Group #25008475 — subgroups

Subgroups — #25008475

Order 11008.

214 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 79 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	#19136464	C₂	1	1	yes	maximal, characteristic
2	#19133415	C₂	1	1	yes	maximal, characteristic
3	#19132660	C₂	1	1	yes	maximal, characteristic
4	#19132565	C₂	1	1	yes	maximal, characteristic
5	#19132564	C₂	1	1	yes	maximal, characteristic
6	#19132516	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1579	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063271	C₄	1	1	yes	characteristic
9	C₈×Dic₈₆	—	2	1	no	—
10	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
11	#14582726	C₄	1	1	yes	characteristic
12	#14582703	C₄	1	1	yes	characteristic
13	#14582702	C₄	1	1	yes	characteristic
14	#14582675	C₂²	1	1	yes	characteristic
15	#14582675	C₂²	1	1	yes	characteristic
16	#14582650	C₂²	1	1	yes	characteristic
17	#14582631	—	2	1	no	—
18	C₈×Q₈×C₄₃	—	2	1	no	—
19	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#334	C₂²	1	1	yes	characteristic
21	C₄₃×Q₈⋊C₈	—	2	1	no	—
22	C₄₃×#322	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₈₆	—	2	1	no	—
25	#13063271	C₈	1	4	yes	complemented
26	C₂×C₄₃⋊C₁₆	—	2	1	no	—
27	C₂×C₄₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
29	#13063259	C₂×C₄	1	1	yes	characteristic
30	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
31	Dic₄₃⋊C₈	—	4	1	no	—
32	#13063250	—	2	1	no	—
33	C₈×Dic₄₃	—	4	1	no	—
34	C₂×Q₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₄×Q₈×C₄₃	—	2	1	no	—
36	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₄₃	—	2	1	no	—
38	C₄⋊C₈×C₄₃	—	4	1	no	—
39	#103×C₄₃	—	2	1	no	—
40	C₄×C₈×C₄₃	—	4	1	no	—
41	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₄₃	—	2	1	no	—
45	C₂×Dic₈₆	—	2	1	no	—
46	Q₁₆×C₄₃	—	2	1	no	—
47	Q₁₆×C₄₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₄₃	—	2	1	no	—
49	C₂×C₈×C₄₃	—	2	1	no	—
50	C₂×C₈×C₄₃	—	4	1	no	—
51	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₄₃	—	2	1	no	—
54	C₄⋊C₄×C₄₃	—	4	1	no	—
55	C₄²×C₄₃	—	4	1	no	—
56	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₄⋊Dic₄₃	—	2	1	no	—
58	Dic₄₃⋊C₄	—	4	1	no	—
59	C₄×Dic₄₃	—	4	1	no	—
60	C₂×C₄₃⋊C₈	—	4	1	no	—
61	Dic₁₇₂	—	2	1	no	—
62	Dic₁₇₂	C₂×C₈	1	2	yes	—
63	C₄₃⋊C₁₆	—	2	1	no	—
64	C₄₃⋊C₁₆	—	4	1	no	—
65	C₄₃⋊C₁₆	C₂×C₈	1	2	yes	—
66	Q₈×C₄₃	—	2	2	no	—
67	Q₈×C₄₃	—	4	1	no	—
68	C₂×C₄×C₄₃	—	4	1	no	—
69	C₂×C₄×C₄₃	—	4	1	no	—
70	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×Dic₄₃	—	4	1	no	—
74	C₂×Dic₄₃	—	4	1	no	—
75	Dic₈₆	—	2	2	no	—
76	Dic₈₆	—	4	1	no	—
77	C₈×C₄₃	—	2	1	no	—
78	C₈×C₄₃	—	4	1	no	—
79	C₈×C₄₃	—	4	1	no	—
80	C₈×C₄₃	—	8	1	no	—
81	C₈×C₄₃	C₂²×C₈	1	1	yes	characteristic
82	C₈×C₄₃	#131	1	1	yes	characteristic
83	C₄₃⋊C₈	—	8	1	no	—
84	#12066	—	43	1	no	maximal, Sylow
85	C₂²×C₄₃	#442	1	1	yes	characteristic
86	C₄×C₄₃	—	2	1	no	—
87	C₄×C₄₃	—	2	1	no	—
88	C₄×C₄₃	—	4	2	no	—
89	C₄×C₄₃	—	8	1	no	—
