Group #19133243 — subgroups

Subgroups — #19133243

Order 5504.

256 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 73 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	#14583450	C₂	1	1	yes	maximal, characteristic
2	#14582951	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₈×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582645	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582642	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582642	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621902	—	2	1	no	—
9	C₂×C₈×D₄₃	—	2	1	no	—
10	#13063322	C₂²	1	1	yes	characteristic
11	#13063314	—	2	1	no	—
12	#13063300	—	2	1	no	—
13	#13063300	C₄	1	2	yes	—
14	#13063287	—	2	1	no	—
15	#13063287	C₄	1	2	yes	—
16	C₄²×D₄₃	C₂²	1	1	yes	characteristic
17	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	Dic₄₃⋊C₈	—	2	1	no	—
20	(C₂×D₄₃)⋊C₈	—	2	1	no	—
21	C₈×Dic₄₃	—	2	1	no	—
22	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₃	—	2	1	no	—
26	#10621902	—	4	2	no	—
27	#10621902	—	4	2	no	—
28	C₂×D₁₇₂	—	2	1	no	—
29	C₂×D₁₇₂	—	2	1	no	—
30	C₂×D₁₇₂	—	2	2	no	—
31	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₄₃	—	2	2	no	—
33	C₂×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
35	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₄⋊C₄×C₄₃	—	2	1	no	—
39	C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
40	C₄²×C₄₃	C₂³	1	1	yes	characteristic
41	(C₂×D₄₃)⋊C₄	—	2	2	no	—
42	(C₂×D₄₃)⋊C₄	—	2	2	no	—
43	C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₄₃	D₄	1	1	yes	characteristic
45	C₄×Dic₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄₃⋊C₈	—	2	1	no	—
47	C₂×C₄₃⋊C₈	—	2	1	no	—
48	C₈×D₄₃	—	4	2	no	—
49	C₂²×D₄₃	—	2	2	no	—
50	C₂²×D₄₃	C₂²⋊C₄	1	1	yes	characteristic
51	Q₈×C₄₃	—	4	1	no	—
52	Q₈×C₄₃	—	4	1	no	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
60	D₁₇₂	—	4	2	no	—
61	D₁₇₂	—	4	4	no	—
62	D₁₇₂	—	4	1	no	—
63	D₁₇₂	—	4	1	no	—
64	D₁₇₂	—	4	1	no	—
65	D₁₇₂	—	4	1	no	—
66	C₄×D₄₃	—	2	1	no	—
67	C₄×D₄₃	—	2	1	no	—
68	C₄×D₄₃	—	4	1	no	—
69	C₄×D₄₃	—	4	4	no	—
70	C₄×D₄₃	D₈	1	2	yes	—
71	C₄×D₄₃	SD₁₆	1	2	yes	—
72	C₈×C₄₃	—	4	1	no	—
73	C₈×C₄₃	—	4	1	no	—
74	C₄₃⋊C₈	—	4	1	no	—
75	C₄₃⋊C₈	—	4	1	no	—
76	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×D₄₃	—	2	1	no	—
78	C₂×D₄₃	—	2	1	no	—
79	C₂×D₄₃	—	4	2	no	—
80	C₂×D₄₃	—	4	2	no	—
81	C₂×D₄₃	—	4	2	no	—
82	C₂×D₄₃	#102	1	2	yes	—
83	C₄×C₄₃	—	2	1	no	—
84	C₄×C₄₃	—	4	2	no	—
85	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
86	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₄×C₄₃	#135	1	1	yes	characteristic
88	C₄×C₄₃	#135	1	1	yes	characteristic
89	Dic₄₃	—	2	1	no	—
90	Dic₄₃	—	2	1	no	—
91	Dic₄₃	#102	1	2	yes	—
92	#1540	—	43	1	no	maximal, Sylow
93	C₂×C₄₃	#413	1	1	yes	characteristic
