Group #10695144 — subgroups

Subgroups — #10695144

Order 768.

390 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 119 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	#78442	C₂	1	1	yes	maximal, characteristic
2	C₃×#1837	C₂	1	1	yes	maximal, characteristic
3	#70501	C₂	1	2	yes	maximal
4	#70501	C₂	1	2	yes	maximal
5	C₂×#4454	C₂	1	1	yes	maximal, characteristic
6	#9859	—	3	1	no	maximal, Sylow
7	#5470	C₂²	1	2	yes	—
8	C₃×#423	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	#5106	—	2	2	no	—
11	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	#4683	—	2	2	no	—
14	#4682	C₄	1	1	yes	characteristic
15	#4682	C₄	1	1	yes	characteristic
16	#4680	C₄	1	2	yes	—
17	C₃×#341	—	2	2	no	—
18	#4484	—	2	4	no	—
19	#4484	—	2	4	no	—
20	#4454	—	2	2	no	—
21	#3160	—	3	1	no	—
22	#1837	D₃	1	1	yes	characteristic
23	#1493	—	3	2	no	—
24	#1493	—	3	2	no	—
25	C₂×#322	—	3	1	no	—
26	C₂×C₃×M₁₆	—	2	2	no	—
27	C₂²×C₃×C₈	D₄	1	2	yes	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	#926	—	2	2	no	—
30	#924	—	2	2	no	—
31	#924	—	2	2	no	—
32	C₂²×C₃⋊C₈	—	2	1	no	—
33	C₂²×C₃⋊C₈	—	2	1	no	—
34	#870	—	2	2	no	—
35	#869	C₂×C₄	1	2	yes	—
36	#869	C₂×C₄	1	2	yes	—
37	C₄×Dic₆	—	2	2	no	—
38	C₄×Dic₆	—	2	2	no	—
39	C₃×C₄⋊C₈	—	2	2	no	—
40	C₃×C₄⋊C₈	—	2	2	no	—
41	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×C₈	D₄	1	2	yes	—
44	#818	—	2	2	no	—
45	#818	—	2	2	no	—
46	#817	—	2	4	no	—
47	C₄×C₃⋊C₈	—	2	1	no	—
48	C₄×C₃⋊C₈	—	2	1	no	—
49	C₄×C₃⋊C₈	D₄	1	2	yes	—
50	C₄×C₃⋊C₈	D₄	1	2	yes	—
51	#533	—	3	2	no	—
52	#500	—	3	1	no	—
53	#500	—	3	1	no	—
54	#499	—	6	2	no	—
55	#498	—	3	2	no	—
56	#427	—	6	2	no	—
57	#423	C₂×D₃	1	2	yes	—
58	C₂×C₄×C₈	—	3	1	no	—
59	C₂×C₄×C₈	—	3	1	no	—
60	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
61	#341	—	2	2	no	—
62	Q₈⋊C₈	—	6	4	no	—
63	Q₈⋊C₈	—	6	4	no	—
64	#322	—	6	2	no	—
65	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₃×C₄	#41	1	1	yes	characteristic
67	C₂²×C₃×C₄	#41	1	1	yes	characteristic
68	C₂×Dic₆	—	2	2	no	—
69	C₂×Dic₆	—	2	2	no	—
70	C₃×M₁₆	—	4	4	no	—
71	C₂×C₃×C₈	—	2	4	no	—
72	C₂×C₃×C₈	—	4	2	no	—
73	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
75	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
79	#217	—	4	2	no	—
80	C₄⋊Dic₃	—	2	2	no	—
81	C₄⋊Dic₃	—	2	2	no	—
82	C₄⋊Dic₃	—	2	2	no	—
83	C₄⋊Dic₃	—	2	2	no	—
84	Dic₃⋊C₄	—	4	2	no	—
85	Dic₃⋊C₄	—	4	2	no	—
86	C₄×Dic₃	—	4	2	no	—
87	C₂×C₃⋊C₈	—	2	4	no	—
88	C₂×C₃⋊C₈	—	2	2	no	—
89	C₂×C₃⋊C₈	—	2	2	no	—
90	C₂×C₃⋊C₈	—	4	2	no	—
91	C₂×M₁₆	—	2	2	no	—
92	C₂²×C₈	—	6	1	no	—
93	C₂²×C₈	—	6	1	no	—
94	C₂²×C₈	#67	1	2	yes	—
95	C₄⋊Q₈	—	3	2	no	—
96	C₄⋊Q₈	—	3	2	no	—
97	#125	—	6	2	no	—
98	C₂²⋊Q₈	—	6	2	no	—
99	C₂²⋊Q₈	—	6	2	no	—
100	C₄×Q₈	—	6	2	no	—
101	C₄×Q₈	—	6	2	no	—
102	#117	—	6	2	no	—
103	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
104	#107	—	6	2	no	—
105	#107	—	6	2	no	—
106	#106	—	6	4	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄×C₈	—	3	2	no	—
110	C₄×C₈	—	3	2	no	—
