Group #10686113 — subgroups

Subgroups — #10686113

Order 768.

444 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 151 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×C₄×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×#4543	C₂	1	2	yes	maximal
3	#69389	C₂	1	2	yes	maximal
4	#67630	C₂	1	2	yes	maximal
5	#8225	—	3	1	no	maximal, Sylow
6	C₂×C₃×#125	C₂²	1	1	yes	characteristic
7	C₃×#389	C₄	1	2	yes	—
8	C₃×#387	—	2	2	no	—
9	C₃×#382	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₄⋊C₄	—	2	2	no	—
11	C₃×C₄×C₄⋊C₄	C₄	1	2	yes	—
12	C₃×C₄³	C₂²	1	1	yes	characteristic
13	C₂×#803	C₂²	1	4	yes	—
14	#4543	C₄	1	8	yes	complemented
15	#4524	—	2	2	no	—
16	#4524	—	2	2	no	—
17	C₄×#125	D₃	1	1	yes	characteristic
18	#1745	—	3	2	no	—
19	C₂×#330	—	3	2	no	—
20	#1295	—	3	2	no	—
21	C₃×#125	—	2	2	no	—
22	C₃×#125	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄²	—	2	2	no	—
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	D₄	1	2	yes	—
31	C₂×C₃×C₄²	D₄	1	2	yes	—
32	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
33	C₂²×C₃⋊C₈	—	2	4	no	—
34	C₃×#95	—	2	2	no	—
35	C₃×#95	—	2	2	no	—
36	C₃×#95	—	2	2	no	—
37	#803	—	2	4	no	—
38	#803	C₂×C₄	1	8	yes	—
39	C₂×#125	C₂×D₃	1	1	yes	characteristic
40	C₂×#97	—	3	4	no	—
41	#389	Dic₃	1	2	yes	—
42	#387	—	2	2	no	—
43	#382	C₂×D₃	1	1	yes	characteristic
44	C₄×C₄⋊C₄	—	2	2	no	—
45	C₄×C₄⋊C₄	Dic₃	1	2	yes	—
46	C₄³	C₂×D₃	1	1	yes	characteristic
47	#336	—	6	2	no	—
48	#336	—	6	2	no	—
49	#330	—	3	8	no	—
50	C₂²×C₃×C₄	—	2	2	no	—
51	C₂²×C₃×C₄	—	2	2	no	—
52	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	2	8	no	—
59	C₃×C₄⋊C₄	—	4	2	no	—
60	C₃×C₄⋊C₄	—	4	4	no	—
61	C₃×C₄⋊C₄	C₄²	1	4	yes	—
62	C₃×C₄²	—	2	2	no	—
63	C₃×C₄²	—	4	1	no	—
64	C₃×C₄²	—	4	2	no	—
65	C₃×C₄²	—	4	2	no	—
66	C₃×C₄²	—	4	4	no	—
67	C₃×C₄²	C₂²×C₄	1	2	yes	—
68	C₃×C₄²	#41	1	2	yes	—
69	C₂×C₃⋊C₈	—	2	8	no	—
70	C₂×C₃⋊C₈	—	4	4	no	—
71	C₂×C₃⋊C₈	—	4	4	no	—
72	C₂²×C₈	—	6	4	no	—
73	#125	—	2	2	no	—
74	#125	C₄×D₃	1	4	yes	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
78	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
79	C₂×C₄²	—	2	2	no	—
80	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
81	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₄²	#67	1	2	yes	—
83	C₂×C₄²	#67	1	2	yes	—
84	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
85	#97	—	3	8	no	—
86	#97	—	6	4	no	—
87	#95	—	2	2	no	—
88	#95	—	2	2	no	—
89	#95	—	2	2	no	—
90	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃×C₄	—	2	1	no	—
92	C₂×C₃×C₄	—	2	1	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	4	no	—
97	C₂×C₃×C₄	—	2	1	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	4	2	no	—
103	C₂×C₃×C₄	—	4	2	no	—
104	C₂×C₃×C₄	—	4	2	no	—
105	C₂×C₃×C₄	—	4	2	no	—
106	C₂×C₃×C₄	—	4	2	no	—
107	C₂×C₃×C₄	#101	1	2	yes	—
108	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₃×C₄	#117	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄×D₄	1	4	yes	—
111	C₃⋊C₈	—	4	8	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	#217	1	1	yes	characteristic
115	C₂²×C₄	#217	1	2	yes	—
116	C₂²×C₄	#217	1	1	yes	characteristic
117	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
118	C₂²×C₄	C₂×#67	1	2	yes	—
119	C₂×C₈	—	6	8	no	—
120	C₂×C₈	—	12	4	no	—
121	C₂×C₈	—	12	4	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	8	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	4	no	—
126	C₄⋊C₄	C₄×Dic₃	1	4	yes	—
127	C₄²	—	2	2	no	—
128	C₄²	—	4	1	no	—
129	C₄²	—	4	2	no	—
130	C₄²	—	4	2	no	—
131	C₄²	—	4	4	no	—
132	C₄²	C₂×C₄×D₃	1	2	yes	—
133	C₄²	#235	1	2	yes	—
134	C₂²×C₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₃	C₂×#101	1	1	yes	characteristic
136	C₂²×C₃	#413	1	1	yes	characteristic
137	C₂²×C₃	#421	1	2	yes	—
138	C₂²×C₃	#421	1	2	yes	—
139	C₃×C₄	—	2	1	no	—
140	C₃×C₄	—	2	1	no	—
141	C₃×C₄	—	2	4	no	—
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	4	1	no	—
144	C₃×C₄	—	4	4	no	—
145	C₃×C₄	—	4	2	no	—
146	C₃×C₄	—	8	2	no	—
147	C₂³	C₂×#217	1	1	yes	characteristic
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	1	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	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	#836	1	2	yes	—
165	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
166	C₂×C₄	#926	1	1	yes	characteristic
167	C₂×C₄	C₄×#67	1	4	yes	—
168	C₈	—	12	8	no	—
169	C₂×C₃	#1534	1	1	yes	characteristic
170	C₂×C₃	#1535	1	2	yes	—
171	C₂×C₃	#1537	1	1	yes	characteristic
172	C₂×C₃	C₄×#101	1	1	yes	characteristic
173	C₂×C₃	#1759	1	2	yes	—
174	C₂²	C₄×#217	1	1	yes	characteristic
175	C₂²	C₂×#836	1	1	yes	characteristic
176	C₂²	#5236	1	2	yes	—
177	C₂²	#5236	1	2	yes	—
178	C₂²	#5237	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	4	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	4	no	—
186	C₄	—	8	2	no	—
187	C₃	#8225	1	1	yes	characteristic, complemented, Sylow
188	C₂	C₄×#836	1	1	yes	characteristic
189	C₂	#71643	1	2	yes	—
190	C₂	#71736	1	1	yes	characteristic
191	C₂	#71749	1	2	yes	—
192	C₂	#71823	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄×#125

