Group #19135182 — subgroups

Subgroups — #19135182

Order 5504.

230 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14582798	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#14582662	C₂	1	1	yes	maximal, characteristic
4	#14582661	C₂	1	1	yes	maximal, characteristic
5	#14582661	C₂	1	1	yes	maximal, characteristic
6	#14582653	C₂	1	1	yes	maximal, characteristic
7	C₂×#106×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₄₃	—	2	1	no	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₄₃	—	2	1	no	—
13	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
14	C₄₃⋊#95	—	2	1	no	—
15	C₄₃⋊#95	—	2	1	no	—
16	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
19	#106×C₄₃	C₄	1	4	yes	complemented
20	C₂²×C₄×C₄₃	—	2	1	no	—
21	C₂²×C₄×C₄₃	—	2	1	no	—
22	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₄₃	—	2	1	no	—
24	C₂²×Dic₄₃	—	2	1	no	—
25	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
26	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
27	C₂×C₈×C₄₃	—	2	1	no	—
28	C₂×C₈×C₄₃	—	2	1	no	—
29	C₂×C₈×C₄₃	C₂×C₄	1	2	yes	—
30	C₄⋊C₄×C₄₃	—	2	1	no	—
31	C₄⋊C₄×C₄₃	—	2	1	no	—
32	C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
33	C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
34	C₄⋊Dic₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	—	2	1	no	—
36	C₄⋊Dic₄₃	—	2	1	no	—
37	C₄⋊Dic₄₃	—	4	2	no	—
38	C₄⋊Dic₄₃	Q₈	1	2	yes	—
39	C₄×Dic₄₃	—	2	1	no	—
40	C₄×Dic₄₃	—	2	1	no	—
41	C₄×Dic₄₃	—	4	2	no	—
42	C₂×C₄₃⋊C₈	—	2	1	no	—
43	C₂×C₄₃⋊C₈	—	2	1	no	—
44	C₂×C₄₃⋊C₈	—	2	1	no	—
45	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	2	2	no	—
48	C₂×C₄×C₄₃	—	4	1	no	—
49	C₂×C₄×C₄₃	—	4	1	no	—
50	C₂×C₄×C₄₃	—	4	1	no	—
51	C₂×C₄×C₄₃	—	4	1	no	—
52	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
57	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
58	C₂×Dic₄₃	—	2	1	no	—
59	C₂×Dic₄₃	—	2	1	no	—
60	C₂×Dic₄₃	—	2	2	no	—
61	C₂×Dic₄₃	—	2	1	no	—
62	C₂×Dic₄₃	—	2	1	no	—
63	C₂×Dic₄₃	—	2	1	no	—
64	C₂×Dic₄₃	—	4	1	no	—
65	C₂×Dic₄₃	—	4	1	no	—
66	C₂×Dic₄₃	—	4	1	no	—
67	C₂×Dic₄₃	—	4	1	no	—
68	C₂×Dic₄₃	—	4	1	no	—
69	C₂×Dic₄₃	SD₁₆	1	2	yes	—
70	C₈×C₄₃	—	2	2	no	—
71	C₄₃⋊C₈	—	4	1	no	—
72	C₂²×C₄₃	C₄×D₄	1	1	yes	characteristic
73	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₄₃	#123	1	1	yes	characteristic
75	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂²×C₄₃	#135	1	1	yes	characteristic
77	C₂²×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂²×C₄₃	#137	1	1	yes	characteristic
79	C₄×C₄₃	—	4	2	no	—
80	C₄×C₄₃	—	4	2	no	—
81	C₄×C₄₃	#117	1	1	yes	characteristic
82	C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
83	C₄×C₄₃	#125	1	1	yes	characteristic
84	C₄×C₄₃	#126	1	1	yes	characteristic
85	Dic₄₃	—	2	2	no	—
86	Dic₄₃	—	4	2	no	—
87	Dic₄₃	—	4	1	no	—
88	Dic₄₃	—	8	1	no	—
89	#1920	—	43	1	no	maximal, Sylow
90	C₂×C₄₃	#387	1	1	yes	characteristic
91	C₂×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
92	C₂×C₄₃	#440	1	1	yes	characteristic
93	C₂×C₄₃	#476	1	1	yes	characteristic
94	C₂×C₄₃	#479	1	1	yes	characteristic
95	C₂×C₄₃	#482	1	1	yes	characteristic
96	C₂×C₄₃	#483	1	1	yes	characteristic
97	C₂×#106	D₄₃	1	1	yes	characteristic
98	#389	—	43	1	no	—
99	C₄×C₄⋊C₄	—	43	1	no	—
100	#340	—	43	1	no	—
101	#340	—	43	1	no	—
102	#340	—	43	1	no	—
103	#336	—	43	1	no	—
104	C₄₃	#1920	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	—	43	1	no	—
106	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	43	1	no	—
108	C₂×C₄⋊C₄	—	43	1	no	—
109	C₂×C₄⋊C₄	—	86	1	no	—
110	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
112	C₂×C₄²	—	43	1	no	—
113	C₂×C₄²	—	86	1	no	—
114	#106	Dic₄₃	1	4	yes	complemented
115	#95	—	86	1	no	—
116	#95	—	86	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	43	1	no	—
120	C₂²×C₄	—	43	1	no	—
121	C₂²×C₄	—	86	1	no	—
122	C₂²×C₄	—	86	1	no	—
123	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	86	1	no	—
127	C₂×C₈	—	86	1	no	—
128	C₂×C₈	—	86	1	no	—
129	C₂×C₈	C₂×Dic₄₃	1	2	yes	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	43	2	no	—
133	C₄⋊C₄	—	86	1	no	—
134	C₄⋊C₄	—	86	1	no	—
135	C₄⋊C₄	—	86	1	no	—
136	C₄⋊C₄	—	172	2	no	—
137	C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
138	C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
139	C₄²	—	86	1	no	—
140	C₄²	—	86	1	no	—
141	C₄²	—	172	2	no	—
142	C₂³	D₄×D₄₃	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	43	2	no	—
150	C₂×C₄	—	86	1	no	—
151	C₂×C₄	—	86	1	no	—
152	C₂×C₄	—	86	2	no	—
153	C₂×C₄	—	86	1	no	—
154	C₂×C₄	—	86	1	no	—
155	C₂×C₄	—	86	1	no	—
156	C₂×C₄	—	172	1	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	#10621898	1	1	yes	characteristic
162	C₂×C₄	#10621898	1	1	yes	characteristic
163	C₂×C₄	#10621900	1	1	yes	characteristic
164	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
165	C₂×C₄	#10621902	1	1	yes	characteristic
166	C₂×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	172	1	no	—
169	C₂²	D₄×Dic₄₃	1	1	yes	characteristic
170	C₂²	#13063303	1	1	yes	characteristic
171	C₂²	#13063310	1	1	yes	characteristic
172	C₂²	SD₁₆×D₄₃	1	1	yes	characteristic
173	C₂²	#13063346	1	1	yes	characteristic
174	C₂²	#13063352	1	1	yes	characteristic
175	C₂²	#13063354	1	1	yes	characteristic
176	C₄	—	4	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	86	2	no	—
179	C₄	—	344	1	no	—
180	C₄	—	172	2	no	—
181	C₄	—	172	1	no	—
182	C₄	#13063284	1	1	yes	characteristic
183	C₄	Q₈×Dic₄₃	1	1	yes	characteristic
184	C₄	#13063315	1	1	yes	characteristic
185	C₄	#13063320	1	1	yes	characteristic
186	C₂	#14582785	1	1	yes	characteristic
187	C₂	SD₁₆×Dic₄₃	1	1	yes	characteristic
188	C₂	#14583000	1	1	yes	characteristic
189	C₂	#14583188	1	1	yes	characteristic
190	C₂	#14583209	1	1	yes	characteristic
191	C₂	#14583225	1	1	yes	characteristic
192	C₂	#14583231	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14582798