94	C₂×C₄₃	C₂×#102	1	1	yes	characteristic
95	C₂×C₄₃	#416	1	1	yes	characteristic
96	D₄₃	—	2	2	no	—
97	D₄₃	—	8	2	no	—
98	#532	—	43	1	no	—
99	#431	—	43	1	no	—
100	C₂×C₄⋊C₈	—	43	1	no	—
101	#332	—	43	1	no	—
102	#332	D₄₃	1	1	yes	characteristic, complemented
103	#331	—	43	1	no	—
104	#331	—	43	1	no	—
105	C₄₃	#1540	1	1	yes	characteristic, complemented, Sylow
106	C₂×#41	—	86	1	no	—
107	C₂²×C₈	—	86	1	no	—
108	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
109	#127	—	43	1	no	—
110	#124	—	86	1	no	—
111	#121	—	43	2	no	—
112	#121	—	86	1	no	—
113	C₄×D₄	—	43	2	no	—
114	C₄×D₄	—	86	1	no	—
115	C₂×C₄²	—	43	1	no	—
116	C₄⋊C₈	—	43	1	no	—
117	C₄⋊C₈	—	43	1	no	—
118	C₄⋊C₈	—	86	1	no	—
119	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
120	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
121	C₂²⋊C₈	—	86	1	no	—
122	C₄×C₈	—	86	1	no	—
123	#41	—	172	2	no	—
124	#41	—	172	2	no	—
125	C₂×Q₈	—	2	1	no	—
126	C₂×D₄	—	43	2	no	—
127	C₂×D₄	—	86	2	no	—
128	C₂×D₄	—	86	1	no	—
129	C₂×D₄	—	86	1	no	—
130	C₂²×C₄	—	43	1	no	—
131	C₂²×C₄	—	43	1	no	—
132	C₂²×C₄	—	43	1	no	—
133	C₂²×C₄	—	86	2	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	86	1	no	—
137	C₂×C₈	—	86	1	no	—
138	C₂×C₈	—	172	2	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	C₄×D₄₃	1	2	yes	—
141	C₂²⋊C₄	—	86	2	no	—
142	C₂²⋊C₄	—	86	2	no	—
143	C₄²	—	43	1	no	—
144	C₄²	—	43	1	no	—
145	C₄²	—	43	1	no	—
146	C₄²	C₂²×D₄₃	1	1	yes	characteristic
147	C₂³	—	43	1	no	—
148	C₂³	—	86	2	no	—
149	Q₈	—	4	1	no	—
150	Q₈	—	4	1	no	—
151	D₄	—	172	2	no	—
152	D₄	—	172	4	no	—
153	D₄	—	172	1	no	—
154	D₄	—	172	1	no	—
155	D₄	—	172	1	no	—
156	D₄	—	172	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	43	1	no	—
159	C₂×C₄	—	43	2	no	—
160	C₂×C₄	—	43	2	no	—
161	C₂×C₄	—	43	1	no	—
162	C₂×C₄	—	43	1	no	—
163	C₂×C₄	—	86	1	no	—
164	C₂×C₄	—	86	1	no	—
165	C₂×C₄	—	172	1	no	—
166	C₂×C₄	—	172	4	no	—
167	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
168	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
169	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
170	C₈	—	4	1	no	—
171	C₈	—	4	1	no	—
172	C₈	—	172	1	no	—
173	C₈	—	172	1	no	—
174	C₂²	—	43	2	no	—
175	C₂²	—	86	1	no	—
176	C₂²	—	86	1	no	—
177	C₂²	—	172	2	no	—
178	C₂²	—	172	2	no	—
179	C₂²	—	172	2	no	—
180	C₂²	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	43	2	no	—
184	C₄	—	86	1	no	—
185	C₄	—	86	1	no	—
186	C₄	D₈×D₄₃	1	1	yes	characteristic
187	C₄	SD₁₆×D₄₃	1	1	yes	characteristic
188	C₄	#13063346	1	1	yes	characteristic
189	C₄	#13063348	1	1	yes	characteristic
190	C₂	—	86	2	no	—
191	C₂	—	344	2	no	—
192	C₂	#14582882	1	1	yes	characteristic
193	C₂	#102×D₄₃	1	1	yes	characteristic
194	C₂	#14582894	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583450

