Group #25016748 — subgroups

Subgroups — #25016748

Order 11008.

268 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19135343	C₂	1	1	yes	maximal, characteristic
2	#19134752	C₂	1	1	yes	maximal, characteristic
3	#19132701	C₂	1	2	yes	maximal
4	#19132694	C₂	1	2	yes	maximal
5	C₄₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063261	C₂²	1	1	yes	characteristic
7	#14582937	C₂²	1	1	yes	characteristic
8	C₂×#13063257	—	2	1	no	—
9	C₂×#13063257	C₂²	1	2	yes	—
10	#14582925	—	2	1	no	—
11	#14582769	—	2	1	no	—
12	#14582659	—	2	1	no	—
13	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×#10621877	—	2	1	no	—
16	#13063284	—	2	1	no	—
17	C₂×C₄⋊Dic₄₃	—	2	2	no	—
18	C₂×C₄⋊Dic₄₃	—	4	2	no	—
19	#13063261	—	2	2	no	—
20	#13063259	Q₈	1	2	yes	—
21	#13063257	—	2	1	no	—
22	#13063257	—	2	4	no	—
23	#13063257	—	4	2	no	—
24	Dic₄₃⋊C₈	—	4	2	no	—
25	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
28	M₃₂×C₄₃	—	2	2	no	—
29	M₃₂×C₄₃	—	2	2	no	—
30	C₂×C₁₆×C₄₃	—	2	2	no	—
31	C₄×C₈×C₄₃	—	2	2	no	—
32	C₂²×C₄×C₄₃	—	2	1	no	—
33	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×Dic₄₃	—	4	2	no	—
35	C₂×C₈×C₄₃	—	2	1	no	—
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
38	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₄₃	#41	1	2	yes	—
40	C₂×C₈×C₄₃	#41	1	2	yes	—
41	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
42	C₄²×C₄₃	—	2	1	no	—
43	C₄²×C₄₃	Q₁₆	1	2	yes	—
44	C₂²⋊Dic₄₃	—	4	1	no	—
45	C₄⋊Dic₄₃	—	2	2	no	—
46	C₄⋊Dic₄₃	—	4	2	no	—
47	C₄⋊Dic₄₃	—	4	2	no	—
48	C₄⋊Dic₄₃	—	8	4	no	—
49	C₄×Dic₄₃	—	4	1	no	—
50	#10621877	—	4	1	no	—
51	#10621877	—	4	1	no	—
52	C₂×C₄₃⋊C₈	—	4	1	no	—
53	C₁₆×C₄₃	—	4	2	no	—
54	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×C₄×C₄₃	—	2	4	no	—
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×Dic₄₃	—	4	2	no	—
64	C₂×Dic₄₃	—	8	4	no	—
65	C₂×Dic₄₃	—	8	2	no	—
66	C₈×C₄₃	—	2	2	no	—
67	C₈×C₄₃	#125	1	2	yes	—
68	C₈×C₄₃	#126	1	2	yes	—
69	C₄₃⋊C₈	—	8	1	no	—
70	#13543	—	43	1	no	maximal, Sylow
71	C₂²×C₄₃	—	2	2	no	—
72	C₂²×C₄₃	#476	1	1	yes	characteristic
73	C₂²×C₄₃	#484	1	1	yes	characteristic
74	C₂²×C₄₃	#490	1	1	yes	characteristic
75	C₄×C₄₃	—	2	4	no	—
76	C₄×C₄₃	#400	1	1	yes	characteristic
77	C₄×C₄₃	#477	1	1	yes	characteristic
78	C₄×C₄₃	#481	1	1	yes	characteristic
79	C₄×C₄₃	#487	1	1	yes	characteristic
80	Dic₄₃	—	8	2	no	—
81	Dic₄₃	—	16	2	no	—
82	#1945	—	43	1	no	—
83	#1844	—	43	1	no	—
84	#1386	—	43	2	no	—
85	#1383	—	43	2	no	—
86	#1376	D₄₃	1	1	yes	characteristic
87	C₂×C₄₃	—	2	1	no	—
88	C₂×C₄₃	—	2	1	no	—
89	C₂×C₄₃	#2100	1	1	yes	characteristic
90	C₂×C₄₃	#2230	1	2	yes	—
91	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
92	C₂×#108	—	43	1	no	—
93	#427	—	43	1	no	—
94	C₂×#106	—	43	2	no	—
95	C₂×#106	—	86	1	no	—
96	#424	—	86	1	no	—
97	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
98	#384	—	86	1	no	—
99	#339	—	86	1	no	—
100	C₄₃	#13543	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	86	1	no	—
102	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
104	#117	—	86	1	no	—
105	C₂×C₄⋊C₄	—	86	2	no	—
106	C₂×C₄⋊C₄	—	172	2	no	—
107	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	#108	—	86	2	no	—
112	#107	—	43	2	no	—
113	#106	—	86	1	no	—
114	#106	—	86	4	no	—
115	#106	—	172	2	no	—
116	C₄⋊C₈	—	172	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	172	2	no	—
120	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
121	M₁₆	—	172	1	no	—
122	M₁₆	—	172	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	172	1	no	—
126	C₂×C₈	#10621873	1	2	yes	—
127	C₂×C₈	#10621898	1	2	yes	—
128	C₂×C₈	#10621900	1	2	yes	—
129	C₂×C₈	#10621900	1	1	yes	characteristic
130	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
131	C₄⋊C₄	—	86	2	no	—
132	C₄⋊C₄	—	344	4	no	—
133	C₄⋊C₄	—	172	2	no	—
134	C₄⋊C₄	—	172	2	no	—
135	C₂²⋊C₄	—	172	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	172	1	no	—
138	C₄²	Dic₁₇₂	1	2	yes	—
139	C₁₆	—	4	2	no	—
140	C₂³	#13063353	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	344	4	no	—
144	C₂×C₄	—	344	2	no	—
145	C₂×C₄	—	172	2	no	—
146	C₂×C₄	#13063305	1	1	yes	characteristic
147	C₂×C₄	#13063309	1	1	yes	characteristic
148	C₂×C₄	#13063313	1	1	yes	characteristic
149	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
150	C₂×C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
151	C₂×C₄	#13063353	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	—	344	1	no	—
154	C₈	#13063315	1	2	yes	—
155	C₈	#13063318	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	#14583190	1	1	yes	characteristic
158	C₂²	#14583235	1	1	yes	characteristic
159	C₂²	#14583264	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	344	2	no	—
162	C₄	—	688	2	no	—
163	C₄	#14582839	1	1	yes	characteristic
164	C₄	#14583195	1	1	yes	characteristic
165	C₄	#14583218	1	1	yes	characteristic
166	C₄	#14583250	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#19136159	1	1	yes	characteristic
170	C₂	#19136767	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135343

