Group #24991924 — subgroups

Subgroups — #24991924

Order 11008.

426 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 111 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	#19143697	C₂	1	1	yes	maximal, characteristic, complemented
2	#19133190	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133012	C₂	1	2	yes	maximal
4	#19132432	C₂	1	2	yes	maximal, complemented
5	C₄₃×#3131	C₂	1	1	yes	maximal, characteristic
6	#14583030	—	2	2	no	—
7	Q₈×C₄₃⋊C₈	—	2	2	no	—
8	#14582960	—	2	2	no	—
9	#14582952	C₂²	1	2	yes	—
10	C₄×#10621877	C₂²	1	2	yes	—
11	#14582641	—	2	2	no	—
12	D₄⋊(C₄₃⋊C₈)	—	2	2	no	—
13	D₄⋊(C₄₃⋊C₈)	—	2	2	no	—
14	C₄₃×#533	C₂²	1	1	yes	characteristic
15	C₄₃×#532	C₂²	1	1	yes	characteristic
16	C₄₃×#488	—	2	2	no	—
17	C₄₃×#487	C₄	1	2	yes	—
18	C₄₃×#486	C₄	1	2	yes	—
19	C₄₃×#416	—	2	2	no	—
20	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×#10621877	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
23	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
24	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
25	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
26	#13063236	—	2	2	no	—
27	#13063236	—	2	2	no	—
28	C₄×C₄₃⋊C₈	—	2	2	no	—
29	C₄×C₄₃⋊C₈	—	2	2	no	—
30	C₄×C₄₃⋊C₈	—	2	2	no	—
31	C₂×#41×C₄₃	—	2	1	no	—
32	C₂²×C₈×C₄₃	—	2	1	no	—
33	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
35	#127×C₄₃	C₂×C₄	1	1	yes	characteristic
36	#125×C₄₃	D₄	1	2	yes	—
37	#124×C₄₃	—	2	1	no	—
38	C₂²⋊Q₈×C₄₃	—	2	2	no	—
39	#121×C₄₃	—	2	2	no	—
40	C₄×Q₈×C₄₃	—	2	2	no	—
41	C₄×D₄×C₄₃	—	2	2	no	—
42	#117×C₄₃	D₄	1	2	yes	—
43	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
44	#103×C₄₃	—	2	4	no	—
45	#102×C₄₃	—	2	4	no	—
46	C₄×C₈×C₄₃	—	2	1	no	—
47	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
48	#41×C₄₃	—	4	4	no	—
49	C₂×Q₈×C₄₃	—	2	2	no	—
50	C₂×Q₈×C₄₃	—	2	1	no	—
51	C₂×D₄×C₄₃	—	2	1	no	—
52	C₂×D₄×C₄₃	—	2	2	no	—
53	C₂×D₄×C₄₃	—	4	1	no	—
54	C₂²×C₄×C₄₃	—	4	1	no	—
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	#41	1	2	yes	—
57	C₂×C₈×C₄₃	—	2	2	no	—
58	C₂×C₈×C₄₃	—	4	2	no	—
59	C₄⋊C₄×C₄₃	—	2	2	no	—
60	C₄⋊C₄×C₄₃	—	2	4	no	—
61	C₄⋊C₄×C₄₃	—	2	2	no	—
62	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
63	C₂²⋊C₄×C₄₃	—	2	2	no	—
64	C₂²⋊C₄×C₄₃	—	4	2	no	—
65	C₄²×C₄₃	—	2	2	no	—
66	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₄₃	C₂×D₄	1	2	yes	—
68	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
69	#10621877	—	2	4	no	—
70	#10621877	—	4	2	no	—
71	C₂×C₄₃⋊C₈	—	2	2	no	—
72	C₂×C₄₃⋊C₈	—	2	2	no	—
73	C₂×C₄₃⋊C₈	—	4	2	no	—
74	C₂³×C₄₃	—	4	1	no	—
75	C₂³×C₄₃	#123	1	1	yes	characteristic
76	Q₈×C₄₃	—	4	2	no	—
77	Q₈×C₄₃	—	4	2	no	—
78	Q₈×C₄₃	—	4	2	no	—
79	D₄×C₄₃	—	4	2	no	—
80	D₄×C₄₃	—	4	2	no	—
81	D₄×C₄₃	—	4	2	no	—
82	D₄×C₄₃	—	8	2	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	2	4	no	—
87	C₂×C₄×C₄₃	—	4	1	no	—
88	C₂×C₄×C₄₃	—	8	2	no	—
89	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
92	C₂×C₄×C₄₃	#121	1	2	yes	—
93	C₈×C₄₃	—	4	2	no	—
94	C₄₃⋊C₈	—	4	4	no	—
95	C₄₃⋊C₈	—	4	2	no	—
96	C₄₃⋊C₈	—	4	2	no	—
97	#9070	—	43	1	no	maximal, Sylow
98	C₂²×C₄₃	—	2	2	no	—
