Group #19133720 — subgroups

Subgroups — #19133720

Order 5504.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14583571	C₂	1	1	yes	maximal, characteristic
2	#14583273	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583186	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582642	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582626	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621902	—	2	1	no	—
9	C₂×#10621884	—	2	2	no	—
10	#13063322	—	2	1	no	—
11	#13063322	C₂²	1	1	yes	characteristic
12	#13063300	—	2	2	no	—
13	#13063300	—	2	1	no	—
14	C₄×D₁₇₂	—	2	1	no	—
15	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
16	#13063256	—	2	1	no	—
17	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
18	#13063245	—	2	1	no	—
19	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
22	#107×C₄₃	—	2	1	no	—
23	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	#103×C₄₃	—	2	1	no	—
25	C₂×Q₈×C₄₃	—	2	1	no	—
26	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
27	#10621902	—	4	2	no	—
28	#10621902	—	4	2	no	—
29	C₂×D₁₇₂	—	2	2	no	—
30	C₂×D₁₇₂	—	2	1	no	—
31	C₂×D₁₇₂	—	2	1	no	—
32	C₂×D₁₇₂	—	2	2	no	—
33	C₂×D₁₇₂	D₄	1	1	yes	characteristic
34	C₂×D₁₇₂	D₄	1	2	yes	—
35	C₂×C₄×D₄₃	—	2	2	no	—
36	C₂×C₄×D₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₄⋊C₄×C₄₃	—	2	1	no	—
39	C₄⋊C₄×C₄₃	—	2	1	no	—
40	C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
42	C₄²×C₄₃	—	2	1	no	—
43	C₄²×C₄₃	C₂³	1	1	yes	characteristic
44	#10621884	—	4	4	no	—
45	(C₂×D₄₃)⋊C₄	—	2	2	no	—
46	(C₂×D₄₃)⋊C₄	—	2	1	no	—
47	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄₃⋊C₈	—	2	1	no	—
49	C₂×C₄₃⋊C₈	—	2	1	no	—
50	C₂²×D₄₃	—	2	1	no	—
51	C₂²×D₄₃	—	2	2	no	—
52	Q₈×C₄₃	—	2	1	no	—
53	Q₈×C₄₃	—	4	2	no	—
54	Q₈×C₄₃	SD₁₆	1	2	yes	complemented
55	C₂×C₄×C₄₃	—	2	1	no	—
56	C₂×C₄×C₄₃	—	2	1	no	—
57	C₂×C₄×C₄₃	—	4	1	no	—
58	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₄₃	—	2	1	no	—
62	D₁₇₂	—	2	1	no	—
63	D₁₇₂	—	2	1	no	—
64	D₁₇₂	—	4	2	no	—
65	D₁₇₂	—	4	4	no	—
66	D₁₇₂	—	4	1	no	—
67	D₁₇₂	—	4	1	no	—
68	D₁₇₂	—	4	1	no	—
69	D₁₇₂	—	4	1	no	—
70	D₁₇₂	—	8	2	no	—
71	C₄×D₄₃	—	4	4	no	—
72	C₄×D₄₃	—	4	1	no	—
73	C₄×D₄₃	—	4	1	no	—
74	C₈×C₄₃	—	4	1	no	—
75	C₄₃⋊C₈	—	2	2	no	—
76	C₄₃⋊C₈	—	4	1	no	—
77	C₂²×C₄₃	#120	1	1	yes	characteristic
78	C₂×D₄₃	—	4	2	no	—
79	C₂×D₄₃	—	4	1	no	—
80	C₂×D₄₃	—	4	1	no	—
81	C₂×D₄₃	—	4	2	no	—
82	C₂×D₄₃	—	4	2	no	—
83	C₂×D₄₃	—	4	1	no	—
84	C₄×C₄₃	—	2	2	no	—
85	C₄×C₄₃	—	2	1	no	—
86	C₄×C₄₃	—	4	1	no	—
87	C₄×C₄₃	—	8	1	no	—
88	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₄×C₄₃	#135	1	1	yes	characteristic
90	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₄×C₄₃	#137	1	1	yes	characteristic
92	Dic₄₃	—	4	1	no	—
93	#1631	—	43	1	no	maximal, Sylow
94	C₂×C₄₃	#450	1	1	yes	characteristic
95	C₂×C₄₃	#452	1	1	yes	characteristic
