Group #10693977 — subgroups

Subgroups — #10693977

Order 768.

530 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 171 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	C₄×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×#4456	C₂	1	4	yes	maximal
4	#69392	C₂	1	1	yes	maximal, characteristic
5	#9641	—	3	1	no	maximal, Sylow
6	C₂×#869	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
9	C₃×C₄³	C₂²	1	1	yes	characteristic
10	#4935	C₄	1	1	yes	characteristic
11	C₄×C₄⋊Dic₃	—	2	2	no	—
12	#4929	C₄	1	2	yes	—
13	#4928	—	2	2	no	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₃×#336	—	2	2	no	—
16	#4524	—	2	2	no	—
17	#4456	—	2	8	no	—
18	C₄×C₄⋊Q₈	—	3	1	no	—
19	#1831	—	3	1	no	—
20	#1744	D₃	1	1	yes	characteristic
21	C₂×Q₈⋊C₈	—	3	4	no	—
22	C₂²×Dic₆	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₈	—	2	2	no	—
24	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄²	D₄	1	2	yes	—
27	C₂×C₃×C₄²	D₄	1	2	yes	—
28	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₃	—	2	2	no	—
30	C₂×C₄⋊Dic₃	—	2	2	no	—
31	C₂×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
32	C₂×Dic₃⋊C₄	—	2	2	no	—
33	C₂×C₄×Dic₃	—	2	2	no	—
34	C₂²×C₃⋊C₈	—	2	2	no	—
35	#869	—	2	2	no	—
36	#869	C₈	1	4	yes	complemented
37	C₄×Dic₆	—	2	4	no	—
38	C₄×Dic₆	—	2	4	no	—
39	C₃×C₄×C₈	—	2	2	no	—
40	C₃×C₄×C₈	D₄	1	4	yes	—
41	C₃⋊#95	—	2	4	no	—
42	C₄⋊(C₃⋊C₈)	—	2	4	no	—
43	C₂×C₄⋊Q₈	—	3	1	no	—
44	C₂×C₄×Q₈	—	3	2	no	—
45	C₂×C₄⋊C₈	—	3	2	no	—
46	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
47	#391	—	3	2	no	—
48	#389	—	6	2	no	—
49	#384	—	3	1	no	—
50	C₄×C₄⋊C₄	—	6	2	no	—
51	C₄³	C₂×D₃	1	1	yes	characteristic
52	#336	—	2	2	no	—
53	#336	—	6	2	no	—
54	Q₈⋊C₈	—	6	8	no	—
55	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₃	—	2	2	no	—
61	C₂²×Dic₃	—	2	2	no	—
62	C₂×Dic₆	—	2	4	no	—
63	C₂×Dic₆	—	2	4	no	—
64	C₂×Dic₆	—	2	2	no	—
65	C₂×Dic₆	C₂×C₈	1	4	yes	—
66	C₂×C₃×C₈	—	2	4	no	—
67	C₂×C₃×C₈	—	4	4	no	—
68	C₃×C₄²	—	2	2	no	—
69	C₃×C₄²	—	2	4	no	—
70	C₃×C₄²	—	4	2	no	—
71	C₃×C₄²	C₂×C₈	1	2	yes	—
72	C₃×C₄²	M₁₆	1	2	yes	—
73	C₃×C₄²	C₂×D₄	1	2	yes	—
74	C₃×C₄²	#41	1	2	yes	—
75	C₄⋊Dic₃	—	2	8	no	—
76	C₄⋊Dic₃	—	2	4	no	—
77	C₄⋊Dic₃	—	4	4	no	—
78	Dic₃⋊C₄	—	4	4	no	—
79	C₄×Dic₃	—	4	2	no	—
80	C₄×Dic₃	—	4	2	no	—
81	C₂×C₃⋊C₈	—	4	2	no	—
82	C₂×C₃⋊C₈	—	4	4	no	—
83	C₂²×Q₈	—	3	2	no	—
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	—	6	2	no	—
86	C₄⋊Q₈	—	3	4	no	—
87	C₄⋊Q₈	—	6	2	no	—
88	C₄×Q₈	—	6	4	no	—
89	C₄×Q₈	—	6	4	no	—
90	C₂×C₄⋊C₄	—	3	2	no	—
91	C₂×C₄⋊C₄	—	6	2	no	—
92	C₂×C₄⋊C₄	—	6	2	no	—
93	C₂×C₄⋊C₄	—	6	2	no	—
94	C₂×C₄²	—	6	2	no	—
95	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
96	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
97	C₂×C₄²	D₁₂	1	2	yes	—
98	C₂×C₄²	#67	1	2	yes	—
99	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
100	C₄⋊C₈	—	6	4	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	D₁₂	1	4	yes	—
103	#95	—	6	4	no	—
104	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	C₂²⋊C₈	1	4	yes	—
