Group #24996086 — subgroups

Subgroups — #24996086

Order 11008.

506 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14583272	C₂	1	1	yes	maximal, characteristic
2	#19133268	C₂	1	2	yes	maximal, complemented
3	#19133264	C₂	1	2	yes	maximal
4	#19132887	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10621873	—	2	2	no	—
7	C₂×#13063325	C₂²	1	1	yes	characteristic
8	C₂×#13063322	C₂²	1	1	yes	characteristic
9	#14583272	—	2	2	no	—
10	#14583272	C₄	1	4	yes	—
11	#14582952	C₂²	1	2	yes	—
12	#14582646	—	2	4	no	—
13	#14582642	—	2	4	no	—
14	#14582621	—	2	2	no	—
15	C₄₃×#423	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#341	—	2	2	no	—
18	C₂²×Dic₈₆	—	2	1	no	—
19	C₂²×D₁₇₂	—	2	1	no	—
20	C₂²×D₁₇₂	—	2	2	no	—
21	C₂×#10621873	—	2	8	no	—
22	C₂×#10621873	—	4	8	no	—
23	#13063325	—	2	2	no	—
24	#13063325	—	2	1	no	—
25	#13063325	C₂×C₄	1	2	yes	—
26	#13063322	—	2	2	no	—
27	#13063322	—	2	1	no	—
28	#13063322	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₄₃	—	2	2	no	—
30	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
31	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
32	C₄×C₄₃⋊C₈	—	2	2	no	—
33	C₂×M₁₆×C₄₃	—	2	2	no	—
34	C₂²×C₈×C₄₃	D₄	1	2	yes	—
35	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₃	—	2	2	no	—
37	C₄⋊C₈×C₄₃	—	2	2	no	—
38	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₃	D₄	1	2	yes	—
40	C₂³×D₄₃	—	4	1	no	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	#41	1	2	yes	—
43	C₂²×Dic₄₃	—	4	1	no	—
44	C₂×D₁₇₂	—	2	2	no	—
45	C₂×D₁₇₂	—	2	4	no	—
46	C₂×D₁₇₂	—	4	2	no	—
47	C₂×D₁₇₂	—	4	1	no	—
48	C₂×D₁₇₂	—	4	4	no	—
49	C₂×D₁₇₂	—	8	4	no	—
50	C₂×Dic₈₆	—	2	2	no	—
51	C₂×Dic₈₆	—	4	4	no	—
52	C₂×Dic₈₆	—	4	1	no	—
53	M₁₆×C₄₃	—	4	4	no	—
54	C₂×C₈×C₄₃	—	2	4	no	—
55	C₂×C₈×C₄₃	—	4	2	no	—
56	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₄₃	C₂×D₄	1	2	yes	—
60	C₄⋊Dic₄₃	—	2	4	no	—
61	C₄⋊Dic₄₃	—	4	2	no	—
62	C₂×C₄₃⋊C₈	—	2	4	no	—
63	#10621873	—	4	16	no	—
64	C₂³×C₄₃	#124	1	1	yes	characteristic
65	C₂²×D₄₃	—	4	2	no	—
66	C₂²×D₄₃	—	8	4	no	—
67	C₂²×D₄₃	—	8	2	no	—
68	C₂×C₄×C₄₃	—	2	4	no	—
69	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₄₃	#121	1	2	yes	—
72	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
73	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
74	C₂×Dic₄₃	—	4	2	no	—
75	C₂×Dic₄₃	—	8	2	no	—
76	D₁₇₂	—	4	4	no	—
77	D₁₇₂	—	8	4	no	—
78	D₁₇₂	—	8	4	no	—
79	D₁₇₂	—	8	2	no	—
80	Dic₈₆	—	4	4	no	—
81	Dic₈₆	—	8	2	no	—
82	C₈×C₄₃	—	2	4	no	—
83	C₈×C₄₃	—	8	2	no	—
84	C₄₃⋊C₈	—	4	4	no	—
85	#9777	—	43	1	no	maximal, Sylow
86	C₂²×C₄₃	—	2	2	no	—
87	C₂²×C₄₃	#390	1	1	yes	characteristic
88	C₂²×C₄₃	#490	1	2	yes	—
89	C₂×D₄₃	—	8	2	no	—
90	C₂×D₄₃	—	8	4	no	—
91	C₂×D₄₃	—	16	4	no	—
92	C₄×C₄₃	—	2	2	no	—
93	C₄×C₄₃	#418	1	2	yes	—
94	C₄×C₄₃	#471	1	4	yes	—
95	C₄×C₄₃	#496	1	2	yes	—
96	Dic₄₃	—	8	2	no	—
97	C₂×#492	—	43	1	no	—
98	#1837	D₄₃	1	1	yes	characteristic, complemented
