Group #19144078 — subgroups

Subgroups — #19144078

Order 5504.

480 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13063356	C₂	1	2	yes	maximal
2	C₄×#10621898	C₂	1	1	yes	maximal, characteristic
3	#14583106	C₂	1	4	yes	maximal, complemented
4	#14583101	C₂	1	4	yes	maximal, complemented
5	#14582923	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582908	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10621898	C₂²	1	1	yes	characteristic
9	#13063356	—	2	8	no	—
10	C₂×#10621885	C₂²	1	4	yes	—
11	C₂×#10621884	C₂²	1	4	yes	—
12	C₂×#10621877	C₂²	1	2	yes	—
13	C₄×Dic₈₆	C₂²	1	2	yes	—
14	C₄×D₁₇₂	C₂²	1	2	yes	—
15	C₄×C₂²⋊D₄₃	—	2	2	no	—
16	#13063284	C₂²	1	1	yes	characteristic
17	#13063282	—	2	1	no	—
18	C₄²×D₄₃	—	2	1	no	—
19	C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
20	#13063249	C₂²	1	8	yes	—
21	C₂²×Q₈×C₄₃	C₂²	1	2	yes	—
22	C₄⋊Q₈×C₄₃	—	2	1	no	—
23	C₄⋊Q₈×C₄₃	—	2	1	no	—
24	C₄⋊Q₈×C₄₃	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
26	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₄₃	—	2	2	no	—
28	C₂×Q₈×C₄₃	—	2	4	no	—
29	C₂×Q₈×C₄₃	—	2	4	no	—
30	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₄₃	—	2	2	no	—
32	C₂²×C₄×C₄₃	D₄	1	2	yes	—
33	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₄₃	—	2	1	no	—
35	#10621898	—	2	2	no	—
36	#10621898	D₄	1	4	yes	—
37	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₄₃	—	2	1	no	—
39	C₂×C₄×D₄₃	—	2	2	no	—
40	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₄₃	—	2	4	no	—
42	C₄⋊C₄×C₄₃	—	2	4	no	—
43	C₄²×C₄₃	D₄	1	1	yes	characteristic
44	C₄²×C₄₃	D₄	1	1	yes	characteristic
45	C₄²×C₄₃	C₂³	1	2	yes	—
46	C₂²⋊Dic₄₃	—	2	1	no	—
47	#10621885	—	2	8	no	—
48	#10621884	—	2	8	no	—
49	(C₂×D₄₃)⋊C₄	—	2	2	no	—
50	C₄⋊Dic₄₃	C₂³	1	2	yes	—
51	Dic₄₃⋊C₄	—	2	2	no	—
52	C₄×Dic₄₃	—	2	1	no	—
53	C₄×Dic₄₃	—	2	2	no	—
54	#10621877	—	2	4	no	—
55	C₂×C₄₃⋊C₈	C₂³	1	4	yes	—
56	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×D₄₃	—	2	1	no	—
58	Q₈×C₄₃	—	2	4	no	—
59	Q₈×C₄₃	—	2	4	no	—
60	Q₈×C₄₃	—	4	4	no	—
61	C₂×C₄×C₄₃	—	2	4	no	—
62	C₂×C₄×C₄₃	—	2	4	no	—
63	C₂×C₄×C₄₃	—	4	4	no	—
64	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₄₃	#41	1	2	yes	—
69	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
70	C₂²⋊D₄₃	—	2	4	no	—
71	C₂×Dic₄₃	—	2	1	no	—
72	C₂×Dic₄₃	—	2	2	no	—
73	D₁₇₂	—	2	1	no	—
74	D₁₇₂	C₂×D₄	1	2	yes	—
75	C₄×D₄₃	—	2	4	no	—
76	C₄×D₄₃	—	4	4	no	—
77	Dic₈₆	—	2	1	no	—
78	Dic₈₆	C₂×D₄	1	2	yes	—
79	C₄₃⋊C₈	—	2	4	no	—
80	C₂²×C₄₃	—	2	2	no	—
81	C₂²×C₄₃	#121	1	2	yes	—
82	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₄₃	—	2	2	no	—
84	C₂×D₄₃	—	4	2	no	—
85	C₄×C₄₃	—	2	2	no	—
86	C₄×C₄₃	—	4	4	no	—
87	C₄×C₄₃	#121	1	2	yes	—
88	C₄×C₄₃	#137	1	4	yes	complemented
89	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
91	Dic₄₃	—	2	2	no	—
92	Dic₄₃	—	4	2	no	—
93	#3039	—	43	1	no	maximal, Sylow