112	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
113	C₂×C₃×C₄	#121	1	2	yes	—
114	C₂×C₃×C₄	#124	1	1	yes	characteristic
115	C₂×C₃×C₄	#127	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₃×C₄	#131	1	2	yes	—
119	C₂×C₃×C₄	#137	1	4	yes	—
120	C₂×Dic₃	—	2	4	no	—
121	C₂×Dic₃	—	4	2	no	—
122	C₂×Dic₃	—	4	2	no	—
123	C₂×Dic₃	—	4	2	no	—
124	C₂×Dic₃	—	4	4	no	—
125	Dic₆	—	2	8	no	—
126	Dic₆	—	4	4	no	—
127	C₃×C₈	—	4	4	no	—
128	C₃⋊C₈	—	8	2	no	—
129	C₂×Q₈	—	3	4	no	—
130	C₂×Q₈	—	6	2	no	—
131	C₂×Q₈	—	6	4	no	—
132	C₂×Q₈	—	6	4	no	—
133	C₂²×C₄	—	6	2	no	—
134	C₂²×C₄	—	6	2	no	—
135	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
136	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
139	C₂²×C₄	C₂×#67	1	1	yes	characteristic
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	4	4	no	—
142	C₂×C₈	—	12	2	no	—
143	C₂×C₈	—	12	4	no	—
144	C₄⋊C₄	—	6	8	no	—
145	C₄⋊C₄	—	6	4	no	—
146	C₄⋊C₄	—	12	4	no	—
147	C₄⋊C₄	—	12	4	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	4	2	no	—
151	C₄²	—	12	2	no	—
152	C₄²	—	12	2	no	—
153	C₄²	D₃×C₈	1	2	yes	—
154	C₄²	ΓL(1,𝔽₂₅)	1	2	yes	—
155	C₄²	C₂×D₁₂	1	2	yes	—
156	C₄²	#235	1	2	yes	—
157	C₂²×C₃	#390	1	1	yes	characteristic
158	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₃	#408	1	1	yes	characteristic
160	C₂²×C₃	#419	1	2	yes	—
161	C₂²×C₃	#421	1	2	yes	—
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	2	4	no	—
164	C₃×C₄	—	4	2	no	—
165	C₃×C₄	C₈×D₄	1	2	yes	—
166	C₃×C₄	#436	1	2	yes	—
167	C₃×C₄	#491	1	2	yes	—
168	C₃×C₄	#497	1	2	yes	—
169	Dic₃	—	4	4	no	—
170	Dic₃	—	8	2	no	—
171	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	Q₈	—	6	8	no	—
173	Q₈	—	12	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	6	4	no	—
181	C₂×C₄	—	12	2	no	—
182	C₂×C₄	—	12	2	no	—
183	C₂×C₄	—	12	2	no	—
184	C₂×C₄	—	12	4	no	—
185	C₂×C₄	(C₂×D₃)⋊C₈	1	4	yes	—
186	C₂×C₄	C₄×D₁₂	1	2	yes	—
187	C₂×C₄	#874	1	1	yes	characteristic
188	C₂×C₄	#875	1	1	yes	characteristic
189	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
190	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
191	C₂×C₄	#901	1	2	yes	—
192	C₂×C₄	#909	1	4	yes	—
193	C₂×C₄	#930	1	2	yes	—
194	C₈	—	4	4	no	—
195	C₈	—	24	2	no	—
196	C₂×C₃	#1479	1	2	yes	—
197	C₂×C₃	#1497	1	2	yes	—
198	C₂×C₃	#1954	1	1	yes	characteristic
199	C₂×C₃	#1960	1	1	yes	characteristic
200	C₂×C₃	#1963	1	1	yes	characteristic
201	C₂²	#4939	1	1	yes	characteristic
202	C₂²	#5005	1	2	yes	—
203	C₂²	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
204	C₂²	#5110	1	1	yes	characteristic
205	C₂²	#5120	1	2	yes	—
206	C₄	—	2	2	no	—
207	C₄	—	2	4	no	—
208	C₄	—	4	2	no	—
209	C₄	—	12	4	no	—
210	C₄	—	24	2	no	—
211	C₄	C₈×D₁₂	1	2	yes	—
212	C₄	#4688	1	2	yes	—
213	C₄	#4703	1	2	yes	—
214	C₄	#4715	1	2	yes	—
215	C₃	#9641	1	1	yes	characteristic, complemented, Sylow
216	C₂	#69440	1	1	yes	characteristic
217	C₂	#69787	1	1	yes	characteristic
218	C₂	#69822	1	1	yes	characteristic
219	C₂	#70767	1	2	yes	—
220	C₂	#70771	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#869

