Group #25303020 — subgroups

Subgroups — #25303020

Order 11008.

1778 conjugacy classes of subgroups; only the 539 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 539 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	#19142859	C₂	1	3	yes	maximal
2	#19142855	C₂	1	12	yes	maximal, complemented
3	#19137862	C₂	1	12	yes	maximal, complemented
4	#19136973	C₂	1	3	yes	maximal
5	C₄₃×#2840	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₄₃	C₂²	1	1	yes	characteristic
7	C₂×Q₈×Dic₄₃	C₂²	1	4	yes	complemented
8	C₂×#13063281	C₂²	1	6	yes	—
9	#14582839	C₂²	1	12	yes	complemented
10	#14582835	C₂²	1	12	yes	—
11	#14582833	C₂²	1	24	yes	complemented
12	#14582822	C₂²	1	24	yes	—
13	#14582818	C₂²	1	12	yes	complemented
14	#14582760	C₂²	1	6	yes	—
15	#14582757	C₂²	1	12	yes	—
16	#14582755	C₂²	1	3	yes	—
17	#14582750	C₂²	1	12	yes	complemented
18	C₂²⋊C₄×Dic₄₃	C₂²	1	12	yes	—
19	C₄₃×#393	C₂²	1	12	yes	—
20	C₄₃×#379	C₂²	1	3	yes	—
21	C₂³×Dic₄₃	—	4	4	no	—
22	C₂³×Dic₄₃	Q₈	1	8	yes	complemented
23	C₂³×Dic₄₃	C₂³	1	3	yes	—
24	C₂³×Dic₄₃	C₂³	1	3	yes	—
25	Q₈×Dic₄₃	—	4	8	no	—
26	#13063281	—	4	12	no	—
27	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
28	C₂×Dic₄₃⋊C₄	C₂³	1	24	yes	—
29	C₂×Dic₄₃⋊C₄	C₂³	1	12	yes	—
30	C₂×C₂²⋊Dic₄₃	C₂³	1	6	yes	—
31	C₂×C₂²⋊Dic₄₃	C₂³	1	12	yes	—
32	C₂×C₄×Dic₄₃	C₂³	1	12	yes	—
33	C₄₃⋊#95	C₂³	1	12	yes	—
34	C₄₃⋊#95	C₂³	1	24	yes	—
35	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²×Q₈×C₄₃	C₂³	1	4	yes	—
37	C₂×C₂²⋊C₄×C₄₃	C₂³	1	12	yes	—
38	C₂×C₂²⋊C₄×C₄₃	C₂³	1	6	yes	—
39	#95×C₄₃	C₂³	1	12	yes	—
40	C₂⁴×C₄₃	—	4	4	no	—
41	C₂⁴×C₄₃	—	4	2	no	—
42	C₂⁴×C₄₃	C₂×Q₈	1	4	yes	—
43	C₂⁴×C₄₃	C₂⁴	1	3	yes	—
44	C₂×Q₈×C₄₃	—	4	8	no	—
45	C₂×Q₈×C₄₃	—	4	4	no	—
46	C₂²×C₄×C₄₃	C₂⁴	1	12	yes	—
47	C₂²×Dic₄₃	—	2	12	no	—
48	C₂²×Dic₄₃	—	4	6	no	—
49	C₂²×Dic₄₃	—	4	16	no	—
50	C₂²×Dic₄₃	C₂×Q₈	1	12	yes	—
51	C₂²×Dic₄₃	C₂×Q₈	1	12	yes	—
52	C₂²×Dic₄₃	C₂⁴	1	12	yes	—
53	C₂²×Dic₄₃	C₂⁴	1	3	yes	—
54	C₂²×Dic₄₃	C₂⁴	1	1	yes	characteristic
55	C₂²⋊C₄×C₄₃	—	4	12	no	—
56	C₂²⋊C₄×C₄₃	—	4	6	no	—
57	C₂²⋊Dic₄₃	—	4	12	no	—
58	C₂²⋊Dic₄₃	—	4	6	no	—
59	C₄⋊Dic₄₃	—	4	12	no	—
60	Dic₄₃⋊C₄	—	4	24	no	—
61	Dic₄₃⋊C₄	—	4	12	no	—
62	C₄×Dic₄₃	—	2	24	no	—
63	C₂³×C₄₃	—	2	6	no	—
64	C₂³×C₄₃	—	2	12	no	—
65	C₂³×C₄₃	—	4	4	no	—
66	C₂³×C₄₃	—	4	8	no	—
67	C₂³×C₄₃	—	4	16	no	—
68	C₂³×C₄₃	C₂²×Q₈	1	6	yes	—
69	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
70	Q₈×C₄₃	—	4	16	no	—
71	C₂×C₄×C₄₃	—	2	12	no	—
72	C₂×C₄×C₄₃	—	2	24	no	—
73	C₂×Dic₄₃	—	2	6	no	—
74	C₂×Dic₄₃	—	2	24	no	—
75	C₂×Dic₄₃	—	2	12	no	—
76	C₂×Dic₄₃	—	2	24	no	—
77	C₂×Dic₄₃	—	4	12	no	—
78	C₂×Dic₄₃	C₂²×Q₈	1	6	yes	—
79	C₂×Dic₄₃	C₂²×Q₈	1	2	yes	—
80	C₂×Dic₄₃	H(5,𝔽₂)	1	4	yes	—
81	#36674	—	43	1	no	maximal, Sylow
82	C₂²×C₄₃	—	2	6	no	—
83	C₂²×C₄₃	—	2	12	no	—
84	C₂²×C₄₃	—	4	16	no	—
85	C₂²×C₄₃	—	4	12	no	—
86	C₂²×C₄₃	C₂³×Q₈	1	1	yes	characteristic
