Group #25001404 — subgroups

Subgroups — #25001404

Order 11008.

346 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 85 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	#19135419	C₂	1	1	yes	maximal, characteristic
2	#19134736	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133262	C₂	1	2	yes	maximal
4	#19132454	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1281	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063322	C₂²	1	1	yes	characteristic
7	#14582923	C₂²	1	2	yes	—
8	C₂×#13063247	—	2	2	no	—
9	#14582765	C₂²	1	1	yes	characteristic
10	#14582664	—	2	2	no	—
11	#14582642	—	2	4	no	—
12	C₄₃×#431	C₂²	1	2	yes	—
13	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×D₁₇₂	—	2	1	no	—
15	C₂²×D₁₇₂	—	2	2	no	—
16	C₂×#10621877	—	2	2	no	—
17	#13063322	—	2	2	no	—
18	#13063322	—	2	1	no	—
19	#13063322	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₄₃	—	2	1	no	—
21	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	#13063247	—	4	4	no	—
24	C₄₃⋊#95	—	2	2	no	—
25	C₂²×C₈×C₄₃	D₄	1	2	yes	—
26	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₄₃	—	2	2	no	—
28	C₂²⋊C₈×C₄₃	—	2	2	no	—
29	#97×C₄₃	—	2	2	no	—
30	C₄×C₈×C₄₃	—	2	2	no	—
31	C₂³×D₄₃	—	4	1	no	—
32	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	#41	1	2	yes	—
34	C₂²×Dic₄₃	—	4	1	no	—
35	C₂×D₁₇₂	—	2	2	no	—
36	C₂×D₁₇₂	—	2	4	no	—
37	C₂×D₁₇₂	—	4	1	no	—
38	C₂×D₁₇₂	—	4	2	no	—
39	C₂×D₁₇₂	—	4	2	no	—
40	C₂×D₁₇₂	—	4	2	no	—
41	C₂×D₁₇₂	—	8	4	no	—
42	C₂×C₈×C₄₃	—	2	2	no	—
43	C₂×C₈×C₄₃	—	2	2	no	—
44	C₂×C₈×C₄₃	—	2	2	no	—
45	C₂×C₈×C₄₃	—	4	2	no	—
46	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₄₃	#41	1	2	yes	—
48	C₄²×C₄₃	#41	1	1	yes	characteristic
49	C₄⋊Dic₄₃	—	4	2	no	—
50	#10621877	—	4	2	no	—
51	#10621877	—	4	2	no	—
52	C₂×C₄₃⋊C₈	—	4	2	no	—
53	C₂³×C₄₃	#124	1	1	yes	characteristic
54	C₂²×D₄₃	—	4	2	no	—
55	C₂²×D₄₃	—	8	4	no	—
56	C₂²×D₄₃	—	8	1	no	—
57	C₂²×D₄₃	—	8	1	no	—
58	C₂×C₄×C₄₃	—	4	2	no	—
59	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#123	1	2	yes	—
61	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#126	1	2	yes	—
63	C₂×C₄×C₄₃	#135	1	2	yes	—
64	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
65	C₂×Dic₄₃	—	8	2	no	—
66	C₂×Dic₄₃	—	8	1	no	—
67	D₁₇₂	—	4	4	no	—
68	D₁₇₂	—	8	4	no	—
69	D₁₇₂	—	8	4	no	—
70	D₁₇₂	—	8	2	no	—
71	C₈×C₄₃	—	4	2	no	—
72	C₈×C₄₃	—	8	2	no	—
73	C₄₃⋊C₈	—	8	2	no	—
74	#10634	—	43	1	no	maximal, Sylow
75	C₂²×C₄₃	—	2	2	no	—
76	C₂²×C₄₃	#397	1	1	yes	characteristic
77	C₂²×C₄₃	#488	1	1	yes	characteristic
78	C₂²×C₄₃	#490	1	1	yes	characteristic
79	C₂×D₄₃	—	8	2	no	—
80	C₂×D₄₃	—	8	4	no	—
81	C₂×D₄₃	—	16	2	no	—
82	C₂×D₄₃	—	16	2	no	—
83	C₄×C₄₃	—	2	2	no	—