99	C₂²×C₄₃	—	2	1	no	—
100	C₂²×C₄₃	—	8	2	no	—
101	C₂²×C₄₃	—	8	1	no	—
102	C₂²×C₄₃	#386	1	1	yes	characteristic
103	C₄×C₄₃	—	2	2	no	—
104	C₄×C₄₃	—	2	1	no	—
105	C₄×C₄₃	—	4	4	no	—
106	C₄×C₄₃	—	8	1	no	—
107	C₄×C₄₃	#443	1	2	yes	—
108	C₄×C₄₃	#444	1	2	yes	—
109	C₄×C₄₃	#454	1	2	yes	—
110	#3131	D₄₃	1	1	yes	characteristic
111	#2986	—	43	1	no	—
112	#1526	—	43	1	no	—
113	#1484	—	43	2	no	—
114	#1278	—	43	2	no	—
115	C₂×C₄₃	—	4	1	no	—
116	C₂×C₄₃	—	16	1	no	—
117	C₂×C₄₃	#1876	1	2	yes	—
118	C₂×C₄₃	#1888	1	1	yes	characteristic
119	#533	C₂×D₄₃	1	1	yes	characteristic
120	#532	C₂×D₄₃	1	1	yes	characteristic
121	#488	—	2	2	no	—
122	#487	Dic₄₃	1	2	yes	—
123	#486	Dic₄₃	1	2	yes	—
124	#446	—	86	2	no	—
125	C₈×Q₈	—	86	2	no	—
126	#433	—	86	2	no	—
127	#431	—	43	2	no	—
128	#416	—	2	2	no	—
129	C₄×M₁₆	—	43	2	no	—
130	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
131	#330	—	86	2	no	—
132	D₄⋊C₈	—	86	2	no	—
133	D₄⋊C₈	—	86	2	no	—
134	C₄₃	#9070	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	—	2	1	no	—
136	C₂×M₁₆	—	86	2	no	—
137	C₂²×C₈	—	2	1	no	—
138	C₄⋊Q₈	C₂×Dic₄₃	1	1	yes	characteristic
139	C₄⋊Q₈	C₂×Dic₄₃	1	2	yes	—
140	#127	C₂×Dic₄₃	1	1	yes	characteristic
141	#125	C₂²⋊D₄₃	1	2	yes	—
142	#124	—	2	1	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	#121	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	#117	C₂²⋊D₄₃	1	2	yes	—
148	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
149	C₄⋊C₈	—	43	2	no	—
150	C₄⋊C₈	—	43	2	no	—
151	C₄⋊C₈	—	86	4	no	—
152	#103	—	2	4	no	—
153	#102	—	2	4	no	—
154	C₂²⋊C₈	—	86	2	no	—
155	#97	—	86	2	no	—
156	#97	—	86	2	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	—	86	2	no	—
159	C₄×C₈	—	86	2	no	—
160	C₄×C₈	—	86	2	no	—
161	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
162	#41	—	4	4	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	#10621900	1	2	yes	—
170	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
171	M₁₆	—	86	4	no	—
172	M₁₆	—	172	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	86	2	no	—
176	C₂×C₈	—	86	2	no	—
177	C₂×C₈	—	172	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₂²⋊Dic₄₃	1	4	yes	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	D₄×D₄₃	1	2	yes	—
186	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
188	C₂³	—	4	1	no	—
189	C₂³	#13063306	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	4	2	no	—
193	D₄	—	4	2	no	—
194	D₄	—	4	2	no	—
195	D₄	—	4	2	no	—
196	D₄	—	8	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
204	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
205	C₂×C₄	#13063293	1	1	yes	characteristic
206	C₂×C₄	#13063297	1	2	yes	—
207	C₈	—	4	2	no	—
208	C₈	—	172	4	no	—
209	C₈	—	172	2	no	—
210	C₈	—	172	2	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	8	1	no	—
214	C₂²	—	8	2	no	—
215	C₂²	#14582779	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	1	no	—
220	C₄	#14583016	1	2	yes	—
221	C₄	#14583021	1	2	yes	—
222	C₄	#14583071	1	2	yes	—
223	C₂	—	4	1	no	—
224	C₂	—	16	1	no	—
225	C₂	#19134924	1	2	yes	—
226	C₂	#19134999	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19143697

