Group #19136096 — subgroups

Subgroups — #19136096

Order 5504.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂×#13063256	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14582797	C₂	1	1	yes	maximal, characteristic
4	#14582661	C₂	1	1	yes	maximal, characteristic
5	#14582661	C₂	1	1	yes	maximal, characteristic
6	#14582661	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#389	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₃⋊C₄	—	2	1	no	—
13	#13063256	—	2	2	no	—
14	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
15	C₄₃⋊#95	—	2	1	no	—
16	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
17	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
20	#95×C₄₃	—	2	1	no	—
21	C₂²×C₄×C₄₃	—	2	1	no	—
22	C₂²×C₄×C₄₃	—	2	1	no	—
23	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₃	—	2	1	no	—
27	C₂²×Dic₄₃	—	2	1	no	—
28	C₄⋊C₄×C₄₃	—	2	1	no	—
29	C₄⋊C₄×C₄₃	—	2	1	no	—
30	C₄⋊C₄×C₄₃	—	2	1	no	—
31	C₄⋊C₄×C₄₃	—	4	2	no	—
32	C₄⋊C₄×C₄₃	Q₈	1	2	yes	—
33	C₄²×C₄₃	—	2	1	no	—
34	C₄²×C₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	—	2	1	no	—
36	C₄⋊Dic₄₃	—	2	1	no	—
37	C₄⋊Dic₄₃	—	2	1	no	—
38	C₄⋊Dic₄₃	Q₈	1	2	yes	—
39	Dic₄₃⋊C₄	—	4	2	no	—
40	C₂×C₄₃⋊C₈	—	2	1	no	—
41	C₂×C₄₃⋊C₈	—	2	1	no	—
42	C₂×C₄₃⋊C₈	—	2	1	no	—
43	C₂×C₄₃⋊C₈	—	2	1	no	—
44	C₂×C₄₃⋊C₈	—	2	1	no	—
45	C₂×C₄₃⋊C₈	D₄	1	2	yes	—
46	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	—	2	1	no	—
48	C₂×C₄×C₄₃	—	2	1	no	—
49	C₂×C₄×C₄₃	—	2	1	no	—
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	—	4	1	no	—
54	C₂×C₄×C₄₃	—	4	1	no	—
55	C₂×C₄×C₄₃	—	4	1	no	—
56	C₂×C₄×C₄₃	—	4	1	no	—
57	C₂×C₄×C₄₃	—	4	1	no	—
58	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
59	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
65	C₂×Dic₄₃	—	2	2	no	—
66	C₂×Dic₄₃	—	4	1	no	—
67	C₂×Dic₄₃	—	4	1	no	—
68	C₂×Dic₄₃	—	4	1	no	—
69	C₂×Dic₄₃	—	4	1	no	—
70	C₂×Dic₄₃	—	4	1	no	—
71	C₄₃⋊C₈	—	2	2	no	—
72	C₄₃⋊C₈	—	4	1	no	—
73	C₂²×C₄₃	#121	1	1	yes	characteristic
74	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂²×C₄₃	#135	1	1	yes	characteristic
78	C₂²×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂²×C₄₃	#137	1	1	yes	characteristic
80	C₄×C₄₃	—	2	2	no	—
81	C₄×C₄₃	—	4	1	no	—
82	C₄×C₄₃	—	4	2	no	—
83	C₄×C₄₃	—	8	1	no	—
84	C₄×C₄₃	#123	1	1	yes	characteristic
85	C₄×C₄₃	#125	1	1	yes	characteristic
86	C₄×C₄₃	#125	1	1	yes	characteristic
87	C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
88	Dic₄₃	—	4	2	no	—
89	Dic₄₃	—	8	1	no	—
90	#2088	—	43	1	no	maximal, Sylow
91	C₂×C₄₃	#398	1	1	yes	characteristic
92	C₂×C₄₃	#465	1	1	yes	characteristic
93	C₂×C₄₃	#468	1	1	yes	characteristic
94	C₂×C₄₃	#475	1	1	yes	characteristic
95	C₂×C₄₃	#476	1	1	yes	characteristic
96	C₂×C₄₃	#478	1	1	yes	characteristic
97	C₂×C₄₃	#479	1	1	yes	characteristic
98	C₂×#106	—	43	1	no	—
99	C₂×C₄⋊C₈	—	43	1	no	—
100	#389	—	43	1	no	—
101	#389	D₄₃	1	1	yes	characteristic
102	#340	—	43	1	no	—
103	#340	—	43	1	no	—
104	#340	—	43	1	no	—
105	C₄₃	#2088	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	43	1	no	—
107	C₂²×C₈	—	43	1	no	—
108	C₂×C₄⋊C₄	—	2	1	no	—
109	C₂×C₄⋊C₄	—	43	1	no	—
110	C₂×C₄⋊C₄	—	43	1	no	—
111	C₂×C₄⋊C₄	—	86	1	no	—
112	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
114	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
115	#106	—	86	2	no	—
116	C₄⋊C₈	—	86	2	no	—
117	#95	—	2	1	no	—
118	#95	—	86	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	86	1	no	—
122	C₂²×C₄	—	86	1	no	—
123	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
126	C₂×C₈	—	43	2	no	—
127	C₂×C₈	—	86	1	no	—
128	C₂×C₈	—	86	1	no	—
129	C₂×C₈	—	86	1	no	—
130	C₂×C₈	—	86	1	no	—
131	C₂×C₈	—	86	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	43	2	no	—
137	C₄⋊C₄	—	86	1	no	—
138	C₄⋊C₄	—	86	1	no	—
139	C₄⋊C₄	—	86	1	no	—
140	C₄⋊C₄	—	172	2	no	—
141	C₄⋊C₄	Dic₈₆	1	2	yes	—
142	C₄²	—	2	1	no	—
143	C₄²	—	2	1	no	—
144	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	86	2	no	—
157	C₂×C₄	—	172	1	no	—
158	C₂×C₄	—	172	1	no	—
159	C₂×C₄	—	172	1	no	—
160	C₂×C₄	—	172	1	no	—
161	C₂×C₄	—	172	1	no	—
162	C₂×C₄	#10621883	1	2	yes	—
163	C₂×C₄	C₂×Dic₈₆	1	1	yes	characteristic
164	C₂×C₄	#10621898	1	1	yes	characteristic
165	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
166	C₂×C₄	#10621900	1	1	yes	characteristic
167	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
168	C₂×C₄	#10621902	1	1	yes	characteristic
169	C₈	—	86	2	no	—
170	C₈	—	172	1	no	—
171	C₂²	#13063297	1	1	yes	characteristic
172	C₂²	#13063301	1	1	yes	characteristic
173	C₂²	#13063304	1	1	yes	characteristic
174	C₂²	C₂×#10621883	1	1	yes	characteristic
175	C₂²	#13063345	1	1	yes	characteristic
176	C₂²	#13063350	1	1	yes	characteristic
177	C₂²	#13063356	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	4	1	no	—
180	C₄	—	4	2	no	—
181	C₄	—	8	1	no	—
182	C₄	—	344	1	no	—
183	C₄	—	172	2	no	—
184	C₄	#13063310	1	1	yes	characteristic
185	C₄	#13063315	1	1	yes	characteristic
186	C₄	#13063316	1	1	yes	characteristic
187	C₄	#13063324	1	1	yes	characteristic
188	C₂	#14582832	1	1	yes	characteristic
189	C₂	#14583126	1	1	yes	characteristic
190	C₂	#14583147	1	1	yes	characteristic
191	C₂	#14583181	1	1	yes	characteristic
192	C₂	#14583189	1	1	yes	characteristic
193	C₂	#14583196	1	1	yes	characteristic
194	C₂	#14583206	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063256