111	C₄×C₈	—	6	1	no	—
112	C₄×C₈	—	6	1	no	—
113	C₄×C₈	C₄×D₃	1	1	yes	characteristic
114	C₄×C₈	C₄×D₃	1	1	yes	characteristic
115	C₄×C₈	D₁₂	1	2	yes	—
116	C₂³×C₃	#124	1	1	yes	characteristic
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
121	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
122	C₂×C₃×C₄	#121	1	1	yes	characteristic
123	C₂×C₃×C₄	#121	1	1	yes	characteristic
124	C₂×C₃×C₄	#127	1	1	yes	characteristic
125	C₂×C₃×C₄	#137	1	4	yes	—
126	C₂×Dic₃	—	4	2	no	—
127	C₂×Dic₃	—	4	2	no	—
128	Dic₆	—	4	4	no	—
129	Dic₆	—	4	4	no	—
130	C₃×C₈	—	2	2	no	—
131	C₃×C₈	—	2	2	no	—
132	C₃×C₈	—	8	2	no	—
133	C₃⋊C₈	—	2	4	no	—
134	C₃⋊C₈	—	4	2	no	—
135	C₂×Q₈	—	6	2	no	—
136	C₂×Q₈	—	6	2	no	—
137	C₂²×C₄	#239	1	1	yes	characteristic
138	C₂²×C₄	#239	1	1	yes	characteristic
139	C₂²×C₄	C₂×#67	1	1	yes	characteristic
140	M₁₆	—	4	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	6	4	no	—
144	C₂×C₈	—	6	2	no	—
145	C₂×C₈	—	6	2	no	—
146	C₂×C₈	—	12	2	no	—
147	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
148	C₂×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
149	C₄⋊C₄	—	6	2	no	—
150	C₄⋊C₄	—	6	2	no	—
151	C₄⋊C₄	—	6	2	no	—
152	C₄⋊C₄	—	6	2	no	—
153	C₄⋊C₄	—	12	2	no	—
154	C₄⋊C₄	—	12	2	no	—
155	C₂²⋊C₄	—	12	2	no	—
156	C₄²	—	12	2	no	—
157	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
158	C₄²	C₂×D₁₂	1	1	yes	characteristic
159	C₄²	D₃×D₄	1	1	yes	characteristic
160	C₄²	D₃×D₄	1	1	yes	characteristic
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	#390	1	1	yes	characteristic
164	C₂²×C₃	#491	1	2	yes	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	#419	1	2	yes	—
168	C₃×C₄	#443	1	2	yes	—
169	C₃×C₄	#444	1	2	yes	—
170	C₃×C₄	#497	1	2	yes	—
171	Dic₃	—	8	2	no	—
172	Dic₃	—	8	2	no	—
173	C₂³	#947	1	1	yes	characteristic
174	Q₈	—	12	4	no	—
175	Q₈	—	12	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	12	2	no	—
179	C₂×C₄	—	12	2	no	—
180	C₂×C₄	#893	1	1	yes	characteristic
181	C₂×C₄	#893	1	1	yes	characteristic
182	C₂×C₄	#895	1	1	yes	characteristic
183	C₂×C₄	#895	1	1	yes	characteristic
184	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₄	#940	1	1	yes	characteristic
186	C₂×C₄	#951	1	4	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	2	2	no	—
189	C₈	—	6	4	no	—
190	C₈	—	8	2	no	—
191	C₈	—	12	2	no	—
192	C₂×C₃	—	2	2	no	—
193	C₂×C₃	#1960	1	1	yes	characteristic
194	C₂×C₃	#1974	1	1	yes	characteristic
195	C₂×C₃	#1975	1	1	yes	characteristic
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	#4988	1	1	yes	characteristic
199	C₂²	#5069	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	24	2	no	—
203	C₄	—	24	2	no	—
204	C₄	#4894	1	2	yes	—
205	C₄	#4895	1	2	yes	—
206	C₄	#5004	1	2	yes	—
207	C₄	#5060	1	2	yes	—
208	C₃	#9859	1	1	yes	characteristic, complemented, Sylow
209	C₂	—	2	2	no	—
210	C₂	#69552	1	1	yes	characteristic
211	C₂	#70696	1	1	yes	characteristic
212	C₂	#70697	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#78442

