no	—
142	C₃×C₄	—	4	2	no	—
143	C₃×C₄	—	8	4	no	—
144	C₃×C₄	#474	1	2	yes	—
145	C₃×C₄	#475	1	2	yes	—
146	C₃×C₄	#488	1	2	yes	—
147	C₃×C₄	#499	1	2	yes	—
148	C₂³	C₂×#217	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	#832	1	4	yes	—
159	C₂×C₄	C₂×#213	1	1	yes	characteristic
160	C₂×C₄	C₂×#214	1	1	yes	characteristic
161	C₂×C₄	#942	1	2	yes	—
162	C₂×C₄	#944	1	1	yes	characteristic
163	C₂×C₄	Q₈×Dic₃	1	2	yes	—
164	C₂×C₄	#946	1	2	yes	—
165	C₂×C₄	#947	1	1	yes	characteristic
166	C₂×C₄	#950	1	4	yes	—
167	C₈	—	12	4	no	—
168	C₈	—	24	2	no	—
169	C₂×C₃	#1540	1	2	yes	—
170	C₂×C₃	#1554	1	2	yes	—
171	C₂×C₃	#1988	1	1	yes	characteristic
172	C₂×C₃	#1991	1	1	yes	characteristic
173	C₂×C₃	#1998	1	1	yes	characteristic
174	C₂²	C₂×#832	1	1	yes	characteristic
175	C₂²	#5225	1	1	yes	characteristic
176	C₂²	#5229	1	1	yes	characteristic
177	C₂²	#5233	1	2	yes	—
178	C₂²	#5237	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	2	4	no	—
181	C₄	—	4	2	no	—
182	C₄	—	8	4	no	—
183	C₄	#5065	1	2	yes	—
184	C₄	#5068	1	2	yes	—
185	C₄	#5087	1	2	yes	—
186	C₄	#5094	1	2	yes	—
187	C₃	#9899	1	1	yes	characteristic, complemented, Sylow
188	C₂	#71740	1	1	yes	characteristic
189	C₂	#71742	1	1	yes	characteristic
190	C₂	#71780	1	1	yes	characteristic
191	C₂	#71804	1	2	yes	—
192	C₂	#71832	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#2859

C₂

yes

maximal, characteristic

#69392

C₂

yes

maximal, characteristic

C₄×C₄⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#67634

C₂

yes

maximal

#9899

—

maximal, Sylow

C₂×C₃×#125

—

C₃×#389

—

C₃×#389

C₄

yes

—

C₃×#384

C₂²

yes

—

C₃×#382

—

C₃×C₄³

C₂²

yes

characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₂×C₄×C₃⋊C₈

—

#4524

—

#4524

—

#2859

D₃

yes

characteristic

#1831

—

C₄×C₄⋊C₈

—

#1296

—

C₃×#125

—

C₃×#125

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₃×C₄²

C₂×C₄

yes

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂²×C₃⋊C₈

—

C₂²×C₃⋊C₈

—

C₃×#95

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

Q₈

yes

—

C₄×C₃⋊C₈

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#389

—

#389

Dic₃

yes

—

#384

C₂×D₃

yes

—

#382

—

C₄³

C₂×D₃

yes

characteristic

#336

—

#336

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂²⋊C₄

yes

—

C₂²×C₃×C₄

C₂²⋊C₄

yes

—

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

D₈

yes

—

C₃×C₄²

SD₁₆

yes

—

C₃×C₄²

C₂×Q₈

yes

—

C₃×C₄²

#41

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₃

yes

—

C₂×C₄²

C₂×Dic₃

yes

—

C₂×C₄²

#67

yes

characteristic

C₂×C₄²

#67

yes

—

C₂×C₄²

#67

yes

characteristic

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

#95

—

C₂³×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

#102

yes

—

102

C₂×C₃×C₄

C₄×Q₈

yes

—

103

C₂×C₃×C₄

C₂²⋊Q₈

yes

—

104

C₂×C₃×C₄

#124

yes

characteristic

105

C₂×C₃×C₄

C₄⋊Q₈

yes

characteristic

106

C₂×C₃×C₄

C₂×D₈

yes

characteristic

107

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

108

C₂×C₃×C₄

#135

yes

—

109

C₂×C₃×C₄

#137

yes

—

110

C₃⋊C₈

—

111

C₃⋊C₈

—

112

C₂²×C₄

—

113

C₂²×C₄

#217

yes

—

114

C₂²×C₄

#217

yes

—

115

C₂²×C₄

C₂²×Dic₃

yes

characteristic

116

C₂²×C₄

C₂×#67

yes

characteristic

117

C₂²×C₄

C₂×#67

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

#213

yes

—

132

C₄²

#214

yes

—

133

C₄²

D₃×Q₈

yes

—

134

C₄²

#239

yes

—

135

C₂²×C₃

#391

yes

characteristic

136

C₂²×C₃

#413

yes

—

137

C₂²×C₃

C₂×#102

yes

characteristic

138

C₂²×C₃

#416

yes

—

139

C₂²×C₃

#419

yes

characteristic

140

C₃×C₄

—

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

#474

yes

—

145

C₃×C₄

#475

yes

—

146

C₃×C₄

#488

yes

—

147

C₃×C₄

#499

yes

—

148

C₂³

C₂×#217

yes

characteristic

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#832

yes

—

159

C₂×C₄

C₂×#213

yes

characteristic

160

C₂×C₄

C₂×#214

yes

characteristic

161

C₂×C₄

#942

yes

—

162

C₂×C₄

#944

yes

characteristic

163

C₂×C₄

Q₈×Dic₃

yes

—

164

C₂×C₄

#946

yes

—

165

C₂×C₄

#947

yes

characteristic

166

C₂×C₄

#950

yes

—

167

C₈

—

168

C₈

—

169

C₂×C₃

#1540

yes

—

170

C₂×C₃

#1554

yes

—

171

C₂×C₃

#1988

yes

characteristic

172

C₂×C₃

#1991

yes

characteristic

173

C₂×C₃

#1998

yes

characteristic

174

C₂²

C₂×#832

yes

characteristic

175

C₂²

#5225

yes

characteristic

176

C₂²

#5229

yes

characteristic

177

C₂²

#5233

yes

—

178

C₂²

#5237

yes

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

#5065

yes

—

184

C₄

#5068

yes

—

185

C₄

#5087

yes

—

186

C₄

#5094

yes

—

187

C₃

#9899

yes

characteristic, complemented, Sylow

188

C₂

#71740

yes

characteristic

189

C₂

#71742

yes

characteristic

190

C₂

#71780

yes

characteristic

191

C₂

#71804

yes

—

192

C₂

#71832

yes

—

	subgroup	G/H
center	C₂³	C₂×#217
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#2859	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#2859	D₃
hypocenter	C₃	#9899
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×#217

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×#2859

C₂

socle, prime radical

C₂³×C₃

C₂×C₂²⋊C₄

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
187. C₃	#9899	semidirect

H ≅ G/N

type

187. C₃

#9899

semidirect

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

subgroup	index
C₃×#2859	2
#69392	2
C₄×C₄⋊(C₃⋊C₈)	2
#67634	2
#9899	3

subgroup

index

C₄×C₄⋊(C₃⋊C₈)

Subgroups — #10695317

Conjugacy classes of subgroups

Complements

Maximal subgroups