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

#14582797

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, characteristic

C₄₃×#389

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

—

#13063256

—

C₄⋊(C₄₃⋊C₈)

—

C₄₃⋊#95

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

Q₈

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

Q₈

yes

—

Dic₄₃⋊C₄

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

D₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

#121

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₄⋊Q₈

yes

characteristic

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄⋊C₈

—

100

#389

—

101

#389

D₄₃

yes

characteristic

102

#340

—

103

#340

—

104

#340

—

105

C₄₃

#2088

yes

characteristic, complemented, Sylow

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

113

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

114

C₂×C₄²

C₂×D₄₃

yes

characteristic

115

#106

—

116

C₄⋊C₈

—

117

#95

—

118

#95

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

124

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

125

C₂²×C₄

C₂²×D₄₃

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

172

—

141

C₄⋊C₄

Dic₈₆

yes

—

142

C₄²

—

143

C₄²

—

144

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

172

—

158

C₂×C₄

—

172

—

159

C₂×C₄

—

172

—

160

C₂×C₄

—

172

—

161

C₂×C₄

—

172

—

162

C₂×C₄

#10621883

yes

—

163

C₂×C₄

C₂×Dic₈₆

yes

characteristic

164

C₂×C₄

#10621898

yes

characteristic

165

C₂×C₄

D₄×D₄₃

yes

characteristic

166

C₂×C₄

#10621900

yes

characteristic

167

C₂×C₄

Q₈×D₄₃

yes

characteristic

168

C₂×C₄

#10621902

yes

characteristic

169

C₈

—

170

C₈

—

172

—

171

C₂²

#13063297

yes

characteristic

172

C₂²

#13063301

yes

characteristic

173

C₂²

#13063304

yes

characteristic

174

C₂²

C₂×#10621883

yes

characteristic

175

C₂²

#13063345

yes

characteristic

176

C₂²

#13063350

yes

characteristic

177

C₂²

#13063356

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

344

—

183

C₄

—

172

—

184

C₄

#13063310

yes

characteristic

185

C₄

#13063315

yes

characteristic

186

C₄

#13063316

yes

characteristic

187

C₄

#13063324

yes

characteristic

188

C₂

#14582832

yes

characteristic

189

C₂

#14583126

yes

characteristic

190

C₂

#14583147

yes

characteristic

191

C₂

#14583181

yes

characteristic

192

C₂

#14583189

yes

characteristic

193

C₂

#14583196

yes

characteristic

194

C₂

#14583206

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#389	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	#389	D₄₃
hypocenter	C₄₃	#2088
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#389

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
105. C₄₃	#2088	semidirect

H ≅ G/N

type

105. C₄₃

#2088

semidirect

Subgroups — #19136096

Conjugacy classes of subgroups

Complements