94	C₂×C₄₃	—	2	2	no	—
95	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
96	C₂×C₄₃	C₂×#137	1	2	yes	—
97	D₄₃	—	4	2	no	—
98	C₂×#137	—	43	2	no	—
99	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
100	C₄×#41	—	43	1	no	—
101	#462	—	43	4	no	—
102	#461	—	43	4	no	—
103	#423	—	43	1	no	—
104	#419	—	43	2	no	—
105	C₄₃	#3039	1	1	yes	characteristic, complemented, Sylow
106	C₂×#41	—	43	1	no	—
107	C₂²×Q₈	C₂×D₄₃	1	2	yes	—
108	#137	—	86	8	no	—
109	C₂×Q₁₆	—	43	4	no	—
110	C₂×SD₁₆	—	43	4	no	—
111	C₂×M₁₆	—	43	2	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	C₂×D₄₃	1	4	yes	—
115	C₄×Q₈	—	43	2	no	—
116	C₄×D₄	—	43	2	no	—
117	C₄×D₄	—	86	2	no	—
118	#117	—	43	1	no	—
119	#117	—	86	1	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	—	86	1	no	—
122	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
123	C₄⋊C₈	—	43	4	no	—
124	#103	—	43	8	no	—
125	#41	—	43	4	no	—
126	#41	—	86	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	—	43	1	no	—
131	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
132	C₂×D₄	—	43	1	no	—
133	C₂×D₄	—	86	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	86	1	no	—
136	C₂²×C₄	—	86	2	no	—
137	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
138	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
139	Q₁₆	—	86	8	no	—
140	SD₁₆	—	86	8	no	—
141	M₁₆	—	86	4	no	—
142	C₂×C₈	—	43	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	43	2	no	—
146	C₄⋊C₄	—	86	2	no	—
147	C₂²⋊C₄	—	86	1	no	—
148	C₂²⋊C₄	—	86	2	no	—
149	C₄²	—	86	1	no	—
150	C₄²	—	86	2	no	—
151	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
152	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
153	C₄²	C₂²×D₄₃	1	2	yes	—
154	C₂³	—	86	1	no	—
155	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
156	Q₈	—	2	4	no	—
157	Q₈	—	2	4	no	—
158	Q₈	—	4	4	no	—
159	Q₈	—	43	2	no	—
160	Q₈	—	86	1	no	—
161	D₄	—	43	2	no	—
162	D₄	—	86	4	no	—
163	D₄	—	86	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	86	1	no	—
168	C₂×C₄	—	86	4	no	—
169	C₂×C₄	—	86	2	no	—
170	C₂×C₄	—	172	4	no	—
171	C₂×C₄	D₄×D₄₃	1	2	yes	—
172	C₂×C₄	#10621900	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
175	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
176	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
177	C₈	—	86	4	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	86	2	no	—
180	C₂²	—	172	2	no	—
181	C₂²	#13063296	1	2	yes	—
182	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	86	2	no	—
186	C₄	—	172	2	no	—
187	C₄	#13063296	1	2	yes	—
188	C₄	#13063356	1	4	yes	complemented
189	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
190	C₄	C₂×#10621900	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	172	2	no	—
193	C₂	C₂×#13063296	1	1	yes	characteristic
194	C₂	C₂×#13063356	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063356

