Group #25015205 — subgroups

Subgroups — #25015205

Order 11008.

374 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₂×#14583301	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582675	C₂	1	1	yes	maximal, characteristic
3	#19132686	C₂	1	4	yes	maximal
4	C₄₃×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063317	—	2	1	no	—
6	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
7	#14583301	—	2	2	no	—
8	#14583301	C₄	1	4	yes	complemented
9	C₂×#13063259	C₂²	1	2	yes	—
10	C₂×#13063257	—	2	1	no	—
11	#14582675	—	2	2	no	—
12	C₂×#107×C₄₃	—	2	1	no	—
13	C₂×#107×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#384	—	2	1	no	—
16	#13063317	—	2	4	no	—
17	#13063317	—	4	2	no	—
18	C₂×C₄⋊Dic₄₃	—	2	2	no	—
19	C₂×Dic₄₃⋊C₄	—	4	2	no	—
20	C₂×C₄₃⋊C₁₆	—	2	2	no	—
21	C₂×C₄₃⋊C₁₆	—	2	2	no	—
22	C₂×C₄₃⋊C₁₆	Q₈	1	4	yes	—
23	#13063259	—	2	2	no	—
24	#13063259	C₂×C₄	1	4	yes	—
25	#13063257	—	2	4	no	—
26	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
30	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
31	#107×C₄₃	—	2	4	no	—
32	#107×C₄₃	—	4	2	no	—
33	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₄₃	D₄	1	2	yes	—
35	C₂²×C₄×C₄₃	—	4	2	no	—
36	C₂²×C₄×C₄₃	D₈	1	2	yes	—
37	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₃	—	4	2	no	—
39	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₄₃	#41	1	2	yes	—
46	C₄⋊C₄×C₄₃	—	4	2	no	—
47	C₄⋊C₄×C₄₃	—	4	2	no	—
48	C₄⋊C₄×C₄₃	—	8	4	no	—
49	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₄₃	SD₁₆	1	2	yes	—
51	C₄⋊Dic₄₃	—	2	4	no	—
52	C₄⋊Dic₄₃	—	4	2	no	—
53	Dic₄₃⋊C₄	—	4	8	no	—
54	C₄₃⋊C₁₆	—	2	4	no	—
55	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	—	2	2	no	—
58	C₂×C₄×C₄₃	—	8	2	no	—
59	C₂×C₄×C₄₃	—	8	4	no	—
60	C₂×C₄×C₄₃	#102	1	4	yes	—
61	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
63	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₄₃	—	4	4	no	—
68	C₂×Dic₄₃	—	8	2	no	—
69	C₂×Dic₄₃	—	8	2	no	—
70	C₈×C₄₃	#103	1	4	yes	—
71	C₈×C₄₃	C₄×Q₈	1	2	yes	—
72	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
73	#13284	—	43	1	no	maximal, Sylow
74	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
75	C₂²×C₄₃	#486	1	1	yes	characteristic
76	C₂²×C₄₃	#500	1	1	yes	characteristic
77	C₂²×C₄₃	#509	1	2	yes	—
78	C₂²×C₄₃	#510	1	2	yes	—
79	C₄×C₄₃	—	2	4	no	—
80	C₄×C₄₃	—	16	2	no	—
81	C₄×C₄₃	#391	1	1	yes	characteristic
82	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
83	C₄×C₄₃	#487	1	1	yes	characteristic
84	C₄×C₄₃	#498	1	1	yes	characteristic
85	Dic₄₃	—	8	4	no	—
86	C₂×#499	—	43	1	no	—
87	C₂×#346	—	43	1	no	—
88	#1845	D₄₃	1	1	yes	characteristic
89	#1380	—	43	4	no	—
90	C₂×C₄₃	#1990	1	1	yes	characteristic
91	C₂×C₄₃	#2139	1	2	yes	—
92	C₂×C₄₃	#2244	1	1	yes	characteristic
93	C₂×C₄₃	#2245	1	1	yes	characteristic
