Group #25016471 — subgroups

Subgroups — #25016471

Order 11008.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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	#19136208	C₂	1	1	yes	maximal, characteristic
2	#19134660	C₂	1	2	yes	maximal
3	#19132703	C₂	1	2	yes	maximal, complemented
4	#19132689	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063331	C₂²	1	1	yes	characteristic
7	C₂×#13063265	C₂²	1	1	yes	characteristic
8	C₂×#13063261	C₂²	1	2	yes	—
9	#14582898	—	2	2	no	—
10	(C₂×D₄₃)⋊C₁₆	—	2	2	no	—
11	#14582668	—	2	2	no	—
12	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×#107×C₄₃	C₂²	1	2	yes	—
14	C₄₃×#366	—	2	2	no	—
15	C₂×#10621898	D₄	1	1	yes	characteristic
16	#13063331	—	2	2	no	—
17	#13063331	—	2	2	no	—
18	C₂×#10621877	—	2	2	no	—
19	C₂×#10621877	D₄	1	1	yes	characteristic
20	C₂×#10621872	—	2	1	no	—
21	C₂×C₈×D₄₃	—	2	1	no	—
22	#13063265	—	2	2	no	—
23	C₂×C₄₃⋊C₁₆	—	2	1	no	—
24	#13063261	—	2	2	no	—
25	#13063261	—	2	2	no	—
26	#13063248	—	4	2	no	—
27	#13063245	—	4	2	no	—
28	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂×C₁₆×C₄₃	—	2	2	no	—
31	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
32	#107×C₄₃	—	2	2	no	—
33	#107×C₄₃	—	2	2	no	—
34	C₂²×C₄×C₄₃	—	4	2	no	—
35	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₂²⋊D₄₃	—	2	1	no	—
37	#10621898	—	2	2	no	—
38	#10621898	—	2	2	no	—
39	C₂×D₁₇₂	D₈	1	1	yes	characteristic
40	C₂×C₄×D₄₃	—	2	1	no	—
41	C₂×Dic₈₆	D₈	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₄₃	#41	1	2	yes	—
44	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₄₃	#41	1	2	yes	—
46	C₄⋊C₄×C₄₃	—	4	2	no	—
47	C₄⋊C₄×C₄₃	—	4	2	no	—
48	#10621877	—	2	2	no	—
49	#10621877	—	4	2	no	—
50	#10621877	—	4	2	no	—
51	C₂×C₄₃⋊C₈	—	2	1	no	—
52	C₂×C₄₃⋊C₈	—	4	2	no	—
53	#10621872	—	4	2	no	—
54	C₈×D₄₃	—	4	1	no	—
55	C₈×D₄₃	—	4	1	no	—
56	C₁₆×C₄₃	—	2	2	no	—
57	C₄₃⋊C₁₆	—	4	1	no	—
58	C₂³×C₄₃	#135	1	1	yes	characteristic
59	C₂²×D₄₃	—	2	1	no	—
60	C₂×C₄×C₄₃	—	8	2	no	—
61	C₂×C₄×C₄₃	—	16	2	no	—
62	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#123	1	2	yes	—
64	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₄₃	#137	1	2	yes	—
66	C₂²⋊D₄₃	—	4	2	no	—
67	C₂×Dic₄₃	—	2	1	no	—
68	D₁₇₂	—	2	2	no	—
69	C₄×D₄₃	—	4	1	no	—
70	C₄×D₄₃	—	4	1	no	—
71	Dic₈₆	—	2	2	no	—
72	C₈×C₄₃	#123	1	1	yes	characteristic
73	C₈×C₄₃	#124	1	1	yes	characteristic
74	C₈×C₄₃	#126	1	2	yes	—
75	C₄₃⋊C₈	—	4	1	no	—
76	C₄₃⋊C₈	—	8	2	no	—
77	#13495	—	43	1	no	maximal, Sylow
78	C₂²×C₄₃	—	2	1	no	—
79	C₂²×C₄₃	—	2	1	no	—
80	C₂²×C₄₃	#466	1	1	yes	characteristic
81	C₂²×C₄₃	#485	1	2	yes	—
82	C₂×D₄₃	—	4	1	no	—
83	C₂×D₄₃	—	4	1	no	—
84	C₂×D₄₃	—	4	1	no	—
85	C₄×C₄₃	—	16	2	no	—
86	C₄×C₄₃	#397	1	1	yes	characteristic
87	C₄×C₄₃	#470	1	1	yes	characteristic
88	C₄×C₄₃	#480	1	2	yes	—
89	Dic₄₃	—	4	1	no	—