90	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
91	C₄×C₄₃	#444	1	1	yes	characteristic
92	Dic₄₃	—	4	2	no	—
93	Dic₄₃	—	8	1	no	—
94	#2176	—	43	1	no	—
95	#1579	—	43	1	no	—
96	#1579	D₄₃	1	1	yes	characteristic
97	#1368	—	43	1	no	—
98	#1334	—	43	1	no	—
99	#1334	—	43	1	no	—
100	#1313	—	43	1	no	—
101	C₂×C₄₃	#1580	1	1	yes	characteristic
102	C₂×C₄₃	#2180	1	1	yes	characteristic
103	C₂×C₄₃	#2180	1	1	yes	characteristic
104	C₂×Q₃₂	—	43	1	no	—
105	C₈×Q₈	—	2	1	no	—
106	C₈×Q₈	—	86	1	no	—
107	C₄×Q₁₆	—	43	1	no	—
108	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
109	#367	—	43	1	no	—
110	#358	—	43	1	no	—
111	#358	—	43	1	no	—
112	#346	—	43	1	no	—
113	#346	—	43	1	no	—
114	#334	—	43	1	no	—
115	#334	C₂×D₄₃	1	1	yes	characteristic
116	Q₈⋊C₈	—	2	1	no	—
117	Q₈⋊C₈	—	86	1	no	—
118	#322	C₂×D₄₃	1	1	yes	characteristic
119	C₄₃	#12066	1	1	yes	characteristic, complemented, Sylow
120	C₂×Q₁₆	—	43	1	no	—
121	C₂×Q₁₆	C₄×D₄₃	1	1	yes	characteristic
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	—	86	1	no	—
124	Q₃₂	—	43	4	no	—
125	C₂×C₁₆	—	43	1	no	—
126	C₂×C₁₆	—	43	1	no	—
127	C₂×C₁₆	—	86	1	no	—
128	#107	—	43	1	no	—
129	#107	C₄×D₄₃	1	1	yes	characteristic
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	—	4	1	no	—
132	C₄⋊C₈	—	86	1	no	—
133	C₄⋊C₈	—	172	1	no	—
134	#103	—	2	1	no	—
135	#103	—	86	1	no	—
136	C₄×C₈	—	4	1	no	—
137	C₄×C₈	—	172	1	no	—
138	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
140	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	86	1	no	—
143	Q₁₆	—	2	1	no	—
144	Q₁₆	—	43	2	no	—
145	Q₁₆	—	86	1	no	—
146	Q₁₆	C₈×D₄₃	1	2	yes	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	172	1	no	—
151	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
152	C₂×C₈	#10621898	1	1	yes	characteristic
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	86	1	no	—
156	C₄⋊C₄	—	172	1	no	—
157	C₄²	—	4	1	no	—
158	C₄²	—	172	1	no	—
159	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
160	C₁₆	—	43	2	no	—
161	C₁₆	—	86	1	no	—
162	C₁₆	—	172	1	no	—
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	86	2	no	—
166	Q₈	—	172	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	172	1	no	—
170	C₂×C₄	—	172	1	no	—
171	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
172	C₂×C₄	#13063350	1	1	yes	characteristic
173	C₂×C₄	#13063351	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	4	1	no	—
176	C₈	—	4	1	no	—
177	C₈	—	8	1	no	—
178	C₈	—	344	1	no	—
179	C₈	C₂×C₈×D₄₃	1	1	yes	characteristic
180	C₈	#13063331	1	1	yes	characteristic
181	C₂²	#14583009	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	8	1	no	—
186	C₄	—	344	1	no	—
187	C₄	—	172	2	no	—
188	C₄	C₈×C₂²⋊D₄₃	1	1	yes	characteristic
189	C₄	#14583019	1	1	yes	characteristic
190	C₂	#19133417	1	1	yes	characteristic
191	C₂	#19136484	1	1	yes	characteristic
192	C₂	#19136486	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136464

