Group #24996214 — subgroups

Subgroups — #24996214

Order 11008.

442 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 139 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×#14583304	C₂	1	1	yes	maximal, characteristic
2	#19135439	C₂	1	2	yes	maximal
3	#19133354	C₂	1	1	yes	maximal, characteristic
4	#19133266	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063325	C₂²	1	2	yes	—
7	#14583304	—	2	2	no	—
8	#14583304	—	2	1	no	—
9	#14583304	—	2	1	no	—
10	C₂×#13063259	C₄	1	4	yes	—
11	#14582923	C₂²	1	2	yes	—
12	C₂×#13063243	—	2	4	no	—
13	#14582652	—	2	2	no	—
14	#14582644	—	2	4	no	—
15	C₄₃×#431	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#322	—	2	1	no	—
19	C₈²×C₄₃	—	2	1	no	—
20	C₂²×Dic₈₆	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	#13063325	—	2	2	no	—
23	#13063325	—	2	2	no	—
24	#13063325	—	2	2	no	—
25	#13063325	—	2	2	no	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
28	#13063259	—	2	4	no	—
29	#13063259	—	2	4	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
32	#13063243	—	4	8	no	—
33	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
34	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
37	C₂²⋊C₈×C₄₃	—	2	1	no	—
38	#97×C₄₃	—	2	1	no	—
39	C₄×C₈×C₄₃	—	2	1	no	—
40	C₄×C₈×C₄₃	—	2	1	no	—
41	C₄×C₈×C₄₃	D₄	1	2	yes	—
42	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
47	C₂²×Dic₄₃	—	2	4	no	—
48	C₂×Dic₈₆	—	4	2	no	—
49	C₂×Dic₈₆	—	4	2	no	—
50	C₂×Dic₈₆	—	4	4	no	—
51	C₂×Dic₈₆	—	4	4	no	—
52	M₁₆×C₄₃	—	2	2	no	—
53	M₁₆×C₄₃	D₈	1	4	yes	—
54	C₂×C₈×C₄₃	—	2	2	no	—
55	C₂×C₈×C₄₃	—	2	4	no	—
56	C₂×C₈×C₄₃	—	2	2	no	—
57	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₄₃	#41	1	1	yes	characteristic
63	C₄⋊Dic₄₃	—	2	4	no	—
64	C₄⋊Dic₄₃	—	2	4	no	—
65	C₄⋊Dic₄₃	—	4	2	no	—
66	C₄⋊Dic₄₃	—	4	2	no	—
67	#10621877	—	4	4	no	—
68	C₂×C₄₃⋊C₈	—	4	2	no	—
69	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₄₃	#135	1	2	yes	—
80	C₂×Dic₄₃	—	4	4	no	—
81	C₂×Dic₄₃	—	4	8	no	—
82	Dic₈₆	—	8	4	no	—
83	Dic₈₆	—	8	2	no	—
84	Dic₈₆	—	8	2	no	—
85	C₈×C₄₃	—	2	2	no	—
86	C₈×C₄₃	—	2	4	no	—
87	C₈×C₄₃	—	4	2	no	—
88	C₈×C₄₃	#101	1	4	yes	complemented
89	C₄₃⋊C₈	—	8	2	no	—
90	#9799	—	43	1	no	maximal, Sylow
91	C₂²×C₄₃	—	2	1	no	—
92	C₂²×C₄₃	—	2	1	no	—
93	C₂²×C₄₃	#390	1	1	yes	characteristic
94	C₂²×C₄₃	C₄×D₈	1	2	yes	—
95	C₄×C₄₃	—	2	1	no	—
96	C₄×C₄₃	—	2	1	no	—
97	C₄×C₄₃	C₂×#101	1	2	yes	—
98	C₄×C₄₃	#466	1	2	yes	—
99	C₄×C₄₃	#473	1	2	yes	—
100	C₄×C₄₃	#493	1	1	yes	characteristic
101	C₄×C₄₃	#495	1	1	yes	characteristic
102	Dic₄₃	—	8	4	no	—
103	C₂×#500	—	43	1	no	—
104	#1966	—	43	2	no	—
105	#1567	—	43	1	no	—
106	#1546	—	43	2	no	—
107	C₈×M₁₆	D₄₃	1	1	yes	characteristic
108	C₂×C₄₃	—	2	2	no	—
109	C₂×C₄₃	#1966	1	1	yes	characteristic
110	C₂×C₄₃	#1970	1	1	yes	characteristic
111	C₂×C₄₃	#1979	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	43	2	no	—
113	#500	—	86	2	no	—
114	#500	—	86	1	no	—
