Group #10695215 — subgroups

Subgroups — #10695215

Order 768.

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

normal subgroups only

This group has 434 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	C₃×C₄×#125	C₂	1	1	yes	maximal, characteristic
2	#69392	C₂	1	1	yes	maximal, characteristic
3	#69388	C₂	1	1	yes	maximal, characteristic
4	#67619	C₂	1	4	yes	maximal
5	#9875	—	3	1	no	maximal, Sylow
6	C₂×C₃×#125	C₄	1	1	yes	characteristic
7	C₃×#389	C₄	1	2	yes	—
8	C₃×#387	—	2	2	no	—
9	C₃×#382	C₄	1	1	yes	characteristic
10	C₃×C₄×C₄⋊C₄	—	2	2	no	—
11	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₄³	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
15	#4524	—	2	2	no	—
16	#4524	—	2	2	no	—
17	C₄×#125	D₃	1	1	yes	characteristic
18	#1831	—	3	1	no	—
19	#1744	—	3	1	no	—
20	#1279	—	3	4	no	—
21	C₃×#125	—	2	4	no	—
22	C₂×C₃×C₄⋊C₄	—	2	2	no	—
23	C₂×C₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄²	—	2	2	no	—
27	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	D₄	1	2	yes	—
30	C₂×C₃×C₄²	D₄	1	2	yes	—
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	2	no	—
35	C₃×#95	—	2	2	no	—
36	C₃×#95	—	2	2	no	—
37	C₄⋊(C₃⋊C₈)	—	2	4	no	—
38	C₄×C₃⋊C₈	—	2	2	no	—
39	C₄×C₃⋊C₈	Q₈	1	4	yes	—
40	C₂×#125	Dic₃	1	1	yes	characteristic
41	C₂×C₄⋊C₈	—	3	2	no	—
42	C₂×C₄×C₈	—	3	2	no	—
43	#389	Dic₃	1	2	yes	—
44	#387	—	2	2	no	—
45	#382	Dic₃	1	1	yes	characteristic
46	C₄×C₄⋊C₄	—	2	2	no	—
47	C₄×C₄⋊C₄	C₂×D₃	1	2	yes	—
48	C₄³	C₂×D₃	1	1	yes	characteristic
49	#336	—	6	2	no	—
50	#336	—	6	2	no	—
51	C₂²×C₃×C₄	—	2	2	no	—
52	C₂²×C₃×C₄	—	2	2	no	—
53	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
58	C₃×C₄⋊C₄	—	2	8	no	—
59	C₃×C₄⋊C₄	—	2	2	no	—
60	C₃×C₄⋊C₄	—	2	4	no	—
61	C₃×C₄⋊C₄	—	4	4	no	—
62	C₃×C₄⋊C₄	M₁₆	1	4	yes	—
63	C₃×C₄²	—	2	2	no	—
64	C₃×C₄²	—	2	2	no	—
65	C₃×C₄²	—	2	4	no	—
66	C₃×C₄²	—	4	2	no	—
67	C₃×C₄²	—	4	4	no	—
68	C₃×C₄²	C₂×Q₈	1	2	yes	—
69	C₃×C₄²	#41	1	2	yes	—
70	C₂×C₃⋊C₈	—	2	4	no	—
71	C₂×C₃⋊C₈	—	4	4	no	—
72	C₂×C₃⋊C₈	—	4	2	no	—
73	C₂×C₃⋊C₈	—	4	4	no	—
74	C₂²×C₈	—	6	2	no	—
75	C₂²×C₈	—	6	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
81	C₂×C₄²	—	2	2	no	—
82	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
83	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
84	C₂×C₄²	#67	1	2	yes	—
85	C₂×C₄²	#67	1	2	yes	—
86	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
87	C₄⋊C₈	—	6	4	no	—
88	C₄×C₈	—	3	4	no	—
89	C₄×C₈	—	6	2	no	—
90	#95	—	2	2	no	—
