Group #24996718 — subgroups

Subgroups — #24996718

Order 11008.

430 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 163 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	#19142958	C₂	1	1	yes	maximal, characteristic
2	#19134712	C₂	1	1	yes	maximal, characteristic
3	#19132479	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063317	—	2	1	no	—
6	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
7	#14582797	—	2	2	no	—
8	#14582797	C₄	1	4	yes	—
9	#14582769	C₂²	1	2	yes	—
10	#14582765	—	2	1	no	—
11	#14582654	—	2	2	no	—
12	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₃	—	2	1	no	—
14	C₄³×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#336	—	2	1	no	—
16	C₂²×C₄₃⋊C₈	—	2	2	no	—
17	#13063317	—	4	4	no	—
18	C₂×C₄⋊Dic₄₃	—	2	2	no	—
19	C₂×C₄⋊Dic₄₃	—	2	2	no	—
20	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
21	C₂×Dic₄₃⋊C₄	—	2	4	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
23	C₄₃⋊#95	—	2	4	no	—
24	C₂²×C₈×C₄₃	—	2	2	no	—
25	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₄₃	D₄	1	2	yes	—
27	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₃	—	2	2	no	—
31	C₄⋊C₈×C₄₃	Q₈	1	4	yes	—
32	C₄×C₈×C₄₃	—	4	2	no	—
33	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₃	—	2	4	no	—
39	C₂×C₈×C₄₃	—	2	4	no	—
40	C₂×C₈×C₄₃	—	4	2	no	—
41	C₂×C₈×C₄₃	—	4	2	no	—
42	C₄²×C₄₃	—	2	4	no	—
43	C₄²×C₄₃	—	2	2	no	—
44	C₄²×C₄₃	—	4	2	no	—
45	C₄²×C₄₃	SD₁₆	1	2	yes	—
46	C₄²×C₄₃	Q₁₆	1	2	yes	—
47	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
48	C₄²×C₄₃	#41	1	2	yes	—
49	C₄⋊Dic₄₃	—	4	4	no	—
50	C₄⋊Dic₄₃	—	4	4	no	—
51	C₄⋊Dic₄₃	—	4	4	no	—
52	Dic₄₃⋊C₄	—	4	8	no	—
53	C₂×C₄₃⋊C₈	—	4	2	no	—
54	C₂×C₄₃⋊C₈	—	4	4	no	—
55	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	—	2	2	no	—
58	C₂×C₄×C₄₃	—	2	2	no	—
59	C₂×C₄×C₄₃	—	2	2	no	—
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	#103	1	4	yes	—
63	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#135	1	4	yes	—
70	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₄₃	—	4	4	no	—
72	C₂×Dic₄₃	—	4	4	no	—
73	C₂×Dic₄₃	—	4	4	no	—
74	C₈×C₄₃	—	4	4	no	—
75	C₄₃⋊C₈	—	8	2	no	—
76	#9897	—	43	1	no	maximal, Sylow
77	C₂²×C₄₃	#391	1	1	yes	characteristic
78	C₂²×C₄₃	#413	1	2	yes	—
79	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
80	C₂²×C₄₃	#416	1	2	yes	—
81	C₂²×C₄₃	#418	1	1	yes	characteristic
82	C₄×C₄₃	—	2	4	no	—
83	C₄×C₄₃	—	2	2	no	—
84	C₄×C₄₃	—	4	2	no	—
85	C₄×C₄₃	#476	1	2	yes	—
86	C₄×C₄₃	#477	1	2	yes	—
87	C₄×C₄₃	#488	1	2	yes	—
88	C₄×C₄₃	#499	1	2	yes	—
89	Dic₄₃	—	8	4	no	—
