Group #25249171 — subgroups

Subgroups — #25249171

Order 11008.

1006 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 395 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×#14583302	C₂	1	4	yes	maximal
2	#19142993	C₂	1	1	yes	maximal, characteristic
3	#19134754	C₂	1	8	yes	maximal
4	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#2868	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063325	—	2	2	no	—
7	C₂×#13063325	C₂²	1	2	yes	—
8	#14583302	—	2	16	no	—
9	C₂×#13063258	C₂²	1	16	yes	—
10	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
12	#14582769	—	2	2	no	—
13	#14582769	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₄₃	—	2	2	no	—
15	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
16	C₄₃×#384	—	2	2	no	—
17	C₄₃×#384	C₂²	1	4	yes	—
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₂²×Dic₈₆	—	2	2	no	—
20	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
21	#13063325	—	2	8	no	—
22	#13063325	—	4	8	no	—
23	C₂×C₄⋊Dic₄₃	—	2	8	no	—
24	C₂×C₄⋊Dic₄₃	—	2	2	no	—
25	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
26	#13063258	—	2	32	no	—
27	C₄×C₄₃⋊C₈	D₄	1	8	yes	—
28	C₄×C₄₃⋊C₈	Q₈	1	16	yes	complemented
29	C₂²×Q₈×C₄₃	—	2	2	no	—
30	C₄⋊Q₈×C₄₃	—	2	8	no	—
31	C₄⋊Q₈×C₄₃	—	4	8	no	—
32	C₂×C₄⋊C₄×C₄₃	—	2	8	no	—
33	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
35	C₂×C₄²×C₄₃	D₄	1	2	yes	—
36	C₂×C₄²×C₄₃	D₄	1	2	yes	—
37	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
39	C₂×Q₈×C₄₃	—	4	4	no	—
40	C₂×Q₈×C₄₃	—	4	8	no	—
41	C₂²×C₄×C₄₃	—	2	4	no	—
42	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
45	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
46	C₂²×Dic₄₃	—	2	4	no	—
47	C₂×Dic₈₆	—	4	4	no	—
48	C₂×Dic₈₆	—	4	8	no	—
49	C₄⋊C₄×C₄₃	—	2	8	no	—
50	C₄⋊C₄×C₄₃	—	4	4	no	—
51	C₄⋊C₄×C₄₃	—	4	16	no	—
52	C₄²×C₄₃	D₈	1	8	yes	—
53	C₄²×C₄₃	Q₁₆	1	8	yes	—
54	C₄²×C₄₃	C₂×D₄	1	4	yes	—
55	C₄²×C₄₃	C₂×Q₈	1	8	yes	—
56	C₄⋊Dic₄₃	—	2	8	no	—
57	C₄⋊Dic₄₃	—	4	4	no	—
58	C₄⋊Dic₄₃	—	4	16	no	—
59	C₂×C₄₃⋊C₈	C₂×D₄	1	8	yes	—
60	C₂×C₄₃⋊C₈	C₂×Q₈	1	16	yes	—
61	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₄₃	—	8	8	no	—
63	C₂×C₄×C₄₃	—	4	4	no	—
64	C₂×C₄×C₄₃	—	4	8	no	—
65	C₂×C₄×C₄₃	#127	1	2	yes	—
66	C₂×C₄×C₄₃	#127	1	2	yes	—
67	C₂×C₄×C₄₃	C₄⋊Q₈	1	8	yes	—
68	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
69	C₂×C₄×C₄₃	C₂×D₈	1	8	yes	—
70	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
71	C₂×C₄×C₄₃	C₂×Q₁₆	1	8	yes	—
72	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
73	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
74	C₂×Dic₄₃	—	4	4	no	—
75	C₂×Dic₄₃	—	4	8	no	—
76	Dic₈₆	—	8	8	no	—
77	C₄₃⋊C₈	C₄⋊Q₈	1	16	yes	complemented
78	#33312	—	43	1	no	maximal, Sylow
79	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
80	C₂²×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
81	C₂²×C₄₃	C₂²×D₈	1	2	yes	—
82	C₂²×C₄₃	C₂²×Q₁₆	1	2	yes	—
83	C₄×C₄₃	—	8	4	no	—
84	C₄×C₄₃	#493	1	4	yes	—
85	C₄×C₄₃	#494	1	4	yes	—
86	C₄×C₄₃	#500	1	16	yes	complemented
