Group #25016256 — subgroups

Subgroups — #25016256

Order 11008.

318 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 101 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	#19136368	C₂	1	2	yes	maximal
3	#19135348	C₂	1	1	yes	maximal, characteristic
4	#19132697	C₂	1	2	yes	maximal
5	C₄₃×#1375	C₂	1	1	yes	maximal, characteristic
6	#14583456	—	2	1	no	—
7	#14583308	—	2	4	no	—
8	C₂×#13063261	C₂²	1	1	yes	characteristic
9	#14582937	—	2	1	no	—
10	#14582937	C₂²	1	2	yes	—
11	#14582937	C₂²	1	1	yes	characteristic
12	#14582925	—	2	1	no	—
13	#14582722	—	2	4	no	—
14	#14582659	—	2	1	no	—
15	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
16	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#10621877	—	2	1	no	—
18	#13063325	—	2	2	no	—
19	#13063317	—	2	2	no	—
20	#13063304	—	4	2	no	—
21	C₄×Dic₈₆	—	4	2	no	—
22	#13063284	—	2	1	no	—
23	#13063284	—	2	2	no	—
24	#13063261	—	2	2	no	—
25	#13063259	—	2	2	no	—
26	#13063259	—	2	2	no	—
27	#13063259	—	2	1	no	—
28	#13063257	—	2	2	no	—
29	#13063257	D₄	1	4	yes	—
30	#13063257	Q₈	1	2	yes	complemented
31	Dic₄₃⋊C₈	—	4	2	no	—
32	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
35	M₃₂×C₄₃	—	2	2	no	—
36	M₃₂×C₄₃	—	2	2	no	—
37	C₂×C₁₆×C₄₃	Q₈	1	4	yes	complemented
38	#97×C₄₃	—	2	2	no	—
39	C₂²×C₄×C₄₃	—	2	1	no	—
40	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₈₆	—	4	2	no	—
42	C₂×C₈×C₄₃	—	2	1	no	—
43	C₂×C₈×C₄₃	—	2	1	no	—
44	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
45	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
49	C₄²×C₄₃	—	2	1	no	—
50	C₄²×C₄₃	SD₁₆	1	2	yes	—
51	C₂²⋊Dic₄₃	—	4	1	no	—
52	C₂²⋊Dic₄₃	—	4	2	no	—
53	C₄⋊Dic₄₃	—	2	4	no	—
54	C₄⋊Dic₄₃	—	2	2	no	—
55	C₄⋊Dic₄₃	—	4	2	no	—
56	Dic₄₃⋊C₄	—	4	4	no	—
57	C₄×Dic₄₃	—	4	1	no	—
58	C₄×Dic₄₃	—	4	2	no	—
59	#10621877	—	4	1	no	—
60	#10621877	—	4	1	no	—
61	C₂×C₄₃⋊C₈	—	4	1	no	—
62	C₁₆×C₄₃	—	2	4	no	—
63	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	—	8	1	no	—
66	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
72	C₂×Dic₄₃	—	4	4	no	—
73	C₂×Dic₄₃	—	4	2	no	—
74	Dic₈₆	—	8	4	no	—
75	C₈×C₄₃	—	2	2	no	—
76	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₄₃	C₄⋊Q₈	1	2	yes	—
78	C₄₃⋊C₈	—	8	1	no	—
79	#13459	—	43	1	no	maximal, Sylow
80	C₂²×C₄₃	—	4	1	no	—
81	C₂²×C₄₃	#450	1	1	yes	characteristic
82	C₂²×C₄₃	#476	1	1	yes	characteristic
83	C₂²×C₄₃	#501	1	1	yes	characteristic
84	C₄×C₄₃	—	4	2	no	—
85	C₄×C₄₃	#395	1	1	yes	characteristic
86	C₄×C₄₃	#448	1	1	yes	characteristic
87	C₄×C₄₃	#475	1	1	yes	characteristic
88	C₄×C₄₃	#498	1	1	yes	characteristic
89	Dic₄₃	—	8	4	no	—
90	Dic₄₃	—	8	2	no	—
91	#3162	—	43	1	no	—
92	#2155	—	43	2	no	—
93	#1946	—	43	1	no	—
94	#1384	—	43	2	no	—
