Group #19133660 — subgroups

Subgroups — #19133660

Order 5504.

278 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14583275	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583099	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583094	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582643	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621884	—	2	1	no	—
9	C₂×#10621882	—	2	1	no	—
10	C₂×#10621882	—	2	2	no	—
11	#13063322	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
14	#13063245	—	2	1	no	—
15	#13063245	—	2	1	no	—
16	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×#41×C₄₃	—	2	1	no	—
18	#127×C₄₃	—	2	1	no	—
19	#127×C₄₃	C₂²	1	1	yes	characteristic
20	#121×C₄₃	—	2	2	no	—
21	#121×C₄₃	—	2	1	no	—
22	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₄₃	—	2	1	no	—
24	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
25	#41×C₄₃	—	4	2	no	—
26	#41×C₄₃	—	4	2	no	—
27	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
28	C₂×D₄×C₄₃	—	2	2	no	—
29	C₂×D₄×C₄₃	—	2	2	no	—
30	C₂×D₄×C₄₃	—	2	1	no	—
31	C₂×D₄×C₄₃	—	2	1	no	—
32	C₂×D₄×C₄₃	D₄	1	1	yes	characteristic
33	C₂×D₄×C₄₃	D₄	1	2	yes	—
34	C₂²×C₄×C₄₃	—	2	2	no	—
35	C₂²×C₄×C₄₃	—	2	1	no	—
36	C₂×D₁₇₂	—	2	1	no	—
37	C₂×D₁₇₂	—	2	1	no	—
38	C₂×D₁₇₂	—	2	1	no	—
39	C₄⋊C₄×C₄₃	—	2	1	no	—
40	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
42	C₂²⋊C₄×C₄₃	—	2	2	no	—
43	C₂²⋊C₄×C₄₃	—	2	1	no	—
44	C₄²×C₄₃	—	2	1	no	—
45	C₄²×C₄₃	C₂³	1	1	yes	characteristic
46	#10621884	—	4	2	no	—
47	#10621882	—	4	4	no	—
48	#10621882	—	4	2	no	—
49	C₂×C₄₃⋊C₈	—	2	1	no	—
50	C₂×C₄₃⋊C₈	—	2	1	no	—
51	C₂×C₄₃⋊C₈	—	2	1	no	—
52	C₂³×C₄₃	—	2	1	no	—
53	C₂³×C₄₃	—	2	2	no	—
54	C₂²×D₄₃	—	4	1	no	—
55	Q₈×C₄₃	—	2	1	no	—
56	Q₈×C₄₃	D₈	1	2	yes	complemented
57	D₄×C₄₃	—	2	1	no	—
58	D₄×C₄₃	—	2	1	no	—
59	D₄×C₄₃	—	4	2	no	—
60	D₄×C₄₃	—	4	4	no	—
61	D₄×C₄₃	—	4	1	no	—
62	D₄×C₄₃	—	4	1	no	—
63	D₄×C₄₃	—	4	1	no	—
64	D₄×C₄₃	—	4	1	no	—
65	D₄×C₄₃	—	8	2	no	—
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	1	no	—
68	C₂×C₄×C₄₃	—	2	1	no	—
69	C₂×C₄×C₄₃	—	4	4	no	—
70	C₂×C₄×C₄₃	—	4	1	no	—
71	C₂×C₄×C₄₃	—	4	1	no	—
72	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	D₁₇₂	—	4	2	no	—
76	D₁₇₂	—	4	1	no	—
77	D₁₇₂	—	4	1	no	—
78	D₁₇₂	—	4	1	no	—
79	D₁₇₂	—	4	1	no	—
80	C₄₃⋊C₈	—	2	2	no	—
81	C₄₃⋊C₈	—	4	1	no	—
82	C₄₃⋊C₈	—	4	1	no	—
83	C₂²×C₄₃	—	4	2	no	—
84	C₂²×C₄₃	—	4	1	no	—
85	C₂²×C₄₃	—	4	1	no	—
86	C₂²×C₄₃	—	4	2	no	—
87	C₂²×C₄₃	—	4	2	no	—
88	C₂²×C₄₃	—	4	1	no	—
89	C₂²×C₄₃	#120	1	1	yes	characteristic
90	C₂×D₄₃	—	8	1	no	—
91	C₂×D₄₃	—	8	1	no	—
92	C₂×D₄₃	—	8	1	no	—
93	C₄×C₄₃	—	2	2	no	—
94	C₄×C₄₃	—	2	1	no	—
95	C₄×C₄₃	—	4	1	no	—
96	C₄×C₄₃	—	4	1	no	—
97	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
98	C₄×C₄₃	#135	1	1	yes	characteristic