115	#500	—	86	1	no	—
116	#431	C₂×D₄₃	1	1	yes	characteristic
117	C₂×#107	—	43	4	no	—
118	#423	—	43	2	no	—
119	C₂×#101	—	86	4	no	—
120	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
122	#335	—	86	2	no	—
123	#332	—	86	4	no	—
124	#322	—	2	1	no	—
125	C₈²	—	2	1	no	—
126	C₄₃	#9799	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	86	2	no	—
128	C₂×M₁₆	—	86	2	no	—
129	C₂×M₁₆	D₁₇₂	1	2	yes	—
130	C₂²×C₈	C₄×D₄₃	1	2	yes	—
131	C₄⋊Q₈	—	86	2	no	—
132	C₄⋊Q₈	—	86	2	no	—
133	C₄⋊Q₈	—	86	2	no	—
134	C₄⋊Q₈	—	86	2	no	—
135	C₂×C₄⋊C₄	—	43	4	no	—
136	C₂×C₄⋊C₄	—	86	2	no	—
137	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
138	#107	—	86	4	no	—
139	#107	—	86	4	no	—
140	C₄⋊C₈	—	86	2	no	—
141	C₄⋊C₈	—	86	2	no	—
142	C₄⋊C₈	D₁₇₂	1	2	yes	—
143	#101	—	172	8	no	—
144	C₂²⋊C₈	—	2	1	no	—
145	#97	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	D₁₇₂	1	2	yes	—
149	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
151	C₂×Q₈	—	172	2	no	—
152	C₂×Q₈	—	172	2	no	—
153	C₂×Q₈	—	172	4	no	—
154	C₂×Q₈	—	172	4	no	—
155	C₂²×C₄	—	86	4	no	—
156	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
158	C₂²×C₄	#10621898	1	1	yes	characteristic
159	M₁₆	—	2	2	no	—
160	M₁₆	—	172	4	no	—
161	M₁₆	D₃₄₄	1	4	yes	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	172	2	no	—
166	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
167	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
168	C₄⋊C₄	—	86	4	no	—
169	C₄⋊C₄	—	86	4	no	—
170	C₄⋊C₄	—	172	2	no	—
171	C₄⋊C₄	—	172	2	no	—
172	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
173	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
174	C₄²	#10621898	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
176	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
177	Q₈	—	344	4	no	—
178	Q₈	—	344	2	no	—
179	Q₈	—	344	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	172	4	no	—
183	C₂×C₄	—	172	8	no	—
184	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
186	C₂×C₄	#13063299	1	1	yes	characteristic
187	C₂×C₄	#13063299	1	1	yes	characteristic
188	C₂×C₄	#13063313	1	1	yes	characteristic
189	C₂×C₄	#13063322	1	1	yes	characteristic
190	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
191	C₂×C₄	#13063347	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	—	2	4	no	—
194	C₈	—	4	2	no	—
195	C₈	—	344	2	no	—
196	C₈	#13063243	1	4	yes	complemented
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#14582801	1	1	yes	characteristic
200	C₂²	C₄×D₃₄₄	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	344	4	no	—
204	C₄	C₂×#13063243	1	2	yes	—
205	C₄	#14583135	1	2	yes	—
206	C₄	#14583174	1	2	yes	—
207	C₄	#14583276	1	1	yes	characteristic
208	C₄	#14583283	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	#19135439	1	1	yes	characteristic
211	C₂	#19135461	1	1	yes	characteristic
212	C₂	#19135510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583304

