Group #19135728 — subgroups

Subgroups — #19135728

Order 5504.

326 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13063245	C₂	1	2	yes	maximal
2	#14582801	C₂	1	1	yes	maximal, characteristic
3	#14582661	C₂	1	2	yes	maximal, complemented
4	#14582654	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
7	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
10	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
11	#13063245	—	2	4	no	—
12	C₂²×Q₈×C₄₃	—	2	1	no	—
13	C₄⋊Q₈×C₄₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂³×D₄₃	—	2	1	no	—
18	C₂×Q₈×C₄₃	—	2	4	no	—
19	C₂×Q₈×C₄₃	—	4	2	no	—
20	C₂×Q₈×C₄₃	—	4	2	no	—
21	C₂²×C₄×C₄₃	—	2	2	no	—
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₃	—	2	1	no	—
26	C₂×D₁₇₂	—	2	1	no	—
27	C₂×D₁₇₂	—	2	2	no	—
28	C₂×D₁₇₂	—	2	2	no	—
29	C₂×D₁₇₂	Q₈	1	2	yes	complemented
30	C₄⋊C₄×C₄₃	—	2	4	no	—
31	C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₄⋊C₄×C₄₃	D₄	1	4	yes	—
33	C₄²×C₄₃	—	2	2	no	—
34	(C₂×D₄₃)⋊C₄	—	4	2	no	—
35	(C₂×D₄₃)⋊C₄	—	4	2	no	—
36	C₄⋊Dic₄₃	—	2	2	no	—
37	C₂×C₄₃⋊C₈	—	2	2	no	—
38	C₂×C₄₃⋊C₈	—	2	2	no	—
39	C₂×C₄₃⋊C₈	—	2	2	no	—
40	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₄₃	—	2	2	no	—
42	C₂²×D₄₃	—	4	1	no	—
43	C₂²×D₄₃	—	4	1	no	—
44	C₂²×D₄₃	—	4	1	no	—
45	C₂²×D₄₃	—	4	1	no	—
46	C₂²×D₄₃	—	4	1	no	—
47	C₂²×D₄₃	—	4	1	no	—
48	Q₈×C₄₃	—	4	8	no	—
49	C₂×C₄×C₄₃	—	2	1	no	—
50	C₂×C₄×C₄₃	—	2	1	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	—	2	4	no	—
54	C₂×C₄×C₄₃	—	4	2	no	—
55	C₂×C₄×C₄₃	—	4	2	no	—
56	C₂×C₄×C₄₃	D₈	1	2	yes	—
57	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
58	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#41	1	2	yes	—
61	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
62	C₂×Dic₄₃	—	4	1	no	—
63	C₂×Dic₄₃	—	4	1	no	—
64	C₂×Dic₄₃	—	4	1	no	—
65	D₁₇₂	—	2	4	no	—
66	D₁₇₂	—	4	2	no	—
67	C₄₃⋊C₈	—	4	2	no	—
68	C₂²×C₄₃	#120	1	1	yes	characteristic
69	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₄₃	#124	1	1	yes	characteristic
71	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
72	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂²×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂²×C₄₃	#137	1	1	yes	characteristic
75	C₂×D₄₃	—	4	2	no	—
76	C₂×D₄₃	—	4	2	no	—
77	C₂×D₄₃	—	4	2	no	—
78	C₂×D₄₃	—	8	1	no	—
79	C₂×D₄₃	—	8	1	no	—
80	C₂×D₄₃	—	8	1	no	—
81	C₂×D₄₃	—	8	1	no	—
82	C₄×C₄₃	—	2	2	no	—
83	C₄×C₄₃	—	2	2	no	—
84	C₄×C₄₃	—	4	4	no	—
85	C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
86	C₄×C₄₃	#124	1	2	yes	—
87	Dic₄₃	—	8	1	no	—
88	#2023	—	43	1	no	maximal, Sylow
89	C₂×C₄₃	#393	1	1	yes	characteristic
90	C₂×C₄₃	#447	1	1	yes	characteristic
91	C₂×C₄₃	#448	1	1	yes	characteristic
92	C₂×C₄₃	#474	1	1	yes	characteristic
93	C₂×C₄₃	#476	1	1	yes	characteristic