99	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	#1621	—	43	1	no	maximal, Sylow
102	C₂×C₄₃	—	8	1	no	—
103	C₂×C₄₃	—	8	2	no	—
104	C₂×C₄₃	#447	1	1	yes	characteristic
105	C₂×C₄₃	#449	1	1	yes	characteristic
106	C₂×C₄₃	#453	1	1	yes	characteristic
107	D₄₃	—	16	1	no	—
108	#550	D₄₃	1	1	yes	characteristic, complemented
109	#493	—	43	1	no	—
110	#460	—	43	1	no	—
111	#459	—	43	1	no	—
112	#331	—	43	1	no	—
113	Q₈⋊C₈	—	43	1	no	—
114	D₄⋊C₈	—	43	1	no	—
115	C₄₃	#1621	1	1	yes	characteristic, complemented, Sylow
116	C₂×#41	—	2	1	no	—
117	C₂×SD₁₆	—	86	1	no	—
118	C₂×D₈	—	86	1	no	—
119	C₂×D₈	—	86	2	no	—
120	#127	—	2	1	no	—
121	#127	—	43	1	no	—
122	#127	C₂×D₄₃	1	1	yes	characteristic
123	#121	—	2	2	no	—
124	#121	—	2	1	no	—
125	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
126	C₄×D₄	—	2	1	no	—
127	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
128	C₄⋊C₈	—	43	1	no	—
129	C₄⋊C₈	—	43	1	no	—
130	#102	—	86	1	no	—
131	#102	—	86	1	no	—
132	C₄×C₈	—	43	1	no	—
133	#41	—	4	2	no	—
134	#41	—	4	2	no	—
135	C₂×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	1	no	—
139	C₂×D₄	—	2	1	no	—
140	C₂×D₄	—	86	1	no	—
141	C₂×D₄	—	86	1	no	—
142	C₂×D₄	—	86	1	no	—
143	C₂×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
144	C₂×D₄	C₂²⋊D₄₃	1	2	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	1	no	—
147	SD₁₆	—	172	2	no	—
148	D₈	—	172	4	no	—
149	D₈	—	172	2	no	—
150	C₂×C₈	—	86	1	no	—
151	C₂×C₈	—	86	1	no	—
152	C₂×C₈	—	86	1	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
155	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	C₂²×D₄₃	1	1	yes	characteristic
160	C₂³	—	2	1	no	—
161	C₂³	—	2	2	no	—
162	C₂³	—	172	1	no	—
163	Q₈	—	2	1	no	—
164	Q₈	#10621882	1	2	yes	complemented
165	D₄	—	2	1	no	—
166	D₄	—	2	1	no	—
167	D₄	—	4	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	4	1	no	—
170	D₄	—	4	1	no	—
171	D₄	—	4	1	no	—
172	D₄	—	4	1	no	—
173	D₄	—	8	2	no	—
174	D₄	—	172	2	no	—
175	D₄	—	172	1	no	—
176	D₄	—	172	1	no	—
177	D₄	—	172	1	no	—
178	D₄	—	172	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
188	C₈	—	86	2	no	—
189	C₈	—	172	1	no	—
190	C₈	—	172	1	no	—
191	C₂²	—	4	1	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	—	4	1	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	344	1	no	—
198	C₂²	—	344	1	no	—
199	C₂²	—	344	1	no	—
200	C₂²	#13063294	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	1	no	—
204	C₄	—	4	1	no	—
205	C₄	C₂×#10621882	1	1	yes	characteristic
206	C₄	#13063345	1	1	yes	characteristic
207	C₄	#13063350	1	1	yes	characteristic
208	C₄	#13063351	1	1	yes	characteristic
209	C₂	—	8	1	no	—
210	C₂	—	8	2	no	—
211	C₂	—	688	1	no	—
212	C₂	#14583033	1	1	yes	characteristic
213	C₂	#14583044	1	1	yes	characteristic
214	C₂	#14583067	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583275

