Group #24996524 — subgroups

Subgroups — #24996524

Order 11008.

422 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 123 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	#19135572	C₂	1	2	yes	maximal
3	#19133351	C₂	1	1	yes	maximal, characteristic
4	#19133051	C₂	1	2	yes	maximal
5	C₄₃×#1744	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	#14582923	C₂²	1	2	yes	—
13	#14582659	—	2	4	no	—
14	#14582650	—	2	2	no	—
15	#14582631	—	2	4	no	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#336	—	2	2	no	—
20	C₂×Dic₁₇₂	—	2	2	no	—
21	C₂×#10621877	—	2	2	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	#13063259	—	2	2	no	—
31	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
33	#13063250	—	2	4	no	—
34	C₂²×C₈×C₄₃	—	2	1	no	—
35	C₂²×C₈×C₄₃	D₄	1	2	yes	—
36	C₂×C₄²×C₄₃	—	2	1	no	—
37	C₂×C₄²×C₄₃	—	2	1	no	—
38	C₂×C₄²×C₄₃	D₄	1	2	yes	—
39	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₄₃	—	2	1	no	—
41	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₄₃	D₄	1	2	yes	—
44	C₄×C₈×C₄₃	D₄	1	2	yes	—
45	C₂²×C₄×C₄₃	—	2	2	no	—
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	C₂×Dic₈₆	—	2	2	no	—
50	C₂×Dic₈₆	—	2	2	no	—
51	C₂×C₈×C₄₃	—	2	2	no	—
52	C₂×C₈×C₄₃	—	4	2	no	—
53	C₂×C₈×C₄₃	—	4	2	no	—
54	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₄₃	—	2	4	no	—
57	C₄²×C₄₃	—	2	4	no	—
58	C₄²×C₄₃	—	4	2	no	—
59	C₄²×C₄₃	C₂²×C₄	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	C₂²⋊Dic₄₃	—	4	2	no	—
64	C₄⋊Dic₄₃	—	2	2	no	—
65	C₄⋊Dic₄₃	—	2	2	no	—
66	C₄⋊Dic₄₃	—	2	2	no	—
67	C₄⋊Dic₄₃	—	2	2	no	—
68	Dic₄₃⋊C₄	—	4	2	no	—
69	Dic₄₃⋊C₄	—	4	2	no	—
70	C₄×Dic₄₃	—	4	2	no	—
71	#10621877	—	4	4	no	—
72	C₂×C₄₃⋊C₈	—	4	2	no	—
73	Dic₁₇₂	—	4	4	no	—
74	C₂³×C₄₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₄₃	—	2	2	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	2	4	no	—
78	C₂×C₄×C₄₃	—	4	2	no	—
79	C₂×C₄×C₄₃	—	4	2	no	—
80	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₄₃	#137	1	4	yes	—
87	C₂×Dic₄₃	—	4	2	no	—
88	C₂×Dic₄₃	—	4	2	no	—
89	Dic₈₆	—	4	4	no	—
90	Dic₈₆	—	4	4	no	—
91	C₈×C₄₃	—	2	2	no	—
92	C₈×C₄₃	—	2	2	no	—
93	C₈×C₄₃	—	4	2	no	—
94	C₄₃⋊C₈	—	8	2	no	—
95	#9853	—	43	1	no	maximal, Sylow
96	C₂²×C₄₃	—	2	1	no	—
97	C₂²×C₄₃	—	2	1	no	—
98	C₂²×C₄₃	#390	1	1	yes	characteristic
99	C₂²×C₄₃	#470	1	2	yes	—
100	C₄×C₄₃	—	2	1	no	—
101	C₄×C₄₃	—	2	1	no	—
102	C₄×C₄₃	—	4	4	no	—
103	C₄×C₄₃	#421	1	2	yes	—
104	C₄×C₄₃	#441	1	2	yes	—
105	C₄×C₄₃	#443	1	2	yes	—
106	C₄×C₄₃	#497	1	2	yes	—
107	Dic₄₃	—	8	2	no	—
108	Dic₄₃	—	8	2	no	—
109	#3170	—	43	1	no	—
110	#1994	—	43	2	no	—
111	#1744	D₄₃	1	1	yes	characteristic
112	#1566	—	43	1	no	—
113	#1493	—	43	2	no	—
114	C₂×C₄₃	—	2	2	no	—
115	C₂×C₄₃	#1963	1	1	yes	characteristic
116	C₂×C₄₃	#1971	1	1	yes	characteristic
117	C₂×C₄₃	#1974	1	1	yes	characteristic
118	#533	—	43	2	no	—
119	#500	—	43	1	no	—
120	#494	—	43	1	no	—
121	#487	—	43	2	no	—
122	#467	—	86	2	no	—