C₂

yes

maximal, characteristic

#19134752

C₂

yes

maximal, characteristic

#19132701

C₂

yes

maximal

#19132694

C₂

yes

maximal

C₄₃×#1376

C₂

yes

maximal, characteristic

C₂×#13063261

C₂²

yes

characteristic

#14582937

C₂²

yes

characteristic

C₂×#13063257

—

C₂×#13063257

C₂²

yes

—

#14582925

—

#14582769

—

#14582659

—

C₂×M₃₂×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

#13063284

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063261

—

#13063259

Q₈

yes

—

#13063257

—

#13063257

—

#13063257

—

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

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

Q₁₆

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

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

C₂²⋊Q₈

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#125

yes

—

C₈×C₄₃

#126

yes

—

C₄₃⋊C₈

—

#13543

—

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

#2100

yes

characteristic

C₂×C₄₃

#2230

yes

—

C₂×M₃₂

C₂×D₄₃

yes

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#384

—

#339

—

100

C₄₃

#13543

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

102

C₂²×C₈

D₁₇₂

yes

characteristic

103

C₂²×C₈

C₂²×D₄₃

yes

characteristic

104

#117

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

172

—

107

C₂×C₄²

D₁₇₂

yes

characteristic

108

M₃₂

—

109

M₃₂

—

110

C₂×C₁₆

—

111

#108

—

112

#107

—

113

#106

—

114

#106

—

115

#106

—

172

—

116

C₄⋊C₈

—

172

—

117

C₄×C₈

—

118

C₂²×C₄

—

119

C₂²×C₄

—

172

—

120

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

121

M₁₆

—

172

—

122

M₁₆

—

172

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

172

—

126

C₂×C₈

#10621873

yes

—

127

C₂×C₈

#10621898

yes

—

128

C₂×C₈

#10621900

yes

—

129

C₂×C₈

#10621900

yes

characteristic

130

C₂×C₈

Q₈×D₄₃

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

344

—

133

C₄⋊C₄

—

172

—

134

C₄⋊C₄

—

172

—

135

C₂²⋊C₄

—

172

—

136

C₄²

—

137

C₄²

—

172

—

138

C₄²

Dic₁₇₂

yes

—

139

C₁₆

—

140

C₂³

#13063353

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

344

—

144

C₂×C₄

—

344

—

145

C₂×C₄

—

172

—

146

C₂×C₄

#13063305

yes

characteristic

147

C₂×C₄

#13063309

yes

characteristic

148

C₂×C₄

#13063313

yes

characteristic

149

C₂×C₄

C₂×#10621873

yes

characteristic

150

C₂×C₄

C₂×Dic₁₇₂

yes

characteristic

151

C₂×C₄

#13063353

yes

characteristic

152

C₈

—

153

C₈

—

344

—

154

C₈

#13063315

yes

—

155

C₈

#13063318

yes

—

156

C₂²

—

157

C₂²

#14583190

yes

characteristic

158

C₂²

#14583235

yes

characteristic

159

C₂²

#14583264

yes

characteristic

160

C₄

—

161

C₄

—

344

—

162

C₄

—

688

—

163

C₄

#14582839

yes

characteristic

164

C₄

#14583195

yes

characteristic

165

C₄

#14583218

yes

characteristic

166

C₄

#14583250

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#19136159

yes

characteristic

170

C₂

#19136767

yes

—

	subgroup	G/H
center	C₂²	#14583190
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1376	C₂
socle	C₂²×C₄₃	#476
prime radical	C₂³×C₄₃	AGL(1,ℤ/8)
hypercenter	#1376	D₄₃
hypocenter	C₄₃	#13543
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#14583190

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1376

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
100. C₄₃	#13543	semidirect

H ≅ G/N

type

100. C₄₃

#13543

semidirect

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

subgroup	index
#19135343	2
#19134752	2
#19132701	2
#19132694	2
C₄₃×#1376	2
#13543	43

subgroup

index

C₄₃×#1376

Subgroups — #25016748

Conjugacy classes of subgroups

Complements

Maximal subgroups