C₂

yes

maximal, characteristic

#19135439

C₂

yes

maximal

#19133354

C₂

yes

maximal, characteristic

#19133266

C₂

yes

maximal

C₈×M₁₆×C₄₃

C₂

yes

maximal, characteristic

C₂×#13063325

C₂²

yes

—

—

—

—

C₄

yes

—

#14582923

C₂²

yes

—

C₂×#13063243

—

#14582652

—

#14582644

—

C₄₃×#431

C₂²

yes

characteristic

C₄×M₁₆×C₄₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#322

—

C₈²×C₄₃

—

C₂²×Dic₈₆

—

—

—

—

—

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂×C₄

yes

—

#13063259

—

#13063259

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063243

—

C₂×M₁₆×C₄₃

D₄

yes

—

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

D₄

yes

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

M₁₆×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

#41

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₄×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#101

yes

complemented

C₄₃⋊C₈

—

#9799

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#390

yes

characteristic

C₂²×C₄₃

C₄×D₈

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#101

yes

—

C₄×C₄₃

#466

yes

—

C₄×C₄₃

#473

yes

—

100

C₄×C₄₃

#493

yes

characteristic

101

C₄×C₄₃

#495

yes

characteristic

102

Dic₄₃

—

103

C₂×#500

—

104

#1966

—

105

#1567

—

106

#1546

—

107

C₈×M₁₆

D₄₃

yes

characteristic

108

C₂×C₄₃

—

109

C₂×C₄₃

#1966

yes

characteristic

110

C₂×C₄₃

#1970

yes

characteristic

111

C₂×C₄₃

#1979

yes

characteristic

112

C₂×C₄⋊Q₈

—

113

#500

—

114

#500

—

115

#500

—

116

#431

C₂×D₄₃

yes

characteristic

117

C₂×#107

—

118

#423

—

119

C₂×#101

—

120

C₄×M₁₆

C₂×D₄₃

yes

characteristic

121

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

122

#335

—

123

#332

—

124

#322

—

125

C₈²

—

126

C₄₃

#9799

yes

characteristic, complemented, Sylow

127

C₂²×Q₈

—

128

C₂×M₁₆

—

129

C₂×M₁₆

D₁₇₂

yes

—

130

C₂²×C₈

C₄×D₄₃

yes

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

133

C₄⋊Q₈

—

134

C₄⋊Q₈

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄²

C₂²×D₄₃

yes

characteristic

138

#107

—

139

#107

—

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

C₄⋊C₈

D₁₇₂

yes

—

143

#101

—

172

—

144

C₂²⋊C₈

—

145

#97

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

D₁₇₂

yes

—

149

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

150

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

151

C₂×Q₈

—

172

—

152

C₂×Q₈

—

172

—

153

C₂×Q₈

—

172

—

154

C₂×Q₈

—

172

—

155

C₂²×C₄

—

156

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

157

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

158

C₂²×C₄

#10621898

yes

characteristic

159

M₁₆

—

160

M₁₆

—

172

—

161

M₁₆

D₃₄₄

yes

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

172

—

166

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

167

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

172

—

171

C₄⋊C₄

—

172

—

172

C₄²

C₂×D₁₇₂

yes

characteristic

173

C₄²

C₂×D₁₇₂

yes

characteristic

174

C₄²

#10621898

yes

characteristic

175

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

176

C₂³

C₄×D₁₇₂

yes

characteristic

177

Q₈

—

344

—

178

Q₈

—

344

—

179

Q₈

—

344

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

172

—

183

C₂×C₄

—

172

—

184

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

185

C₂×C₄

C₄×D₁₇₂

yes

characteristic

186

C₂×C₄

#13063299

yes

characteristic

187

C₂×C₄

#13063299

yes

characteristic

188

C₂×C₄

#13063313

yes

characteristic

189

C₂×C₄

#13063322

yes

characteristic

190

C₂×C₄

C₂×D₃₄₄

yes

—

191

C₂×C₄

#13063347

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

344

—

196

C₈

#13063243

yes

complemented

197

C₂²

—

198

C₂²

—

199

C₂²

#14582801

yes

characteristic

200

C₂²

C₄×D₃₄₄

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

344

—

204

C₄

C₂×#13063243

yes

—

205

C₄

#14583135

yes

—

206

C₄

#14583174

yes

—

207

C₄

#14583276

yes

characteristic

208

C₄

#14583283

yes

characteristic

209

C₂

—

210

C₂

#19135439

yes

characteristic

211

C₂

#19135461

yes

characteristic

212

C₂

#19135510

yes

characteristic

	subgroup	G/H
center	C₂²	#14582801
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₈×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₈×M₁₆	D₄₃
hypocenter	C₄₃	#9799
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

subgroup

G/H

center

C₂²

#14582801

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₈×M₁₆×C₄₃

C₂

socle

C₂²×C₄₃

#390

prime radical

C₂³×C₄₃

C₄×D₄

cosocle

C₂×C₄²×C₄₃

C₂³

hypercenter

C₈×M₁₆

D₄₃

hypocenter

C₄₃

#9799

LCS

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

N	H ≅ G/N	type
88. C₈×C₄₃	#101	semidirect
126. C₄₃	#9799	semidirect
196. C₈	#13063243	semidirect

H ≅ G/N

type

88. C₈×C₄₃

#101

semidirect

126. C₄₃

#9799

semidirect

196. C₈

#13063243

semidirect

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

subgroup	index
C₂×#14583304	2
#19135439	2
#19133354	2
#19133266	2
C₈×M₁₆×C₄₃	2
#9799	43

subgroup

index

C₈×M₁₆×C₄₃

Subgroups — #24996214

Conjugacy classes of subgroups

Complements

Maximal subgroups