123	C₄×Q₁₆	—	86	2	no	—
124	#423	—	43	2	no	—
125	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
127	C₄³	C₂×D₄₃	1	1	yes	characteristic
128	#339	—	86	4	no	—
129	#336	—	2	2	no	—
130	#334	—	86	2	no	—
131	Q₈⋊C₈	—	86	4	no	—
132	C₄₃	#9853	1	1	yes	characteristic, complemented, Sylow
133	C₂×Q₁₆	—	86	2	no	—
134	C₂×M₁₆	—	86	2	no	—
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
137	C₄⋊Q₈	—	43	2	no	—
138	C₄⋊Q₈	—	43	2	no	—
139	#125	—	86	2	no	—
140	C₂²⋊Q₈	—	86	2	no	—
141	C₂²⋊Q₈	—	86	2	no	—
142	C₄×Q₈	—	86	2	no	—
143	C₄×Q₈	—	86	2	no	—
144	#117	—	86	2	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	D₁₇₂	1	2	yes	—
148	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
149	#107	—	86	2	no	—
150	C₄⋊C₈	—	86	2	no	—
151	C₄⋊C₈	—	86	2	no	—
152	#103	—	86	4	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
155	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
156	C₄×C₈	D₁₇₂	1	2	yes	—
157	C₄×C₈	D₁₇₂	1	2	yes	—
158	C₂×Q₈	—	86	2	no	—
159	C₂×Q₈	—	86	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
162	C₂²×C₄	#10621898	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
164	Q₁₆	—	172	4	no	—
165	M₁₆	—	172	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	172	2	no	—
170	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
171	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
172	C₄⋊C₄	—	86	2	no	—
173	C₄⋊C₄	—	86	2	no	—
174	C₄⋊C₄	—	86	2	no	—
175	C₄⋊C₄	—	86	2	no	—
176	C₄⋊C₄	—	172	2	no	—
177	C₄⋊C₄	—	172	2	no	—
178	C₂²⋊C₄	—	172	2	no	—
179	C₄²	—	2	4	no	—
180	C₄²	—	2	4	no	—
181	C₄²	—	4	2	no	—
182	C₄²	—	172	2	no	—
183	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
184	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
185	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
186	C₄²	#10621898	1	1	yes	characteristic
187	C₂³	#13063299	1	1	yes	characteristic
188	Q₈	—	172	4	no	—
189	Q₈	—	172	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	172	2	no	—
196	C₂×C₄	—	172	2	no	—
197	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
198	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
199	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
200	C₂×C₄	#13063299	1	1	yes	characteristic
201	C₂×C₄	#13063313	1	1	yes	characteristic
202	C₂×C₄	#13063322	1	1	yes	characteristic
203	C₂×C₄	#13063358	1	4	yes	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	4	2	no	—
207	C₈	—	344	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#14582801	1	1	yes	characteristic
211	C₂²	#14583157	1	2	yes	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	344	2	no	—
216	C₄	—	344	2	no	—
217	C₄	#14582917	1	2	yes	—
218	C₄	#14583007	1	2	yes	—
219	C₄	#14583017	1	2	yes	—
220	C₄	#14583293	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	#19135427	1	1	yes	characteristic
223	C₂	#19135466	1	1	yes	characteristic
224	C₂	#19135483	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145181

