Group #25254274 — subgroups

Subgroups — #25254274

Order 11008.

528 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 165 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	Q₈×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136818	C₂	1	3	yes	maximal, complemented
3	#19136742	C₂	1	3	yes	maximal, complemented
4	#19136632	C₂	1	3	yes	maximal, complemented
5	Q₈×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13063269	C₂	1	3	yes	maximal, complemented
7	C₄₃×#3391	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₈₆	—	2	1	no	—
9	C₂×#13063269	C₂²	1	3	yes	—
10	#14583304	C₂²	1	3	yes	—
11	#14583195	—	2	3	no	—
12	C₄×Dic₁₇₂	C₂²	1	3	yes	—
13	#14582726	C₂²	1	3	yes	—
14	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	3	yes	—
15	#14582702	C₂²	1	1	yes	characteristic
16	#14582700	C₂²	1	3	yes	—
17	#14582700	C₂²	1	6	yes	—
18	C₄×C₄₃⋊C₁₆	C₂²	1	3	yes	—
19	C₄₃×#550	—	2	1	no	—
20	C₄₃×#493	C₂²	1	3	yes	—
21	C₄₃×#459	—	2	3	no	—
22	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄×D₈×C₄₃	C₂²	1	3	yes	—
24	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
25	#13063325	—	2	3	no	—
26	#13063324	—	4	3	no	—
27	Q₈×Dic₄₃	—	4	1	no	—
28	C₄×Dic₈₆	—	2	3	no	—
29	#13063269	—	2	6	no	—
30	C₂×C₄₃⋊C₁₆	C₂³	1	3	yes	—
31	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
32	#13063259	—	2	3	no	—
33	#13063259	C₂³	1	3	yes	—
34	#13063250	—	2	3	no	—
35	C₂×#41×C₄₃	—	4	1	no	—
36	C₂×D₈×C₄₃	—	2	3	no	—
37	C₂×D₈×C₄₃	C₂³	1	3	yes	—
38	#127×C₄₃	—	2	3	no	—
39	#121×C₄₃	—	4	3	no	—
40	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
41	C₄×D₄×C₄₃	—	2	3	no	—
42	#107×C₄₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₃	D₄	1	3	yes	—
44	#102×C₄₃	—	2	3	no	—
45	C₄×C₈×C₄₃	C₂³	1	3	yes	—
46	#41×C₄₃	—	8	4	no	—
47	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
48	C₂×D₄×C₄₃	—	2	3	no	—
49	C₂×D₄×C₄₃	—	4	3	no	—
50	C₂×D₄×C₄₃	—	4	3	no	—
51	C₂²×C₄×C₄₃	—	4	3	no	—
52	C₂×Dic₈₆	—	2	1	no	—
53	C₂×Dic₈₆	—	4	3	no	—
54	D₈×C₄₃	—	2	3	no	—
55	D₈×C₄₃	—	2	3	no	—
56	D₈×C₄₃	—	8	3	no	—
57	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
58	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₄₃	—	2	1	no	—
60	C₄⋊C₄×C₄₃	D₈	1	3	yes	—
61	C₂²⋊C₄×C₄₃	—	4	3	no	—
62	C₄²×C₄₃	C₂×D₄	1	3	yes	—
63	C₄⋊Dic₄₃	—	2	3	no	—
64	C₄⋊Dic₄₃	—	4	3	no	—
65	Dic₄₃⋊C₄	—	4	3	no	—
66	C₄×Dic₄₃	—	4	3	no	—
67	Dic₁₇₂	—	2	1	no	—
68	Dic₁₇₂	#41	1	2	yes	complemented
69	C₄₃⋊C₁₆	—	2	3	no	—
70	C₄₃⋊C₁₆	—	2	1	no	—
71	C₂³×C₄₃	—	4	3	no	—
72	Q₈×C₄₃	SD₃₂	1	4	yes	complemented
73	D₄×C₄₃	—	4	3	no	—
74	D₄×C₄₃	—	4	3	no	—
75	D₄×C₄₃	—	8	6	no	—
76	D₄×C₄₃	—	16	3	no	—
77	C₂×C₄×C₄₃	—	4	1	no	—
78	C₂×C₄×C₄₃	—	8	6	no	—
79	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
80	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
81	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₄₃	—	4	1	no	—
83	C₂×Dic₄₃	—	4	3	no	—
84	Dic₈₆	—	2	2	no	—
85	Dic₈₆	—	4	1	no	—
86	Dic₈₆	—	16	3	no	—
87	C₈×C₄₃	—	2	3	no	—
88	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
89	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
90	#33929	—	43	1	no	maximal, Sylow
91	C₂²×C₄₃	—	8	3	no	—
92	C₂²×C₄₃	—	8	3	no	—
93	C₂²×C₄₃	—	8	3	no	—