99	#1546	—	43	2	no	—
100	#1545	—	43	2	no	—
101	#1452	—	43	1	no	—
102	C₂×C₄₃	—	2	2	no	—
103	C₂×C₄₃	#1961	1	1	yes	characteristic
104	C₂×C₄₃	#1977	1	2	yes	—
105	D₄₃	—	16	2	no	—
106	C₂²×SD₁₆	—	86	2	no	—
107	C₂×C₄⋊Q₈	—	43	1	no	—
108	C₂×#127	—	43	1	no	—
109	#492	—	43	4	no	—
110	#492	—	86	2	no	—
111	#431	—	43	2	no	—
112	#423	C₂×D₄₃	1	2	yes	—
113	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
114	#341	—	2	2	no	—
115	#332	—	86	4	no	—
116	#331	—	86	4	no	—
117	#322	—	86	2	no	—
118	C₄₃	#9777	1	1	yes	characteristic, complemented, Sylow
119	C₂²×Q₈	—	86	1	no	—
120	C₂²×D₄	—	86	2	no	—
121	C₂²×D₄	—	86	1	no	—
122	C₂×SD₁₆	—	86	8	no	—
123	C₂×SD₁₆	—	172	8	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
126	C₄⋊Q₈	—	43	2	no	—
127	C₄⋊Q₈	—	86	1	no	—
128	C₄⋊Q₈	—	86	2	no	—
129	#127	—	43	2	no	—
130	#127	—	86	1	no	—
131	#127	—	86	2	no	—
132	C₂×C₄⋊C₄	—	86	2	no	—
133	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	43	4	no	—
137	C₂²⋊C₈	—	86	2	no	—
138	C₄×C₈	—	86	2	no	—
139	C₄×C₈	C₄×D₄₃	1	2	yes	—
140	C₄×C₈	D₁₇₂	1	2	yes	—
141	C₂⁴	—	172	1	no	—
142	C₂×Q₈	—	86	2	no	—
143	C₂×Q₈	—	172	4	no	—
144	C₂×Q₈	—	172	1	no	—
145	C₂×D₄	—	86	2	no	—
146	C₂×D₄	—	86	4	no	—
147	C₂×D₄	—	344	4	no	—
148	C₂×D₄	—	172	2	no	—
149	C₂×D₄	—	172	1	no	—
150	C₂×D₄	—	172	4	no	—
151	C₂²×C₄	—	172	1	no	—
152	C₂²×C₄	#10621902	1	2	yes	—
153	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
154	SD₁₆	—	172	16	no	—
155	M₁₆	—	4	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	86	4	no	—
159	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
160	C₄⋊C₄	—	86	4	no	—
161	C₄⋊C₄	—	172	2	no	—
162	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
163	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
164	C₄²	D₄×D₄₃	1	2	yes	—
165	C₂³	—	344	4	no	—
166	C₂³	—	344	2	no	—
167	C₂³	—	172	2	no	—
168	C₂³	#13063314	1	1	yes	characteristic
169	Q₈	—	344	2	no	—
170	Q₈	—	172	4	no	—
171	D₄	—	344	4	no	—
172	D₄	—	344	4	no	—
173	D₄	—	344	2	no	—
174	D₄	—	172	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	344	2	no	—
177	C₂×C₄	—	172	2	no	—
178	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
179	C₂×C₄	#13063287	1	2	yes	—
180	C₂×C₄	#13063300	1	2	yes	—
181	C₂×C₄	#13063321	1	1	yes	characteristic
182	C₂×C₄	#13063350	1	4	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	8	2	no	—
185	C₈	—	172	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	344	4	no	—
188	C₂²	—	344	2	no	—
189	C₂²	—	688	4	no	—
190	C₂²	#14582800	1	1	yes	characteristic
191	C₂²	#14583265	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	344	2	no	—
194	C₄	#14582903	1	2	yes	—
195	C₄	#14583163	1	4	yes	—
196	C₄	#14583289	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	—	688	2	no	—
199	C₂	#19135418	1	1	yes	characteristic
200	C₂	#19135496	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583272