87	C₂²×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
88	C₂²×C₄₃	#585	1	4	yes	—
89	C₄×C₄₃	—	4	12	no	—
90	Dic₄₃	—	2	6	no	—
91	Dic₄₃	—	4	12	no	—
92	Dic₄₃	#543	1	4	yes	—
93	#2842	—	43	3	no	—
94	#2841	—	43	12	no	—
95	#2840	D₄₃	1	1	yes	characteristic
96	#2396	—	43	12	no	—
97	#2280	—	43	3	no	—
98	C₂×C₄₃	—	4	6	no	—
99	C₂×C₄₃	C₂×#543	1	1	yes	characteristic
100	C₂×C₄₃	#3458	1	2	yes	—
101	C₂×C₄₃	#3529	1	4	yes	—
102	C₂⁴×C₄	—	43	1	no	—
103	C₂×C₄×Q₈	—	43	4	no	—
104	C₂×#117	—	43	6	no	—
105	#400	—	43	12	no	—
106	#399	—	43	12	no	—
107	#398	—	43	24	no	—
108	#396	—	43	24	no	—
109	#395	—	43	12	no	—
110	#393	C₂×D₄₃	1	12	yes	—
111	#381	—	43	6	no	—
112	#380	—	43	12	no	—
113	#379	—	43	3	no	—
114	#379	C₂×D₄₃	1	3	yes	—
115	C₄×C₄⋊C₄	—	43	12	no	—
116	C₄×C₂²⋊C₄	—	43	12	no	—
117	C₄₃	#36674	1	1	yes	characteristic, complemented, Sylow
118	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
119	C₂²×Q₈	C₂²×D₄₃	1	4	yes	—
120	C₂³×C₄	—	43	3	no	—
121	C₂³×C₄	—	43	8	no	—
122	C₂³×C₄	—	43	3	no	—
123	C₂³×C₄	—	172	4	no	—
124	C₄×Q₈	—	172	8	no	—
125	#117	—	172	12	no	—
126	C₂×C₄⋊C₄	—	43	12	no	—
127	C₂×C₄⋊C₄	—	43	12	no	—
128	C₂×C₄⋊C₄	—	43	24	no	—
129	C₂×C₂²⋊C₄	—	43	6	no	—
130	C₂×C₂²⋊C₄	—	43	12	no	—
131	C₂×C₂²⋊C₄	C₂²×D₄₃	1	6	yes	—
132	C₂×C₂²⋊C₄	C₂²×D₄₃	1	12	yes	—
133	C₂×C₄²	—	43	12	no	—
134	#95	—	43	24	no	—
135	#95	—	43	12	no	—
136	#95	C₂²×D₄₃	1	12	yes	—
137	C₂⁴	—	4	2	no	—
138	C₂⁴	—	4	4	no	—
139	C₂⁴	Q₈×D₄₃	1	4	yes	—
140	C₂⁴	C₂³×D₄₃	1	3	yes	—
141	C₂×Q₈	—	4	4	no	—
142	C₂×Q₈	—	4	8	no	—
143	C₂²×C₄	—	43	1	no	—
144	C₂²×C₄	—	43	12	no	—
145	C₂²×C₄	—	43	3	no	—
146	C₂²×C₄	—	43	12	no	—
147	C₂²×C₄	—	43	12	no	—
148	C₂²×C₄	—	86	12	no	—
149	C₂²×C₄	—	172	16	no	—
150	C₂²×C₄	—	172	6	no	—
151	C₂²×C₄	C₂³×D₄₃	1	12	yes	—
152	C₄⋊C₄	—	172	12	no	—
153	C₄⋊C₄	—	172	12	no	—
154	C₄⋊C₄	—	172	24	no	—
155	C₂²⋊C₄	—	4	6	no	—
156	C₂²⋊C₄	—	4	12	no	—
157	C₂²⋊C₄	—	172	6	no	—
158	C₂²⋊C₄	—	172	12	no	—
159	C₄²	—	86	24	no	—
160	C₂³	—	2	6	no	—
161	C₂³	—	2	12	no	—
162	C₂³	—	4	4	no	—
163	C₂³	—	4	8	no	—
164	C₂³	—	4	16	no	—
165	C₂³	C₂×Q₈×D₄₃	1	6	yes	—
166	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
167	Q₈	—	4	16	no	—
168	C₂×C₄	—	2	12	no	—
169	C₂×C₄	—	2	24	no	—
170	C₂×C₄	—	43	2	no	—
171	C₂×C₄	—	43	4	no	—
172	C₂×C₄	—	43	6	no	—
173	C₂×C₄	—	86	24	no	—
174	C₂×C₄	—	86	6	no	—
175	C₂×C₄	—	86	12	no	—
176	C₂×C₄	—	86	24	no	—
177	C₂×C₄	—	172	12	no	—
178	C₂²	—	2	6	no	—
179	C₂²	—	2	12	no	—
180	C₂²	—	4	16	no	—
181	C₂²	—	4	12	no	—
182	C₂²	C₂²×Q₈×D₄₃	1	1	yes	characteristic
183	C₂²	H(5,𝔽₂)×D₄₃	1	2	yes	—
184	C₂²	#14583734	1	4	yes	—
185	C₄	—	4	12	no	—
186	C₄	—	43	4	no	—
187	C₄	—	86	6	no	—
188	C₄	—	172	12	no	—
189	C₂	—	4	6	no	—
190	C₂	#543×D₄₃	1	1	yes	characteristic
191	C₂	#19147997	1	2	yes	—
192	C₂	#19148735	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19142859

