Group #24999332 — subgroups

Subgroups — #24999332

Order 11008.

528 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 111 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	#19145540	C₂	1	2	yes	maximal
2	#19142973	C₂	1	1	yes	maximal, characteristic
3	#19133277	C₂	1	2	yes	maximal
4	#19132428	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1994	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063325	C₂²	1	2	yes	—
7	#14583278	—	2	2	no	—
8	#14583251	—	2	2	no	—
9	#14583109	—	2	4	no	—
10	#14583054	—	2	4	no	—
11	#14582952	C₂²	1	2	yes	—
12	#14582805	—	2	4	no	—
13	#14582765	—	2	2	no	—
14	#14582647	—	2	4	no	—
15	C₄₃×#533	C₂²	1	1	yes	characteristic
16	C₄₃×#423	C₂²	1	1	yes	characteristic
17	C₄³×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#339	—	2	2	no	—
19	C₂²×Dic₈₆	D₄	1	4	yes	—
20	C₂×#10621885	—	2	8	no	—
21	#13063325	—	2	2	no	—
22	#13063325	—	2	2	no	—
23	#13063325	—	2	2	no	—
24	#13063325	—	2	2	no	—
25	C₂×C₄⋊Dic₄₃	—	2	2	no	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
28	#13063248	—	2	8	no	—
29	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
30	C₄×C₄₃⋊C₈	—	2	2	no	—
31	C₄₃⋊#95	—	2	8	no	—
32	C₂×M₁₆×C₄₃	—	2	1	no	—
33	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
34	#125×C₄₃	—	2	1	no	—
35	C₂²⋊Q₈×C₄₃	—	2	2	no	—
36	C₄×Q₈×C₄₃	—	2	2	no	—
37	#117×C₄₃	—	2	1	no	—
38	C₂×C₄²×C₄₃	—	2	2	no	—
39	C₂×C₄²×C₄₃	D₄	1	2	yes	—
40	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₃	—	2	2	no	—
42	C₂×Q₈×C₄₃	—	2	2	no	—
43	C₂²×C₄×C₄₃	—	2	2	no	—
44	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₄₃	—	2	4	no	—
47	C₂×Dic₈₆	—	2	4	no	—
48	C₂×Dic₈₆	—	2	4	no	—
49	C₂×Dic₈₆	—	4	8	no	—
50	M₁₆×C₄₃	—	4	2	no	—
51	C₂×C₈×C₄₃	—	4	1	no	—
52	C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₄⋊C₄×C₄₃	—	2	2	no	—
54	C₄⋊C₄×C₄₃	—	4	2	no	—
55	C₂²⋊C₄×C₄₃	—	4	1	no	—
56	C₄²×C₄₃	—	2	2	no	—
57	C₄²×C₄₃	—	4	4	no	—
58	C₄²×C₄₃	—	4	1	no	—
59	C₄²×C₄₃	Q₁₆	1	4	yes	complemented
60	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₄₃	C₂×D₄	1	2	yes	—
62	C₄²×C₄₃	#41	1	1	yes	characteristic
63	#10621885	—	8	8	no	—
64	C₄⋊Dic₄₃	—	4	2	no	—
65	C₄⋊Dic₄₃	—	4	2	no	—
66	C₄⋊Dic₄₃	—	4	2	no	—
67	C₄⋊Dic₄₃	—	4	2	no	—
68	C₂×C₄₃⋊C₈	—	2	4	no	—
69	C₂³×C₄₃	#120	1	1	yes	characteristic
70	Q₈×C₄₃	—	4	4	no	—
71	C₂×C₄×C₄₃	—	2	4	no	—
72	C₂×C₄×C₄₃	—	2	4	no	—
73	C₂×C₄×C₄₃	—	4	4	no	—
74	C₂×C₄×C₄₃	—	4	2	no	—
75	C₂×C₄×C₄₃	#120	1	2	yes	—
76	C₂×C₄×C₄₃	#121	1	2	yes	—
77	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
79	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
80	C₂×C₄×C₄₃	#137	1	2	yes	—
81	C₂×Dic₄₃	—	4	8	no	—
82	C₂×Dic₄₃	—	4	4	no	—
83	Dic₈₆	—	4	8	no	—
84	Dic₈₆	—	8	4	no	—
85	C₈×C₄₃	—	8	1	no	—
86	C₄₃⋊C₈	—	4	4	no	—
87	#10297	—	43	1	no	maximal, Sylow
88	C₂²×C₄₃	—	2	2	no	—
89	C₂²×C₄₃	#392	1	1	yes	characteristic
90	C₂²×C₄₃	#451	1	2	yes	—
91	C₄×C₄₃	—	2	2	no	—
92	C₄×C₄₃	—	4	4	no	—
93	C₄×C₄₃	—	8	2	no	—
94	C₄×C₄₃	#455	1	4	yes	—