87	C₄×C₄₃	C₂×C₄⋊Q₈	1	4	yes	—
88	Dic₄₃	—	8	4	no	—
89	C₂×#500	—	43	4	no	—
90	#2868	—	43	1	no	—
91	#2868	D₄₃	1	1	yes	characteristic
92	#1845	—	43	8	no	—
93	C₄²×C₈	—	43	1	no	—
94	C₂×C₄₃	#2868	1	1	yes	characteristic
95	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
96	C₂×C₄₃	C₂×#494	1	1	yes	characteristic
97	C₂×C₄₃	C₂×#500	1	4	yes	—
98	C₂×C₄⋊Q₈	—	2	2	no	—
99	C₂×C₄⋊Q₈	—	43	2	no	—
100	C₂×C₄⋊Q₈	—	86	2	no	—
101	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
102	#500	—	86	16	no	—
103	C₂×#107	—	43	16	no	—
104	C₂×C₄×C₈	—	43	4	no	—
105	C₂×C₄×C₈	—	43	2	no	—
106	#384	—	2	2	no	—
107	#384	—	43	4	no	—
108	#384	—	86	2	no	—
109	#384	C₂×D₄₃	1	4	yes	—
110	C₄³	C₂×D₄₃	1	1	yes	characteristic
111	C₄₃	#33312	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	2	2	no	—
113	C₂²×Q₈	—	86	2	no	—
114	C₂²×C₈	—	43	4	no	—
115	C₄⋊Q₈	—	2	8	no	—
116	C₄⋊Q₈	—	4	8	no	—
117	C₄⋊Q₈	—	86	8	no	—
118	C₄⋊Q₈	—	172	8	no	—
119	C₂×C₄⋊C₄	—	2	8	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	43	4	no	—
122	C₂×C₄⋊C₄	—	86	2	no	—
123	C₂×C₄⋊C₄	—	86	8	no	—
124	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
125	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
126	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
129	#107	—	86	32	no	—
130	C₄×C₈	—	43	8	no	—
131	C₄×C₈	—	43	16	no	—
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	4	8	no	—
134	C₂×Q₈	—	172	4	no	—
135	C₂×Q₈	—	172	8	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	86	4	no	—
138	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
141	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
142	C₂×C₈	—	43	8	no	—
143	C₂×C₈	—	43	16	no	—
144	C₄⋊C₄	—	2	8	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	16	no	—
147	C₄⋊C₄	—	86	8	no	—
148	C₄⋊C₄	—	172	4	no	—
149	C₄⋊C₄	—	172	16	no	—
150	C₄²	#10621882	1	8	yes	—
151	C₄²	#10621885	1	8	yes	—
152	C₄²	D₄×D₄₃	1	4	yes	—
153	C₄²	Q₈×D₄₃	1	8	yes	—
154	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
155	Q₈	—	8	8	no	—
156	Q₈	—	344	8	no	—
157	C₂×C₄	—	4	8	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	172	8	no	—
160	C₂×C₄	—	172	4	no	—
161	C₂×C₄	#13063321	1	2	yes	—
162	C₂×C₄	#13063321	1	2	yes	—
163	C₂×C₄	#13063323	1	8	yes	—
164	C₂×C₄	C₂×#10621882	1	8	yes	—
165	C₂×C₄	C₂×#10621882	1	4	yes	—
166	C₂×C₄	C₂×#10621885	1	8	yes	—
167	C₂×C₄	C₂×#10621885	1	4	yes	—
168	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
169	C₂×C₄	C₂×Q₈×D₄₃	1	4	yes	—
170	C₈	—	43	16	no	—
171	C₂²	C₂×#13063321	1	1	yes	characteristic
172	C₂²	C₂×#13063323	1	2	yes	—
173	C₂²	C₂²×#10621882	1	2	yes	—
174	C₂²	C₂²×#10621885	1	2	yes	—
175	C₄	—	8	4	no	—
176	C₄	—	344	4	no	—
177	C₄	#14583275	1	4	yes	—
178	C₄	#14583278	1	4	yes	—
179	C₄	#14583302	1	16	yes	complemented
180	C₄	C₄⋊Q₈×D₄₃	1	4	yes	—
181	C₂	#19142991	1	1	yes	characteristic
182	C₂	C₂×#14583275	1	1	yes	characteristic
183	C₂	C₂×#14583278	1	1	yes	characteristic
184	C₂	C₂×#14583302	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583302