90	#2859	—	43	1	no	—
91	#1831	—	43	1	no	—
92	C₄×C₄⋊C₈	D₄₃	1	1	yes	characteristic
93	#1296	—	43	4	no	—
94	C₂×C₄₃	#1542	1	2	yes	—
95	C₂×C₄₃	#1553	1	2	yes	—
96	C₂×C₄₃	#1987	1	1	yes	characteristic
97	C₂×C₄₃	#1991	1	1	yes	characteristic
98	C₂×C₄₃	#1998	1	1	yes	characteristic
99	C₂×#125	—	86	1	no	—
100	C₂×C₄⋊C₈	—	43	2	no	—
101	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
102	C₂×C₄×C₈	—	2	1	no	—
103	#389	—	43	4	no	—
104	#389	—	86	2	no	—
105	#384	—	43	2	no	—
106	#382	—	86	1	no	—
107	C₄³	C₂×D₄₃	1	1	yes	characteristic
108	#336	—	2	1	no	—
109	#336	—	86	2	no	—
110	C₄₃	#9897	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	86	2	no	—
113	#125	—	172	4	no	—
114	C₂×C₄⋊C₄	—	43	4	no	—
115	C₂×C₄⋊C₄	—	86	4	no	—
116	C₂×C₄⋊C₄	—	86	2	no	—
117	C₂×C₄⋊C₄	—	86	2	no	—
118	C₂×C₄²	C₄×D₄₃	1	2	yes	—
119	C₂×C₄²	D₁₇₂	1	2	yes	—
120	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	86	4	no	—
125	C₄⋊C₈	Dic₈₆	1	4	yes	—
126	C₄×C₈	—	4	2	no	—
127	#95	—	86	4	no	—
128	C₂²×C₄	—	86	4	no	—
129	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
130	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	172	2	no	—
138	C₂×C₈	—	172	4	no	—
139	C₄⋊C₄	—	172	4	no	—
140	C₄⋊C₄	—	172	8	no	—
141	C₄⋊C₄	—	172	4	no	—
142	C₄⋊C₄	—	172	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#10621883	1	2	yes	—
147	C₄²	#10621885	1	2	yes	—
148	C₄²	C₂×Dic₈₆	1	2	yes	—
149	C₄²	#10621898	1	2	yes	—
150	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	172	4	no	—
158	C₂×C₄	—	172	4	no	—
159	C₂×C₄	—	172	4	no	—
160	C₂×C₄	#13063248	1	4	yes	—
161	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
162	C₂×C₄	#13063304	1	2	yes	—
163	C₂×C₄	#13063313	1	1	yes	characteristic
164	C₂×C₄	#13063325	1	1	yes	characteristic
165	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
166	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
167	C₂×C₄	#13063347	1	4	yes	—
168	C₂×C₄	#13063350	1	2	yes	—
169	C₈	—	4	4	no	—
170	C₈	—	344	2	no	—
171	C₂²	#14582805	1	1	yes	characteristic
172	C₂²	#14582881	1	2	yes	—
173	C₂²	C₂×#13063248	1	1	yes	characteristic
174	C₂²	#14582896	1	2	yes	—
175	C₂²	#14582903	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	344	4	no	—
180	C₄	#14583189	1	2	yes	—
181	C₄	#14583194	1	2	yes	—
182	C₄	#14583254	1	2	yes	—
183	C₄	#14583301	1	2	yes	—
184	C₂	#19133253	1	2	yes	—
185	C₂	#19133298	1	2	yes	—
186	C₂	#19135543	1	1	yes	characteristic
187	C₂	#19135560	1	1	yes	characteristic
188	C₂	#19135588	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19142958