91	#95	—	2	2	no	—
92	#95	—	2	2	no	—
93	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	2	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	C₄×Q₈	1	2	yes	—
108	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₃×C₄	#124	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
112	C₂×C₃×C₄	#131	1	2	yes	—
113	C₂×C₃×C₄	#135	1	4	yes	—
114	C₃⋊C₈	—	4	4	no	—
115	C₃⋊C₈	—	8	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	#217	1	2	yes	—
119	C₂²×C₄	#217	1	2	yes	—
120	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
121	C₂²×C₄	C₂×#67	1	1	yes	characteristic
122	C₂²×C₄	C₂×#67	1	1	yes	characteristic
123	C₂×C₈	—	6	4	no	—
124	C₂×C₈	—	12	4	no	—
125	C₂×C₈	—	12	2	no	—
126	C₂×C₈	—	12	4	no	—
127	C₄⋊C₄	—	2	8	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	4	4	no	—
131	C₄⋊C₄	#208	1	4	yes	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	4	no	—
135	C₄²	—	4	2	no	—
136	C₄²	—	4	4	no	—
137	C₄²	D₃×Q₈	1	2	yes	—
138	C₄²	#239	1	2	yes	—
139	C₂²×C₃	#391	1	1	yes	characteristic
140	C₂²×C₃	#407	1	1	yes	characteristic
141	C₂²×C₃	#408	1	1	yes	characteristic
142	C₂²×C₃	#416	1	2	yes	—
143	C₂²×C₃	#421	1	2	yes	—
144	C₃×C₄	—	2	2	no	—
145	C₃×C₄	—	4	2	no	—
146	C₃×C₄	—	4	2	no	—
147	C₃×C₄	—	4	4	no	—
148	C₃×C₄	—	8	2	no	—
149	C₃×C₄	#446	1	4	yes	—
150	C₃×C₄	#488	1	2	yes	—
151	C₃×C₄	#499	1	2	yes	—
152	C₂³	C₂×#217	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×#208	1	2	yes	—
167	C₂×C₄	#944	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₃	1	2	yes	—
169	C₂×C₄	#946	1	2	yes	—
170	C₂×C₄	#947	1	1	yes	characteristic
171	C₂×C₄	#948	1	2	yes	—
172	C₂×C₄	#950	1	4	yes	—
173	C₈	—	12	4	no	—
174	C₈	—	24	2	no	—
175	C₂×C₃	#1482	1	2	yes	—
176	C₂×C₃	#1488	1	2	yes	—
177	C₂×C₃	#1985	1	1	yes	characteristic
178	C₂×C₃	#1991	1	1	yes	characteristic
179	C₂×C₃	#1995	1	1	yes	characteristic
180	C₂²	#5207	1	1	yes	characteristic
181	C₂²	#5229	1	1	yes	characteristic
182	C₂²	#5233	1	2	yes	—
183	C₂²	#5234	1	1	yes	characteristic
184	C₂²	#5236	1	2	yes	—
185	C₄	—	2	2	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	4	no	—
189	C₄	—	8	2	no	—
190	C₄	#5024	1	4	yes	—
191	C₄	#5065	1	2	yes	—
192	C₄	#5068	1	2	yes	—
193	C₃	#9875	1	1	yes	characteristic, complemented, Sylow
194	C₂	#71635	1	1	yes	characteristic
195	C₂	#71640	1	2	yes	—
196	C₂	#71740	1	1	yes	characteristic
197	C₂	#71750	1	2	yes	—
198	C₂	#71840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄×#125