C₂

yes

maximal, characteristic

#19133415

C₂

yes

maximal, characteristic

#19132660

C₂

yes

maximal, characteristic

#19132565

C₂

yes

maximal, characteristic

#19132564

C₂

yes

maximal, characteristic

#19132516

C₂

yes

maximal, characteristic

C₄₃×#1579

C₂

yes

maximal, characteristic

C₂×#13063271

C₄

yes

characteristic

C₈×Dic₈₆

—

C₄×Dic₁₇₂

C₂²

yes

characteristic

#14582726

C₄

yes

characteristic

#14582703

C₄

yes

characteristic

#14582702

C₄

yes

characteristic

#14582675

C₂²

yes

characteristic

#14582675

C₂²

yes

characteristic

#14582650

C₂²

yes

characteristic

#14582631

—

C₈×Q₈×C₄₃

—

C₄×Q₁₆×C₄₃

C₂²

yes

characteristic

C₄₃×#334

C₂²

yes

characteristic

C₄₃×Q₈⋊C₈

—

C₄₃×#322

C₂²

yes

characteristic

C₂×Dic₁₇₂

C₂×C₄

yes

characteristic

C₄×Dic₈₆

—

#13063271

C₈

yes

complemented

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

characteristic

#13063259

C₂×C₄

yes

characteristic

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

#13063250

—

C₈×Dic₄₃

—

C₂×Q₁₆×C₄₃

C₂×C₄

yes

characteristic

C₄×Q₈×C₄₃

—

#107×C₄₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

#103×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

Dic₁₇₂

C₂×C₈

yes

—

C₄₃⋊C₁₆

—

C₄₃⋊C₁₆

—

C₄₃⋊C₁₆

C₂×C₈

yes

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂²×C₈

yes

characteristic

C₈×C₄₃

#131

yes

characteristic

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₈×D₄

yes

characteristic

C₄×C₄₃

#444

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#2176

—

#1579

—

#1579

D₄₃

yes

—

—

—

100

#1313

—

101

C₂×C₄₃

#1580

yes

characteristic

102

C₂×C₄₃

#2180

yes

characteristic

103

C₂×C₄₃

#2180

yes

characteristic

104

C₂×Q₃₂

—

105

C₈×Q₈

—

106

C₈×Q₈

—

107

C₄×Q₁₆

—

108

C₄×Q₁₆

C₂×D₄₃

yes

characteristic

109

#367

—

110

#358

—

111

#358

—

112

#346

—

113

#346

—

114

#334

—

115

#334

C₂×D₄₃

yes

characteristic

116

Q₈⋊C₈

—

117

Q₈⋊C₈

—

118

#322

C₂×D₄₃

yes

characteristic

119

C₄₃

#12066

yes

characteristic, complemented, Sylow

120

C₂×Q₁₆

—

121

C₂×Q₁₆

C₄×D₄₃

yes

characteristic

122

C₄×Q₈

—

123

C₄×Q₈

—

124

Q₃₂

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

#107

—

129

#107

C₄×D₄₃

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

133

C₄⋊C₈

—

172

—

134

#103

—

135

#103

—

136

C₄×C₈

—

137

C₄×C₈

—

172

—

138

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

139

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

140

C₄×C₈

C₂²×D₄₃

yes

characteristic

141

C₂×Q₈

—

142

C₂×Q₈

—

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

146

Q₁₆

C₈×D₄₃

yes

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

172

—

151

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

152

C₂×C₈

#10621898

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

172

—

157

C₄²

—

158

C₄²

—

172

—

159

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

160

C₁₆

—

161

C₁₆

—

162

C₁₆

—

172

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

172

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

172

—

170

C₂×C₄

—

172

—

171

C₂×C₄

C₄×C₂²⋊D₄₃

yes

characteristic

172

C₂×C₄

#13063350

yes

characteristic

173

C₂×C₄

#13063351

yes

characteristic

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

344

—

179

C₈

C₂×C₈×D₄₃

yes

characteristic

180

C₈

#13063331

yes

characteristic

181

C₂²

#14583009

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

344

—

187

C₄

—

172

—

188

C₄

C₈×C₂²⋊D₄₃

yes

characteristic

189

C₄

#14583019

yes

characteristic

190

C₂

#19133417

yes

characteristic

191

C₂

#19136484

yes

characteristic

192

C₂

#19136486

yes

characteristic

	subgroup	G/H
center	C₂²	#14583009
commutator	C₈×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1579	C₂
socle, prime radical	C₂²×C₄₃	#442
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#1579	D₄₃
hypocenter	C₄₃	#12066
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1579

subgroup

G/H

center

C₂²

#14583009

commutator

C₈×C₄₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1579

C₂

socle, prime radical

C₂²×C₄₃

#442

cosocle

C₄×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1579

N	H ≅ G/N	type
25. #13063271	C₈	semidirect
119. C₄₃	#12066	semidirect

H ≅ G/N

type

25. #13063271

C₈

semidirect

119. C₄₃

#12066

semidirect

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

subgroup	index
#19136464	2
#19133415	2
#19132660	2
#19132565	2
#19132564	2
#19132516	2
C₄₃×#1579	2
#12066	43

subgroup

index

C₄₃×#1579

Subgroups — #25008475

Conjugacy classes of subgroups

Complements

Maximal subgroups