84	C₄×C₄₃	—	2	2	no	—
85	C₄×C₄₃	#463	1	2	yes	—
86	C₄×C₄₃	#482	1	2	yes	—
87	Dic₄₃	—	16	1	no	—
88	#1961	—	43	1	no	—
89	#1838	—	43	1	no	—
90	#1545	—	43	2	no	—
91	#1286	—	43	2	no	—
92	#1281	D₄₃	1	1	yes	characteristic, complemented
93	C₂×C₄₃	—	2	1	no	—
94	C₂×C₄₃	—	2	1	no	—
95	C₂×C₄₃	#2073	1	1	yes	characteristic
96	C₂×C₄₃	#2081	1	2	yes	—
97	D₄₃	—	16	2	no	—
98	C₂×#127	—	43	1	no	—
99	#431	C₂×D₄₃	1	2	yes	—
100	#423	—	43	2	no	—
101	C₂×#102	—	86	2	no	—
102	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
103	#382	—	43	1	no	—
104	#341	—	86	2	no	—
105	#331	—	86	4	no	—
106	C₄₃	#10634	1	1	yes	characteristic, complemented, Sylow
107	C₂²×D₄	—	86	1	no	—
108	C₂²×D₄	—	86	2	no	—
109	C₂×M₁₆	—	86	2	no	—
110	C₂²×C₈	D₁₇₂	1	2	yes	—
111	#127	—	43	2	no	—
112	#127	—	86	2	no	—
113	#127	—	86	1	no	—
114	C₂×C₄⋊C₄	—	86	1	no	—
115	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	86	2	no	—
118	C₄⋊C₈	—	86	2	no	—
119	#102	—	172	4	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	#95	—	86	2	no	—
124	C₂⁴	—	172	1	no	—
125	C₂×D₄	—	86	2	no	—
126	C₂×D₄	—	86	4	no	—
127	C₂×D₄	—	344	4	no	—
128	C₂×D₄	—	172	1	no	—
129	C₂×D₄	—	172	2	no	—
130	C₂×D₄	—	172	2	no	—
131	C₂×D₄	—	172	2	no	—
132	C₂²×C₄	—	172	1	no	—
133	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
134	C₂²×C₄	#10621898	1	2	yes	—
135	M₁₆	—	172	2	no	—
136	M₁₆	—	172	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	172	2	no	—
142	C₄⋊C₄	—	172	2	no	—
143	C₄²	D₄×D₄₃	1	1	yes	characteristic
144	C₄²	#10621900	1	2	yes	—
145	C₄²	#10621902	1	1	yes	characteristic
146	C₂³	—	344	4	no	—
147	C₂³	—	344	1	no	—
148	C₂³	—	344	1	no	—
149	C₂³	—	172	2	no	—
150	C₂³	#13063313	1	1	yes	characteristic
151	D₄	—	344	4	no	—
152	D₄	—	344	4	no	—
153	D₄	—	344	2	no	—
154	D₄	—	172	4	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	344	2	no	—
157	C₂×C₄	—	344	1	no	—
158	C₂×C₄	#13063300	1	1	yes	characteristic
159	C₂×C₄	#13063307	1	2	yes	—
160	C₂×C₄	#13063309	1	1	yes	characteristic
161	C₂×C₄	#13063320	1	2	yes	—
162	C₂×C₄	#13063347	1	2	yes	—
163	C₂×C₄	#13063353	1	2	yes	—
164	C₈	—	4	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	344	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	344	2	no	—
169	C₂²	—	344	4	no	—
170	C₂²	—	688	2	no	—
171	C₂²	—	688	2	no	—
172	C₂²	#14582826	1	1	yes	characteristic
173	C₂²	#14583254	1	1	yes	characteristic
174	C₂²	#14583264	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	688	1	no	—
178	C₄	#14583115	1	2	yes	—
179	C₄	#14583224	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	—	688	2	no	—
183	C₂	#19136010	1	1	yes	characteristic
184	C₂	#19136054	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135419

