Group #25110297 — subgroups

Subgroups — #25110297

Order 11008.

578 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 203 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	#19145109	C₂	1	1	yes	maximal, characteristic
2	#19143532	C₂	1	2	yes	maximal
3	#19135575	C₂	1	2	yes	maximal
4	#19135193	C₂	1	8	yes	maximal
5	#19134346	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#1837	C₂	1	1	yes	maximal, characteristic
7	#14583456	C₂²	1	2	yes	—
8	#14583304	C₄	1	2	yes	complemented
9	#14583301	C₄	1	4	yes	complemented
10	#14583297	C₄	1	2	yes	complemented
11	#14583028	—	2	8	no	—
12	#14582937	C₂²	1	4	yes	—
13	#14582927	C₂²	1	4	yes	—
14	#14582857	C₂²	1	4	yes	—
15	C₂×#13063236	C₂²	1	1	yes	characteristic
16	#14582745	C₂²	1	1	yes	characteristic
17	#14582660	C₂²	1	8	yes	—
18	#14582653	C₂²	1	4	yes	—
19	C₄₃×#423	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#341	C₂²	1	4	yes	—
22	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
23	#13063325	C₂×C₄	1	4	yes	—
24	#13063317	C₂×C₄	1	4	yes	—
25	#13063304	—	2	4	no	—
26	C₄×Dic₈₆	—	2	4	no	—
27	#13063284	C₂³	1	4	yes	—
28	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
29	#13063259	C₂×C₄	1	8	yes	—
30	#13063257	C₂×C₄	1	8	yes	—
31	Dic₄₃⋊C₈	—	2	8	no	—
32	#13063236	D₄	1	4	yes	—
33	#13063235	—	2	4	no	—
34	C₈×Dic₄₃	—	2	4	no	—
35	C₄₃⋊#95	—	2	2	no	—
36	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
37	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₃	—	2	4	no	—
40	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	#41	1	2	yes	—
44	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
45	C₂²×Dic₄₃	—	2	2	no	—
46	C₂×Dic₈₆	—	2	4	no	—
47	M₁₆×C₄₃	—	2	8	no	—
48	C₂×C₈×C₄₃	—	2	4	no	—
49	C₂×C₈×C₄₃	—	2	4	no	—
50	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
52	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₄₃	C₂×D₄	1	2	yes	—
55	C₂²⋊Dic₄₃	—	2	4	no	—
56	C₄⋊Dic₄₃	—	2	4	no	—
57	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
58	Dic₄₃⋊C₄	—	2	8	no	—
59	C₄×Dic₄₃	—	2	4	no	—
60	C₄×Dic₄₃	—	2	4	no	—
61	C₂×C₄₃⋊C₈	—	2	4	no	—
62	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
63	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
64	C₂×C₄×C₄₃	—	4	2	no	—
65	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
66	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₃	2₋¹⁺⁴	1	4	yes	—
70	C₂×Dic₄₃	—	2	8	no	—
71	C₂×Dic₄₃	—	4	4	no	—
72	C₂×Dic₄₃	—	4	2	no	—
73	Dic₈₆	—	8	4	no	—
74	C₈×C₄₃	—	2	2	no	—
75	C₈×C₄₃	—	2	2	no	—
76	C₈×C₄₃	—	4	4	no	—
77	C₄₃⋊C₈	—	2	4	no	—
78	#22258	—	43	1	no	maximal, Sylow
79	C₂²×C₄₃	—	2	2	no	—
80	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
81	C₂²×C₄₃	#541	1	2	yes	—
82	C₄×C₄₃	—	2	2	no	—
83	C₄×C₄₃	—	2	2	no	—
84	C₄×C₄₃	#519	1	2	yes	—
85	C₄×C₄₃	#536	1	2	yes	—
86	Dic₄₃	—	4	8	no	—
