Group #25001623 — subgroups

Subgroups — #25001623

Order 11008.

296 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19135576	C₂	1	1	yes	maximal, characteristic
2	#19134715	C₂	1	1	yes	maximal, characteristic
3	#19133370	C₂	1	1	yes	maximal, characteristic
4	#19133366	C₂	1	1	yes	maximal, characteristic
5	#19132999	C₂	1	1	yes	maximal, characteristic
6	#19132999	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1274	C₂	1	1	yes	maximal, characteristic
8	#14583456	C₂²	1	1	yes	characteristic
9	#14582923	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	#14582659	—	2	2	no	—
13	#14582654	—	2	2	no	—
14	#14582652	—	2	2	no	—
15	#14582650	—	2	2	no	—
16	#14582631	—	2	2	no	—
17	#14582631	—	2	2	no	—
18	C₄₃×#431	C₂²	1	1	yes	characteristic
19	C₄₃×#431	C₂²	1	1	yes	characteristic
20	C₄₃×#376	C₂²	1	1	yes	characteristic
21	C₂×#10621877	—	2	1	no	—
22	C₂²×C₄₃⋊C₈	—	2	1	no	—
23	C₂²×C₄₃⋊C₈	—	2	1	no	—
24	#13063325	Q₈	1	1	yes	characteristic
25	#13063325	Q₈	1	1	yes	characteristic
26	#13063317	—	2	1	no	—
27	#13063304	—	2	1	no	—
28	#13063304	—	2	1	no	—
29	C₄×Dic₈₆	—	2	1	no	—
30	C₄×Dic₈₆	—	2	1	no	—
31	#13063284	—	2	1	no	—
32	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
33	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
34	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
36	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
37	C₄⋊(C₄₃⋊C₈)	—	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	—	2	1	no	—
42	C₂²⋊C₈×C₄₃	—	2	1	no	—
43	C₂²⋊C₈×C₄₃	—	2	1	no	—
44	C₄×C₈×C₄₃	D₄	1	2	yes	—
45	C₄×C₈×C₄₃	D₄	1	2	yes	—
46	#95×C₄₃	—	2	1	no	—
47	#95×C₄₃	—	2	1	no	—
48	C₂²×C₄×C₄₃	—	2	2	no	—
49	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
52	C₂×Dic₈₆	—	2	1	no	—
53	C₂×Dic₈₆	—	2	1	no	—
54	C₂×C₈×C₄₃	—	2	2	no	—
55	C₂×C₈×C₄₃	—	2	2	no	—
56	C₄²×C₄₃	—	2	2	no	—
57	C₄²×C₄₃	—	2	2	no	—
58	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₄₃	#41	1	1	yes	characteristic
62	C₂²⋊Dic₄₃	—	4	1	no	—
63	C₄⋊Dic₄₃	—	2	1	no	—
64	C₄⋊Dic₄₃	—	2	1	no	—
65	C₄⋊Dic₄₃	—	2	1	no	—
66	C₄⋊Dic₄₃	—	2	1	no	—
67	Dic₄₃⋊C₄	—	4	1	no	—
68	Dic₄₃⋊C₄	—	4	1	no	—
69	C₄×Dic₄₃	—	4	1	no	—
70	#10621877	—	4	2	no	—