96	C₂×C₄₃	#453	1	1	yes	characteristic
97	D₄₃	—	8	1	no	—
98	D₄₃	—	8	2	no	—
99	#550	—	43	1	no	—
100	#492	—	43	1	no	—
101	#477	D₄₃	1	1	yes	characteristic, complemented
102	#476	—	43	1	no	—
103	#331	—	43	1	no	—
104	Q₈⋊C₈	—	43	1	no	—
105	D₄⋊C₈	—	43	1	no	—
106	C₄₃	#1631	1	1	yes	characteristic, complemented, Sylow
107	C₂×#41	—	86	1	no	—
108	C₂×SD₁₆	—	86	2	no	—
109	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
110	#127	—	43	1	no	—
111	#127	—	86	1	no	—
112	#121	—	86	2	no	—
113	#121	—	86	1	no	—
114	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
115	C₄×D₄	—	43	1	no	—
116	C₄×D₄	—	86	1	no	—
117	#107	—	2	1	no	—
118	#106	—	86	1	no	—
119	C₄⋊C₈	—	43	1	no	—
120	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
121	#103	—	2	1	no	—
122	#102	—	86	1	no	—
123	C₄×C₈	—	43	1	no	—
124	#41	—	172	2	no	—
125	#41	—	172	2	no	—
126	C₂×Q₈	—	2	1	no	—
127	C₂×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
128	C₂×D₄	—	43	1	no	—
129	C₂×D₄	—	43	2	no	—
130	C₂×D₄	—	86	2	no	—
131	C₂×D₄	—	86	2	no	—
132	C₂×D₄	—	86	1	no	—
133	C₂×D₄	—	86	1	no	—
134	C₂²×C₄	—	86	2	no	—
135	C₂²×C₄	—	86	1	no	—
136	SD₁₆	—	172	4	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	86	1	no	—
139	C₂×C₈	—	86	1	no	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	2	1	no	—
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	43	1	no	—
144	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
145	C₂²⋊C₄	—	86	2	no	—
146	C₂²⋊C₄	—	86	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	C₂²×D₄₃	1	1	yes	characteristic
149	C₂³	—	86	1	no	—
150	C₂³	—	86	2	no	—
151	Q₈	—	2	1	no	—
152	Q₈	—	4	2	no	—
153	Q₈	#10621884	1	2	yes	complemented
154	D₄	—	86	1	no	—
155	D₄	—	86	1	no	—
156	D₄	—	344	2	no	—
157	D₄	—	172	2	no	—
158	D₄	—	172	4	no	—
159	D₄	—	172	1	no	—
160	D₄	—	172	1	no	—
161	D₄	—	172	1	no	—
162	D₄	—	172	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	86	1	no	—
167	C₂×C₄	—	172	4	no	—
168	C₂×C₄	—	172	1	no	—
169	C₂×C₄	—	172	1	no	—
170	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
171	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
172	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
173	C₈	—	4	1	no	—
174	C₈	—	86	2	no	—
175	C₈	—	172	1	no	—
176	C₂²	—	172	2	no	—
177	C₂²	—	172	1	no	—
178	C₂²	—	172	1	no	—
179	C₂²	—	172	2	no	—
180	C₂²	—	172	1	no	—
181	C₂²	—	172	2	no	—
182	C₂²	#13063293	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	1	no	—
187	C₄	—	172	1	no	—
188	C₄	C₂×#10621884	1	1	yes	characteristic
189	C₄	#13063348	1	1	yes	characteristic
190	C₄	#13063350	1	1	yes	characteristic
191	C₄	#13063357	1	1	yes	characteristic
192	C₂	—	344	1	no	—
193	C₂	—	344	2	no	—
194	C₂	#14583049	1	1	yes	characteristic
195	C₂	#14583060	1	1	yes	characteristic
196	C₂	#14583065	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583571