C₂

yes

maximal, characteristic

#14582951

C₂

yes

maximal, characteristic, complemented

C₄⋊C₈×D₄₃

C₂

yes

maximal, characteristic, complemented

#14582645

C₂

yes

maximal, characteristic, complemented

#14582642

C₂

yes

maximal, characteristic, complemented

#14582642

C₂

yes

maximal, characteristic, complemented

C₄₃×#332

C₂

yes

maximal, characteristic, complemented

C₂×#10621902

—

C₂×C₈×D₄₃

—

#13063322

C₂²

yes

—

—

C₄

yes

—

#13063287

—

#13063287

C₄

yes

—

C₄²×D₄₃

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

Dic₄₃⋊C₈

—

(C₂×D₄₃)⋊C₈

—

C₈×Dic₄₃

—

C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

#10621902

—

#10621902

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

C₂×C₄

yes

characteristic

C₂×C₄×D₄₃

D₄

yes

characteristic

C₂×C₄×D₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄×Dic₄₃

C₂×C₄

yes

characteristic

C₄×Dic₄₃

D₄

yes

characteristic

C₄×Dic₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₈×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂²⋊C₄

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₃

C₂²⋊C₄

yes

characteristic

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

D₈

yes

—

C₄×D₄₃

SD₁₆

yes

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

#102

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₈

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

#102

yes

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

D₄₃

—

#532

—

#431

—

100

C₂×C₄⋊C₈

—

101

#332

—

102

#332

D₄₃

yes

characteristic, complemented

103

#331

—

104

#331

—

105

C₄₃

#1540

yes

characteristic, complemented, Sylow

106

C₂×#41

—

107

C₂²×C₈

—

108

C₄⋊Q₈

C₂×D₄₃

yes

characteristic

109

#127

—

110

#124

—

111

#121

—

112

#121

—

113

C₄×D₄

—

114

C₄×D₄

—

115

C₂×C₄²

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

C₂×D₄₃

yes

characteristic

120

C₄⋊C₈

C₂×D₄₃

yes

characteristic

121

C₂²⋊C₈

—

122

C₄×C₈

—

123

#41

—

172

—

124

#41

—

172

—

125

C₂×Q₈

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

172

—

139

C₄⋊C₄

—

140

C₄⋊C₄

C₄×D₄₃

yes

—

141

C₂²⋊C₄

—

142

C₂²⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂²×D₄₃

yes

characteristic

147

C₂³

—

148

C₂³

—

149

Q₈

—

150

Q₈

—

151

D₄

—

172

—

152

D₄

—

172

—

153

D₄

—

172

—

154

D₄

—

172

—

155

D₄

—

172

—

156

D₄

—

172

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

172

—

166

C₂×C₄

—

172

—

167

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

168

C₂×C₄

D₄×D₄₃

yes

characteristic

169

C₂×C₄

D₄×D₄₃

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

—

172

—

173

C₈

—

172

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

172

—

178

C₂²

—

172

—

179

C₂²

—

172

—

180

C₂²

C₂²⋊C₄×D₄₃

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

D₈×D₄₃

yes

characteristic

187

C₄

SD₁₆×D₄₃

yes

characteristic

188

C₄

#13063346

yes

characteristic

189

C₄

#13063348

yes

characteristic

190

C₂

—

191

C₂

—

344

—

192

C₂

#14582882

yes

characteristic

193

C₂

#102×D₄₃

yes

characteristic

194

C₂

#14582894

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#332	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#332	D₄₃
hypocenter	C₄₃	#1540
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#332

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
2. #14582951	C₂	semidirect
3. C₄⋊C₈×D₄₃	C₂	semidirect
4. #14582645	C₂	semidirect
5. #14582642	C₂	semidirect
6. #14582642	C₂	semidirect
7. C₄₃×#332	C₂	semidirect
102. #332	D₄₃	semidirect
105. C₄₃	#1540	semidirect

H ≅ G/N

type

2. #14582951

C₂

semidirect

3. C₄⋊C₈×D₄₃

C₂

semidirect

4. #14582645

C₂

semidirect

5. #14582642

C₂

semidirect

6. #14582642

C₂

semidirect

7. C₄₃×#332

C₂

semidirect

102. #332

D₄₃

semidirect

105. C₄₃

#1540

semidirect

Subgroups — #19133243

Conjugacy classes of subgroups

Complements