71	C₂×C₄₃⋊C₈	—	4	1	no	—
72	C₂×C₄₃⋊C₈	—	4	1	no	—
73	C₂×C₄₃⋊C₈	—	4	1	no	—
74	C₂×C₄₃⋊C₈	—	4	2	no	—
75	C₂×C₄₃⋊C₈	—	4	2	no	—
76	C₂³×C₄₃	#123	1	1	yes	characteristic
77	C₂×C₄×C₄₃	—	4	1	no	—
78	C₂×C₄×C₄₃	—	4	1	no	—
79	C₂×C₄×C₄₃	—	4	2	no	—
80	C₂×C₄×C₄₃	—	4	2	no	—
81	C₂×C₄×C₄₃	—	4	2	no	—
82	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
86	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
87	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₄₃	#135	1	2	yes	—
89	C₂×C₄×C₄₃	#137	1	2	yes	—
90	C₂×Dic₄₃	—	4	1	no	—
91	C₂×Dic₄₃	—	4	1	no	—
92	Dic₈₆	—	4	2	no	—
93	Dic₈₆	—	4	2	no	—
94	C₈×C₄₃	—	4	2	no	—
95	C₈×C₄₃	—	4	2	no	—
96	C₄₃⋊C₈	—	8	1	no	—
97	C₄₃⋊C₈	—	8	1	no	—
98	C₄₃⋊C₈	—	8	1	no	—
99	#10669	—	43	1	no	maximal, Sylow
100	C₂²×C₄₃	—	2	1	no	—
101	C₂²×C₄₃	—	2	1	no	—
102	C₂²×C₄₃	#399	1	1	yes	characteristic
103	C₂²×C₄₃	#485	1	2	yes	—
104	C₄×C₄₃	—	2	1	no	—
105	C₄×C₄₃	—	2	1	no	—
106	C₄×C₄₃	—	2	1	no	—
107	C₄×C₄₃	—	2	1	no	—
108	C₄×C₄₃	—	8	2	no	—
109	C₄×C₄₃	#479	1	2	yes	—
110	C₄×C₄₃	#495	1	1	yes	characteristic
111	C₄×C₄₃	#497	1	1	yes	characteristic
112	Dic₄₃	—	8	1	no	—
113	Dic₄₃	—	8	1	no	—
114	#1995	—	43	1	no	—
115	#1832	—	43	1	no	—
116	#1571	—	43	1	no	—
117	#1570	—	43	1	no	—
118	#1481	—	43	1	no	—
119	#1481	—	43	1	no	—
120	#1274	D₄₃	1	1	yes	characteristic
121	C₂×C₄₃	—	2	2	no	—
122	C₂×C₄₃	#2097	1	1	yes	characteristic
123	C₂×C₄₃	#2098	1	1	yes	characteristic
124	C₂×C₄₃	#2098	1	1	yes	characteristic
125	#533	—	43	1	no	—
126	#431	C₂×D₄₃	1	1	yes	characteristic
127	#431	C₂×D₄₃	1	1	yes	characteristic
128	#423	—	43	1	no	—
129	C₂×C₄⋊C₈	—	43	1	no	—
130	C₂×C₄⋊C₈	—	43	1	no	—
131	#376	C₂×D₄₃	1	1	yes	characteristic
132	#339	—	86	2	no	—
133	#336	—	86	2	no	—
134	#335	—	86	2	no	—
135	#334	—	86	2	no	—
136	Q₈⋊C₈	—	86	2	no	—
137	Q₈⋊C₈	—	86	2	no	—
138	C₄₃	#10669	1	1	yes	characteristic, complemented, Sylow
139	C₂×M₁₆	—	86	1	no	—
140	C₂²×C₈	—	86	1	no	—
141	C₂²×C₈	—	86	1	no	—
142	C₄⋊Q₈	—	43	1	no	—
143	C₄⋊Q₈	—	43	1	no	—
144	#125	—	86	1	no	—
145	C₂²⋊Q₈	—	86	1	no	—
146	C₂²⋊Q₈	—	86	1	no	—
147	C₄×Q₈	—	86	1	no	—
148	C₄×Q₈	—	86	1	no	—
149	#117	—	86	1	no	—
150	C₂×C₄²	D₁₇₂	1	2	yes	—
151	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	86	1	no	—
