Group #24996505 — subgroups

Subgroups — #24996505

Order 11008.

454 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 127 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	#19145181	C₂	1	1	yes	maximal, characteristic
2	#19135571	C₂	1	2	yes	maximal
3	C₄×#13063235	C₂	1	1	yes	maximal, characteristic
4	#19133049	C₂	1	2	yes	maximal
5	C₄₃×#1566	C₂	1	1	yes	maximal, characteristic
6	#14583456	C₂²	1	2	yes	—
7	#14583304	C₄	1	1	yes	characteristic, complemented
8	#14583279	C₄	1	1	yes	characteristic, complemented
9	#14583250	C₄	1	2	yes	complemented
10	#14583140	—	2	2	no	—
11	C₄×Dic₁₇₂	—	2	2	no	—
12	C₂×#13063235	—	2	2	no	—
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
15	#14582660	—	2	4	no	—
16	#14582629	—	2	4	no	—
17	C₄₃×#423	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#334	—	2	2	no	—
20	C₂×Dic₁₇₂	—	2	2	no	—
21	C₂²×C₄₃⋊C₈	—	2	1	no	—
22	#13063325	C₂×C₄	1	2	yes	—
23	#13063325	C₂×C₄	1	2	yes	—
24	#13063317	—	2	2	no	—
25	#13063304	—	2	2	no	—
26	#13063304	—	2	2	no	—
27	C₄×Dic₈₆	—	2	2	no	—
28	C₄×Dic₈₆	—	2	2	no	—
29	#13063284	—	2	2	no	—
30	C₂×C₄×Dic₄₃	—	2	1	no	—
31	C₂×C₄×Dic₄₃	—	2	1	no	—
32	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
33	#13063259	—	2	2	no	—
34	#13063250	—	2	4	no	—
35	#13063235	—	2	8	no	—
36	C₄×C₄₃⋊C₈	—	2	1	no	—
37	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
38	C₂×M₁₆×C₄₃	—	2	2	no	—
39	C₂²×C₈×C₄₃	D₄	1	2	yes	—
40	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₃	—	2	2	no	—
42	C₄⋊C₈×C₄₃	—	2	2	no	—
43	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₄₃	D₄	1	2	yes	—
46	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
49	C₂²×Dic₄₃	—	2	2	no	—
50	C₂×Dic₈₆	—	2	2	no	—
51	C₂×Dic₈₆	—	2	2	no	—
52	M₁₆×C₄₃	—	4	4	no	—
53	C₂×C₈×C₄₃	—	2	4	no	—
54	C₂×C₈×C₄₃	—	4	2	no	—
55	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	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	1	yes	characteristic
60	C₄²×C₄₃	#41	1	1	yes	characteristic
61	C₂²⋊Dic₄₃	—	4	2	no	—
62	C₄⋊Dic₄₃	—	2	2	no	—
63	C₄⋊Dic₄₃	—	2	2	no	—
64	C₄⋊Dic₄₃	—	2	2	no	—
65	C₄⋊Dic₄₃	—	2	2	no	—
66	Dic₄₃⋊C₄	—	4	2	no	—
67	Dic₄₃⋊C₄	—	4	2	no	—
68	C₄×Dic₄₃	—	2	4	no	—
69	C₄×Dic₄₃	—	2	4	no	—
70	C₄×Dic₄₃	—	4	2	no	—