94	C₂×C₄₃	#2255	1	2	yes	—
95	C₂×#125	—	86	1	no	—
96	C₂²×C₁₆	—	43	2	no	—
97	#499	—	43	4	no	—
98	#499	—	86	2	no	—
99	C₂×#107	—	2	1	no	—
100	C₂×#107	—	43	2	no	—
101	C₂×#107	C₂×D₄₃	1	2	yes	—
102	C₂×#106	—	86	1	no	—
103	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
104	#384	—	2	1	no	—
105	#346	—	86	2	no	—
106	C₄₃	#13284	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
109	#125	—	86	4	no	—
110	#125	—	172	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	86	2	no	—
114	C₂×C₄⋊C₄	—	172	2	no	—
115	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
116	C₂×C₁₆	—	43	4	no	—
117	C₂×C₁₆	—	86	2	no	—
118	C₂×C₁₆	—	86	2	no	—
119	#107	—	2	4	no	—
120	#107	—	4	2	no	—
121	#107	—	43	4	no	—
122	#107	—	86	2	no	—
123	#106	—	86	4	no	—
124	C₄×C₈	C₄×D₄₃	1	2	yes	—
125	C₄×C₈	D₁₇₂	1	2	yes	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	172	2	no	—
128	C₂²×C₄	#10621882	1	2	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
130	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
131	C₂×C₈	#10621883	1	2	yes	—
132	C₂×C₈	#10621885	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
134	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
135	C₂×C₈	#10621900	1	2	yes	—
136	C₂×C₈	Q₈×D₄₃	1	2	yes	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	8	4	no	—
140	C₄⋊C₄	—	86	4	no	—
141	C₄⋊C₄	—	172	8	no	—
142	C₄⋊C₄	—	172	2	no	—
143	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
144	C₄²	#10621884	1	2	yes	—
145	C₁₆	—	86	4	no	—
146	C₂³	C₂×#10621882	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	8	4	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	344	2	no	—
152	C₂×C₄	—	344	2	no	—
153	C₂×C₄	—	172	4	no	—
154	C₂×C₄	#13063245	1	4	yes	—
155	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
156	C₂×C₄	#13063312	1	1	yes	characteristic
157	C₂×C₄	#13063323	1	1	yes	characteristic
158	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
159	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
160	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
161	C₈	#13063248	1	4	yes	—
162	C₈	#13063290	1	2	yes	—
163	C₈	#13063305	1	2	yes	—
164	C₂²	C₂×#13063245	1	1	yes	characteristic
165	C₂²	#14583245	1	1	yes	characteristic
166	C₂²	#14583302	1	1	yes	characteristic
167	C₂²	#14583335	1	2	yes	—
168	C₂²	#14583340	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	16	2	no	—
171	C₄	—	344	4	no	—
172	C₄	#14582804	1	1	yes	characteristic
173	C₄	C₂×#13063248	1	1	yes	characteristic
174	C₄	#14583249	1	1	yes	characteristic
175	C₄	#14583295	1	1	yes	characteristic
176	C₂	#19135555	1	1	yes	characteristic
177	C₂	#19136305	1	2	yes	—
178	C₂	#19136836	1	1	yes	characteristic
179	C₂	#19136840	1	1	yes	characteristic
180	C₂	#19136878	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583301

