Group #25105478 — subgroups

Subgroups — #25105478

Order 11008.

662 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 255 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	#19145119	C₂	1	1	yes	maximal, characteristic
2	#19143525	C₂	1	2	yes	maximal
3	#19135575	C₂	1	2	yes	maximal
4	#19135341	C₂	1	4	yes	maximal
5	#19134482	C₂	1	4	yes	maximal
6	#19134283	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1849	C₂	1	1	yes	maximal, characteristic
8	#14583456	C₂²	1	2	yes	—
9	#14583308	C₄	1	8	yes	complemented
10	#14583026	—	2	4	no	—
11	#14583023	—	2	4	no	—
12	#14582937	C₂²	1	4	yes	—
13	#14582927	C₂²	1	4	yes	—
14	#14582857	C₂²	1	4	yes	—
15	C₂×#13063235	C₂²	1	2	yes	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
18	#14582745	C₂²	1	1	yes	characteristic
19	#14582660	C₂²	1	8	yes	—
20	C₂×#108×C₄₃	C₂²	1	4	yes	—
21	C₄₃×#423	C₂²	1	2	yes	—
22	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
24	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
25	#13063325	C₂×C₄	1	4	yes	—
26	#13063317	C₂×C₄	1	4	yes	—
27	#13063304	—	2	4	no	—
28	C₄×Dic₈₆	—	2	4	no	—
29	#13063284	C₂³	1	4	yes	—
30	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
31	#13063259	C₂×C₄	1	8	yes	—
32	#13063257	C₂×C₄	1	8	yes	—
33	Dic₄₃⋊C₈	—	2	8	no	—
34	#13063235	—	2	4	no	—
35	#13063235	Q₈	1	8	yes	complemented
36	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
37	C₈×Dic₄₃	—	2	4	no	—
38	C₈×Dic₄₃	—	2	4	no	—
39	C₄₃⋊#95	—	2	2	no	—
40	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
41	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	#108×C₄₃	—	2	8	no	—
44	C₄⋊C₈×C₄₃	—	2	4	no	—
45	#97×C₄₃	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₄₃	#41	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
48	C₂²×Dic₄₃	—	2	2	no	—
49	C₂×Dic₈₆	—	2	4	no	—
50	M₁₆×C₄₃	—	2	8	no	—
51	C₂×C₈×C₄₃	—	2	4	no	—
52	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
54	C₂×C₈×C₄₃	#41	1	4	yes	—
55	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
56	C₄²×C₄₃	C₂×D₄	1	2	yes	—
57	C₂²⋊Dic₄₃	—	2	4	no	—
58	C₄⋊Dic₄₃	—	2	4	no	—
59	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
60	Dic₄₃⋊C₄	—	2	8	no	—
61	C₄×Dic₄₃	—	2	4	no	—
62	C₄×Dic₄₃	—	2	2	no	—
63	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
64	C₂×C₄₃⋊C₈	—	2	2	no	—
65	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
66	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
67	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
68	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
69	C₂×C₄×C₄₃	—	4	2	no	—
70	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
71	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
74	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
75	C₂×Dic₄₃	—	2	8	no	—
76	C₂×Dic₄₃	—	2	4	no	—
77	C₂×Dic₄₃	—	4	4	no	—
78	Dic₈₆	—	4	8	no	—
79	C₈×C₄₃	—	4	4	no	—
80	C₈×C₄₃	C₄×Q₈	1	8	yes	—
81	C₄₃⋊C₈	—	2	2	no	—
82	C₄₃⋊C₈	C₄⋊Q₈	1	4	yes	—
83	#21737	—	43	1	no	maximal, Sylow
84	C₂²×C₄₃	—	2	2	no	—
85	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
