Group #19146042 — subgroups

Subgroups — #19146042

Order 5504.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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	#14583511	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583504	C₂	1	1	yes	maximal, characteristic
3	#14583207	C₂	1	2	yes	maximal, complemented
4	#14583200	C₂	1	2	yes	maximal, complemented
5	#14583145	C₂	1	2	yes	maximal, complemented
6	#14583128	C₂	1	2	yes	maximal, complemented
7	#14583105	C₂	1	2	yes	maximal, complemented
8	#14583100	C₂	1	2	yes	maximal, complemented
9	C₄₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621873	C₂²	1	2	yes	—
11	C₂×#10621873	C₂²	1	2	yes	—
12	#13063325	C₂²	1	1	yes	characteristic
13	#13063322	C₂²	1	1	yes	characteristic
14	#13063318	—	2	2	no	—
15	#13063317	C₂²	1	1	yes	characteristic
16	#13063317	C₂²	1	1	yes	characteristic
17	#13063315	—	2	2	no	—
18	#13063310	—	2	2	no	—
19	#13063304	C₂²	1	2	yes	—
20	#13063299	C₂²	1	2	yes	—
21	#13063298	—	2	2	no	—
22	C₄×Dic₈₆	C₂²	1	2	yes	—
23	C₄×D₁₇₂	C₂²	1	2	yes	—
24	#13063259	C₂²	1	2	yes	—
25	#13063257	C₂²	1	2	yes	—
26	#13063250	C₂²	1	2	yes	—
27	#13063250	C₂²	1	2	yes	—
28	#13063247	C₂²	1	2	yes	—
29	#13063247	C₂²	1	2	yes	—
30	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	#117×C₄₃	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
34	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
35	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	C₂×C₂²⋊D₄₃	—	2	2	no	—
37	C₂×D₁₇₂	—	2	2	no	—
38	C₂×D₁₇₂	C₂³	1	2	yes	—
39	C₂×C₄×D₄₃	—	2	2	no	—
40	C₂×Dic₈₆	C₂³	1	2	yes	—
41	M₁₆×C₄₃	—	4	1	no	—
42	C₂×C₈×C₄₃	—	4	1	no	—
43	C₂×C₈×C₄₃	C₂³	1	2	yes	—
44	C₂×C₈×C₄₃	C₂³	1	2	yes	—
45	C₄⋊C₄×C₄₃	D₄	1	2	yes	—
46	C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
47	C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₄₃	—	2	2	no	—
50	(C₂×D₄₃)⋊C₄	—	2	2	no	—
51	(C₂×D₄₃)⋊C₄	—	2	2	no	—
52	C₄⋊Dic₄₃	—	2	2	no	—
53	C₄⋊Dic₄₃	C₂³	1	2	yes	—
54	C₄⋊Dic₄₃	C₂³	1	2	yes	—
55	Dic₄₃⋊C₄	—	2	2	no	—
56	Dic₄₃⋊C₄	—	2	2	no	—
57	Dic₄₃⋊C₄	—	2	2	no	—
58	Dic₄₃⋊C₄	—	2	2	no	—
59	C₄×Dic₄₃	—	2	2	no	—
60	#10621873	—	4	2	no	—
61	#10621873	—	4	2	no	—
62	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×D₄₃	—	2	2	no	—
64	C₂×C₄×C₄₃	—	2	1	no	—
65	C₂×C₄×C₄₃	—	2	1	no	—
66	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
70	C₂²⋊D₄₃	—	8	2	no	—
71	C₂×Dic₄₃	—	2	2	no	—
72	C₂×Dic₄₃	—	2	2	no	—
73	C₂×Dic₄₃	—	2	2	no	—
74	D₁₇₂	—	2	2	no	—
75	D₁₇₂	—	2	2	no	—
76	D₁₇₂	—	8	2	no	—
77	C₄×D₄₃	—	8	2	no	—
78	Dic₈₆	—	2	2	no	—
79	Dic₈₆	—	2	2	no	—
80	C₈×C₄₃	—	2	2	no	—
81	C₈×C₄₃	—	2	2	no	—
82	C₂²×C₄₃	—	2	1	no	—
83	C₂²×C₄₃	—	2	2	no	—
84	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
85	C₂×D₄₃	—	4	2	no	—
86	C₂×D₄₃	—	4	2	no	—
87	C₂×D₄₃	—	4	2	no	—
88	C₄×C₄₃	—	2	1	no	—
89	C₄×C₄₃	—	2	2	no	—
90	C₄×C₄₃	—	2	2	no	—
91	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
92	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
93	Dic₄₃	—	4	2	no	—
94	Dic₄₃	—	4	2	no	—
95	Dic₄₃	—	4	2	no	—
96	#3260	—	43	1	no	maximal, Sylow
97	C₂×C₄₃	—	4	1	no	—
98	C₂×C₄₃	#537	1	1	yes	characteristic
99	C₂×C₄₃	#577	1	2	yes	—
100	D₄₃	—	8	2	no	—
101	#541	—	43	1	no	—
102	#540	—	43	1	no	—
103	#479	—	43	2	no	—
104	#478	—	43	2	no	—
105	#468	—	43	2	no	—
106	#465	—	43	2	no	—
107	#461	—	43	2	no	—
108	#460	—	43	2	no	—
109	#424	D₄₃	1	1	yes	characteristic, complemented
110	C₄₃	#3260	1	1	yes	characteristic, complemented, Sylow
111	C₂×SD₁₆	—	43	2	no	—
112	C₂×SD₁₆	—	43	2	no	—
113	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
114	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
115	C₄⋊Q₈	—	43	1	no	—
116	#127	—	43	1	no	—
117	#126	—	86	2	no	—
118	#125	—	43	1	no	—
119	#125	—	43	1	no	—
120	#125	—	86	2	no	—
121	#123	—	86	2	no	—
122	C₂²⋊Q₈	—	43	2	no	—
123	#121	—	43	2	no	—
124	#121	—	86	2	no	—
125	C₄×Q₈	—	43	2	no	—
126	C₄×D₄	—	43	2	no	—
127	#117	C₂×D₄₃	1	1	yes	characteristic
128	#107	—	43	2	no	—
129	#106	—	43	2	no	—
130	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
131	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
132	#103	—	43	2	no	—
133	#103	—	43	2	no	—
134	#102	—	43	2	no	—
135	#102	—	43	2	no	—
136	C₂×Q₈	—	43	2	no	—
137	C₂×D₄	—	43	2	no	—
138	C₂×D₄	—	86	2	no	—
139	C₂×D₄	—	86	2	no	—
140	C₂²×C₄	—	86	2	no	—
141	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
142	SD₁₆	—	172	2	no	—
143	SD₁₆	—	172	2	no	—
144	M₁₆	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	C₂²×D₄₃	1	2	yes	—
147	C₂×C₈	C₂²×D₄₃	1	2	yes	—
148	C₄⋊C₄	—	43	2	no	—
149	C₄⋊C₄	—	43	2	no	—
150	C₄⋊C₄	—	86	2	no	—
151	C₄⋊C₄	—	86	2	no	—
152	C₄⋊C₄	—	86	2	no	—
153	C₄⋊C₄	—	86	2	no	—
154	C₄⋊C₄	—	86	2	no	—
155	C₄⋊C₄	D₁₇₂	1	2	yes	—
156	C₂²⋊C₄	—	86	2	no	—
157	C₂²⋊C₄	—	86	2	no	—
158	C₂²⋊C₄	—	86	2	no	—
159	C₂²⋊C₄	D₁₇₂	1	2	yes	—
160	C₄²	—	86	2	no	—
161	C₄²	C₂²×D₄₃	1	1	yes	characteristic
162	C₄²	C₂²×D₄₃	1	1	yes	characteristic
163	C₂³	—	86	2	no	—
164	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
165	Q₈	—	86	2	no	—
166	Q₈	—	86	2	no	—
167	D₄	—	86	2	no	—
168	D₄	—	86	2	no	—
169	D₄	—	344	2	no	—
170	D₄	—	344	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	86	2	no	—
174	C₂×C₄	—	86	2	no	—
175	C₂×C₄	—	86	2	no	—
176	C₂×C₄	—	344	2	no	—
177	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
178	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
179	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
180	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
181	C₈	—	2	2	no	—
182	C₈	—	2	2	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	172	2	no	—
186	C₂²	—	172	2	no	—
187	C₂²	—	172	2	no	—
188	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	172	2	no	—
193	C₄	—	172	2	no	—
194	C₄	—	172	2	no	—
195	C₄	#13063370	1	1	yes	characteristic
196	C₄	#13063373	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	—	344	2	no	—
199	C₂	#14583480	1	1	yes	characteristic
200	C₂	#14583711	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14583511