95	#1375	D₄₃	1	1	yes	characteristic
96	C₂×C₄₃	—	2	1	no	—
97	C₂×C₄₃	—	2	1	no	—
98	C₂×C₄₃	—	2	1	no	—
99	C₂×C₄₃	#2058	1	1	yes	characteristic
100	#533	—	86	1	no	—
101	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
102	#501	—	86	4	no	—
103	C₂×#108	—	43	1	no	—
104	#427	—	43	2	no	—
105	#427	—	43	1	no	—
106	#427	—	86	1	no	—
107	#424	—	86	1	no	—
108	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
109	#365	—	86	4	no	—
110	#339	—	86	1	no	—
111	C₄₃	#13459	1	1	yes	characteristic, complemented, Sylow
112	C₂×M₁₆	—	86	1	no	—
113	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
115	C₄⋊Q₈	—	86	2	no	—
116	#125	—	86	2	no	—
117	C₂²⋊Q₈	—	172	2	no	—
118	C₄×Q₈	—	172	2	no	—
119	#117	—	86	2	no	—
120	#117	—	86	1	no	—
121	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	—	2	2	no	—
124	C₂×C₁₆	Dic₈₆	1	4	yes	complemented
125	#108	—	86	2	no	—
126	#107	—	86	2	no	—
127	#107	—	86	2	no	—
128	#107	—	86	1	no	—
129	#106	—	43	4	no	—
130	#106	—	43	2	no	—
131	#106	—	86	2	no	—
132	C₄⋊C₈	—	172	2	no	—
133	#97	—	2	2	no	—
134	C₂×Q₈	—	172	2	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
137	M₁₆	—	172	1	no	—
138	M₁₆	—	172	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	172	1	no	—
142	C₂×C₈	#10621873	1	2	yes	—
143	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
144	C₂×C₈	D₄×D₄₃	1	2	yes	—
145	C₂×C₈	#10621900	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
147	C₄⋊C₄	—	86	4	no	—
148	C₄⋊C₄	—	86	2	no	—
149	C₄⋊C₄	—	172	4	no	—
150	C₄⋊C₄	—	172	2	no	—
151	C₂²⋊C₄	—	172	2	no	—
152	C₂²⋊C₄	—	172	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	172	1	no	—
155	C₄²	—	172	2	no	—
156	C₄²	#10621873	1	2	yes	—
157	C₁₆	—	2	4	no	—
158	C₂³	#13063353	1	1	yes	characteristic
159	Q₈	—	344	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	172	4	no	—
163	C₂×C₄	—	172	2	no	—
164	C₂×C₄	#13063295	1	1	yes	characteristic
165	C₂×C₄	#13063305	1	1	yes	characteristic
166	C₂×C₄	#13063325	1	1	yes	characteristic
167	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
168	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
169	C₂×C₄	#13063358	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	344	1	no	—
172	C₈	#13063302	1	2	yes	—
173	C₈	#13063324	1	2	yes	—
174	C₂²	—	4	1	no	—
175	C₂²	#14583052	1	1	yes	characteristic
176	C₂²	#14583190	1	1	yes	characteristic
177	C₂²	#14583308	1	1	yes	characteristic
178	C₄	—	4	2	no	—
179	C₄	—	344	4	no	—
180	C₄	—	344	2	no	—
181	C₄	#14582820	1	1	yes	characteristic
182	C₄	#14583040	1	1	yes	characteristic
183	C₄	#14583185	1	1	yes	characteristic
184	C₄	#14583297	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#19135929	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145119