C₂

yes

maximal, characteristic

C₂×#4543

C₂

yes

maximal

#69389

C₂

yes

maximal

#67630

C₂

yes

maximal

#8225

—

maximal, Sylow

C₂×C₃×#125

C₂²

yes

characteristic

C₃×#389

C₄

yes

—

C₃×#387

—

C₃×#382

C₂²

yes

characteristic

C₃×C₄×C₄⋊C₄

—

C₃×C₄×C₄⋊C₄

C₄

yes

—

C₃×C₄³

C₂²

yes

characteristic

C₂×#803

C₂²

yes

—

#4543

C₄

yes

—

—

D₃

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂²×C₃⋊C₈

—

C₃×#95

—

C₃×#95

—

C₃×#95

—

#803

—

#803

C₂×C₄

yes

—

C₂×#125

C₂×D₃

yes

characteristic

C₂×#97

—

#389

Dic₃

yes

—

#387

—

#382

C₂×D₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

Dic₃

yes

—

C₄³

C₂×D₃

yes

—

—

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₃×C₄

C₂²⋊C₄

yes

—

C₂²×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₄²

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₄

yes

—

C₃×C₄²

#41

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

#125

—

#125

C₄×D₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₃

yes

—

C₂×C₄⋊C₄

C₂×Dic₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₃

yes

characteristic

C₂×C₄²

C₂×Dic₃

yes

characteristic

C₂×C₄²

#67

yes

—

C₂×C₄²

#67

yes

—

C₂×C₄²

C₂²×D₃

yes

—

—

—

—

—

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

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

—

103

C₂×C₃×C₄

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

#101

yes

—

108

C₂×C₃×C₄

C₂×C₄²

yes

characteristic

109

C₂×C₃×C₄

#117

yes

characteristic

110

C₂×C₃×C₄

C₄×D₄

yes

—

111

C₃⋊C₈

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

#217

yes

characteristic

115

C₂²×C₄

#217

yes

—

116

C₂²×C₄

#217

yes

characteristic

117

C₂²×C₄

C₂²×Dic₃

yes

characteristic

118

C₂²×C₄

C₂×#67

yes

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

C₄×Dic₃

yes

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₂×C₄×D₃

yes

—

133

C₄²

#235

yes

—

134

C₂²×C₃

C₄×C₂²⋊C₄

yes

characteristic

135

C₂²×C₃

C₂×#101

yes

characteristic

136

C₂²×C₃

#413

yes

characteristic

137

C₂²×C₃

#421

yes

—

138

C₂²×C₃

#421

yes

—

139

C₃×C₄

—

140

C₃×C₄

—

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

—

147

C₂³

C₂×#217

yes

characteristic

148

C₂×C₄

—

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

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#836

yes

—

165

C₂×C₄

C₂×C₄×Dic₃

yes

characteristic

166

C₂×C₄

#926

yes

characteristic

167

C₂×C₄

C₄×#67

yes

—

168

C₈

—

169

C₂×C₃

#1534

yes

characteristic

170

C₂×C₃

#1535

yes

—

171

C₂×C₃

#1537

yes

characteristic

172

C₂×C₃

C₄×#101

yes

characteristic

173

C₂×C₃

#1759

yes

—

174

C₂²

C₄×#217

yes

characteristic

175

C₂²

C₂×#836

yes

characteristic

176

C₂²

#5236

yes

—

177

C₂²

#5236

yes

—

178

C₂²

#5237

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₃

#8225

yes

characteristic, complemented, Sylow

188

C₂

C₄×#836

yes

characteristic

189

C₂

#71643

yes

—

190

C₂

#71736

yes

characteristic

191

C₂

#71749

yes

—

192

C₂

#71823

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#217
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×#125	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×#125	D₃
hypocenter	C₃	#8225
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×#217

commutator

C₂×C₃×C₄

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×C₄×#125

C₂

socle, prime radical

C₂³×C₃

C₂×C₂²⋊C₄

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
14. #4543	C₄	semidirect
187. C₃	#8225	semidirect

H ≅ G/N

type

14. #4543

C₄

semidirect

187. C₃

#8225

semidirect

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

subgroup	index
C₃×C₄×#125	2
C₂×#4543	2
#69389	2
#67630	2
#8225	3

subgroup

index

C₃×C₄×#125

Subgroups — #10686113

Conjugacy classes of subgroups

Complements

Maximal subgroups