90	C₂×#366	D₄₃	1	1	yes	characteristic, complemented
91	#2113	—	43	1	no	—
92	#1816	—	43	2	no	—
93	#1387	—	43	2	no	—
94	#1381	—	43	1	no	—
95	C₂×C₄₃	—	2	2	no	—
96	C₂×C₄₃	#2075	1	1	yes	characteristic
97	C₂×C₄₃	#2221	1	1	yes	characteristic
98	C₂×C₄₃	#2222	1	1	yes	characteristic
99	D₄₃	—	8	1	no	—
100	C₂×#131	—	43	1	no	—
101	C₂×M₃₂	—	43	1	no	—
102	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
103	C₂×#108	—	43	2	no	—
104	C₂×#107	C₂×D₄₃	1	2	yes	—
105	#417	—	86	2	no	—
106	#366	—	2	2	no	—
107	C₂²⋊C₁₆	—	86	2	no	—
108	#343	—	86	2	no	—
109	C₄₃	#13495	1	1	yes	characteristic, complemented, Sylow
110	C₂×#41	—	43	1	no	—
111	#131	—	86	2	no	—
112	#131	—	86	2	no	—
113	C₂×M₁₆	—	43	1	no	—
114	C₂×M₁₆	—	86	1	no	—
115	C₂×M₁₆	—	86	2	no	—
116	C₂²×C₈	—	86	1	no	—
117	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	2	2	no	—
119	M₃₂	—	86	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	86	1	no	—
122	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
123	#108	—	86	2	no	—
124	#108	—	86	2	no	—
125	#107	—	2	2	no	—
126	#107	—	2	2	no	—
127	#103	—	172	2	no	—
128	#102	—	172	2	no	—
129	#41	—	86	2	no	—
130	#41	—	86	2	no	—
131	C₂×Q₈	—	43	1	no	—
132	C₂×D₄	—	43	1	no	—
133	C₂×D₄	—	86	1	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	86	1	no	—
136	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
137	M₁₆	—	86	2	no	—
138	M₁₆	—	172	2	no	—
139	M₁₆	—	172	2	no	—
140	M₁₆	—	172	2	no	—
141	C₂×C₈	—	86	1	no	—
142	C₂×C₈	—	172	2	no	—
143	C₂×C₈	—	172	1	no	—
144	C₂×C₈	—	172	1	no	—
145	C₂×C₈	#10621898	1	2	yes	—
146	C₂×C₈	#10621900	1	1	yes	characteristic
147	C₂×C₈	#10621902	1	2	yes	—
148	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	172	1	no	—
153	C₂³	—	86	1	no	—
154	C₂³	#13063344	1	1	yes	characteristic
155	Q₈	—	86	2	no	—
156	D₄	—	86	2	no	—
157	D₄	—	172	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	16	2	no	—
160	C₂×C₄	—	86	1	no	—
161	C₂×C₄	—	172	1	no	—
162	C₂×C₄	—	172	1	no	—
163	C₂×C₄	#13063296	1	1	yes	characteristic
164	C₂×C₄	#13063310	1	2	yes	—
165	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
166	C₂×C₄	#13063357	1	2	yes	—
167	C₈	—	344	2	no	—
168	C₈	—	172	1	no	—
169	C₈	#13063308	1	1	yes	characteristic
170	C₈	#13063314	1	1	yes	characteristic
171	C₈	#13063320	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	172	1	no	—
175	C₂²	—	172	1	no	—
176	C₂²	—	172	1	no	—
177	C₂²	#14583132	1	1	yes	characteristic
178	C₂²	#14583242	1	2	yes	—
179	C₄	—	16	2	no	—
180	C₄	—	172	1	no	—
181	C₄	#14582825	1	1	yes	characteristic
182	C₄	#14583154	1	1	yes	characteristic
183	C₄	#14583214	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	—	344	1	no	—
186	C₂	#19136019	1	1	yes	characteristic
187	C₂	#19136712	1	1	yes	characteristic
188	C₂	#19136717	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136208