C₂

yes

maximal, characteristic

C₃×#1837

C₂

yes

maximal, characteristic

#70501

C₂

yes

maximal

#70501

C₂

yes

maximal

C₂×#4454

C₂

yes

maximal, characteristic

#9859

—

maximal, Sylow

#5470

C₂²

yes

—

C₃×#423

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

#5106

—

C₂×C₄×C₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₃⋊C₈

C₂²

yes

characteristic

#4683

—

#4682

C₄

yes

characteristic

#4682

C₄

yes

characteristic

#4680

C₄

yes

—

—

—

—

—

—

D₃

yes

—

—

—

C₂×C₃×M₁₆

—

C₂²×C₃×C₈

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

—

—

—

C₂²×C₃⋊C₈

—

C₂²×C₃⋊C₈

—

#870

—

#869

C₂×C₄

yes

—

#869

C₂×C₄

yes

—

C₄×Dic₆

—

C₄×Dic₆

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

—

#818

—

#818

—

#817

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

D₄

yes

—

C₄×C₃⋊C₈

D₄

yes

—

—

—

—

—

—

—

C₂×D₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

#341

—

Q₈⋊C₈

—

Q₈⋊C₈

—

#322

—

C₂²×C₃×C₄

C₂×D₄

yes

C₂²×C₃×C₄

yes

C₂²×C₃×C₄

yes

C₂×Dic₆

—

C₂×Dic₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

#67

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

C₄×Q₈

—

102

#117

—

103

C₂×C₄²

C₂²×D₃

yes

characteristic

104

#107

—

105

#107

—

106

#106

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₄×D₃

yes

characteristic

114

C₄×C₈

C₄×D₃

yes

characteristic

115

C₄×C₈

D₁₂

yes

—

116

C₂³×C₃

#124

yes

characteristic

117

C₂×C₃×C₄

—

118

C₂×C₃×C₄

—

119

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

120

C₂×C₃×C₄

C₄×D₄

yes

characteristic

121

C₂×C₃×C₄

C₄×D₄

yes

characteristic

122

C₂×C₃×C₄

#121

yes

characteristic

123

C₂×C₃×C₄

#121

yes

characteristic

124

C₂×C₃×C₄

#127

yes

characteristic

125

C₂×C₃×C₄

#137

yes

—

126

C₂×Dic₃

—

127

C₂×Dic₃

—

128

Dic₆

—

129

Dic₆

—

130

C₃×C₈

—

131

C₃×C₈

—

132

C₃×C₈

—

133

C₃⋊C₈

—

134

C₃⋊C₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂²×C₄

#239

yes

characteristic

138

C₂²×C₄

#239

yes

characteristic

139

C₂²×C₄

C₂×#67

yes

characteristic

140

M₁₆

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

148

C₂×C₈

(C₂×D₃)⋊C₄

yes

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₂²⋊C₄

—

156

C₄²

—

157

C₄²

C₂×C₄×D₃

yes

characteristic

158

C₄²

C₂×D₁₂

yes

characteristic

159

C₄²

D₃×D₄

yes

characteristic

160

C₄²

D₃×D₄

yes

characteristic

161

C₂²×C₃

—

162

C₂²×C₃

—

163

C₂²×C₃

#390

yes

characteristic

164

C₂²×C₃

#491

yes

—

165

C₃×C₄

—

166

C₃×C₄

—

167

C₃×C₄

#419

yes

—

168

C₃×C₄

#443

yes

—

169

C₃×C₄

#444

yes

—

170

C₃×C₄

#497

yes

—

171

Dic₃

—

172

Dic₃

—

173

C₂³

#947

yes

characteristic

174

Q₈

—

175

Q₈

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

#893

yes

characteristic

181

C₂×C₄

#893

yes

characteristic

182

C₂×C₄

#895

yes

characteristic

183

C₂×C₄

#895

yes

characteristic

184

C₂×C₄

C₂×(C₂×D₃)⋊C₄

yes

characteristic

185

C₂×C₄

#940

yes

characteristic

186

C₂×C₄

#951

yes

—

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₈

—

192

C₂×C₃

—

193

C₂×C₃

#1960

yes

characteristic

194

C₂×C₃

#1974

yes

characteristic

195

C₂×C₃

#1975

yes

characteristic

196

C₂²

—

197

C₂²

—

198

C₂²

#4988

yes

characteristic

199

C₂²

#5069

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

#4894

yes

—

205

C₄

#4895

yes

—

206

C₄

#5004

yes

—

207

C₄

#5060

yes

—

208

C₃

#9859

yes

characteristic, complemented, Sylow

209

C₂

—

210

C₂

#69552

yes

characteristic

211

C₂

#70696

yes

characteristic

212

C₂

#70697

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#940
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1837	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#124
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1837	D₃
hypocenter	C₃	#9859
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂×C₄

#940

commutator

C₃×C₄²

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
208. C₃	#9859	semidirect

H ≅ G/N

type

208. C₃

#9859

semidirect

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

subgroup	index
#78442	2
C₃×#1837	2
#70501	2
C₂×#4454	2
#9859	3

subgroup

index

Subgroups — #10695144

Conjugacy classes of subgroups

Complements

Maximal subgroups