86	C₂²×C₄₃	#533	1	2	yes	—
87	C₄×C₄₃	—	2	4	no	—
88	C₄×C₄₃	C₂×C₄×Q₈	1	2	yes	—
89	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₄×C₄₃	#532	1	1	yes	characteristic
91	Dic₄₃	—	4	4	no	—
92	Dic₄₃	—	4	8	no	—
93	#3162	—	43	1	no	—
94	#2960	—	43	2	no	—
95	#1995	—	43	2	no	—
96	#1945	—	43	4	no	—
97	#1849	D₄₃	1	1	yes	characteristic
98	#1773	—	43	4	no	—
99	#1726	—	43	1	no	—
100	C₂×C₄₃	—	2	2	no	—
101	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄₃	#2955	1	2	yes	—
103	#533	—	43	2	no	—
104	#501	—	43	8	no	—
105	#446	—	86	4	no	—
106	C₈×Q₈	—	86	4	no	—
107	C₂×#108	C₂×D₄₃	1	4	yes	—
108	#427	—	43	4	no	—
109	#424	—	43	4	no	—
110	#423	C₂×D₄₃	1	2	yes	—
111	#405	—	43	4	no	—
112	C₂×#97	—	43	2	no	—
113	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
114	C₂×C₄×C₈	—	43	1	no	—
115	C₂×C₄×C₈	—	43	2	no	—
116	#376	—	43	1	no	—
117	#339	—	43	8	no	—
118	C₄₃	#21737	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
120	C₂²×C₈	—	43	1	no	—
121	C₂²×C₈	—	43	1	no	—
122	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
123	C₄⋊Q₈	—	43	4	no	—
124	#125	—	43	4	no	—
125	C₂²⋊Q₈	—	86	4	no	—
126	C₄×Q₈	—	86	4	no	—
127	#117	—	43	4	no	—
128	C₂×C₄²	—	43	2	no	—
129	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
130	#108	—	2	8	no	—
131	#107	—	43	8	no	—
132	#106	—	43	8	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	86	8	no	—
135	#97	—	43	8	no	—
136	#97	—	86	4	no	—
137	#97	C₄×D₄₃	1	4	yes	—
138	C₄×C₈	—	43	4	no	—
139	C₄×C₈	—	86	4	no	—
140	C₄×C₈	—	86	4	no	—
141	#95	—	86	2	no	—
142	C₂×Q₈	—	86	4	no	—
143	C₂²×C₄	—	86	2	no	—
144	C₂²×C₄	#10621902	1	2	yes	—
145	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
146	M₁₆	—	2	8	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	43	2	no	—
149	C₂×C₈	—	43	4	no	—
150	C₂×C₈	—	43	2	no	—
151	C₂×C₈	—	86	2	no	—
152	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
153	C₂×C₈	#10621900	1	4	yes	—
154	C₂×C₈	Q₈×D₄₃	1	4	yes	—
155	C₄⋊C₄	—	43	8	no	—
156	C₄⋊C₄	—	86	8	no	—
157	C₄⋊C₄	—	86	4	no	—
158	C₂²⋊C₄	—	86	4	no	—
159	C₄²	—	43	4	no	—
160	C₄²	—	86	4	no	—
161	C₄²	—	86	2	no	—
162	C₄²	C₂×C₄×D₄₃	1	2	yes	—
163	C₄²	D₄×D₄₃	1	2	yes	—
164	C₂³	C₂×#10621902	1	1	yes	characteristic
165	Q₈	—	172	8	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	86	8	no	—
168	C₂×C₄	—	86	4	no	—
169	C₂×C₄	—	172	4	no	—
170	C₂×C₄	#13063287	1	4	yes	—
171	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
172	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
174	C₂×C₄	C₂×#10621900	1	2	yes	—
175	C₈	—	4	4	no	—
176	C₈	—	43	4	no	—
177	C₈	—	86	2	no	—
178	C₈	#13063290	1	8	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	C₂×#13063287	1	1	yes	characteristic
181	C₂²	#14583454	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	172	4	no	—
184	C₄	—	172	8	no	—
185	C₄	C₂×#13063290	1	2	yes	—
186	C₄	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
187	C₄	#14583447	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	#19137162	1	1	yes	characteristic
190	C₂	#19143497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145119