155	C₄⋊C₈	—	86	1	no	—
156	C₄⋊C₈	—	86	1	no	—
157	C₄⋊C₈	—	86	1	no	—
158	C₄⋊C₈	—	86	1	no	—
159	C₄⋊C₈	—	86	1	no	—
160	C₂²⋊C₈	—	2	1	no	—
161	C₂²⋊C₈	—	2	1	no	—
162	C₄×C₈	D₁₇₂	1	2	yes	—
163	C₄×C₈	D₁₇₂	1	2	yes	—
164	#95	—	2	1	no	—
165	#95	—	2	1	no	—
166	C₂×Q₈	—	86	1	no	—
167	C₂×Q₈	—	86	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
170	C₂²×C₄	#10621900	1	1	yes	characteristic
171	C₂²×C₄	#10621900	1	1	yes	characteristic
172	M₁₆	—	172	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	172	1	no	—
176	C₂×C₈	—	172	1	no	—
177	C₂×C₈	—	172	1	no	—
178	C₂×C₈	—	172	2	no	—
179	C₂×C₈	—	172	2	no	—
180	C₄⋊C₄	—	86	1	no	—
181	C₄⋊C₄	—	86	1	no	—
182	C₄⋊C₄	—	86	1	no	—
183	C₄⋊C₄	—	86	1	no	—
184	C₄⋊C₄	—	172	1	no	—
185	C₄⋊C₄	—	172	1	no	—
186	C₂²⋊C₄	—	172	1	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	172	1	no	—
190	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
191	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
192	C₄²	#10621900	1	1	yes	characteristic
193	C₄²	Q₈×D₄₃	1	1	yes	characteristic
194	C₂³	#13063309	1	1	yes	characteristic
195	Q₈	—	172	2	no	—
196	Q₈	—	172	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	172	1	no	—
203	C₂×C₄	—	172	1	no	—
204	C₂×C₄	#13063305	1	1	yes	characteristic
205	C₂×C₄	#13063305	1	1	yes	characteristic
206	C₂×C₄	#13063305	1	1	yes	characteristic
207	C₂×C₄	#13063309	1	1	yes	characteristic
208	C₂×C₄	#13063309	1	1	yes	characteristic
209	C₂×C₄	#13063322	1	1	yes	characteristic
210	C₂×C₄	#13063347	1	2	yes	—
211	C₂×C₄	#13063358	1	2	yes	—
212	C₈	—	4	2	no	—
213	C₈	—	4	2	no	—
214	C₈	—	344	1	no	—
215	C₈	—	344	1	no	—
216	C₈	—	344	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	#14582836	1	1	yes	characteristic
220	C₂²	#14583241	1	2	yes	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	8	2	no	—
226	C₄	—	344	1	no	—
227	C₄	—	344	1	no	—
228	C₄	#14583207	1	2	yes	—
229	C₄	#14583283	1	1	yes	characteristic
230	C₄	#14583293	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	#19136142	1	1	yes	characteristic
233	C₂	#19136149	1	1	yes	characteristic
234	C₂	#19136149	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19135576