94	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₄₃	—	2	3	no	—
96	C₄×C₄₃	—	8	1	no	—
97	C₄×C₄₃	C₂×SD₃₂	1	6	yes	—
98	C₄×C₄₃	#548	1	1	yes	characteristic
99	C₄×C₄₃	#577	1	1	yes	characteristic
100	Dic₄₃	—	4	2	no	—
101	Dic₄₃	—	8	3	no	—
102	#3391	D₄₃	1	1	yes	characteristic
103	Q₈×Q₁₆	—	43	1	no	—
104	#2240	—	43	3	no	—
105	#2226	—	43	3	no	—
106	#2207	—	43	3	no	—
107	Q₈×C₁₆	—	43	1	no	—
108	C₄×SD₃₂	—	43	3	no	—
109	C₂×C₄₃	—	16	3	no	—
110	C₂×C₄₃	#3358	1	1	yes	characteristic
111	C₂×C₄₃	C₂²×SD₃₂	1	1	yes	characteristic
112	C₂×C₄₃	#3415	1	1	yes	characteristic
113	Q₈²	—	86	1	no	—
114	#550	—	2	1	no	—
115	C₂×SD₃₂	—	43	3	no	—
116	#500	—	43	3	no	—
117	#493	C₂×D₄₃	1	3	yes	—
118	#477	—	86	3	no	—
119	#459	—	2	3	no	—
120	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
121	C₄×Q₁₆	—	43	3	no	—
122	C₄×D₈	C₂×D₄₃	1	3	yes	—
123	#367	—	43	3	no	—
124	C₄⋊C₁₆	—	43	3	no	—
125	#358	—	43	1	no	—
126	#357	—	43	3	no	—
127	#357	—	43	6	no	—
128	C₄×C₁₆	—	43	3	no	—
129	C₄₃	#33929	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	—	4	1	no	—
131	C₂×Q₁₆	—	43	1	no	—
132	C₂×D₈	—	2	3	no	—
133	C₂×D₈	C₂²×D₄₃	1	3	yes	—
134	C₄⋊Q₈	—	86	3	no	—
135	C₄⋊Q₈	—	172	3	no	—
136	#127	—	2	3	no	—
137	#121	—	4	3	no	—
138	C₄×Q₈	—	86	3	no	—
139	C₄×Q₈	—	172	1	no	—
140	C₄×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
141	C₄×D₄	—	2	3	no	—
142	SD₃₂	—	86	6	no	—
143	C₂×C₁₆	—	43	1	no	—
144	C₂×C₁₆	—	43	3	no	—
145	#107	—	43	3	no	—
146	#107	—	86	3	no	—
147	#107	C₂²×D₄₃	1	1	yes	characteristic
148	C₄⋊C₈	C₂²⋊D₄₃	1	3	yes	—
149	#103	—	86	3	no	—
150	#102	—	2	3	no	—
151	C₄×C₈	C₂²×D₄₃	1	3	yes	—
152	#41	—	8	4	no	—
153	C₂×Q₈	—	86	1	no	—
154	C₂×Q₈	—	172	3	no	—
155	C₂×Q₈	#10621882	1	1	yes	characteristic
156	C₂×D₄	—	2	3	no	—
157	C₂×D₄	—	4	3	no	—
158	C₂×D₄	—	4	3	no	—
159	C₂²×C₄	—	4	3	no	—
160	Q₁₆	—	43	2	no	—
161	Q₁₆	—	86	1	no	—
162	D₈	—	2	3	no	—
163	D₈	—	2	3	no	—
164	D₈	—	8	3	no	—
165	C₂×C₈	C₂×C₂²⋊D₄₃	1	3	yes	—
166	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	86	3	no	—
169	C₄⋊C₄	—	172	3	no	—
170	C₄⋊C₄	—	172	3	no	—
171	C₄⋊C₄	#10621882	1	3	yes	—
172	C₂²⋊C₄	—	4	3	no	—
173	C₄²	—	172	3	no	—
174	C₄²	C₂×C₂²⋊D₄₃	1	3	yes	—
175	C₁₆	—	86	3	no	—
176	C₁₆	—	86	1	no	—
177	C₂³	—	4	3	no	—
178	Q₈	—	86	2	no	—
179	Q₈	—	172	1	no	—
180	Q₈	—	688	3	no	—
181	Q₈	#13063269	1	4	yes	complemented
182	D₄	—	4	3	no	—
183	D₄	—	4	3	no	—
184	D₄	—	8	6	no	—
185	D₄	—	16	3	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	8	6	no	—
188	C₂×C₄	—	172	3	no	—
189	C₂×C₄	—	172	1	no	—
190	C₂×C₄	C₂×#10621882	1	3	yes	—
191	C₂×C₄	C₂×#10621882	1	3	yes	—
192	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
193	C₈	—	2	3	no	—
194	C₈	#41×D₄₃	1	1	yes	characteristic
195	C₈	#13063372	1	1	yes	characteristic
196	C₂²	—	8	3	no	—
197	C₂²	—	8	3	no	—
198	C₂²	—	8	3	no	—
199	C₂²	C₂²×#10621882	1	1	yes	characteristic
200	C₄	—	2	3	no	—
201	C₄	—	8	1	no	—
202	C₄	—	344	3	no	—
203	C₄	—	172	2	no	—
204	C₄	C₂×#13063269	1	6	yes	—
205	C₄	#14583556	1	1	yes	characteristic
206	C₄	#14583710	1	1	yes	characteristic
207	C₂	—	16	3	no	—
208	C₂	#19147226	1	1	yes	characteristic
209	C₂	C₂²×#13063269	1	1	yes	characteristic
210	C₂	#19147734	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₇₂