C₂

yes

maximal, characteristic

C₂×#14582675

C₂

yes

maximal, characteristic

#19132686

C₂

yes

maximal

C₄₃×#1845

C₂

yes

maximal, characteristic

C₂×#13063317

—

C₂²×C₄₃⋊C₁₆

C₂²

yes

—

#14583301

—

#14583301

C₄

yes

complemented

C₂×#13063259

C₂²

yes

—

C₂×#13063257

—

#14582675

—

C₂×#107×C₄₃

—

C₂×#107×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#384

—

#13063317

—

#13063317

—

C₂×C₄⋊Dic₄₃

—

C₂×Dic₄₃⋊C₄

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

Q₈

yes

—

#13063259

—

#13063259

C₂×C₄

yes

—

#13063257

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

#107×C₄₃

—

#107×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₈

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

Q₁₆

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

SD₁₆

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#124

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

C₂×Q₁₆

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

#103

yes

—

C₈×C₄₃

C₄×Q₈

yes

—

C₈×C₄₃

C₂²⋊Q₈

yes

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#510

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

C₂×#499

—

C₂×#346

—

#1845

D₄₃

yes

characteristic

#1380

—

C₂×C₄₃

#1990

yes

characteristic

C₂×C₄₃

#2139

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×#125

—

C₂²×C₁₆

—

#499

—

#499

—

C₂×#107

—

100

C₂×#107

—

101

C₂×#107

C₂×D₄₃

yes

—

102

C₂×#106

—

103

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

104

#384

—

105

#346

—

106

C₄₃

#13284

yes

characteristic, complemented, Sylow

107

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

108

C₂²×C₈

C₂²×D₄₃

yes

characteristic

109

#125

—

110

#125

—

172

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

172

—

115

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

116

C₂×C₁₆

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

#107

—

120

#107

—

121

#107

—

122

#107

—

123

#106

—

124

C₄×C₈

C₄×D₄₃

yes

—

125

C₄×C₈

D₁₇₂

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

172

—

128

C₂²×C₄

#10621882

yes

—

129

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

130

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

131

C₂×C₈

#10621883

yes

—

132

C₂×C₈

#10621885

yes

—

133

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

134

C₂×C₈

C₂×D₁₇₂

yes

characteristic

135

C₂×C₈

#10621900

yes

—

136

C₂×C₈

Q₈×D₄₃

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

172

—

142

C₄⋊C₄

—

172

—

143

C₄²

(C₂×D₄₃)⋊C₄

yes

—

144

C₄²

#10621884

yes

—

145

C₁₆

—

146

C₂³

C₂×#10621882

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

344

—

152

C₂×C₄

—

344

—

153

C₂×C₄

—

172

—

154

C₂×C₄

#13063245

yes

—

155

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

156

C₂×C₄

#13063312

yes

characteristic

157

C₂×C₄

#13063323

yes

characteristic

158

C₂×C₄

C₂×#10621884

yes

characteristic

159

C₂×C₄

C₂×#10621883

yes

characteristic

160

C₂×C₄

C₂×#10621885

yes

characteristic

161

C₈

#13063248

yes

—

162

C₈

#13063290

yes

—

163

C₈

#13063305

yes

—

164

C₂²

C₂×#13063245

yes

characteristic

165

C₂²

#14583245

yes

characteristic

166

C₂²

#14583302

yes

characteristic

167

C₂²

#14583335

yes

—

168

C₂²

#14583340

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

344

—

172

C₄

#14582804

yes

characteristic

173

C₄

C₂×#13063248

yes

characteristic

174

C₄

#14583249

yes

characteristic

175

C₄

#14583295

yes

characteristic

176

C₂

#19135555

yes

characteristic

177

C₂

#19136305

yes

—

178

C₂

#19136836

yes

characteristic

179

C₂

#19136840

yes

characteristic

180

C₂

#19136878

yes

—

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1845	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1845	D₄₃
hypocenter	C₄₃	#13284
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂³

C₂×#10621882

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1845

C₂

socle, prime radical

C₂³×C₄₃

C₂×D₈

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
8. #14583301	C₄	semidirect
106. C₄₃	#13284	semidirect

H ≅ G/N

type

8. #14583301

C₄

semidirect

106. C₄₃

#13284

semidirect

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

subgroup	index
C₂×#14583301	2
C₂×#14582675	2
#19132686	2
C₄₃×#1845	2
#13284	43

subgroup

index

C₄₃×#1845

Subgroups — #25015205

Conjugacy classes of subgroups

Complements

Maximal subgroups