C₂

yes

maximal, characteristic

C₃×#1744

C₂

yes

maximal, characteristic

C₂×#4456

C₂

yes

maximal

#69392

C₂

yes

maximal, characteristic

#9641

—

maximal, Sylow

C₂×#869

C₄

yes

characteristic

C₂×C₄×Dic₆

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

—

C₃×C₄³

C₂²

yes

characteristic

#4935

C₄

yes

characteristic

C₄×C₄⋊Dic₃

—

#4929

C₄

yes

—

#4928

—

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₃×#336

—

#4524

—

#4456

—

C₄×C₄⋊Q₈

—

#1831

—

#1744

D₃

yes

characteristic

C₂×Q₈⋊C₈

—

C₂²×Dic₆

C₂×C₄

yes

—

C₂²×C₃×C₈

—

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

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂×C₄

yes

—

C₂×Dic₃⋊C₄

—

C₂×C₄×Dic₃

—

C₂²×C₃⋊C₈

—

#869

—

#869

C₈

yes

complemented

C₄×Dic₆

—

C₄×Dic₆

—

C₃×C₄×C₈

—

C₃×C₄×C₈

D₄

yes

—

C₃⋊#95

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₃

yes

—

#391

—

#389

—

#384

—

C₄×C₄⋊C₄

—

C₄³

C₂×D₃

yes

characteristic

#336

—

#336

—

Q₈⋊C₈

—

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

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

C₂×C₈

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×C₈

yes

—

C₃×C₄²

M₁₆

yes

—

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

#41

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₄×D₃

yes

characteristic

C₂×C₄²

C₄×D₃

yes

characteristic

C₂×C₄²

D₁₂

yes

—

C₂×C₄²

#67

yes

—

C₂×C₄²

C₂²×D₃

yes

characteristic

100

C₄⋊C₈

—

101

C₄×C₈

—

102

C₄×C₈

D₁₂

yes

—

103

#95

—

104

C₂³×C₃

C₂×C₂²⋊C₄

yes

characteristic

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

C₂²⋊C₈

yes

—

112

C₂×C₃×C₄

C₄×D₄

yes

—

113

C₂×C₃×C₄

#121

yes

—

114

C₂×C₃×C₄

#124

yes

characteristic

115

C₂×C₃×C₄

#127

yes

characteristic

116

C₂×C₃×C₄

C₂²×C₈

yes

characteristic

117

C₂×C₃×C₄

C₂×M₁₆

yes

characteristic

118

C₂×C₃×C₄

#131

yes

—

119

C₂×C₃×C₄

#137

yes

—

120

C₂×Dic₃

—

121

C₂×Dic₃

—

122

C₂×Dic₃

—

123

C₂×Dic₃

—

124

C₂×Dic₃

—

125

Dic₆

—

126

Dic₆

—

127

C₃×C₈

—

128

C₃⋊C₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

136

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

137

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

138

C₂²×C₄

C₂×D₁₂

yes

characteristic

139

C₂²×C₄

C₂×#67

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