C₂

yes

maximal

#19142855

C₂

yes

maximal, complemented

#19137862

C₂

yes

maximal, complemented

#19136973

C₂

yes

maximal

C₄₃×#2840

C₂

yes

maximal, characteristic

C₂⁴×Dic₄₃

C₂²

yes

characteristic

C₂×Q₈×Dic₄₃

C₂²

yes

complemented

C₂×#13063281

C₂²

yes

—

#14582839

C₂²

yes

complemented

#14582835

C₂²

yes

—

#14582833

C₂²

yes

complemented

#14582822

C₂²

yes

—

#14582818

C₂²

yes

complemented

#14582760

C₂²

yes

—

#14582757

C₂²

yes

—

#14582755

C₂²

yes

—

#14582750

C₂²

yes

complemented

C₂²⋊C₄×Dic₄₃

C₂²

yes

—

C₄₃×#393

C₂²

yes

—

C₄₃×#379

C₂²

yes

—

C₂³×Dic₄₃

—

C₂³×Dic₄₃

Q₈

yes

complemented

C₂³×Dic₄₃

C₂³

yes

—

C₂³×Dic₄₃

C₂³

yes

—

Q₈×Dic₄₃

—

#13063281

—

C₂×C₄⋊Dic₄₃

C₂³

yes

—

C₂×Dic₄₃⋊C₄

C₂³

yes

—

C₂×Dic₄₃⋊C₄

C₂³

yes

—

C₂×C₂²⋊Dic₄₃

C₂³

yes

—

C₂×C₂²⋊Dic₄₃

C₂³

yes

—

C₂×C₄×Dic₄₃

C₂³

yes

—

C₄₃⋊#95

C₂³

yes

—

C₄₃⋊#95

C₂³

yes

—

C₂⁵×C₄₃

C₂³

yes

characteristic

C₂²×Q₈×C₄₃

C₂³

yes

—

C₂×C₂²⋊C₄×C₄₃

C₂³

yes

—

C₂×C₂²⋊C₄×C₄₃

C₂³

yes

—

#95×C₄₃

C₂³

yes

—

C₂⁴×C₄₃

—

C₂⁴×C₄₃

—

C₂⁴×C₄₃

C₂×Q₈

yes

—

C₂⁴×C₄₃

C₂⁴

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

C₂⁴

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂×Q₈

yes

—

C₂²×Dic₄₃

C₂×Q₈

yes

—

C₂²×Dic₄₃

C₂⁴

yes

—

C₂²×Dic₄₃

C₂⁴

yes

—

C₂²×Dic₄₃

C₂⁴

yes

characteristic

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²×Q₈

yes

—

C₂³×C₄₃

C₂⁵

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂²×Q₈

yes

—

C₂×Dic₄₃

C₂²×Q₈

yes

—

C₂×Dic₄₃

H(5,𝔽₂)