C₂

yes

maximal, characteristic

#69392

C₂

yes

maximal, characteristic

#69388

C₂

yes

maximal, characteristic

#67619

C₂

yes

maximal

#9875

—

maximal, Sylow

C₂×C₃×#125

C₄

yes

characteristic

C₃×#389

C₄

yes

—

C₃×#387

—

C₃×#382

C₄

yes

characteristic

C₃×C₄×C₄⋊C₄

—

C₃×C₄×C₄⋊C₄

C₂²

yes

—

C₃×C₄³

C₂²

yes

characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₂×C₄×C₃⋊C₈

C₂²

yes

—

#4524

—

#4524

—

C₄×#125

D₃

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

—

C₂²×C₃⋊C₈

—

C₃×#95

—

C₃×#95

—

C₃×#95

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

Q₈

yes

—

C₂×#125

Dic₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#389

Dic₃

yes

—

#387

—

#382

Dic₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

C₂×D₃

yes

—

C₄³

C₂×D₃

yes

characteristic

#336

—

#336

—

C₂²×C₃×C₄

—

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

M₁₆

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×Q₈

yes

—

C₃×C₄²

#41

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

#125

—

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

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

#95

—

#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

—

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

C₄×Q₈

yes

—

108

C₂×C₃×C₄

C₂²⋊Q₈

yes

—

109

C₂×C₃×C₄

#124

yes

characteristic

110

C₂×C₃×C₄

C₄⋊Q₈

yes

characteristic

111

C₂×C₃×C₄

C₂×M₁₆

yes

—

112

C₂×C₃×C₄

#131

yes

—

113

C₂×C₃×C₄

#135

yes

—

114

C₃⋊C₈

—

115

C₃⋊C₈

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

#217

yes

—

119

C₂²×C₄

#217

yes

—

120

C₂²×C₄

C₂²×Dic₃

yes

characteristic

121

C₂²×C₄

C₂×#67

yes

characteristic

122

C₂²×C₄

C₂×#67

yes

characteristic

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

#208

yes

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

D₃×Q₈

yes

—

138

C₄²

#239

yes

—

139

C₂²×C₃

#391

yes

characteristic

140

C₂²×C₃

#407

yes

characteristic

141

C₂²×C₃

#408

yes

characteristic

142

C₂²×C₃

#416

yes

—

143

C₂²×C₃

#421

yes

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

—

147

C₃×C₄

—

148

C₃×C₄

—

149

C₃×C₄

#446

yes

—

150

C₃×C₄

#488

yes

—

151

C₃×C₄

#499

yes

—

152

C₂³

C₂×#217

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

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

C₂×#208

yes

—

167

C₂×C₄

#944

yes

characteristic

168

C₂×C₄

Q₈×Dic₃

yes

—

169

C₂×C₄

#946

yes

—

170

C₂×C₄

#947

yes

characteristic

171

C₂×C₄

#948

yes

—

172

C₂×C₄

#950

yes

—

173

C₈

—

174

C₈

—

175

C₂×C₃

#1482

yes

—

176

C₂×C₃

#1488

yes

—

177

C₂×C₃

#1985

yes

characteristic

178

C₂×C₃

#1991

yes

characteristic

179

C₂×C₃

#1995

yes

characteristic

180

C₂²

#5207

yes

characteristic

181

C₂²

#5229

yes

characteristic

182

C₂²

#5233

yes

—

183

C₂²

#5234

yes

characteristic

184

C₂²

#5236

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

#5024

yes

—

191

C₄

#5065

yes

—

192

C₄

#5068

yes

—

193

C₃

#9875

yes

characteristic, complemented, Sylow

194

C₂

#71635

yes

characteristic

195

C₂

#71640

yes

—

196

C₂

#71740

yes

characteristic

197

C₂

#71750

yes

—

198

C₂

#71840

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#67
commutator	C₂²×C₃×C₄	C₂²×C₄
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	#9875
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂×#67

commutator

C₂²×C₃×C₄

C₂²×C₄

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 ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
193. C₃	#9875	semidirect

H ≅ G/N

type

193. C₃

#9875

semidirect

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

subgroup	index
C₃×C₄×#125	2
#69392	2
#69388	2
#67619	2
#9875	3

subgroup

index

C₃×C₄×#125

Subgroups — #10695215

Conjugacy classes of subgroups

Complements

Maximal subgroups