71	C₄×Dic₄₃	—	4	2	no	—
72	C₂×C₄₃⋊C₈	—	2	4	no	—
73	C₂×C₄₃⋊C₈	—	2	2	no	—
74	Dic₁₇₂	—	4	4	no	—
75	C₂³×C₄₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	2	2	no	—
78	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
82	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₄₃	#137	1	4	yes	—
85	C₂×Dic₄₃	—	2	4	no	—
86	C₂×Dic₄₃	—	4	2	no	—
87	C₂×Dic₄₃	—	4	2	no	—
88	C₂×Dic₄₃	—	4	2	no	—
89	C₂×Dic₄₃	—	4	2	no	—
90	Dic₈₆	—	4	4	no	—
91	Dic₈₆	—	4	4	no	—
92	C₈×C₄₃	—	2	2	no	—
93	C₈×C₄₃	—	8	2	no	—
94	C₈×C₄₃	#104	1	4	yes	—
95	C₄₃⋊C₈	—	2	4	no	—
96	#9850	—	43	1	no	maximal, Sylow
97	C₂²×C₄₃	—	2	1	no	—
98	C₂²×C₄₃	—	2	1	no	—
99	C₂²×C₄₃	#390	1	1	yes	characteristic
100	C₂²×C₄₃	#470	1	2	yes	—
101	C₄×C₄₃	—	2	1	no	—
102	C₄×C₄₃	—	2	1	no	—
103	C₄×C₄₃	C₂×#104	1	2	yes	—
104	C₄×C₄₃	#441	1	2	yes	—
105	C₄×C₄₃	#444	1	2	yes	—
106	C₄×C₄₃	#497	1	2	yes	—
107	Dic₄₃	—	4	4	no	—
108	Dic₄₃	—	8	2	no	—
109	Dic₄₃	—	8	2	no	—
110	#3170	—	43	1	no	—
111	#1994	—	43	2	no	—
112	C₄×#97	—	43	1	no	—
113	#1566	D₄₃	1	1	yes	characteristic
114	#1493	—	43	2	no	—
115	C₂×C₄₃	—	2	2	no	—
116	C₂×C₄₃	#1962	1	1	yes	characteristic
117	C₂×C₄₃	#1971	1	1	yes	characteristic
118	C₂×C₄₃	#1975	1	1	yes	characteristic
119	#533	—	43	2	no	—
120	#500	—	43	1	no	—
121	#494	—	43	1	no	—
122	#487	—	43	2	no	—
123	#467	—	86	2	no	—
124	C₄×Q₁₆	—	86	2	no	—
125	#423	C₂×D₄₃	1	2	yes	—
126	C₂×#97	—	86	2	no	—
127	C₂×C₄×C₈	—	43	1	no	—
128	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
129	C₄³	—	43	1	no	—
130	#339	—	86	4	no	—
131	#334	—	2	2	no	—
132	Q₈⋊C₈	—	86	4	no	—
133	C₄₃	#9850	1	1	yes	characteristic, complemented, Sylow
134	C₂×Q₁₆	—	86	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂²×C₈	—	86	1	no	—
137	C₂²×C₈	D₁₇₂	1	2	yes	—
138	C₄⋊Q₈	—	43	2	no	—
139	C₄⋊Q₈	—	43	2	no	—
140	#125	—	86	2	no	—
141	C₂²⋊Q₈	—	86	2	no	—
142	C₂²⋊Q₈	—	86	2	no	—
143	C₄×Q₈	—	86	2	no	—
144	C₄×Q₈	—	86	2	no	—
145	#117	—	86	2	no	—
146	C₂×C₄²	—	43	2	no	—
147	C₂×C₄²	—	86	1	no	—
148	C₂×C₄²	—	86	1	no	—
149	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
150	#107	—	86	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	#103	—	86	4	no	—
154	#97	—	86	8	no	—
155	C₄×C₈	—	43	2	no	—
156	C₄×C₈	—	86	1	no	—