—

153

C₄²

D₃×C₈

yes

—

154

C₄²

ΓL(1,𝔽₂₅)

yes

—

155

C₄²

C₂×D₁₂

yes

—

156

C₄²

#235

yes

—

157

C₂²×C₃

#390

yes

characteristic

158

C₂²×C₃

C₂×C₂²⋊C₈

yes

characteristic

159

C₂²×C₃

#408

yes

characteristic

160

C₂²×C₃

#419

yes

—

161

C₂²×C₃

#421

yes

—

162

C₃×C₄

—

163

C₃×C₄

—

164

C₃×C₄

—

165

C₃×C₄

C₈×D₄

yes

—

166

C₃×C₄

#436

yes

—

167

C₃×C₄

#491

yes

—

168

C₃×C₄

#497

yes

—

169

Dic₃

—

170

Dic₃

—

171

C₂³

C₂×(C₂×D₃)⋊C₄

yes

characteristic

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

(C₂×D₃)⋊C₈

yes

—

186

C₂×C₄

C₄×D₁₂

yes

—

187

C₂×C₄

#874

yes

characteristic

188

C₂×C₄

#875

yes

characteristic

189

C₂×C₄

C₂×D₃×C₈

yes

characteristic

190

C₂×C₄

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

191

C₂×C₄

#901

yes

—

192

C₂×C₄

#909

yes

—

193

C₂×C₄

#930

yes

—

194

C₈

—

195

C₈

—

196

C₂×C₃

#1479

yes

—

197

C₂×C₃

#1497

yes

—

198

C₂×C₃

#1954

yes

characteristic

199

C₂×C₃

#1960

yes

characteristic

200

C₂×C₃

#1963

yes

characteristic

201

C₂²

#4939

yes

characteristic

202

C₂²

#5005

yes

—

203

C₂²

C₂×(C₂×D₃)⋊C₈

yes

characteristic

204

C₂²

#5110

yes

characteristic

205

C₂²

#5120

yes

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

C₈×D₁₂

yes

—

212

C₄

#4688

yes

—

213

C₄

#4703

yes

—

214

C₄

#4715

yes

—

215

C₃

#9641

yes

characteristic, complemented, Sylow

216

C₂

#69440

yes

characteristic

217

C₂

#69787

yes

characteristic

218

C₂

#69822

yes

characteristic

219

C₂

#70767

yes

—

220

C₂

#70771

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#1744	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#1744	D₃
hypocenter	C₃	#9641
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₁₂

commutator

C₂×C₃×C₄

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×#1744

C₂

socle, prime radical

C₂³×C₃

C₂×C₂²⋊C₄

cosocle

C₂×C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
36. #869	C₈	semidirect
215. C₃	#9641	semidirect

H ≅ G/N

type

36. #869

C₈

semidirect

215. C₃

#9641

semidirect

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

subgroup	index
C₄×#869	2
C₃×#1744	2
C₂×#4456	2
#69392	2
#9641	3

subgroup

index

Subgroups — #10693977

Conjugacy classes of subgroups

Complements

Maximal subgroups