87	Dic₄₃	—	8	2	no	—
88	#3160	—	43	1	no	—
89	#2961	—	43	2	no	—
90	#1995	—	43	2	no	—
91	#1922	—	43	8	no	—
92	#1837	D₄₃	1	1	yes	characteristic
93	#1746	—	43	1	no	—
94	C₂×C₄₃	—	2	2	no	—
95	C₂×C₄₃	#2327	1	1	yes	characteristic
96	C₂×C₄₃	#2957	1	2	yes	—
97	#533	—	43	2	no	—
98	#500	—	43	2	no	—
99	#499	—	43	4	no	—
100	#498	—	43	2	no	—
101	#446	—	86	8	no	—
102	#427	—	43	4	no	—
103	#424	—	43	4	no	—
104	#423	C₂×D₄₃	1	2	yes	—
105	#405	—	43	4	no	—
106	C₂×#97	—	43	1	no	—
107	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
108	#376	—	43	1	no	—
109	#341	C₂×D₄₃	1	4	yes	—
110	#339	—	43	8	no	—
111	#336	—	43	4	no	—
112	C₄₃	#22258	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
114	C₂²×C₈	—	43	2	no	—
115	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
116	C₄⋊Q₈	—	43	4	no	—
117	#125	—	43	4	no	—
118	C₂²⋊Q₈	—	86	4	no	—
119	C₄×Q₈	—	86	4	no	—
120	#117	—	43	4	no	—
121	C₂×C₄²	—	43	2	no	—
122	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
123	#107	—	43	8	no	—
124	#106	—	43	8	no	—
125	C₄⋊C₈	—	2	4	no	—
126	C₄⋊C₈	—	86	8	no	—
127	#97	—	43	4	no	—
128	#97	—	86	4	no	—
129	C₄×C₈	—	86	4	no	—
130	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
131	C₄×C₈	C₄×D₄₃	1	2	yes	—
132	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
133	#95	—	86	2	no	—
134	C₂×Q₈	—	86	4	no	—
135	C₂²×C₄	—	86	2	no	—
136	C₂²×C₄	#10621902	1	2	yes	—
137	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
138	M₁₆	—	2	8	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	43	4	no	—
142	C₂×C₈	—	86	4	no	—
143	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
144	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
145	C₄⋊C₄	—	43	8	no	—
146	C₄⋊C₄	—	86	8	no	—
147	C₄⋊C₄	—	86	4	no	—
148	C₂²⋊C₄	—	86	4	no	—
149	C₄²	—	86	4	no	—
150	C₄²	—	86	4	no	—
151	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
152	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
153	C₄²	D₄×D₄₃	1	2	yes	—
154	C₂³	C₂×#10621902	1	1	yes	characteristic
155	Q₈	—	344	4	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	86	8	no	—
158	C₂×C₄	—	172	4	no	—
159	C₂×C₄	—	172	2	no	—
160	C₂×C₄	#13063287	1	2	yes	—
161	C₂×C₄	#13063287	1	2	yes	—
162	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
163	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
164	C₂×C₄	#13063372	1	4	yes	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	4	4	no	—
168	C₈	—	86	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	C₂×#13063287	1	1	yes	characteristic
171	C₂²	#14583512	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	344	2	no	—
175	C₄	—	172	8	no	—
176	C₄	#14583376	1	2	yes	—
177	C₄	#14583469	1	2	yes	—
178	C₂	—	2	2	no	—
179	C₂	#19137312	1	1	yes	characteristic
180	C₂	#19143511	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145109