95	C₄×C₄₃	#462	1	2	yes	—
96	C₄×C₄₃	#494	1	1	yes	characteristic
97	C₄×C₄₃	#497	1	1	yes	characteristic
98	Dic₄₃	—	8	4	no	—
99	#3209	—	43	2	no	—
100	#2862	—	43	1	no	—
101	#1994	D₄₃	1	1	yes	characteristic
102	#1548	—	43	2	no	—
103	#1276	—	43	1	no	—
104	C₂×C₄₃	—	2	2	no	—
105	C₂×C₄₃	#2004	1	2	yes	—
106	C₂×C₄₃	#2016	1	1	yes	characteristic
107	#533	C₂×D₄₃	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	—	43	2	no	—
109	#494	—	86	2	no	—
110	#487	—	86	2	no	—
111	#462	—	86	4	no	—
112	#451	—	86	4	no	—
113	#431	—	43	2	no	—
114	#423	C₂×D₄₃	1	1	yes	characteristic
115	#391	—	86	4	no	—
116	#382	—	86	2	no	—
117	C₄³	C₂×D₄₃	1	1	yes	characteristic
118	#339	—	2	2	no	—
119	#333	—	86	4	no	—
120	C₄₃	#10297	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	43	4	no	—
122	C₂×Q₁₆	—	86	8	no	—
123	C₂×M₁₆	—	2	1	no	—
124	C₄⋊Q₈	—	86	2	no	—
125	C₄⋊Q₈	—	86	2	no	—
126	C₄⋊Q₈	—	86	2	no	—
127	C₄⋊Q₈	—	86	2	no	—
128	C₄⋊Q₈	D₁₇₂	1	2	yes	—
129	#125	—	2	1	no	—
130	C₂²⋊Q₈	—	2	2	no	—
131	C₄×Q₈	—	2	2	no	—
132	#117	—	2	1	no	—
133	C₂×C₄⋊C₄	—	86	2	no	—
134	C₂×C₄⋊C₄	—	86	2	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
137	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	43	4	no	—
140	#103	—	86	8	no	—
141	C₂²⋊C₈	—	86	2	no	—
142	C₄×C₈	—	86	2	no	—
143	#95	—	86	8	no	—
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	—	86	4	no	—
146	C₂×Q₈	—	86	4	no	—
147	C₂×Q₈	—	172	8	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	86	4	no	—
150	C₂²×C₄	D₄×D₄₃	1	2	yes	—
151	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
152	Q₁₆	—	344	8	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	86	4	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	172	2	no	—
160	C₄⋊C₄	—	172	2	no	—
161	C₄⋊C₄	—	172	2	no	—
162	C₄⋊C₄	—	172	2	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	4	no	—
166	C₄²	—	4	1	no	—
167	C₄²	#10621885	1	4	yes	complemented
168	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
169	C₄²	#10621898	1	1	yes	characteristic
170	C₄²	D₄×D₄₃	1	2	yes	—
171	C₂³	#13063293	1	1	yes	characteristic
172	Q₈	—	4	4	no	—
173	Q₈	—	344	4	no	—
174	Q₈	—	172	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	172	8	no	—
180	C₂×C₄	—	172	4	no	—
181	C₂×C₄	#13063295	1	2	yes	—
182	C₂×C₄	#13063298	1	2	yes	—
183	C₂×C₄	#13063299	1	1	yes	characteristic
184	C₂×C₄	#13063321	1	1	yes	characteristic
185	C₂×C₄	C₂×#10621885	1	2	yes	—
186	C₂×C₄	#13063355	1	2	yes	—
187	C₈	—	8	1	no	—
188	C₈	—	172	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	#14582808	1	1	yes	characteristic
191	C₂²	#14583054	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	4	4	no	—
194	C₄	—	8	2	no	—
195	C₄	—	344	4	no	—
196	C₄	#14583078	1	4	yes	—
197	C₄	#14583109	1	2	yes	—
198	C₄	#14583278	1	1	yes	characteristic
199	C₄	#14583294	1	1	yes	characteristic
200	C₂	—	2	2	no	—
201	C₂	#19135623	1	2	yes	—
202	C₂	#19135694	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145540