C₂

yes

maximal, characteristic

#19134660

C₂

yes

maximal

#19132703

C₂

yes

maximal, complemented

#19132689

C₂

yes

maximal, characteristic, complemented

C₂×C₄₃×#366

C₂

yes

maximal, characteristic, complemented

C₂×#13063331

C₂²

yes

characteristic

C₂×#13063265

C₂²

yes

characteristic

C₂×#13063261

C₂²

yes

—

#14582898

—

(C₂×D₄₃)⋊C₁₆

—

#14582668

—

C₂²×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×#107×C₄₃

C₂²

yes

—

C₄₃×#366

—

C₂×#10621898

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#10621872

—

C₂×C₈×D₄₃

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

—

—

—

—

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

—

#107×C₄₃

—

#107×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

D₈

yes

characteristic

C₂×C₄×D₄₃

—

C₂×Dic₈₆

D₈

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

#10621877

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

#10621872

—

C₈×D₄₃

—

C₈×D₄₃

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

#135

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#137

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

yes

C₈×C₄₃

yes

C₈×C₄₃

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#13495

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#466

yes

characteristic

C₂²×C₄₃

#485

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

C₂×#366

D₄₃

yes

characteristic, complemented

—

—

—

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

100

C₂×#131

—

101

C₂×M₃₂

—

102

C₂²×C₁₆

C₂×D₄₃

yes

characteristic

103

C₂×#108

—

104

C₂×#107

C₂×D₄₃

yes

—

105

#417

—

106

#366

—

107

C₂²⋊C₁₆

—

108

#343

—

109

C₄₃

#13495

yes

characteristic, complemented, Sylow

110

C₂×#41

—

111

#131

—

112

#131

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂²×C₈

—

117

C₂²×C₈

C₂²×D₄₃

yes

characteristic

118

C₂×C₄⋊C₄

—

119

M₃₂

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

C₂×C₁₆

C₂²⋊D₄₃

yes

—

123

#108

—

124

#108

—

125

#107

—

126

#107

—

127

#103

—

172

—

128

#102

—

172

—

129

#41

—

130

#41

—

131

C₂×Q₈

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

D₄×D₄₃

yes

characteristic

137

M₁₆

—

138

M₁₆

—

172

—

139

M₁₆

—

172

—

140

M₁₆

—

172

—

141

C₂×C₈

—

142

C₂×C₈

—

172

—

143

C₂×C₈

—

172

—

144

C₂×C₈

—

172

—

145

C₂×C₈

#10621898

yes

—

146

C₂×C₈

#10621900

yes

characteristic

147

C₂×C₈

#10621902

yes

—

148

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₁₆

—

152

C₁₆

—

172

—

153

C₂³

—

154

C₂³

#13063344

yes

characteristic

155

Q₈

—

156

D₄

—

157

D₄

—

172

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

172

—

162

C₂×C₄

—

172

—

163

C₂×C₄

#13063296

yes

characteristic

164

C₂×C₄

#13063310

yes

—

165

C₂×C₄

D₈×D₄₃

yes

characteristic

166

C₂×C₄

#13063357

yes

—

167

C₈

—

344

—

168

C₈

—

172

—

169

C₈

#13063308

yes

characteristic

170

C₈

#13063314

yes

characteristic

171

C₈

#13063320

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

172

—

175

C₂²

—

172

—

176

C₂²

—

172

—

177

C₂²

#14583132

yes

characteristic

178

C₂²

#14583242

yes

—

179

C₄

—

180

C₄

—

172

—

181

C₄

#14582825

yes

characteristic

182

C₄

#14583154

yes

characteristic

183

C₄

#14583214

yes

—

184

C₂

—

185

C₂

—

344

—

186

C₂

#19136019

yes

characteristic

187

C₂

#19136712

yes

characteristic

188

C₂

#19136717

yes

characteristic

	subgroup	G/H
center	C₂²	#14583132
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#366	C₂
socle	C₂²×C₄₃	#466
prime radical	C₂×#10621898	D₄
hypercenter	C₂×#366	D₄₃
hypocenter	C₄₃	#13495
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂²

#14583132

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₂×C₄₃×#366

C₂

C₂²×C₄₃

D₄

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
3. #19132703	C₂	semidirect
4. #19132689	C₂	semidirect
5. C₂×C₄₃×#366	C₂	semidirect
90. C₂×#366	D₄₃	semidirect
109. C₄₃	#13495	semidirect

H ≅ G/N

type

3. #19132703

C₂

semidirect

4. #19132689

C₂

semidirect

5. C₂×C₄₃×#366

C₂

semidirect

90. C₂×#366

D₄₃

semidirect

109. C₄₃

#13495

semidirect

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

subgroup	index
#19136208	2
#19134660	2
#19132703	2
#19132689	2
C₂×C₄₃×#366	2
#13495	43

subgroup

index

C₂×C₄₃×#366

Subgroups — #25016471

Conjugacy classes of subgroups

Complements

Maximal subgroups