C₂

yes

maximal, characteristic

#19133268

C₂

yes

maximal, complemented

#19133264

C₂

yes

maximal

#19132887

C₂

yes

maximal, characteristic, complemented

C₄₃×#1837

C₂

yes

maximal, characteristic, complemented

C₂²×#10621873

—

C₂×#13063325

C₂²

yes

characteristic

C₂×#13063322

C₂²

yes

characteristic

#14583272

—

#14583272

C₄

yes

—

#14582952

C₂²

yes

—

#14582646

—

#14582642

—

#14582621

—

C₄₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#341

—

C₂²×Dic₈₆

—

C₂²×D₁₇₂

—

C₂²×D₁₇₂

—

—

—

—

—

C₂×C₄

yes

—

#13063322

—

#13063322

—

#13063322

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

#10621873

—

C₂³×C₄₃

#124

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#9777

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#390

yes

characteristic

C₂²×C₄₃

#490

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#418

yes

—

C₄×C₄₃

#471

yes

—

C₄×C₄₃

#496

yes

—

Dic₄₃

—

C₂×#492

—

#1837

D₄₃

yes

characteristic, complemented

#1546

—

100

#1545

—

101

#1452

—

102

C₂×C₄₃

—

103

C₂×C₄₃

#1961

yes

characteristic

104

C₂×C₄₃

#1977

yes

—

105

D₄₃

—

106

C₂²×SD₁₆

—

107

C₂×C₄⋊Q₈

—

108

C₂×#127

—

109

#492

—

110

#492

—

111

#431

—

112

#423

C₂×D₄₃

yes

—

113

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

114

#341

—

115

#332

—

116

#331

—

117

#322

—

118

C₄₃

#9777

yes

characteristic, complemented, Sylow

119

C₂²×Q₈

—

120

C₂²×D₄

—

121

C₂²×D₄

—

122

C₂×SD₁₆

—

123

C₂×SD₁₆

—

172

—

124

C₂×M₁₆

—

125

C₂²×C₈

C₂²⋊D₄₃

yes

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

#127

—

130

#127

—

131

#127

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²×D₄₃

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₂²⋊C₈

—

138

C₄×C₈

—

139

C₄×C₈

C₄×D₄₃

yes

—

140

C₄×C₈

D₁₇₂

yes

—

141

C₂⁴

—

172

—

142

C₂×Q₈

—

143

C₂×Q₈

—

172

—

144

C₂×Q₈

—

172

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

344

—

148

C₂×D₄

—

172

—

149

C₂×D₄

—

172

—

150

C₂×D₄

—

172

—

151

C₂²×C₄

—

172

—

152

C₂²×C₄

#10621902

yes

—

153

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

154

SD₁₆

—

172

—

155

M₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

172

—

162

C₄²

C₂×C₄×D₄₃

yes

characteristic

163

C₄²

C₂×D₁₇₂

yes

characteristic

164

C₄²

D₄×D₄₃

yes

—

165

C₂³

—

344

—

166

C₂³

—

344

—

167

C₂³

—

172

—

168

C₂³

#13063314

yes

characteristic

169

Q₈

—

344

—

170

Q₈

—

172

—

171

D₄

—

344

—

172

D₄

—

344

—

173

D₄

—

344

—

174

D₄

—

172

—

175

C₂×C₄

—

176

C₂×C₄

—

344

—

177

C₂×C₄

—

172

—

178

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

179

C₂×C₄

#13063287

yes

—

180

C₂×C₄

#13063300

yes

—

181

C₂×C₄

#13063321

yes

characteristic

182

C₂×C₄

#13063350

yes

—

183

C₈

—

184

C₈

—

185

C₈

—

172

—

186

C₂²

—

187

C₂²

—

344

—

188

C₂²

—

344

—

189

C₂²

—

688

—

190

C₂²

#14582800

yes

characteristic

191

C₂²

#14583265

yes

—

192

C₄

—

193

C₄

—

344

—

194

C₄

#14582903

yes

—

195

C₄

#14583163

yes

—

196

C₄

#14583289

yes

—

197

C₂

—

198

C₂

—

688

—

199

C₂

#19135418

yes

characteristic

200

C₂

#19135496

yes

—

	subgroup	G/H
center	C₂²	#14582800
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1837	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂×#13063322	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1837	D₄₃
hypocenter	C₄₃	#9777
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#14582800

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1837

C₂

C₂²×C₄₃

C₂²

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
2. #19133268	C₂	semidirect
4. #19132887	C₂	semidirect
5. C₄₃×#1837	C₂	semidirect
98. #1837	D₄₃	semidirect
118. C₄₃	#9777	semidirect

H ≅ G/N

type

2. #19133268

C₂

semidirect

4. #19132887

C₂

semidirect

5. C₄₃×#1837

C₂

semidirect

98. #1837

D₄₃

semidirect

118. C₄₃

#9777

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#14583272	2
#19133268	2
#19133264	2
#19132887	2
C₄₃×#1837	2
#9777	43

subgroup

index

C₄₃×#1837

Subgroups — #24996086

Conjugacy classes of subgroups

Complements

Maximal subgroups