C₂

yes

maximal, characteristic

#19134736

C₂

yes

maximal, characteristic, complemented

#19133262

C₂

yes

maximal

#19132454

C₂

yes

maximal, complemented

C₄₃×#1281

C₂

yes

maximal, characteristic, complemented

C₂×#13063322

C₂²

yes

characteristic

#14582923

C₂²

yes

—

C₂×#13063247

—

#14582765

C₂²

yes

characteristic

#14582664

—

#14582642

—

C₄₃×#431

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂²×D₁₇₂

—

C₂²×D₁₇₂

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063247

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₄×C₈×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄²×C₄₃

#41

yes

characteristic

C₄⋊Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#124

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

#123

yes

characteristic

C₂×C₄×C₄₃

#126

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10634

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#463

yes

—

C₄×C₄₃

#482

yes

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#2073

yes

characteristic

C₂×C₄₃

#2081

yes

—

D₄₃

—

C₂×#127

—

#431

C₂×D₄₃

yes

—

100

#423

—

101

C₂×#102

—

102

C₂×#97

C₂×D₄₃

yes

characteristic

103

#382

—

104

#341

—

105

#331

—

106

C₄₃

#10634

yes

characteristic, complemented, Sylow

107

C₂²×D₄

—

108

C₂²×D₄

—

109

C₂×M₁₆

—

110

C₂²×C₈

D₁₇₂

yes

—

111

#127

—

112

#127

—

113

#127

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₂²×D₄₃

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

#102

—

172

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

—

123

#95

—

124

C₂⁴

—

172

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

344

—

128

C₂×D₄

—

172

—

129

C₂×D₄

—

172

—

130

C₂×D₄

—

172

—

131

C₂×D₄

—

172

—

132

C₂²×C₄

—

172

—

133

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

134

C₂²×C₄

#10621898

yes

—

135

M₁₆

—

172

—

136

M₁₆

—

172

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

172

—

142

C₄⋊C₄

—

172

—

143

C₄²

D₄×D₄₃

yes

characteristic

144

C₄²

#10621900

yes

—

145

C₄²

#10621902

yes

characteristic

146

C₂³

—

344

—

147

C₂³

—

344

—

148

C₂³

—

344

—

149

C₂³

—

172

—

150

C₂³

#13063313

yes

characteristic

151

D₄

—

344

—

152

D₄

—

344

—

153

D₄

—

344

—

154

D₄

—

172

—

155

C₂×C₄

—

156

C₂×C₄

—

344

—

157

C₂×C₄

—

344

—

158

C₂×C₄

#13063300

yes

characteristic

159

C₂×C₄

#13063307

yes

—

160

C₂×C₄

#13063309

yes

characteristic

161

C₂×C₄

#13063320

yes

—

162

C₂×C₄

#13063347

yes

—

163

C₂×C₄

#13063353

yes

—

164

C₈

—

165

C₈

—

166

C₈

—

344

—

167

C₂²

—

168

C₂²

—

344

—

169

C₂²

—

344

—

170

C₂²

—

688

—

171

C₂²

—

688

—

172

C₂²

#14582826

yes

characteristic

173

C₂²

#14583254

yes

characteristic

174

C₂²

#14583264

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

688

—

178

C₄

#14583115

yes

—

179

C₄

#14583224

yes

—

180

C₂

—

181

C₂

—

182

C₂

—

688

—

183

C₂

#19136010

yes

characteristic

184

C₂

#19136054

yes

—

	subgroup	G/H
center	C₂²	#14582826
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1281	C₂
socle	C₂²×C₄₃	#397
prime radical	C₂×#13063322	C₂²
hypercenter	#1281	D₄₃
hypocenter	C₄₃	#10634
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂²

#14582826

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1281

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
2. #19134736	C₂	semidirect
4. #19132454	C₂	semidirect
5. C₄₃×#1281	C₂	semidirect
92. #1281	D₄₃	semidirect
106. C₄₃	#10634	semidirect

H ≅ G/N

type

2. #19134736

C₂

semidirect

4. #19132454

C₂

semidirect

5. C₄₃×#1281

C₂

semidirect

92. #1281

D₄₃

semidirect

106. C₄₃

#10634

semidirect

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

subgroup	index
#19135419	2
#19134736	2
#19133262	2
#19132454	2
C₄₃×#1281	2
#10634	43

subgroup

index

C₄₃×#1281

Subgroups — #25001404

Conjugacy classes of subgroups

Complements

Maximal subgroups