C₂

yes

maximal, characteristic

#19135572

C₂

yes

maximal

#19133351

C₂

yes

maximal, characteristic

#19133051

C₂

yes

maximal

C₄₃×#1744

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

—

#14583304

C₄

yes

characteristic, complemented

#14583279

C₄

yes

characteristic, complemented

#14583250

C₄

yes

complemented

#14583140

—

C₄×Dic₁₇₂

—

#14582923

C₂²

yes

—

#14582659

—

#14582650

—

#14582631

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄³×C₄₃

C₂²

yes

characteristic

C₄₃×#336

—

C₂×Dic₁₇₂

—

C₂×#10621877

—

#13063325

C₂×C₄

yes

—

#13063325

C₂×C₄

yes

—

#13063317

—

#13063304

—

#13063304

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

#13063284

—

#13063259

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063250

—

C₂²×C₈×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

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

—

#10621877

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

C₂³×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

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

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#9853

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#390

yes

characteristic

C₂²×C₄₃

#470

yes

—

100

C₄×C₄₃

—

101

C₄×C₄₃

—

102

C₄×C₄₃

—

103

C₄×C₄₃

#421

yes

—

104

C₄×C₄₃

#441

yes

—

105

C₄×C₄₃

#443

yes

—

106

C₄×C₄₃

#497

yes

—

107

Dic₄₃

—

108

Dic₄₃

—

109

#3170

—

110

#1994

—

111

#1744

D₄₃

yes

characteristic

112

#1566

—

113

#1493

—

114

C₂×C₄₃

—

115

C₂×C₄₃

#1963

yes

characteristic

116

C₂×C₄₃

#1971

yes

characteristic

117

C₂×C₄₃

#1974

yes

characteristic

118

#533

—

119

#500

—

120

#494

—

121

#487

—

122

#467

—

123

C₄×Q₁₆

—

124

#423

—

125

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

126

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

127

C₄³

C₂×D₄₃

yes

characteristic

128

#339

—

129

#336

—

130

#334

—

131

Q₈⋊C₈

—

132

C₄₃

#9853

yes

characteristic, complemented, Sylow

133

C₂×Q₁₆

—

134

C₂×M₁₆

—

135

C₂²×C₈

—

136

C₂²×C₈

C₂²⋊D₄₃

yes

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

#125

—

140

C₂²⋊Q₈

—

141

C₂²⋊Q₈

—

142

C₄×Q₈

—

143

C₄×Q₈

—

144

#117

—

145

C₂×C₄²

—

146

C₂×C₄²

—

147

C₂×C₄²

D₁₇₂

yes

—

148

C₂×C₄²

C₂²×D₄₃

yes

characteristic

149

#107

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

#103

—

153

C₄×C₈

—

154

C₄×C₈

C₄×D₄₃

yes

characteristic

155

C₄×C₈

C₄×D₄₃

yes

characteristic

156

C₄×C₈

D₁₇₂

yes

—

157

C₄×C₈

D₁₇₂

yes

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂²×C₄

—

161

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

162

C₂²×C₄

#10621898

yes

characteristic

163

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

164

Q₁₆

—

172

—

165

M₁₆

—

172

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

172

—

170

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

171

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

172

—

177

C₄⋊C₄

—

172

—

178

C₂²⋊C₄

—

172

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

182

C₄²

—

172

—

183

C₄²

C₂×C₄×D₄₃

yes

characteristic

184

C₄²

C₂×D₁₇₂

yes

characteristic

185

C₄²

C₂×D₁₇₂

yes

characteristic

186

C₄²

#10621898

yes

characteristic

187

C₂³

#13063299

yes

characteristic

188

Q₈

—

172

—

189

Q₈

—

172

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

172

—

196

C₂×C₄

—

172

—

197

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

198

C₂×C₄

C₄×D₁₇₂

yes

characteristic

199

C₂×C₄

C₄×D₁₇₂

yes

characteristic

200

C₂×C₄

#13063299

yes

characteristic

201

C₂×C₄

#13063313

yes

characteristic

202

C₂×C₄

#13063322

yes

characteristic

203

C₂×C₄

#13063358

yes

—

204

C₈

—

205

C₈

—

206

C₈

—

207

C₈

—

344

—

208

C₂²

—

209

C₂²

—

210

C₂²

#14582801

yes

characteristic

211

C₂²

#14583157

yes

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

344

—

216

C₄

—

344

—

217

C₄

#14582917

yes

—

218

C₄

#14583007

yes

—

219

C₄

#14583017

yes

—

220

C₄

#14583293

yes

—

221

C₂

—

222

C₂

#19135427

yes

characteristic

223

C₂

#19135466

yes

characteristic

224

C₂

#19135483

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063322
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1744	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂³×C₄₃	#121
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1744	D₄₃
hypocenter	C₄₃	#9853
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center

C₂×C₄

#13063322

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1744

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
7. #14583304	C₄	semidirect
8. #14583279	C₄	semidirect
9. #14583250	C₄	semidirect
132. C₄₃	#9853	semidirect

H ≅ G/N

type

7. #14583304

C₄

semidirect

8. #14583279

C₄

semidirect

9. #14583250

C₄

semidirect

132. C₄₃

#9853

semidirect

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

subgroup	index
#19145181	2
#19135572	2
#19133351	2
#19133051	2
C₄₃×#1744	2
#9853	43

subgroup

index

C₄₃×#1744

Subgroups — #24996524

Conjugacy classes of subgroups

Complements

Maximal subgroups