94	C₂×C₄₃	#486	1	1	yes	characteristic
95	C₂×C₄₃	#489	1	1	yes	characteristic
96	D₄₃	—	8	2	no	—
97	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
98	C₂×#102	—	43	2	no	—
99	#390	—	43	1	no	—
100	#340	—	43	2	no	—
101	#336	—	43	1	no	—
102	C₄₃	#2023	1	1	yes	characteristic, complemented, Sylow
103	C₂²×Q₈	—	2	1	no	—
104	C₂²×D₄	—	43	1	no	—
105	C₂²×C₈	—	43	2	no	—
106	C₄⋊Q₈	—	2	4	no	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	—	43	1	no	—
109	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
110	C₂×C₂²⋊C₄	—	86	1	no	—
111	C₂×C₂²⋊C₄	—	86	1	no	—
112	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
113	#102	—	86	4	no	—
114	C₂⁴	—	86	1	no	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	—	4	2	no	—
117	C₂×Q₈	—	4	2	no	—
118	C₂×D₄	—	43	2	no	—
119	C₂×D₄	—	86	2	no	—
120	C₂×D₄	—	86	1	no	—
121	C₂×D₄	—	86	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	86	1	no	—
124	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
127	C₂×C₈	—	86	2	no	—
128	C₂×C₈	—	86	2	no	—
129	C₂×C₈	—	86	2	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	86	2	no	—
133	C₄⋊C₄	C₂²⋊D₄₃	1	4	yes	—
134	C₂²⋊C₄	—	172	2	no	—
135	C₂²⋊C₄	—	172	2	no	—
136	C₄²	—	2	2	no	—
137	C₂³	—	86	2	no	—
138	C₂³	—	172	1	no	—
139	C₂³	—	172	1	no	—
140	C₂³	—	172	1	no	—
141	C₂³	—	172	1	no	—
142	C₂³	—	172	1	no	—
143	C₂³	—	172	1	no	—
144	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
145	Q₈	—	4	8	no	—
146	D₄	—	86	4	no	—
147	D₄	—	172	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	172	1	no	—
156	C₂×C₄	—	172	1	no	—
157	C₂×C₄	—	172	1	no	—
158	C₂×C₄	#10621882	1	2	yes	—
159	C₂×C₄	#10621884	1	2	yes	—
160	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
161	C₂×C₄	#10621902	1	2	yes	—
162	C₂×C₄	#10621902	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
164	C₈	—	172	2	no	—
165	C₂²	—	344	1	no	—
166	C₂²	—	344	1	no	—
167	C₂²	—	344	1	no	—
168	C₂²	—	344	1	no	—
169	C₂²	—	172	2	no	—
170	C₂²	—	172	2	no	—
171	C₂²	—	172	2	no	—
172	C₂²	#13063294	1	1	yes	characteristic
173	C₂²	#13063301	1	1	yes	characteristic
174	C₂²	#13063314	1	1	yes	characteristic
175	C₂²	C₂×#10621882	1	1	yes	characteristic
176	C₂²	C₂×#10621884	1	1	yes	characteristic
177	C₂²	#13063351	1	1	yes	characteristic
178	C₂²	#13063356	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	4	4	no	—
182	C₄	—	344	1	no	—
183	C₄	#13063301	1	2	yes	—
184	C₄	#13063314	1	2	yes	—
185	C₂	—	344	2	no	—
186	C₂	#14582810	1	1	yes	characteristic
187	C₂	#14583033	1	1	yes	characteristic
188	C₂	#14583038	1	1	yes	characteristic
189	C₂	#14583176	1	1	yes	characteristic
190	C₂	#14583186	1	1	yes	characteristic
191	C₂	#14583247	1	1	yes	characteristic
192	C₂	#14583261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063245

