Group #10695329 — subgroups

Subgroups — #10695329

Order 768.

438 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 159 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	#73307	C₂	1	1	yes	maximal, characteristic
2	C₃×#1842	C₂	1	1	yes	maximal, characteristic
3	#69391	C₂	1	1	yes	maximal, characteristic
4	#67702	C₂	1	4	yes	maximal
5	#9902	—	3	1	no	maximal, Sylow
6	C₂×#870	—	2	1	no	—
7	C₂×C₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₃×C₄³	C₂²	1	1	yes	characteristic
10	#4936	—	2	1	no	—
11	#4935	C₂²	1	2	yes	—
12	#4928	—	2	2	no	—
13	#4928	C₄	1	4	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×#803	—	2	1	no	—
16	#4524	—	2	1	no	—
17	#2859	—	3	1	no	—
18	#1842	D₃	1	1	yes	characteristic
19	#1767	—	3	1	no	—
20	#1296	—	3	4	no	—
21	C₂²×C₃×C₈	—	2	2	no	—
22	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
24	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄²	D₄	1	2	yes	—
26	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₃	—	2	2	no	—
28	C₂×C₄⋊Dic₃	—	2	2	no	—
29	C₂×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
30	C₂×Dic₃⋊C₄	—	2	4	no	—
31	C₂²×C₃⋊C₈	—	2	2	no	—
32	#870	—	4	4	no	—
33	C₃×C₄⋊C₈	—	2	4	no	—
34	C₃×C₄⋊C₈	—	4	4	no	—
35	C₃⋊#95	—	2	4	no	—
36	C₄⋊(C₃⋊C₈)	—	2	2	no	—
37	C₄⋊(C₃⋊C₈)	Q₈	1	4	yes	—
38	#803	—	4	2	no	—
39	C₂×#125	—	6	1	no	—
40	C₂×C₄⋊C₈	—	2	2	no	—
41	C₂×C₄⋊C₈	—	3	2	no	—
42	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
43	C₂×#97	—	6	1	no	—
44	#389	—	3	4	no	—
45	#389	—	6	2	no	—
46	#384	—	3	2	no	—
47	#382	—	6	1	no	—
48	C₄³	C₂×D₃	1	1	yes	characteristic
49	#336	—	6	1	no	—
50	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₃	—	2	4	no	—
56	C₂×C₃×C₈	—	4	2	no	—
57	C₂×C₃×C₈	—	4	4	no	—
58	C₃×C₄²	—	2	2	no	—
59	C₃×C₄²	—	2	2	no	—
60	C₃×C₄²	—	2	2	no	—
61	C₃×C₄²	—	4	2	no	—
62	C₃×C₄²	SD₁₆	1	2	yes	—
63	C₃×C₄²	Q₁₆	1	2	yes	—
64	C₃×C₄²	C₂×Q₈	1	2	yes	—
65	C₃×C₄²	#41	1	2	yes	—
66	C₄⋊Dic₃	—	4	4	no	—
67	C₄⋊Dic₃	—	4	2	no	—
68	C₄⋊Dic₃	—	4	2	no	—
69	C₄⋊Dic₃	—	4	4	no	—
70	Dic₃⋊C₄	—	4	8	no	—
71	C₂×C₃⋊C₈	—	2	4	no	—
72	C₂×C₃⋊C₈	—	4	2	no	—
73	C₂×C₃⋊C₈	—	4	2	no	—
74	C₂²×C₈	—	2	2	no	—
75	C₂²×C₈	—	6	2	no	—
76	#125	—	12	4	no	—
77	C₂×C₄⋊C₄	—	3	4	no	—
78	C₂×C₄⋊C₄	—	6	4	no	—
79	C₂×C₄⋊C₄	—	6	2	no	—
80	C₂×C₄⋊C₄	—	6	2	no	—
81	C₂×C₄²	C₄×D₃	1	2	yes	—
82	C₂×C₄²	D₁₂	1	1	yes	characteristic
83	C₂×C₄²	D₁₂	1	1	yes	characteristic
84	C₂×C₄²	#67	1	2	yes	—
85	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
86	C₄⋊C₈	—	2	4	no	—
87	C₄⋊C₈	—	3	4	no	—
88	C₄⋊C₈	—	4	4	no	—
89	C₄⋊C₈	—	6	2	no	—
90	#97	—	12	2	no	—
91	#95	—	6	4	no	—
92	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
99	C₂×C₃×C₄	#103	1	4	yes	—
100	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