C₂

yes

maximal, characteristic, complemented

#14583099

C₂

yes

maximal, characteristic, complemented

#14583094

C₂

yes

maximal, characteristic, complemented

#14582643

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₄₃×#550

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

#13063245

—

#13063245

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×#41×C₄₃

—

#127×C₄₃

—

#127×C₄₃

C₂²

yes

characteristic

#121×C₄₃

—

#121×C₄₃

—

C₄×Q₈×C₄₃

C₂²

yes

characteristic

C₄×D₄×C₄₃

—

C₄×D₄×C₄₃

C₂²

yes

characteristic

#41×C₄₃

—

#41×C₄₃

—

C₂×Q₈×C₄₃

D₄

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

D₄

yes

characteristic

C₂×D₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂³

yes

—

—

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

D₈

yes

complemented

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#120

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₈

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

100

C₄×C₄₃

yes

101

—

102

C₂×C₄₃

—

103

C₂×C₄₃

—

104

C₂×C₄₃

#447

yes

characteristic

105

C₂×C₄₃

#449

yes

characteristic

106

C₂×C₄₃

#453

yes

characteristic

107

D₄₃

—

108

#550

D₄₃

yes

characteristic, complemented

109

#493

—

110

#460

—

111

#459

—

112

#331

—

113

Q₈⋊C₈

—

114

D₄⋊C₈

—

115

C₄₃

#1621

yes

characteristic, complemented, Sylow

116

C₂×#41

—

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₂×D₈

—

120

#127

—

121

#127

—

122

#127

C₂×D₄₃

yes

characteristic

123

#121

—

124

#121

—

125

C₄×Q₈

C₂×D₄₃

yes

characteristic

126

C₄×D₄

—

127

C₄×D₄

C₂×D₄₃

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

130

#102

—

131

#102

—

132

C₄×C₈

—

133

#41

—

134

#41

—

135

C₂×Q₈

C₂²⋊D₄₃

yes

characteristic

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

C₂²⋊D₄₃

yes

characteristic

144

C₂×D₄

C₂²⋊D₄₃

yes

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

SD₁₆

—

172

—

148

D₈

—

172

—

149

D₈

—

172

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₄⋊C₄

—

154

C₄⋊C₄

C₂²⋊D₄₃

yes

characteristic

155

C₄⋊C₄

C₂²⋊D₄₃

yes

characteristic

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

—

159

C₄²

C₂²×D₄₃

yes

characteristic

160

C₂³

—

161

C₂³

—

162

C₂³

—

172

—

163

Q₈

—

164

Q₈

#10621882

yes

complemented

165

D₄

—

166

D₄

—

167

D₄

—

168

D₄

—

169

D₄

—

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

174

D₄

—

172

—

175

D₄

—

172

—

176

D₄

—

172

—

177

D₄

—

172

—

178

D₄

—

172

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

186

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

187

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

188

C₈

—

189

C₈

—

172

—

190

C₈

—

172

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

344

—

198

C₂²

—

344

—

199

C₂²

—

344

—

200

C₂²

#13063294

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

C₂×#10621882

yes

characteristic

206

C₄

#13063345

yes

characteristic

207

C₄

#13063350

yes

characteristic

208

C₄

#13063351

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

—

688

—

212

C₂

#14583033

yes

characteristic

213

C₂

#14583044

yes

characteristic

214

C₂

#14583067

yes

characteristic

	subgroup	G/H
center	C₂²	#13063294
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#550	C₂
socle	C₂²×C₄₃	#120
hypercenter	#550	D₄₃
hypocenter	C₄₃	#1621
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #550

subgroup

G/H

center

C₂²

#13063294

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#550

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #550

N	H ≅ G/N	type
1. #14583275	C₂	semidirect
2. #14583099	C₂	semidirect
3. #14583094	C₂	semidirect
4. #14582643	C₂	semidirect
5. Q₈⋊(C₄₃⋊C₈)	C₂	semidirect
6. D₄⋊(C₄₃⋊C₈)	C₂	semidirect
7. C₄₃×#550	C₂	semidirect
56. Q₈×C₄₃	D₈	semidirect
108. #550	D₄₃	semidirect
115. C₄₃	#1621	semidirect
164. Q₈	#10621882	semidirect

H ≅ G/N

type

1. #14583275

C₂

semidirect

2. #14583099

C₂

semidirect

3. #14583094

C₂

semidirect

4. #14582643

C₂

semidirect

5. Q₈⋊(C₄₃⋊C₈)

C₂

semidirect

6. D₄⋊(C₄₃⋊C₈)

C₂

semidirect

7. C₄₃×#550

C₂

semidirect

56. Q₈×C₄₃

D₈

semidirect

108. #550

D₄₃

semidirect

115. C₄₃

#1621

semidirect

164. Q₈

#10621882

semidirect

Subgroups — #19133660

Conjugacy classes of subgroups

Complements