C₂

yes

maximal, characteristic

#19143525

C₂

yes

maximal

#19135575

C₂

yes

maximal

#19135341

C₂

yes

maximal

#19134482

C₂

yes

maximal

#19134283

C₂

yes

maximal, characteristic

C₄₃×#1849

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

—

#14583308

C₄

yes

—

—

C₂²

yes

—

#14582927

C₂²

yes

—

#14582857

C₂²

yes

—

C₂×#13063235

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×C₈×Dic₄₃

C₂²

yes

—

#14582745

C₂²

yes

characteristic

#14582660

C₂²

yes

—

C₂×#108×C₄₃

C₂²

yes

—

C₄₃×#423

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂³

yes

characteristic

C₂²×C₄₃⋊C₈

C₂³

yes

characteristic

#13063325

C₂×C₄

yes

—

#13063317

C₂×C₄

yes

—

#13063304

—

C₄×Dic₈₆

—

#13063284

C₂³

yes

—

C₂×C₄×Dic₄₃

C₂³

yes

—

#13063259

C₂×C₄

yes

—

#13063257

C₂×C₄

yes

—

Dic₄₃⋊C₈

—

#13063235

—

#13063235

Q₈

yes

complemented

C₄×C₄₃⋊C₈

D₄

yes

—

C₈×Dic₄₃

—

C₈×Dic₄₃

—

C₄₃⋊#95

—

C₂×M₁₆×C₄₃

C₂³

yes

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#108×C₄₃

—

C₄⋊C₈×C₄₃

—

#97×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

#41

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂²×C₄

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

C₂×Q₈

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

C₂×C₄₃⋊C₈

C₂×Q₈

yes

—

C₂³×C₄₃

C₂×#41

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂³×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×Q₈

yes

—

C₂×C₄×C₄₃

C₂×#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₄×Q₈

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

C₄⋊Q₈

yes

—

#21737

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₃

#533

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄×Q₈

yes

—

C₄×C₄₃

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₄₃

#532

yes

characteristic

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic

#1773

—

#1726

—

100

C₂×C₄₃

—

101

C₂×C₄₃

C₄×C₄⋊Q₈

yes

characteristic

102

C₂×C₄₃

#2955

yes

—

103

#533

—

104

#501

—

105

#446

—

106

C₈×Q₈

—

107

C₂×#108

C₂×D₄₃

yes

—

108

#427

—

109

#424

—

110

#423

C₂×D₄₃

yes

—

111

#405

—

112

C₂×#97

—

113

C₂×#97

C₂×D₄₃

yes

characteristic

114

C₂×C₄×C₈

—

115

C₂×C₄×C₈

—

116

#376

—

117

#339

—

118

C₄₃

#21737

yes

characteristic, complemented, Sylow

119

C₂×M₁₆

C₂²×D₄₃

yes

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂²×D₄₃

yes

—

123

C₄⋊Q₈

—

124

#125

—

125

C₂²⋊Q₈

—

126

C₄×Q₈

—

127

#117

—

128

C₂×C₄²

—

129

C₂×C₄²

C₂²×D₄₃

yes

characteristic

130

#108

—

131

#107

—

132

#106

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#97

—

136

#97

—

137

#97

C₄×D₄₃

yes

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

#95

—

142

C₂×Q₈

—

143

C₂²×C₄

—

144

C₂²×C₄

#10621902

yes

—

145

C₂²×C₄

C₂³×D₄₃

yes

characteristic

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

C₂×C₄×D₄₃

yes

—

153

C₂×C₈

#10621900

yes

—

154

C₂×C₈

Q₈×D₄₃

yes

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

C₂×C₄×D₄₃

yes

—

163

C₄²

D₄×D₄₃

yes

—

164

C₂³

C₂×#10621902

yes

characteristic

165

Q₈

—

172

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

172

—

170

C₂×C₄

#13063287

yes

—

171

C₂×C₄

C₂²×C₄×D₄₃

yes

characteristic

172

C₂×C₄

C₂×D₄×D₄₃

yes

characteristic

173

C₂×C₄

C₂×Q₈×D₄₃

yes

—

174

C₂×C₄

C₂×#10621900

yes

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

#13063290

yes

—

179

C₂²

—

180

C₂²

C₂×#13063287

yes

characteristic

181

C₂²

#14583454

yes

—

182

C₄

—

183

C₄

—

172

—

184

C₄

—

172

—

185

C₄

C₂×#13063290

yes

—

186

C₄

C₄⋊Q₈×D₄₃

yes

characteristic

187

C₄

#14583447

yes

characteristic

188

C₂

—

189

C₂

#19137162

yes

characteristic

190

C₂

#19143497

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1849	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1849	D₄₃
hypocenter	C₄₃	#21737
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting

C₄₃×#1849

C₂

socle

C₂²×C₄₃

C₂×C₄×D₄

prime radical

C₂³×C₄₃

C₂×#41

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
9. #14583308	C₄	semidirect
35. #13063235	Q₈	semidirect
118. C₄₃	#21737	semidirect

H ≅ G/N

type

9. #14583308

C₄

semidirect

35. #13063235

Q₈

semidirect

118. C₄₃

#21737

semidirect

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

subgroup	index
#19145119	2
#19143525	2
#19135575	2
#19135341	2
#19134482	2
#19134283	2
C₄₃×#1849	2
#21737	43

subgroup

index

C₄₃×#1849

Subgroups — #25105478

Conjugacy classes of subgroups

Complements

Maximal subgroups