C₂

yes

maximal

#14582801

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, complemented

#14582654

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

—

C₂×(C₂×D₄₃)⋊C₄

—

#13063245

—

C₂²×Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂³×D₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

Q₈

yes

complemented

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

—

C₄²×C₄₃

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

Q₈×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

D₈

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄₃⋊C₈

—

C₂²×C₄₃

#120

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₃

#124

yes

characteristic

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊Q₈

yes

—

C₄×C₄₃

#124

yes

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×C₄⋊Q₈

D₄₃

yes

characteristic, complemented

C₂×#102

—

#390

—

100

#340

—

101

#336

—

102

C₄₃

#2023

yes

characteristic, complemented, Sylow

103

C₂²×Q₈

—

104

C₂²×D₄

—

105

C₂²×C₈

—

106

C₄⋊Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

110

C₂×C₂²⋊C₄

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₄²

C₂×D₄₃

yes

characteristic

113

#102

—

114

C₂⁴

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

125

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

126

C₂²×C₄

C₂²×D₄₃

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

C₂²⋊D₄₃

yes

—

134

C₂²⋊C₄

—

172

—

135

C₂²⋊C₄

—

172

—

136

C₄²

—

137

C₂³

—

138

C₂³

—

172

—

139

C₂³

—

172

—

140

C₂³

—

172

—

141

C₂³

—

172

—

142

C₂³

—

172

—

143

C₂³

—

172

—

144

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

145

Q₈

—

146

D₄

—

147

D₄

—

172

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

172

—

156

C₂×C₄

—

172

—

157

C₂×C₄

—

172

—

158

C₂×C₄

#10621882

yes

—

159

C₂×C₄

#10621884

yes

—

160

C₂×C₄

Q₈×D₄₃

yes

characteristic

161

C₂×C₄

#10621902

yes

—

162

C₂×C₄

#10621902

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

164

C₈

—

172

—

165

C₂²

—

344

—

166

C₂²

—

344

—

167

C₂²

—

344

—

168

C₂²

—

344

—

169

C₂²

—

172

—

170

C₂²

—

172

—

171

C₂²

—

172

—

172

C₂²

#13063294

yes

characteristic

173

C₂²

#13063301

yes

characteristic

174

C₂²

#13063314

yes

characteristic

175

C₂²

C₂×#10621882

yes

characteristic

176

C₂²

C₂×#10621884

yes

characteristic

177

C₂²

#13063351

yes

characteristic

178

C₂²

#13063356

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

344

—

183

C₄

#13063301

yes

—

184

C₄

#13063314

yes

—

185

C₂

—

344

—

186

C₂

#14582810

yes

characteristic

187

C₂

#14583033

yes

characteristic

188

C₂

#14583038

yes

characteristic

189

C₂

#14583176

yes

characteristic

190

C₂

#14583186

yes

characteristic

191

C₂

#14583247

yes

characteristic

192

C₂

#14583261

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#2023
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₄⋊Q₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₃

hypocenter

C₄₃

#2023

LCS

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #14582661	C₂	semidirect
4. #14582654	C₂	semidirect
5. C₂×C₄⋊Q₈×C₄₃	C₂	semidirect
29. C₂×D₁₇₂	Q₈	semidirect
97. C₂×C₄⋊Q₈	D₄₃	semidirect
102. C₄₃	#2023	semidirect

H ≅ G/N

type

3. #14582661

C₂

semidirect

4. #14582654

C₂

semidirect

5. C₂×C₄⋊Q₈×C₄₃

C₂

semidirect

29. C₂×D₁₇₂

Q₈

semidirect

97. C₂×C₄⋊Q₈

D₄₃

semidirect

102. C₄₃

#2023

semidirect

Subgroups — #19135728

Conjugacy classes of subgroups

Complements