C₂

yes

maximal

#19142973

C₂

yes

maximal, characteristic

#19133277

C₂

yes

maximal

#19132428

C₂

yes

maximal, characteristic

C₄₃×#1994

C₂

yes

maximal, characteristic

C₂×#13063325

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#14582805

—

#14582765

—

#14582647

—

C₄₃×#533

C₂²

yes

characteristic

C₄₃×#423

C₂²

yes

characteristic

C₄³×C₄₃

C₂²

yes

characteristic

C₄₃×#339

—

C₂²×Dic₈₆

D₄

yes

—

—

—

—

—

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

#13063248

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₄₃⋊#95

—

C₂×M₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

D₄

yes

—

#125×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

#117×C₄₃

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

Q₁₆

yes

complemented

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

characteristic

#10621885

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#120

yes

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10297

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#392

yes

characteristic

C₂²×C₄₃

#451

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#455

yes

—

C₄×C₄₃

#462

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

#3209

—

100

#2862

—

101

#1994

D₄₃

yes

characteristic

102

#1548

—

103

#1276

—

104

C₂×C₄₃

—

105

C₂×C₄₃

#2004

yes

—

106

C₂×C₄₃

#2016

yes

characteristic

107

#533

C₂×D₄₃

yes

characteristic

108

C₂×C₄⋊Q₈

—

109

#494

—

110

#487

—

111

#462

—

112

#451

—

113

#431

—

114

#423

C₂×D₄₃

yes

characteristic

115

#391

—

116

#382

—

117

C₄³

C₂×D₄₃

yes

characteristic

118

#339

—

119

#333

—

120

C₄₃

#10297

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂×Q₁₆

—

123

C₂×M₁₆

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

D₁₇₂

yes

—

129

#125

—

130

C₂²⋊Q₈

—

131

C₄×Q₈

—

132

#117

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²⋊D₄₃

yes

—

137

C₂×C₄²

C₂²×D₄₃

yes

characteristic

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

#103

—

141

C₂²⋊C₈

—

142

C₄×C₈

—

143

#95

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

172

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₄₃

yes

—

151

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

152

Q₁₆

—

344

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

172

—

160

C₄⋊C₄

—

172

—

161

C₄⋊C₄

—

172

—

162

C₄⋊C₄

—

172

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

#10621885

yes

complemented

168

C₄²

C₂×D₁₇₂

yes

characteristic

169

C₄²

#10621898

yes

characteristic

170

C₄²

D₄×D₄₃

yes

—

171

C₂³

#13063293

yes

characteristic

172

Q₈

—

173

Q₈

—

344

—

174

Q₈

—

172

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

172

—

180

C₂×C₄

—

172

—

181

C₂×C₄

#13063295

yes

—

182

C₂×C₄

#13063298

yes

—

183

C₂×C₄

#13063299

yes

characteristic

184

C₂×C₄

#13063321

yes

characteristic

185

C₂×C₄

C₂×#10621885

yes

—

186

C₂×C₄

#13063355

yes

—

187

C₈

—

188

C₈

—

172

—

189

C₂²

—

190

C₂²

#14582808

yes

characteristic

191

C₂²

#14583054

yes

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

344

—

196

C₄

#14583078

yes

—

197

C₄

#14583109

yes

—

198

C₄

#14583278

yes

characteristic

199

C₄

#14583294

yes

characteristic

200

C₂

—

201

C₂

#19135623

yes

—

202

C₂

#19135694

yes

characteristic

	subgroup	G/H
center	C₂²	#14582808
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1994	C₂
socle	C₂²×C₄₃	#392
prime radical	C₂³×C₄₃	#120
hypercenter	#1994	D₄₃
hypocenter	C₄₃	#10297
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

subgroup

G/H

center

C₂²

#14582808

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1994

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

N	H ≅ G/N	type
59. C₄²×C₄₃	Q₁₆	semidirect
120. C₄₃	#10297	semidirect
167. C₄²	#10621885	semidirect

H ≅ G/N

type

59. C₄²×C₄₃

Q₁₆

semidirect

120. C₄₃

#10297

semidirect

167. C₄²

#10621885

semidirect

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

subgroup	index
#19145540	2
#19142973	2
#19133277	2
#19132428	2
C₄₃×#1994	2
#10297	43

subgroup

index

C₄₃×#1994

Subgroups — #24999332

Conjugacy classes of subgroups

Complements

Maximal subgroups