C₂

yes

maximal, characteristic, complemented

#19136818

C₂

yes

maximal, complemented

#19136742

C₂

yes

maximal, complemented

#19136632

C₂

yes

maximal, complemented

Q₈×C₄₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄×#13063269

C₂

yes

maximal, complemented

C₄₃×#3391

C₂

yes

maximal, characteristic

Q₈×Dic₈₆

—

C₂×#13063269

C₂²

yes

—

#14583304

C₂²

yes

—

#14583195

—

C₄×Dic₁₇₂

C₂²

yes

—

#14582726

C₂²

yes

—

C₄⋊(C₄₃⋊C₁₆)

C₂²

yes

—

#14582702

C₂²

yes

characteristic

#14582700

C₂²

yes

—

#14582700

C₂²

yes

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₄₃×#550

—

C₄₃×#493

C₂²

yes

—

C₄₃×#459

—

C₈×Q₈×C₄₃

C₂²

yes

characteristic

C₄×D₈×C₄₃

C₂²

yes

—

C₂×Dic₁₇₂

C₂³

yes

characteristic

#13063325

—

#13063324

—

Q₈×Dic₄₃

—

C₄×Dic₈₆

—

#13063269

—

C₂×C₄₃⋊C₁₆

C₂³

yes

—

C₂×C₄₃⋊C₁₆

C₂³

yes

characteristic

#13063259

—

#13063259

C₂³

yes

—

#13063250

—

C₂×#41×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

C₂³

yes

—

#127×C₄₃

—

#121×C₄₃

—

C₄×Q₈×C₄₃

D₄

yes

characteristic

C₄×D₄×C₄₃

—

#107×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

D₄

yes

—

#102×C₄₃

—

C₄×C₈×C₄₃

C₂³

yes

—

#41×C₄₃

—

C₂×Q₈×C₄₃

D₈

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

D₈×C₄₃

—

D₈×C₄₃

—

D₈×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₈

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

#41

yes

complemented

C₄₃⋊C₁₆

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

—

Q₈×C₄₃

SD₃₂

yes

complemented

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×#41

yes

characteristic

C₈×C₄₃

2₋¹⁺⁴

yes

characteristic

#33929

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₈

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×SD₃₂

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

100

Dic₄₃

—

101

Dic₄₃

—

102

#3391

D₄₃

yes

characteristic

103

Q₈×Q₁₆

—

104

#2240

—

105

#2226

—

106

#2207

—

107

Q₈×C₁₆

—

108

C₄×SD₃₂

—

109

C₂×C₄₃

—

110

C₂×C₄₃

#3358

yes

characteristic

111

C₂×C₄₃

C₂²×SD₃₂

yes

characteristic

112

C₂×C₄₃

#3415

yes

characteristic

113

Q₈²

—

114

#550

—

115

C₂×SD₃₂

—

116

#500

—

117

#493

C₂×D₄₃

yes

—

118

#477

—

119

#459

—

120

C₈×Q₈

C₂×D₄₃

yes

characteristic

121

C₄×Q₁₆

—

122

C₄×D₈

C₂×D₄₃

yes

—

123

#367

—

124

C₄⋊C₁₆

—

125

#358

—

126

#357

—

127

#357

—

128

C₄×C₁₆

—

129

C₄₃

#33929

yes

characteristic, complemented, Sylow

130

C₂×#41

—

131

C₂×Q₁₆

—

132

C₂×D₈

—

133

C₂×D₈

C₂²×D₄₃

yes

—

134

C₄⋊Q₈

—

135

C₄⋊Q₈

—

172

—

136

#127

—

137

#121

—

138

C₄×Q₈

—

139

C₄×Q₈

—

172

—

140

C₄×Q₈

C₂²⋊D₄₃

yes

characteristic

141

C₄×D₄

—

142

SD₃₂

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

145

#107

—

146

#107