101	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₃×C₄	#124	1	1	yes	characteristic
103	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
108	C₂×C₃×C₄	#137	1	2	yes	—
109	C₂×Dic₃	—	4	4	no	—
110	C₂×Dic₃	—	4	4	no	—
111	C₂×Dic₃	—	4	4	no	—
112	C₃×C₈	—	8	2	no	—
113	C₃⋊C₈	—	4	4	no	—
114	C₂²×C₄	—	6	4	no	—
115	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
116	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
117	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
118	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
119	C₂²×C₄	C₂×#67	1	1	yes	characteristic
120	C₂×C₈	—	4	4	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	6	4	no	—
123	C₂×C₈	—	12	2	no	—
124	C₂×C₈	—	12	2	no	—
125	C₄⋊C₄	—	12	4	no	—
126	C₄⋊C₄	—	12	8	no	—
127	C₄⋊C₄	—	12	4	no	—
128	C₄⋊C₄	—	12	2	no	—
129	C₄⋊C₄	—	12	2	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	4	2	no	—
134	C₄²	#204	1	2	yes	—
135	C₄²	Dic₁₂	1	2	yes	—
136	C₄²	#237	1	2	yes	—
137	C₄²	D₃×Q₈	1	2	yes	—
138	C₂²×C₃	#391	1	1	yes	characteristic
139	C₂²×C₃	#413	1	2	yes	—
140	C₂²×C₃	C₂×#103	1	1	yes	characteristic
141	C₂²×C₃	#417	1	2	yes	—
142	C₂²×C₃	#418	1	1	yes	characteristic
143	C₃×C₄	—	2	4	no	—
144	C₃×C₄	—	2	2	no	—
145	C₃×C₄	—	4	2	no	—
146	C₃×C₄	#476	1	2	yes	—
147	C₃×C₄	#477	1	2	yes	—
148	C₃×C₄	#490	1	1	yes	characteristic
149	C₃×C₄	#491	1	1	yes	characteristic
150	C₃×C₄	#501	1	2	yes	—
151	Dic₃	—	8	4	no	—
152	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	12	4	no	—
160	C₂×C₄	—	12	4	no	—
161	C₂×C₄	—	12	4	no	—
162	C₂×C₄	#816	1	4	yes	—
163	C₂×C₄	#887	1	2	yes	—
164	C₂×C₄	#897	1	2	yes	—
165	C₂×C₄	C₂×#204	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₁₂	1	1	yes	characteristic
167	C₂×C₄	#908	1	2	yes	—
168	C₂×C₄	#932	1	2	yes	—
169	C₂×C₄	#936	1	1	yes	characteristic
170	C₂×C₄	#942	1	2	yes	—
171	C₂×C₄	#944	1	1	yes	characteristic
172	C₈	—	8	2	no	—
173	C₈	—	12	4	no	—
174	C₂×C₃	#1550	1	2	yes	—
175	C₂×C₃	#1557	1	2	yes	—
176	C₂×C₃	#1987	1	1	yes	characteristic
177	C₂×C₃	#1992	1	1	yes	characteristic
178	C₂×C₃	#1998	1	1	yes	characteristic
179	C₂²	#4966	1	2	yes	—
180	C₂²	#4973	1	1	yes	characteristic
181	C₂²	C₂×#816	1	1	yes	characteristic
182	C₂²	#5131	1	1	yes	characteristic
183	C₂²	#5132	1	2	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	2	2	no	—
186	C₄	—	4	2	no	—
187	C₄	—	24	4	no	—
188	C₄	#4842	1	2	yes	—
189	C₄	#4850	1	2	yes	—
190	C₄	#5071	1	1	yes	characteristic
191	C₄	#5081	1	2	yes	—
192	C₄	#5082	1	1	yes	characteristic
193	C₃	#9902	1	1	yes	characteristic, complemented, Sylow
194	C₂	#69583	1	1	yes	characteristic
195	C₂	#70438	1	2	yes	—
196	C₂	#70459	1	1	yes	characteristic
197	C₂	#70504	1	2	yes	—
198	C₂	#70525	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#73307