157	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
158	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
160	C₂×Q₈	—	86	2	no	—
161	C₂×Q₈	—	86	2	no	—
162	C₂²×C₄	—	86	2	no	—
163	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
164	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
165	C₂²×C₄	#10621902	1	1	yes	characteristic
166	Q₁₆	—	172	4	no	—
167	M₁₆	—	4	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	86	4	no	—
171	C₂×C₈	—	86	2	no	—
172	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
173	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
174	C₄⋊C₄	—	86	2	no	—
175	C₄⋊C₄	—	86	2	no	—
176	C₄⋊C₄	—	86	2	no	—
177	C₄⋊C₄	—	86	2	no	—
178	C₄⋊C₄	—	172	2	no	—
179	C₄⋊C₄	—	172	2	no	—
180	C₂²⋊C₄	—	172	2	no	—
181	C₄²	—	86	4	no	—
182	C₄²	—	86	4	no	—
183	C₄²	—	172	2	no	—
184	C₄²	—	172	2	no	—
185	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
186	C₄²	D₄×D₄₃	1	1	yes	characteristic
187	C₄²	#10621902	1	1	yes	characteristic
188	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
189	C₂³	#13063300	1	1	yes	characteristic
190	Q₈	—	172	4	no	—
191	Q₈	—	172	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	86	4	no	—
195	C₂×C₄	—	172	2	no	—
196	C₂×C₄	—	172	2	no	—
197	C₂×C₄	—	172	2	no	—
198	C₂×C₄	—	172	2	no	—
199	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
200	C₂×C₄	#13063287	1	1	yes	characteristic
201	C₂×C₄	#13063287	1	1	yes	characteristic
202	C₂×C₄	#13063300	1	1	yes	characteristic
203	C₂×C₄	#13063314	1	1	yes	characteristic
204	C₂×C₄	#13063321	1	1	yes	characteristic
205	C₂×C₄	#13063354	1	4	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	8	2	no	—
208	C₈	—	86	4	no	—
209	C₈	#13063251	1	4	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	#14582800	1	1	yes	characteristic
213	C₂²	#14583158	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	344	2	no	—
217	C₄	—	344	2	no	—
218	C₄	—	172	4	no	—
219	C₄	C₂×#13063251	1	2	yes	—
220	C₄	#14583005	1	2	yes	—
221	C₄	#14583020	1	2	yes	—
222	C₄	#14583290	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	#19135422	1	1	yes	characteristic
225	C₂	#19135464	1	1	yes	characteristic
226	C₂	#19135486	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145181