C₂

yes

maximal, characteristic

#19134712

C₂

yes

maximal, characteristic

#19132479

C₂

yes

maximal

C₄×C₄⋊C₈×C₄₃

C₂

yes

maximal, characteristic

C₂×#13063317

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

#14582797

—

#14582797

C₄

yes

—

#14582769

C₂²

yes

—

#14582765

—

#14582654

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

—

C₄³×C₄₃

C₂²

yes

characteristic

C₄₃×#336

—

C₂²×C₄₃⋊C₈

—

#13063317

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×Dic₄₃⋊C₄

—

C₄⋊(C₄₃⋊C₈)

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

Q₈

yes

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

Q₁₆

yes

—

C₄²×C₄₃

C₂×Q₈

yes

—

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#103

yes

—

C₂×C₄×C₄₃

C₄×Q₈

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

C₂×#103

yes

characteristic

C₂²×C₄₃

#416

yes

—

C₂²×C₄₃

#418

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#476

yes

—

C₄×C₄₃

#477

yes

—

C₄×C₄₃

#488

yes

—

C₄×C₄₃

#499

yes

—

Dic₄₃

—

#2859

—

#1831

—

C₄×C₄⋊C₈

D₄₃

yes

characteristic

#1296

—

C₂×C₄₃

#1542

yes

—

C₂×C₄₃

#1553

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

C₂×D₄₃

yes

—

102

C₂×C₄×C₈

—

103

#389

—

104

#389

—

105

#384

—

106

#382

—

107

C₄³

C₂×D₄₃

yes

characteristic

108

#336

—

109

#336

—

110

C₄₃

#9897

yes

characteristic, complemented, Sylow

111

C₂²×C₈

—

112

C₂²×C₈

—

113

#125

—

172

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

C₄×D₄₃

yes

—

119

C₂×C₄²

D₁₇₂

yes

—

120

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

121

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

122

C₂×C₄²

C₂²×D₄₃

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

Dic₈₆

yes

—

126

C₄×C₈

—

127

#95

—

128

C₂²×C₄

—

129

C₂²×C₄

(C₂×D₄₃)⋊C₄

yes

—

130

C₂²×C₄

(C₂×D₄₃)⋊C₄

yes

—

131

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

132

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

133

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

172

—

138

C₂×C₈

—

172

—

139

C₄⋊C₄

—

172

—

140

C₄⋊C₄

—

172

—

141

C₄⋊C₄

—

172

—

142

C₄⋊C₄

—

172

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#10621883

yes

—

147

C₄²

#10621885

yes

—

148

C₄²

C₂×Dic₈₆

yes

—

149

C₄²

#10621898

yes

—

150

C₂³

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

172

—

158

C₂×C₄

—

172

—

159

C₂×C₄

—

172

—

160

C₂×C₄

#13063248

yes

—

161

C₂×C₄

C₄×Dic₈₆

yes

—

162

C₂×C₄

#13063304

yes

—

163

C₂×C₄

#13063313

yes

characteristic

164

C₂×C₄

#13063325

yes

characteristic

165

C₂×C₄

C₂×#10621883

yes

characteristic

166

C₂×C₄

C₂×#10621885

yes

characteristic

167

C₂×C₄

#13063347

yes

—

168

C₂×C₄

#13063350

yes

—

169

C₈

—

170

C₈

—

344

—

171

C₂²

#14582805

yes

characteristic

172

C₂²

#14582881

yes

—

173

C₂²

C₂×#13063248

yes

characteristic

174

C₂²

#14582896

yes

—

175

C₂²

#14582903

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

344

—

180

C₄

#14583189

yes

—

181

C₄

#14583194

yes

—

182

C₄

#14583254

yes

—

183

C₄

#14583301

yes

—

184

C₂

#19133253

yes

—

185

C₂

#19133298

yes

—

186

C₂

#19135543

yes

characteristic

187

C₂

#19135560

yes

characteristic

188

C₂

#19135588

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×C₄⋊C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#9897
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×(C₂×D₄₃)⋊C₄

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄×C₄⋊C₈×C₄₃

C₂

socle, prime radical

C₂³×C₄₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₃

C₂³

hypercenter

C₄×C₄⋊C₈

D₄₃

hypocenter

C₄₃

#9897

LCS

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

N	H ≅ G/N	type
110. C₄₃	#9897	semidirect

H ≅ G/N

type

110. C₄₃

#9897

semidirect

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

subgroup	index
#19142958	2
#19134712	2
#19132479	2
C₄×C₄⋊C₈×C₄₃	2
#9897	43

subgroup

index

C₄×C₄⋊C₈×C₄₃

Subgroups — #24996718

Conjugacy classes of subgroups

Complements

Maximal subgroups