C₂

yes

maximal, characteristic

#19136368

C₂

yes

maximal

#19135348

C₂

yes

maximal, characteristic

#19132697

C₂

yes

maximal

C₄₃×#1375

C₂

yes

maximal, characteristic

#14583456

—

#14583308

—

C₂×#13063261

C₂²

yes

characteristic

#14582937

—

#14582937

C₂²

yes

—

#14582937

C₂²

yes

—

—

—

C₂×M₃₂×C₄₃

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

—

—

—

—

C₄×Dic₈₆

—

—

—

—

—

—

—

—

D₄

yes

—

#13063257

Q₈

yes

complemented

Dic₄₃⋊C₈

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

M₃₂×C₄₃

—

M₃₂×C₄₃

—

C₂×C₁₆×C₄₃

Q₈

yes

complemented

#97×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

SD₁₆

yes

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₂³×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

#137

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊Q₈

yes

—

C₈×C₄₃

C₄⋊Q₈

yes

—

C₄₃⋊C₈

—

#13459

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#2058

yes

characteristic

100

#533

—

101

C₂×M₃₂

C₂×D₄₃

yes

—

102

#501

—

103

C₂×#108

—

104

#427

—

105

#427

—

106

#427

—

107

#424

—

108

C₂×#97

C₂×D₄₃

yes

characteristic

109

#365

—

110

#339

—

111

C₄₃

#13459

yes

characteristic, complemented, Sylow

112

C₂×M₁₆

—

113

C₂²×C₈

D₁₇₂

yes

characteristic

114

C₂²×C₈

C₂²×D₄₃

yes

characteristic

115

C₄⋊Q₈

—

116

#125

—

117

C₂²⋊Q₈

—

172

—

118

C₄×Q₈

—

172

—

119

#117

—

120

#117

—

121

C₂×C₄²

D₁₇₂

yes

characteristic

122

M₃₂

—

123

M₃₂

—

124

C₂×C₁₆

Dic₈₆

yes

complemented

125

#108

—

126

#107

—

127

#107

—

128

#107

—

129

#106

—

130

#106

—

131

#106

—

132

C₄⋊C₈

—

172

—

133

#97

—

134

C₂×Q₈

—

172

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

137

M₁₆

—

172

—

138

M₁₆

—

172

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

172

—

142

C₂×C₈

#10621873

yes

—

143

C₂×C₈

C₂×Dic₈₆

yes

—

144

C₂×C₈

D₄×D₄₃

yes

—

145

C₂×C₈

#10621900

yes

characteristic

146

C₂×C₈

Q₈×D₄₃

yes

characteristic

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

172

—

150

C₄⋊C₄

—

172

—

151

C₂²⋊C₄

—

172

—

152

C₂²⋊C₄

—

172

—

153

C₄²

—

154

C₄²

—

172

—

155

C₄²

—

172

—

156

C₄²

#10621873

yes

—

157

C₁₆

—

158

C₂³

#13063353

yes

characteristic

159

Q₈

—

344

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

172

—

163

C₂×C₄

—

172

—

164

C₂×C₄

#13063295

yes

characteristic

165

C₂×C₄

#13063305

yes

characteristic

166

C₂×C₄

#13063325

yes

characteristic

167

C₂×C₄

C₂×#10621873

yes

characteristic

168

C₂×C₄

C₂×#10621873

yes

characteristic

169

C₂×C₄

#13063358

yes

characteristic

170

C₈

—

171

C₈

—

344

—

172

C₈

#13063302

yes

—

173

C₈

#13063324

yes

—

174

C₂²

—

175

C₂²

#14583052

yes

characteristic

176

C₂²

#14583190

yes

characteristic

177

C₂²

#14583308

yes

characteristic

178

C₄

—

179

C₄

—

344

—

180

C₄

—

344

—

181

C₄

#14582820

yes

characteristic

182

C₄

#14583040

yes

characteristic

183

C₄

#14583185

yes

characteristic

184

C₄

#14583297

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

—

188

C₂

#19135929

yes

characteristic

	subgroup	G/H
center	C₄	#14583297
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1375	C₂
socle	C₂×C₄₃	#2058
prime radical	C₂³×C₄₃	AGL(1,ℤ/8)
hypercenter	#1375	D₄₃
hypocenter	C₄₃	#13459
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

subgroup

G/H

center

C₄

#14583297

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1375

C₂

C₂×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

N	H ≅ G/N	type
30. #13063257	Q₈	semidirect
37. C₂×C₁₆×C₄₃	Q₈	semidirect
111. C₄₃	#13459	semidirect
124. C₂×C₁₆	Dic₈₆	semidirect

H ≅ G/N

type

30. #13063257

Q₈

semidirect

37. C₂×C₁₆×C₄₃

Q₈

semidirect

111. C₄₃

#13459

semidirect

124. C₂×C₁₆

Dic₈₆

semidirect

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

subgroup	index
#19145119	2
#19136368	2
#19135348	2
#19132697	2
C₄₃×#1375	2
#13459	43

subgroup

index

C₄₃×#1375

Subgroups — #25016256

Conjugacy classes of subgroups

Complements

Maximal subgroups