C₂

yes

maximal, characteristic

#19143532

C₂

yes

maximal

#19135575

C₂

yes

maximal

#19135193

C₂

yes

maximal

#19134346

C₂

yes

maximal, characteristic

C₄₃×#1837

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

—

#14583304

C₄

yes

complemented

#14583301

C₄

yes

complemented

#14583297

C₄

yes

complemented

#14583028

—

#14582937

C₂²

yes

—

#14582927

C₂²

yes

—

#14582857

C₂²

yes

—

C₂×#13063236

C₂²

yes

characteristic

#14582745

C₂²

yes

characteristic

#14582660

C₂²

yes

—

#14582653

C₂²

yes

—

C₄₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#341

C₂²

yes

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

#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

—

#13063236

D₄

yes

—

#13063235

—

C₈×Dic₄₃

—

C₄₃⋊#95

—

C₂×M₁₆×C₄₃

C₂³

yes

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂²×C₄

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

C₂³×C₄₃

C₂×#41

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂³×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₃

2₋¹⁺⁴

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#22258

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₃

#541

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#519

yes

—

C₄×C₄₃

#536

yes

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic

#1746

—

C₂×C₄₃

—

C₂×C₄₃

#2327

yes

characteristic

C₂×C₄₃

#2957

yes

—

#533

—

#500

—

#499

—

100

#498

—

101

#446

—

102

#427

—

103

#424

—

104

#423

C₂×D₄₃

yes

—

105

#405

—

106

C₂×#97

—

107

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

108

#376

—

109

#341

C₂×D₄₃

yes

—

110

#339

—

111

#336

—

112

C₄₃

#22258

yes

characteristic, complemented, Sylow

113

C₂×M₁₆

C₂²×D₄₃

yes

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²×D₄₃

yes

—

116

C₄⋊Q₈

—

117

#125

—

118

C₂²⋊Q₈

—

119

C₄×Q₈

—

120

#117

—

121

C₂×C₄²

—

122

C₂×C₄²

C₂²×D₄₃

yes

characteristic

123

#107

—

124

#106

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

#97

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

C₄×D₄₃

yes

characteristic

131

C₄×C₈

C₄×D₄₃

yes

—

132

C₄×C₈

C₄×D₄₃

yes

characteristic

133

#95

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

#10621902

yes

—

137

C₂²×C₄

C₂³×D₄₃

yes

characteristic

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

C₂×C₄×D₄₃

yes

—

144

C₂×C₈

C₂×C₄×D₄₃

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₂²⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂×C₄×D₄₃

yes

characteristic

152

C₄²

C₂×C₄×D₄₃

yes

characteristic

153

C₄²

D₄×D₄₃

yes

—

154

C₂³

C₂×#10621902

yes

characteristic

155

Q₈

—

344

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

172

—

159

C₂×C₄

—

172

—

160

C₂×C₄

#13063287

yes

—

161

C₂×C₄

#13063287

yes

—

162

C₂×C₄

C₂²×C₄×D₄₃

yes

characteristic

163

C₂×C₄

C₂×D₄×D₄₃

yes

characteristic

164

C₂×C₄

#13063372

yes

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₂²

—

170

C₂²

C₂×#13063287

yes

characteristic

171

C₂²

#14583512

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

344

—

175

C₄

—

172

—

176

C₄

#14583376

yes

—

177

C₄

#14583469

yes

—

178

C₂

—

179

C₂

#19137312

yes

characteristic

180

C₂

#19143511

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1837	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1837	D₄₃
hypocenter	C₄₃	#22258
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting

C₄₃×#1837

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 ◁ #1837

N	H ≅ G/N	type
8. #14583304	C₄	semidirect
9. #14583301	C₄	semidirect
10. #14583297	C₄	semidirect
112. C₄₃	#22258	semidirect

H ≅ G/N

type

8. #14583304

C₄

semidirect

9. #14583301

C₄

semidirect

10. #14583297

C₄

semidirect

112. C₄₃

#22258

semidirect

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

subgroup	index
#19145109	2
#19143532	2
#19135575	2
#19135193	2
#19134346	2
C₄₃×#1837	2
#22258	43

subgroup

index

C₄₃×#1837

Subgroups — #25110297

Conjugacy classes of subgroups

Complements

Maximal subgroups