C₂

yes

maximal, characteristic

C₃×#1842

C₂

yes

maximal, characteristic

#69391

C₂

yes

maximal, characteristic

#67702

C₂

yes

—

—

C₂×C₃×C₄⋊C₈

—

C₂×C₃×C₄⋊C₈

C₂²

yes

—

C₃×C₄³

C₂²

yes

characteristic

#4936

—

#4935

C₂²

yes

—

#4928

—

#4928

C₄

yes

—

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

—

—

—

D₃

yes

characteristic

#1767

—

#1296

—

C₂²×C₃×C₈

—

C₂×C₃×C₄²

C₂×C₄

yes

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂×C₄

yes

—

C₂×Dic₃⋊C₄

—

C₂²×C₃⋊C₈

—

#870

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

C₃⋊#95

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

Q₈

yes

—

#803

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₃

yes

—

C₂×#97

—

—

—

—

—

C₄³

C₂×D₃

yes

characteristic

#336

—

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

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

SD₁₆

yes

—

C₃×C₄²

Q₁₆

yes

—

C₃×C₄²

C₂×Q₈

yes

—

C₃×C₄²

#41

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₄×D₃

yes

—

C₂×C₄²

D₁₂

yes

characteristic

C₂×C₄²

D₁₂

yes

characteristic

C₂×C₄²

#67

yes

—

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#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

#103

yes

—

100

C₂×C₃×C₄

C₄×Q₈

yes

—

101

C₂×C₃×C₄

C₂²⋊Q₈

yes

—

102

C₂×C₃×C₄

#124

yes

characteristic

103

C₂×C₃×C₄

C₄⋊Q₈

yes

characteristic

104

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

105

C₂×C₃×C₄

C₂×Q₁₆

yes

characteristic

106

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

107

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

108

C₂×C₃×C₄

#137

yes

—

109

C₂×Dic₃

—

110

C₂×Dic₃

—

111

C₂×Dic₃

—

112

C₃×C₈

—

113

C₃⋊C₈

—

114

C₂²×C₄

—

115

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

116

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

117

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

118

C₂²×C₄

C₂×D₁₂

yes

characteristic

119

C₂²×C₄

C₂×#67

yes

characteristic

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

#204

yes

—

135

C₄²

Dic₁₂

yes

—

136

C₄²

#237

yes

—

137

C₄²

D₃×Q₈

yes

—

138

C₂²×C₃

#391

yes

characteristic

139

C₂²×C₃

#413

yes

—

140

C₂²×C₃

C₂×#103

yes

characteristic

141

C₂²×C₃

#417

yes

—

142

C₂²×C₃

#418

yes

characteristic

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

#476

yes

—

147

C₃×C₄

#477

yes

—

148

C₃×C₄

#490

yes

characteristic

149

C₃×C₄

#491

yes

characteristic

150

C₃×C₄

#501

yes

—

151

Dic₃

—

152

C₂³

C₂×(C₂×D₃)⋊C₄

yes

characteristic

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#816

yes

—

163

C₂×C₄

#887

yes

—

164

C₂×C₄

#897

yes

—

165

C₂×C₄

C₂×#204

yes

characteristic

166

C₂×C₄

C₂×Dic₁₂

yes

characteristic

167

C₂×C₄

#908

yes

—

168

C₂×C₄

#932

yes

—

169

C₂×C₄

#936

yes

characteristic

170

C₂×C₄

#942

yes

—

171

C₂×C₄

#944

yes

characteristic

172

C₈

—

173

C₈

—

174

C₂×C₃

#1550

yes

—

175

C₂×C₃

#1557

yes

—

176

C₂×C₃

#1987

yes

characteristic

177

C₂×C₃

#1992

yes

characteristic

178

C₂×C₃

#1998

yes

characteristic

179

C₂²

#4966

yes

—

180

C₂²

#4973

yes

characteristic

181

C₂²

C₂×#816

yes

characteristic

182

C₂²

#5131

yes

characteristic

183

C₂²

#5132

yes

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

#4842

yes

—

189

C₄

#4850

yes

—

190

C₄

#5071

yes

characteristic

191

C₄

#5081

yes

—

192

C₄

#5082

yes

characteristic

193

C₃

#9902

yes

characteristic, complemented, Sylow

194

C₂

#69583

yes

characteristic

195

C₂

#70438

yes

—

196

C₂

#70459

yes

characteristic

197

C₂

#70504

yes

—

198

C₂

#70525

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₃)⋊C₄
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1842	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1842	D₃
hypocenter	C₃	#9902
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₂×(C₂×D₃)⋊C₄

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×#1842

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 ◁ #1842

N	H ≅ G/N	type
193. C₃	#9902	semidirect

H ≅ G/N

type

193. C₃

#9902

semidirect

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

subgroup	index
#73307	2
C₃×#1842	2
#69391	2
#67702	2
#9902	3

subgroup

index

Subgroups — #10695329

Conjugacy classes of subgroups

Complements

Maximal subgroups