C₂

yes

maximal, characteristic, complemented

#14583504

C₂

yes

maximal, characteristic

#14583207

C₂

yes

maximal, complemented

#14583200

C₂

yes

maximal, complemented

#14583145

C₂

yes

maximal, complemented

#14583128

C₂

yes

maximal, complemented

#14583105

C₂

yes

maximal, complemented

#14583100

C₂

yes

maximal, complemented

C₄₃×#424

C₂

yes

maximal, characteristic, complemented

C₂×#10621873

C₂²

yes

—

C₂×#10621873

C₂²

yes

—

#13063325

C₂²

yes

characteristic

#13063322

C₂²

yes

characteristic

#13063318

—

#13063317

C₂²

yes

characteristic

#13063317

C₂²

yes

—

—

C₂²

yes

—

#13063299

C₂²

yes

—

#13063298

—

C₄×Dic₈₆

C₂²

yes

—

C₄×D₁₇₂

C₂²

yes

—

#13063259

C₂²

yes

—

#13063257

C₂²

yes

—

#13063250

C₂²

yes

—

#13063250

C₂²

yes

—

#13063247

C₂²

yes

—

#13063247

C₂²

yes

—

C₂×M₁₆×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

#117×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₂²

yes

—

C₄⋊C₈×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₂²⋊D₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂³