C₂

yes

maximal, characteristic, complemented

#19133190

C₂

yes

maximal, characteristic, complemented

#19133012

C₂

yes

maximal

#19132432

C₂

yes

maximal, complemented

C₄₃×#3131

C₂

yes

maximal, characteristic

#14583030

—

Q₈×C₄₃⋊C₈

—

#14582960

—

#14582952

C₂²

yes

—

C₄×#10621877

C₂²

yes

—

#14582641

—

D₄⋊(C₄₃⋊C₈)

—

D₄⋊(C₄₃⋊C₈)

—

C₄₃×#533

C₂²

yes

characteristic

C₄₃×#532

C₂²

yes

characteristic

C₄₃×#488

—

C₄₃×#487

C₄

yes

—

C₄₃×#486

C₄

yes

—

C₄₃×#416

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₂×#41×C₄₃

—

C₂²×C₈×C₄₃

—

C₄⋊Q₈×C₄₃

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₄₃

C₂×C₄

yes

—

#127×C₄₃

C₂×C₄

yes

characteristic

#125×C₄₃

D₄

yes

—

#124×C₄₃

—

C₂²⋊Q₈×C₄₃

—

#121×C₄₃

—

C₄×Q₈×C₄₃

—

C₄×D₄×C₄₃

—

#117×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#103×C₄₃

—

#102×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

#41×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

#123

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

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

—

C₂×C₄×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#9070

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

100

C₂²×C₄₃

—

101

C₂²×C₄₃

—

102

C₂²×C₄₃

#386

yes

characteristic

103

C₄×C₄₃

—

104

C₄×C₄₃

—

105

C₄×C₄₃

—

106

C₄×C₄₃

—

107

C₄×C₄₃

#443

yes

—

108

C₄×C₄₃

#444

yes

—

109

C₄×C₄₃

#454

yes

—

110

#3131

D₄₃

yes

characteristic

111

#2986

—

112

#1526

—

113

#1484

—

114

#1278

—

115

C₂×C₄₃

—

116

C₂×C₄₃

—

117

C₂×C₄₃

#1876

yes

—

118

C₂×C₄₃

#1888

yes

characteristic

119

#533

C₂×D₄₃

yes

characteristic

120

#532

C₂×D₄₃

yes

characteristic

121

#488

—

122

#487

Dic₄₃

yes

—

123

#486

Dic₄₃

yes

—

124

#446

—

125

C₈×Q₈

—

126

#433

—

127

#431

—

128

#416

—

129

C₄×M₁₆

—

130

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

131

#330

—

132

D₄⋊C₈

—

133

D₄⋊C₈

—

134

C₄₃

#9070

yes

characteristic, complemented, Sylow

135

C₂×#41

—

136

C₂×M₁₆

—

137

C₂²×C₈

—

138

C₄⋊Q₈

C₂×Dic₄₃

yes

characteristic

139

C₄⋊Q₈

C₂×Dic₄₃

yes

—

140

#127

C₂×Dic₄₃

yes

characteristic

141

#125

C₂²⋊D₄₃

yes

—

142

#124

—

143

C₂²⋊Q₈

—

144

#121

—

145

C₄×Q₈

—

146

C₄×D₄

—

147

#117

C₂²⋊D₄₃

yes

—

148

C₂×C₄²

C₂²×D₄₃

yes

characteristic

149

C₄⋊C₈

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

#103

—

153

#102

—

154

C₂²⋊C₈

—

155

#97

—

156

#97

—

157

C₄×C₈

—

158

C₄×C₈

—

159

C₄×C₈

—

160

C₄×C₈

—

161

C₄×C₈

C₂×Dic₄₃

yes

—

162

#41

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×D₄

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂²×C₄

—

169

C₂²×C₄

#10621900

yes

—

170

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

171

M₁₆

—

172

M₁₆

—

172

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

172

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

C₂²⋊Dic₄₃

yes

—

182

C₂²⋊C₄

—

183

C₂²⋊C₄

—

184

C₄²

—

185

C₄²

D₄×D₄₃

yes

—

186

C₄²

C₂²×Dic₄₃

yes

characteristic

187

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

188

C₂³

—

189

C₂³

#13063306

yes

characteristic

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

D₄

—

194

D₄

—

195

D₄

—

196

D₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

204

C₂×C₄

D₄×Dic₄₃

yes

—

205

C₂×C₄

#13063293

yes

characteristic

206

C₂×C₄

#13063297

yes

—

207

C₈

—

208

C₈

—

172

—

209

C₈

—

172

—

210

C₈

—

172

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

#14582779

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

#14583016

yes

—

221

C₄

#14583021

yes

—

222

C₄

#14583071

yes

—

223

C₂

—

224

C₂

—

225

C₂

#19134924

yes

—

226

C₂

#19134999

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063293
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#3131	C₂
socle	C₂²×C₄₃	#386
prime radical	C₄₃×#532	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#3131	D₄₃
hypocenter	C₄₃	#9070
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

subgroup

G/H

center

C₂×C₄

#13063293

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#3131

C₂

socle

C₂²×C₄₃

#386

prime radical

C₄₃×#532

C₂²

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

N	H ≅ G/N	type
1. #19143697	C₂	semidirect
2. #19133190	C₂	semidirect
4. #19132432	C₂	semidirect
134. C₄₃	#9070	semidirect

H ≅ G/N

type

1. #19143697

C₂

semidirect

2. #19133190

C₂

semidirect

4. #19132432

C₂

semidirect

134. C₄₃

#9070

semidirect

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

subgroup	index
#19143697	2
#19133190	2
#19133012	2
#19132432	2
C₄₃×#3131	2
#9070	43

subgroup

index

C₄₃×#3131

Subgroups — #24991924

Conjugacy classes of subgroups

Complements

Maximal subgroups