C₂

yes

maximal, characteristic

#19135571

C₂

yes

maximal

C₄×#13063235

C₂

yes

maximal, characteristic

#19133049

C₂

yes

maximal

C₄₃×#1566

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

—

#14583304

C₄

yes

characteristic, complemented

#14583279

C₄

yes

characteristic, complemented

#14583250

C₄

yes

complemented

#14583140

—

C₄×Dic₁₇₂

—

C₂×#13063235

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₄²×Dic₄₃

C₂²

yes

characteristic

#14582660

—

#14582629

—

C₄₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#334

—

C₂×Dic₁₇₂

—

C₂²×C₄₃⋊C₈

—

#13063325

C₂×C₄

yes

—

#13063325

C₂×C₄

yes

—

#13063317

—

#13063304

—

#13063304

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

#13063284

—

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

D₄

yes

—

#13063259

—

#13063250

—

#13063235

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

D₄

yes

—

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

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

#41

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

C₂³×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

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×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#104

yes

—

C₄₃⋊C₈

—

#9850

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#390

yes

characteristic

100

C₂²×C₄₃

#470

yes

—

101

C₄×C₄₃

—

102

C₄×C₄₃

—

103

C₄×C₄₃

C₂×#104

yes

—

104

C₄×C₄₃

#441

yes

—

105

C₄×C₄₃

#444

yes

—

106

C₄×C₄₃

#497

yes

—

107

Dic₄₃

—

108

Dic₄₃

—

109

Dic₄₃

—

110

#3170

—

111

#1994

—

112

C₄×#97

—

113

#1566

D₄₃

yes

characteristic

114

#1493

—

115

C₂×C₄₃

—

116

C₂×C₄₃

#1962

yes

characteristic

117

C₂×C₄₃

#1971

yes

characteristic

118

C₂×C₄₃

#1975

yes

characteristic

119

#533

—

120

#500

—

121

#494

—

122

#487

—

123

#467

—

124

C₄×Q₁₆

—

125

#423

C₂×D₄₃

yes

—

126

C₂×#97

—

127

C₂×C₄×C₈

—

128

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

129

C₄³

—

130

#339

—

131

#334

—

132

Q₈⋊C₈

—

133

C₄₃

#9850

yes

characteristic, complemented, Sylow

134

C₂×Q₁₆

—

135

C₂×M₁₆

—

136

C₂²×C₈

—

137

C₂²×C₈

D₁₇₂

yes

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

—

140

#125

—

141

C₂²⋊Q₈

—

142

C₂²⋊Q₈

—

143

C₄×Q₈

—

144

C₄×Q₈

—

145

#117

—

146

C₂×C₄²

—

147

C₂×C₄²

—

148

C₂×C₄²

—

149

C₂×C₄²

C₂²×D₄₃

yes

characteristic

150

#107

—

151

C₄⋊C₈

—

152

C₄⋊C₈

—

153

#103

—

154

#97

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

C₄×D₄₃

yes

characteristic

158

C₄×C₈

C₄×D₄₃

yes

characteristic

159

C₄×C₈

C₂²⋊D₄₃

yes

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂²×C₄

—

163

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

164

C₂²×C₄

D₄×D₄₃

yes

characteristic

165

C₂²×C₄

#10621902

yes

characteristic

166

Q₁₆

—

172

—

167

M₁₆

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

173

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

178

C₄⋊C₄

—

172

—

179

C₄⋊C₄

—

172

—

180

C₂²⋊C₄

—

172

—

181

C₄²

—

182

C₄²

—

183

C₄²

—

172

—

184

C₄²

—

172

—

185

C₄²

C₂×C₄×D₄₃

yes

characteristic

186

C₄²

D₄×D₄₃

yes

characteristic

187

C₄²

#10621902

yes

characteristic

188

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

189

C₂³

#13063300

yes

characteristic

190

Q₈

—

172

—

191

Q₈

—

172

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

172

—

196

C₂×C₄

—

172

—

197

C₂×C₄

—

172

—

198

C₂×C₄

—

172

—

199

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

200

C₂×C₄

#13063287

yes

characteristic

201

C₂×C₄

#13063287

yes

characteristic

202

C₂×C₄

#13063300

yes

characteristic

203

C₂×C₄

#13063314

yes

characteristic

204

C₂×C₄

#13063321

yes

characteristic

205

C₂×C₄

#13063354

yes

—

206

C₈

—

207

C₈

—

208

C₈

—

209

C₈

#13063251

yes

—

210

C₂²

—

211

C₂²

—

212

C₂²

#14582800

yes

characteristic

213

C₂²

#14583158

yes

—

214

C₄

—

215

C₄

—

216

C₄

—

344

—

217

C₄

—

344

—

218

C₄

—

172

—

219

C₄

C₂×#13063251

yes

—

220

C₄

#14583005

yes

—

221

C₄

#14583020

yes

—

222

C₄

#14583290

yes

—

223

C₂

—

224

C₂

#19135422

yes

characteristic

225

C₂

#19135464

yes

characteristic

226

C₂

#19135486

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063321
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1566	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂³×C₄₃	#121
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1566	D₄₃
hypocenter	C₄₃	#9850
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#13063321

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1566

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
7. #14583304	C₄	semidirect
8. #14583279	C₄	semidirect
9. #14583250	C₄	semidirect
133. C₄₃	#9850	semidirect

H ≅ G/N

type

7. #14583304

C₄

semidirect

8. #14583279

C₄

semidirect

9. #14583250

C₄

semidirect

133. C₄₃

#9850

semidirect

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

subgroup	index
#19145181	2
#19135571	2
C₄×#13063235	2
#19133049	2
C₄₃×#1566	2
#9850	43

subgroup

index

C₄₃×#1566

Subgroups — #24996505

Conjugacy classes of subgroups

Complements

Maximal subgroups