C₂

yes

maximal, characteristic

#19134715

C₂

yes

maximal, characteristic

#19133370

C₂

yes

maximal, characteristic

#19133366

C₂

yes

maximal, characteristic

#19132999

C₂

yes

maximal, characteristic

#19132999

C₂

yes

maximal, characteristic

C₄₃×#1274

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

characteristic

#14582923

C₂²

yes

characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

—

—

—

—

—

C₄₃×#431

C₂²

yes

characteristic

C₄₃×#431

C₂²

yes

characteristic

C₄₃×#376

C₂²

yes

characteristic

C₂×#10621877

—

C₂²×C₄₃⋊C₈

—

C₂²×C₄₃⋊C₈

—

#13063325

Q₈

yes

characteristic

#13063325

Q₈

yes

—

—

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

#13063284

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

#95×C₄₃

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

C₂²×C₄×C₄₃

yes

C₂²×C₄×C₄₃

yes

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

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

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#123

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#10669

—

maximal, Sylow

100

C₂²×C₄₃

—

101

C₂²×C₄₃

—

102

C₂²×C₄₃

#399

yes

characteristic

103

C₂²×C₄₃

#485

yes

—

104

C₄×C₄₃

—

105

C₄×C₄₃

—

106

C₄×C₄₃

—

107

C₄×C₄₃

—

108

C₄×C₄₃

—

109

C₄×C₄₃

#479

yes

—

110

C₄×C₄₃

#495

yes

characteristic

111

C₄×C₄₃

#497

yes

characteristic

112

Dic₄₃

—

113

Dic₄₃

—

114

#1995

—

115

#1832

—

116

#1571

—

117

#1570

—

118

#1481

—

119

#1481

—

120

#1274

D₄₃

yes

characteristic

121

C₂×C₄₃

—

122

C₂×C₄₃

#2097

yes

characteristic

123

C₂×C₄₃

#2098

yes

characteristic

124

C₂×C₄₃

#2098

yes

characteristic

125

#533

—

126

#431

C₂×D₄₃

yes

characteristic

127

#431

C₂×D₄₃

yes

characteristic

128

#423

—

129

C₂×C₄⋊C₈

—

130

C₂×C₄⋊C₈

—

131

#376

C₂×D₄₃

yes

characteristic

132

#339

—

133

#336

—

134

#335

—

135

#334

—

136

Q₈⋊C₈

—

137

Q₈⋊C₈

—

138

C₄₃

#10669

yes

characteristic, complemented, Sylow

139

C₂×M₁₆

—

140

C₂²×C₈

—

141

C₂²×C₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

#125

—

145

C₂²⋊Q₈

—

146

C₂²⋊Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

#117

—

150

C₂×C₄²

D₁₇₂

yes

—

151

C₂×C₄²

C₂²×D₄₃

yes

characteristic

152

C₄⋊C₈

—

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

C₄⋊C₈

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

C₂²⋊C₈

—

161

C₂²⋊C₈

—

162

C₄×C₈

D₁₇₂

yes

—

163

C₄×C₈

D₁₇₂

yes

—

164

#95

—

165

#95

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂²×C₄

—

169

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

170

C₂²×C₄

#10621900

yes

characteristic

171

C₂²×C₄

#10621900

yes

characteristic

172

M₁₆

—

172

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

172

—

176

C₂×C₈

—

172

—

177

C₂×C₈

—

172

—

178

C₂×C₈

—

172

—

179

C₂×C₈

—

172

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

172

—

185

C₄⋊C₄

—

172

—

186

C₂²⋊C₄

—

172

—

187

C₄²

—

188

C₄²

—

189

C₄²

—

172

—

190

C₄²

C₂×D₁₇₂

yes

characteristic

191

C₄²

C₂×D₁₇₂

yes

characteristic

192

C₄²

#10621900

yes

characteristic

193

C₄²

Q₈×D₄₃

yes

characteristic

194

C₂³

#13063309

yes

characteristic

195

Q₈

—

172

—

196

Q₈

—

172

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

172

—

203

C₂×C₄

—

172

—

204

C₂×C₄

#13063305

yes

characteristic

205

C₂×C₄

#13063305

yes

characteristic

206

C₂×C₄

#13063305

yes

characteristic

207

C₂×C₄

#13063309

yes

characteristic

208

C₂×C₄

#13063309

yes

characteristic

209

C₂×C₄

#13063322

yes

characteristic

210

C₂×C₄

#13063347

yes

—

211

C₂×C₄

#13063358

yes

—

212

C₈

—

213

C₈

—

214

C₈

—

344

—

215

C₈

—

344

—

216

C₈

—

344

—

217

C₂²

—

218

C₂²

—

219

C₂²

#14582836

yes

characteristic

220

C₂²

#14583241

yes

—

221

C₄

—

222

C₄

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

344

—

227

C₄

—

344

—

228

C₄

#14583207

yes

—

229

C₄

#14583283

yes

characteristic

230

C₄

#14583293

yes

characteristic

231

C₂

—

232

C₂

#19136142

yes

characteristic

233

C₂

#19136149

yes

characteristic

234

C₂

#19136149

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063322
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1274	C₂
socle	C₂²×C₄₃	#399
prime radical	C₂³×C₄₃	#123
hypercenter	#1274	D₄₃
hypocenter	C₄₃	#10669
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#13063322

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1274

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
138. C₄₃	#10669	semidirect

H ≅ G/N

type

138. C₄₃

#10669

semidirect

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

subgroup	index
#19135576	2
#19134715	2
#19133370	2
#19133366	2
#19132999	2
C₄₃×#1274	2
#10669	43

subgroup

index

C₄₃×#1274

Subgroups — #25001623

Conjugacy classes of subgroups

Complements

Maximal subgroups