yes

—

#36674

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂³×Q₈

yes

characteristic

C₂²×C₄₃

C₂×H(5,𝔽₂)

yes

—

C₂²×C₄₃

#585

yes

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

#543

yes

—

#2842

—

#2841

—

#2840

D₄₃

yes

characteristic

#2396

—

#2280

—

C₂×C₄₃

—

C₂×C₄₃

C₂×#543

yes

characteristic

100

C₂×C₄₃

#3458

yes

—

101

C₂×C₄₃

#3529

yes

—

102

C₂⁴×C₄

—

103

C₂×C₄×Q₈

—

104

C₂×#117

—

105

#400

—

106

#399

—

107

#398

—

108

#396

—

109

#395

—

110

#393

C₂×D₄₃

yes

—

111

#381

—

112

#380

—

113

#379

—

114

#379

C₂×D₄₃

yes

—

115

C₄×C₄⋊C₄

—

116

C₄×C₂²⋊C₄

—

117

C₄₃

#36674

yes

characteristic, complemented, Sylow

118

C₂⁵

C₂²×D₄₃

yes

characteristic

119

C₂²×Q₈

C₂²×D₄₃

yes

—

120

C₂³×C₄

—

121

C₂³×C₄

—

122

C₂³×C₄

—

123

C₂³×C₄

—

172

—

124

C₄×Q₈

—

172

—

125

#117

—

172

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₂²⋊C₄

—

130

C₂×C₂²⋊C₄

—

131

C₂×C₂²⋊C₄

C₂²×D₄₃

yes

—

132

C₂×C₂²⋊C₄

C₂²×D₄₃

yes

—

133

C₂×C₄²

—

134

#95

—

135

#95

—

136

#95

C₂²×D₄₃

yes

—

137

C₂⁴

—

138

C₂⁴

—

139

C₂⁴

Q₈×D₄₃

yes

—

140

C₂⁴

C₂³×D₄₃

yes

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

172

—

150

C₂²×C₄

—

172

—

151

C₂²×C₄

C₂³×D₄₃

yes

—

152

C₄⋊C₄

—

172

—

153

C₄⋊C₄

—

172

—

154

C₄⋊C₄

—

172

—

155

C₂²⋊C₄

—

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

172

—

158

C₂²⋊C₄

—

172

—

159

C₄²

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

C₂×Q₈×D₄₃

yes

—

166

C₂³

C₂⁴×D₄₃

yes

characteristic

167

Q₈

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

172

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

C₂²×Q₈×D₄₃

yes

characteristic

183

C₂²

H(5,𝔽₂)×D₄₃

yes

—

184

C₂²

#14583734

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

172

—

189

C₂

—

190

C₂

#543×D₄₃

yes

characteristic

191

C₂

#19147997

yes

—

192

C₂

#19148735

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₃
commutator, socle, cosocle	C₂³×C₄₃	C₂⁵
Fitting	C₄₃×#2840	C₂
prime radical	C₂⁵×C₄₃	C₂³
hypercenter	#2840	D₄₃
hypocenter	C₄₃	#36674
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #2840

subgroup

G/H

center, Frattini

C₂³

C₂⁴×D₄₃

commutator, socle, cosocle

C₂³×C₄₃

C₂⁵

Fitting

C₄₃×#2840

C₂

prime radical

C₂⁵×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ #2840

N	H ≅ G/N	type
2. #19142855	C₂	semidirect
3. #19137862	C₂	semidirect
7. C₂×Q₈×Dic₄₃	C₂²	semidirect
9. #14582839	C₂²	semidirect
11. #14582833	C₂²	semidirect
13. #14582818	C₂²	semidirect
17. #14582750	C₂²	semidirect
22. C₂³×Dic₄₃	Q₈	semidirect
117. C₄₃	#36674	semidirect

H ≅ G/N

type

2. #19142855

C₂

semidirect

3. #19137862

C₂

semidirect

7. C₂×Q₈×Dic₄₃

C₂²

semidirect

9. #14582839

C₂²

semidirect

11. #14582833

C₂²

semidirect

13. #14582818

C₂²

semidirect

17. #14582750

C₂²

semidirect

22. C₂³×Dic₄₃

Q₈

semidirect

117. C₄₃

#36674

semidirect

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

subgroup	index
#19142859	2
#19142855	2
#19137862	2
#19136973	2
C₄₃×#2840	2
#36674	43

subgroup

index

C₄₃×#2840

Subgroups — #25303020

Conjugacy classes of subgroups

Complements

Maximal subgroups