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

C₂³

yes

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂³

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

C₄⋊C₄×C₄₃

D₄

yes

—

C₂²⋊C₄×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₂²⋊Dic₄₃

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

#10621873

—

#10621873

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

H(5,𝔽₂)

yes

characteristic

C₄×C₄₃

2₋¹⁺⁴

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#3260

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

#537

yes

characteristic

C₂×C₄₃

#577

yes

—

100

D₄₃

—

101

#541

—

102

#540

—

103

#479

—

104

#478

—

105

#468

—

106

#465

—

107

#461

—

108

#460

—

109

#424

D₄₃

yes

characteristic, complemented

110

C₄₃

#3260

yes

characteristic, complemented, Sylow

111

C₂×SD₁₆

—

112

C₂×SD₁₆

—

113

C₂×M₁₆

C₂×D₄₃

yes

characteristic

114

C₂²×C₈

C₂×D₄₃

yes

characteristic

115

C₄⋊Q₈

—

116

#127

—

117

#126

—

118

#125

—

119

#125

—

120

#125

—

121

#123

—

122

C₂²⋊Q₈

—

123

#121

—

124

#121

—

125

C₄×Q₈

—

126

C₄×D₄

—

127

#117

C₂×D₄₃

yes

characteristic

128

#107

—

129

#106

—

130

C₄⋊C₈

C₂×D₄₃

yes

—

131

C₄⋊C₈

C₂×D₄₃

yes

—

132

#103

—

133

#103

—

134

#102

—

135

#102

—

136

C₂×Q₈

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²×D₄₃

yes

characteristic

142

SD₁₆

—

172

—

143

SD₁₆

—

172

—

144

M₁₆

—

145

C₂×C₈

—

146

C₂×C₈

C₂²×D₄₃

yes

—

147

C₂×C₈

C₂²×D₄₃

yes

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

D₁₇₂

yes

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

C₂²⋊C₄

—

159

C₂²⋊C₄

D₁₇₂

yes

—

160

C₄²

—

161

C₄²

C₂²×D₄₃

yes

characteristic

162

C₄²

C₂²×D₄₃

yes

characteristic

163

C₂³

—

164

C₂³

C₂×D₁₇₂

yes

characteristic

165

Q₈

—

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

344

—

170

D₄

—

344

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

344

—

177

C₂×C₄

C₂×D₁₇₂

yes

—

178

C₂×C₄

C₂×D₁₇₂

yes

—

179

C₂×C₄

C₂×D₁₇₂

yes

characteristic

180

C₂×C₄

C₂³×D₄₃

yes

characteristic

181

C₈

—

182

C₈

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

172

—

186

C₂²

—

172

—

187

C₂²

—

172

—

188

C₂²

C₂²×D₁₇₂

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

172

—

193

C₄

—

172

—

194

C₄

—

172

—

195

C₄

#13063370

yes

characteristic

196

C₄

#13063373

yes

characteristic

197

C₂

—

198

C₂

—

344

—

199

C₂

#14583480

yes

characteristic

200

C₂

#14583711

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#424	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#424	D₄₃
hypocenter	C₄₃	#3260
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#424

C₂

socle

C₂²×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #14583511	C₂	semidirect
3. #14583207	C₂	semidirect
4. #14583200	C₂	semidirect
5. #14583145	C₂	semidirect
6. #14583128	C₂	semidirect
7. #14583105	C₂	semidirect
8. #14583100	C₂	semidirect
9. C₄₃×#424	C₂	semidirect
109. #424	D₄₃	semidirect
110. C₄₃	#3260	semidirect

H ≅ G/N

type

1. #14583511

C₂

semidirect

3. #14583207

C₂

semidirect

4. #14583200

C₂

semidirect

5. #14583145

C₂

semidirect

6. #14583128

C₂

semidirect

7. #14583105

C₂

semidirect

8. #14583100

C₂

semidirect

9. C₄₃×#424

C₂

semidirect

109. #424

D₄₃

semidirect

110. C₄₃

#3260

semidirect

Subgroups — #19146042

Conjugacy classes of subgroups

Complements