—

147

#107

C₂²×D₄₃

yes

characteristic

148

C₄⋊C₈

C₂²⋊D₄₃

yes

—

149

#103

—

150

#102

—

151

C₄×C₈

C₂²×D₄₃

yes

—

152

#41

—

153

C₂×Q₈

—

154

C₂×Q₈

—

172

—

155

C₂×Q₈

#10621882

yes

characteristic

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂²×C₄

—

160

Q₁₆

—

161

Q₁₆

—

162

D₈

—

163

D₈

—

164

D₈

—

165

C₂×C₈

C₂×C₂²⋊D₄₃

yes

—

166

C₂×C₈

C₂³×D₄₃

yes

characteristic

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

172

—

170

C₄⋊C₄

—

172

—

171

C₄⋊C₄

#10621882

yes

—

172

C₂²⋊C₄

—

173

C₄²

—

172

—

174

C₄²

C₂×C₂²⋊D₄₃

yes

—

175

C₁₆

—

176

C₁₆

—

177

C₂³

—

178

Q₈

—

179

Q₈

—

172

—

180

Q₈

—

688

—

181

Q₈

#13063269

yes

complemented

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

172

—

189

C₂×C₄

—

172

—

190

C₂×C₄

C₂×#10621882

yes

—

191

C₂×C₄

C₂×#10621882

yes

—

192

C₂×C₄

C₂²×C₂²⋊D₄₃

yes

characteristic

193

C₈

—

194

C₈

#41×D₄₃

yes

characteristic

195

C₈

#13063372

yes

characteristic

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

C₂²×#10621882

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

344

—

203

C₄

—

172

—

204

C₄

C₂×#13063269

yes

—

205

C₄

#14583556

yes

characteristic

206

C₄

#14583710

yes

characteristic

207

C₂

—

208

C₂

#19147226

yes

characteristic

209

C₂

C₂²×#13063269

yes

characteristic

210

C₂

#19147734

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting, prime radical	C₄₃×#3391	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3391	D₄₃
hypocenter	C₄₃	#33929
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

subgroup

G/H

center

C₂²

C₂²×#10621882

commutator, cosocle

C₂×C₈×C₄₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₄₃

Fitting, prime radical

C₄₃×#3391

C₂

socle

C₂²×C₄₃

C₂²×D₈

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

N	H ≅ G/N	type
1. Q₈×Dic₁₇₂	C₂	semidirect
2. #19136818	C₂	semidirect
3. #19136742	C₂	semidirect
4. #19136632	C₂	semidirect
5. Q₈×C₄₃⋊C₁₆	C₂	semidirect
6. C₄×#13063269	C₂	semidirect
68. Dic₁₇₂	#41	semidirect
72. Q₈×C₄₃	SD₃₂	semidirect
129. C₄₃	#33929	semidirect
181. Q₈	#13063269	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₇₂

C₂

semidirect

2. #19136818

C₂

semidirect

3. #19136742

C₂

semidirect

4. #19136632

C₂

semidirect

5. Q₈×C₄₃⋊C₁₆

C₂

semidirect

6. C₄×#13063269

C₂

semidirect

68. Dic₁₇₂

#41

semidirect

72. Q₈×C₄₃

SD₃₂

semidirect

129. C₄₃

#33929

semidirect

181. Q₈

#13063269

semidirect

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

subgroup	index
Q₈×Dic₁₇₂	2
#19136818	2
#19136742	2
#19136632	2
Q₈×C₄₃⋊C₁₆	2
C₄×#13063269	2
C₄₃×#3391	2
#33929	43

subgroup

index

Q₈×Dic₁₇₂

Q₈×C₄₃⋊C₁₆

C₄₃×#3391

Subgroups — #25254274

Conjugacy classes of subgroups

Complements

Maximal subgroups