C₂

yes

maximal

C₄×#10621898

C₂

yes

maximal, characteristic

#14583106

C₂

yes

maximal, complemented

#14583101

C₂

yes

maximal, complemented

#14582923

C₂

yes

maximal, characteristic, complemented

#14582908

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₂²

yes

characteristic

#13063356

—

C₂×#10621885

C₂²

yes

—

C₂×#10621884

C₂²

yes

—

C₂×#10621877

C₂²

yes

—

C₄×Dic₈₆

C₂²

yes

—

C₄×D₁₇₂

C₂²

yes

—

C₄×C₂²⋊D₄₃

—

#13063284

C₂²

yes

characteristic

#13063282

—

C₄²×D₄₃

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

#13063249

C₂²

yes

—

C₂²×Q₈×C₄₃

C₂²

yes

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

D₄

yes

—

C₂×D₁₇₂

C₂³

yes

characteristic

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

—

C₂²⋊Dic₄₃

—

#10621885

—

#10621884

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

C₂³

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

C₂×D₄

yes

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

C₂×D₄

yes

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

#121

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#121

yes

—

C₄×C₄₃

#137

yes

complemented

C₄×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

C₂×#41

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#3039

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂×#121

yes

characteristic

C₂×C₄₃

C₂×#137

yes

—

D₄₃

—

C₂×#137

—

C₂×C₄⋊Q₈

D₄₃

yes

characteristic, complemented

100

C₄×#41

—

101

#462

—

102

#461

—

103

#423

—

104

#419

—

105

C₄₃

#3039

yes

characteristic, complemented, Sylow

106

C₂×#41

—

107

C₂²×Q₈

C₂×D₄₃

yes

—

108

#137

—

109

C₂×Q₁₆

—

110

C₂×SD₁₆

—

111

C₂×M₁₆

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

C₂×D₄₃

yes

—

115

C₄×Q₈

—

116

C₄×D₄

—

117

C₄×D₄

—

118

#117

—

119

#117

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

—

122

C₂×C₄²

C₂×D₄₃

yes

characteristic

123

C₄⋊C₈

—

124

#103

—

125

#41

—

126

#41

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

C₂²×D₄₃

yes

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂²⋊D₄₃

yes

—

138

C₂²×C₄

C₂²×D₄₃

yes

characteristic

139

Q₁₆

—

140

SD₁₆

—

141

M₁₆

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₂²⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂²⋊D₄₃

yes

characteristic

152

C₄²

C₂²⋊D₄₃

yes

characteristic

153

C₄²

C₂²×D₄₃

yes

—

154

C₂³

—

155

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

156

Q₈

—

157

Q₈

—

158

Q₈

—

159

Q₈

—

160

Q₈

—

161

D₄

—

162

D₄

—

163

D₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

172

—

171

C₂×C₄

D₄×D₄₃

yes

—

172

C₂×C₄

#10621900

yes

—

173

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

174

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

175

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

176

C₂×C₄

C₂³×D₄₃

yes

characteristic

177

C₈

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

172

—

181

C₂²

#13063296

yes

—

182

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

172

—

187

C₄

#13063296

yes

—

188

C₄

#13063356

yes

complemented

189

C₄

C₂×D₄×D₄₃

yes

characteristic

190

C₄

C₂×#10621900

yes

characteristic

191

C₂

—

192

C₂

—

172

—

193

C₂

C₂×#13063296

yes

characteristic

194

C₂

C₂×#13063356

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#3039
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₂×C₄⋊Q₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂²×D₄

hypercenter

C₂×C₄⋊Q₈

D₄₃

hypocenter

C₄₃

#3039

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #14583106	C₂	semidirect
4. #14583101	C₂	semidirect
5. #14582923	C₂	semidirect
6. #14582908	C₂	semidirect
7. C₂×C₄⋊Q₈×C₄₃	C₂	semidirect
88. C₄×C₄₃	#137	semidirect
99. C₂×C₄⋊Q₈	D₄₃	semidirect
105. C₄₃	#3039	semidirect
188. C₄	#13063356	semidirect

H ≅ G/N

type

3. #14583106

C₂

semidirect

4. #14583101

C₂

semidirect

5. #14582923

C₂

semidirect

6. #14582908

C₂

semidirect

7. C₂×C₄⋊Q₈×C₄₃

C₂

semidirect

88. C₄×C₄₃

#137

semidirect

99. C₂×C₄⋊Q₈

D₄₃

semidirect

105. C₄₃

#3039

semidirect

188. C₄

#13063356

semidirect

Subgroups — #19144078

Conjugacy classes of subgroups

Complements