C₂

yes

maximal

#19142993

C₂

yes

maximal, characteristic

#19134754

C₂

yes

maximal

C₄²×C₄₃⋊C₈

C₂

yes

maximal, characteristic

C₄₃×#2868

C₂

yes

maximal, characteristic

C₂×#13063325

—

C₂×#13063325

C₂²

yes

—

#14583302

—

C₂×#13063258

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582769

—

#14582769

C₂²

yes

—

C₂×C₄⋊Q₈×C₄₃

—

C₂×C₄⋊Q₈×C₄₃

C₂²

yes

—

C₄₃×#384

—

C₄₃×#384

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

#13063325

—

#13063325

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂³

yes

—

#13063258

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₄×C₄₃⋊C₈

Q₈

yes

complemented

C₂²×Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

D₈

yes

—

C₄²×C₄₃

Q₁₆

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×Q₈

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

C₂×C₄₃⋊C₈

C₂×Q₈

yes

—

C₂³×C₄₃

C₂²×D₄

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

—

C₂×C₄×C₄₃

#127

yes

—

C₂×C₄×C₄₃

C₄⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

C₂²×D₄

yes

—

C₂×C₄×C₄₃

C₂²×Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₄₃⋊C₈

C₄⋊Q₈

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

C₂×C₄⋊Q₈

yes

—

C₂²×C₄₃

C₂²×D₈

yes

—

C₂²×C₄₃

C₂²×Q₁₆

yes

—

C₄×C₄₃

—

C₄×C₄₃

#493

yes

—

C₄×C₄₃

#494

yes

—

C₄×C₄₃

#500

yes

complemented

C₄×C₄₃

C₂×C₄⋊Q₈

yes

—

Dic₄₃

—

C₂×#500

—

#2868

—

#2868

D₄₃

yes

characteristic

#1845

—

C₄²×C₈

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

100

C₂×C₄⋊Q₈

—

101

C₂×C₄⋊Q₈

C₂×D₄₃

yes

—

102

#500

—

103

C₂×#107

—

104

C₂×C₄×C₈

—

105

C₂×C₄×C₈

—

106

#384

—

107

#384

—

108

#384

—

109

#384

C₂×D₄₃

yes

—

110

C₄³

C₂×D₄₃

yes

characteristic

111

C₄₃

#33312

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂²×Q₈

—

114

C₂²×C₈

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

172

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

C₂²×D₄₃

yes

—

125

C₂×C₄²

C₂²⋊D₄₃

yes

—

126

C₂×C₄²

C₂²⋊D₄₃

yes

—

127

C₂×C₄²

C₂²×D₄₃

yes

characteristic

128

C₂×C₄²

C₂²×D₄₃

yes

—

129

#107

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

172

—

135

C₂×Q₈

—

172

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

139

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

141

C₂²×C₄

C₂³×D₄₃

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

172

—

149

C₄⋊C₄

—

172

—

150

C₄²

#10621882

yes

—

151

C₄²

#10621885

yes

—

152

C₄²

D₄×D₄₃

yes

—

153

C₄²

Q₈×D₄₃

yes

—

154

C₂³

C₂²×C₂²⋊D₄₃

yes

characteristic

155

Q₈

—

156

Q₈

—

344

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

172

—

160

C₂×C₄

—

172

—

161

C₂×C₄

#13063321

yes

—

162

C₂×C₄

#13063321

yes

—

163

C₂×C₄

#13063323

yes

—

164

C₂×C₄

C₂×#10621882

yes

—

165

C₂×C₄

C₂×#10621882

yes

—

166

C₂×C₄

C₂×#10621885

yes

—

167

C₂×C₄

C₂×#10621885

yes

—

168

C₂×C₄

C₂×D₄×D₄₃

yes

—

169

C₂×C₄

C₂×Q₈×D₄₃

yes

—

170

C₈

—

171

C₂²

C₂×#13063321

yes

characteristic

172

C₂²

C₂×#13063323

yes

—

173

C₂²

C₂²×#10621882

yes

—

174

C₂²

C₂²×#10621885

yes

—

175

C₄

—

176

C₄

—

344

—

177

C₄

#14583275

yes

—

178

C₄

#14583278

yes

—

179

C₄

#14583302

yes

complemented

180

C₄

C₄⋊Q₈×D₄₃

yes

—

181

C₂

#19142991

yes

characteristic

182

C₂

C₂×#14583275

yes

characteristic

183

C₂

C₂×#14583278

yes

characteristic

184

C₂

C₂×#14583302

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#2868	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	#2868	D₄₃
hypocenter	C₄₃	#33312
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂²×C₂²⋊D₄₃

commutator, cosocle

C₂²×C₄×C₄₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting

C₄₃×#2868

C₂

socle, prime radical

C₂³×C₄₃

C₂²×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

N	H ≅ G/N	type
28. C₄×C₄₃⋊C₈	Q₈	semidirect
77. C₄₃⋊C₈	C₄⋊Q₈	semidirect
86. C₄×C₄₃	#500	semidirect
111. C₄₃	#33312	semidirect
179. C₄	#14583302	semidirect

H ≅ G/N

type

28. C₄×C₄₃⋊C₈

Q₈

semidirect

77. C₄₃⋊C₈

C₄⋊Q₈

semidirect

86. C₄×C₄₃

#500

semidirect

111. C₄₃

#33312

semidirect

179. C₄

#14583302

semidirect

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

subgroup	index
C₂×#14583302	2
#19142993	2
#19134754	2
C₄²×C₄₃⋊C₈	2
C₄₃×#2868	2
#33312	43

subgroup

index

C₂×#14583302

#19142993

#19134754

C₄²×C₄₃⋊C₈

C₄₃×#2868

#33312

Subgroups — #25249171

Conjugacy classes of subgroups

Complements

Maximal subgroups