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₄₃

C₂

yes

maximal, characteristic

#14582662

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, characteristic

#14582653

C₂

yes

maximal, characteristic

C₂×#106×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

C₂²

yes

characteristic

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

#106×C₄₃

C₄

yes

complemented

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

D₄

yes

characteristic

C₂²×Dic₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

Q₈

yes

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

C₂×C₄×C₄₃

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

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

SD₁₆

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

C₄×D₄

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

#123

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#117

yes

characteristic

C₄×C₄₃

C₄×Q₈

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

C₄×SD₁₆

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

yes

characteristic

#389

—

C₄×C₄⋊C₄

—

100

#340

—

101

#340

—

102

#340

—

103

#336

—

104

C₄₃

#1920

yes

characteristic, complemented, Sylow

105

C₂²×C₈

—

106

C₂²×C₈

C₂×D₄₃

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

111

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

112

C₂×C₄²

—

113

C₂×C₄²

—

114

#106

Dic₄₃

yes

complemented

115

#95

—

116

#95

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂²×D₄₃

yes

characteristic

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂×Dic₄₃

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

172

—

137

C₄⋊C₄

C₂×Dic₄₃

yes

—

138

C₄⋊C₄

C₂×Dic₄₃

yes

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

172

—

142

C₂³

D₄×D₄₃

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

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

—

172

—

157

C₂×C₄

—

172

—

158

C₂×C₄

—

172

—

159

C₂×C₄

—

172

—

160

C₂×C₄

—

172

—

161

C₂×C₄

#10621898

yes

characteristic

162

C₂×C₄

#10621898

yes

characteristic

163

C₂×C₄

#10621900

yes

characteristic

164

C₂×C₄

Q₈×D₄₃

yes

characteristic

165

C₂×C₄

#10621902

yes

characteristic

166

C₂×C₄

C₂²×Dic₄₃

yes

characteristic

167

C₈

—

168

C₈

—

172

—

169

C₂²

D₄×Dic₄₃

yes

characteristic

170

C₂²

#13063303

yes

characteristic

171

C₂²

#13063310

yes

characteristic

172

C₂²

SD₁₆×D₄₃

yes

characteristic

173

C₂²

#13063346

yes

characteristic

174

C₂²

#13063352

yes

characteristic

175

C₂²

#13063354

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

344

—

180

C₄

—

172

—

181

C₄

—

172

—

182

C₄

#13063284

yes

characteristic

183

C₄

Q₈×Dic₄₃

yes

characteristic

184

C₄

#13063315

yes

characteristic

185

C₄

#13063320

yes

characteristic

186

C₂

#14582785

yes

characteristic

187

C₂

SD₁₆×Dic₄₃

yes

characteristic

188

C₂

#14583000

yes

characteristic

189

C₂

#14583188

yes

characteristic

190

C₂

#14583209

yes

characteristic

191

C₂

#14583225

yes

characteristic

192

C₂

#14583231

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#106×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×#106	D₄₃
hypocenter	C₄₃	#1920
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#106×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
19. #106×C₄₃	C₄	semidirect
104. C₄₃	#1920	semidirect
114. #106	Dic₄₃	semidirect

H ≅ G/N

type

19. #106×C₄₃

C₄

semidirect

104. C₄₃

#1920

semidirect

114. #106

Dic₄₃

semidirect

Subgroups — #19135182

Conjugacy classes of subgroups

Complements