C₂

yes

maximal, characteristic

#14583273

C₂

yes

maximal, characteristic, complemented

#14583186

C₂

yes

maximal, characteristic, complemented

#14582642

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

#14582626

C₂

yes

maximal, characteristic, complemented

C₄₃×#477

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#13063300

—

#13063300

—

C₄×D₁₇₂

—

C₄×D₁₇₂

C₂²

yes

characteristic

#13063256

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

#13063245

—

C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄×Q₈×C₄₃

C₂²

yes

characteristic

#107×C₄₃

—

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

#103×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

D₄

yes

characteristic

#10621902

—

#10621902

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

D₄

yes

characteristic

C₂×D₁₇₂

D₄

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

C₂³

yes

characteristic

#10621884

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂²×D₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

SD₁₆

yes

complemented

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

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

#120

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×SD₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

D₄₃

—

#550

—

100

#492

—

101

#477

D₄₃

yes

characteristic, complemented

102

#476

—

103

#331

—

104

Q₈⋊C₈

—

105

D₄⋊C₈

—

106

C₄₃

#1631

yes

characteristic, complemented, Sylow

107

C₂×#41

—

108

C₂×SD₁₆

—

109

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

110

#127

—

111

#127

—

112

#121

—

113

#121

—

114

C₄×Q₈

C₂×D₄₃

yes

characteristic

115

C₄×D₄

—

116

C₄×D₄

—

117

#107

—

118

#106

—

119

C₄⋊C₈

—

120

C₄⋊C₈

C₂×D₄₃

yes

characteristic

121

#103

—

122

#102

—

123

C₄×C₈

—

124

#41

—

172

—

125

#41

—

172

—

126

C₂×Q₈

—

127

C₂×Q₈

C₂²⋊D₄₃

yes

characteristic

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

SD₁₆

—

172

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

C₂²⋊D₄₃

yes

characteristic

145

C₂²⋊C₄

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₄²

C₂²×D₄₃

yes

characteristic

149

C₂³

—

150

C₂³

—

151

Q₈

—

152

Q₈

—

153

Q₈

#10621884

yes

complemented

154

D₄

—

155

D₄

—

156

D₄

—

344

—

157

D₄

—

172

—

158

D₄

—

172

—

159

D₄

—

172

—

160

D₄

—

172

—

161

D₄

—

172

—

162

D₄

—

172

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

172

—

168

C₂×C₄

—

172

—

169

C₂×C₄

—

172

—

170

C₂×C₄

D₄×D₄₃

yes

characteristic

171

C₂×C₄

D₄×D₄₃

yes

characteristic

172

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

173

C₈

—

174

C₈

—

175

C₈

—

172

—

176

C₂²

—

172

—

177

C₂²

—

172

—

178

C₂²

—

172

—

179

C₂²

—

172

—

180

C₂²

—

172

—

181

C₂²

—

172

—

182

C₂²

#13063293

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

172

—

188

C₄

C₂×#10621884

yes

characteristic

189

C₄

#13063348

yes

characteristic

190

C₄

#13063350

yes

characteristic

191

C₄

#13063357

yes

characteristic

192

C₂

—

344

—

193

C₂

—

344

—

194

C₂

#14583049

yes

characteristic

195

C₂

#14583060

yes

characteristic

196

C₂

#14583065

yes

characteristic

	subgroup	G/H
center	C₂²	#13063293
commutator	C₄²×C₄₃	C₂³
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#477	C₂
socle	C₂²×C₄₃	#120
hypercenter	#477	D₄₃
hypocenter	C₄₃	#1631
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#13063293

commutator

C₄²×C₄₃

C₂³

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#477

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
2. #14583273	C₂	semidirect
3. #14583186	C₂	semidirect
4. #14582642	C₂	semidirect
5. Q₈⋊(C₄₃⋊C₈)	C₂	semidirect
6. #14582626	C₂	semidirect
7. C₄₃×#477	C₂	semidirect
54. Q₈×C₄₃	SD₁₆	semidirect
101. #477	D₄₃	semidirect
106. C₄₃	#1631	semidirect
153. Q₈	#10621884	semidirect

H ≅ G/N

type

2. #14583273

C₂

semidirect

3. #14583186

C₂

semidirect

4. #14582642

C₂

semidirect

5. Q₈⋊(C₄₃⋊C₈)

C₂

semidirect

6. #14582626

C₂

semidirect

7. C₄₃×#477

C₂

semidirect

54. Q₈×C₄₃

SD₁₆

semidirect

101. #477

D₄₃

semidirect

106. C₄₃

#1631

semidirect

153. Q₈

#10621884

semidirect

Subgroups — #19133720

Conjugacy classes of subgroups

Complements