Group #25016741 — subgroups

Subgroups — #25016741

Order 11008.

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

normal subgroups only

This group has 252 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	#19132702	C₂	1	2	yes	maximal
3	#19132693	C₂	1	2	yes	maximal
4	#19132677	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063265	C₂²	1	2	yes	—
7	C₂×#13063261	C₂²	1	1	yes	characteristic
8	#14582937	C₂²	1	1	yes	characteristic
9	#14582925	—	2	1	no	—
10	#14582659	—	2	1	no	—
11	C₂×#107×C₄₃	—	2	1	no	—
12	C₂×#106×C₄₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#382	—	2	1	no	—
15	C₂×#10621877	—	2	1	no	—
16	#13063284	—	2	1	no	—
17	#13063265	—	2	2	no	—
18	#13063265	—	2	2	no	—
19	C₂×C₄₃⋊C₁₆	—	2	2	no	—
20	#13063261	—	2	2	no	—
21	#13063259	Q₈	1	2	yes	—
22	#13063257	—	2	1	no	—
23	Dic₄₃⋊C₈	—	4	2	no	—
24	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
27	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
28	#107×C₄₃	—	4	2	no	—
29	#106×C₄₃	—	2	4	no	—
30	C₄×C₈×C₄₃	—	2	2	no	—
31	#95×C₄₃	—	4	2	no	—
32	C₂²×C₄×C₄₃	—	2	1	no	—
33	C₂²×C₄×C₄₃	—	4	2	no	—
34	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₄₃	—	4	2	no	—
43	C₄⋊C₄×C₄₃	—	4	2	no	—
44	C₄²×C₄₃	—	2	1	no	—
45	C₄²×C₄₃	Q₁₆	1	2	yes	—
46	C₂²⋊Dic₄₃	—	4	1	no	—
47	C₄⋊Dic₄₃	—	2	2	no	—
48	C₄×Dic₄₃	—	4	1	no	—
49	#10621877	—	4	1	no	—
50	#10621877	—	4	1	no	—
51	C₂×C₄₃⋊C₈	—	4	1	no	—
52	C₄₃⋊C₁₆	—	4	2	no	—
53	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	—	4	2	no	—
56	C₂×C₄×C₄₃	—	8	4	no	—
57	C₂×C₄×C₄₃	—	8	2	no	—
58	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×Dic₄₃	—	4	2	no	—
65	C₈×C₄₃	—	2	2	no	—
66	C₈×C₄₃	#125	1	2	yes	—
67	C₈×C₄₃	#126	1	2	yes	—
68	C₄₃⋊C₈	—	8	1	no	—
69	#13542	—	43	1	no	maximal, Sylow
70	C₂²×C₄₃	—	2	2	no	—
71	C₂²×C₄₃	#476	1	1	yes	characteristic
72	C₂²×C₄₃	#484	1	1	yes	characteristic
73	C₂²×C₄₃	#490	1	1	yes	characteristic
74	C₄×C₄₃	—	4	2	no	—
75	C₄×C₄₃	—	16	2	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	#1945	—	43	1	no	—
82	#1846	D₄₃	1	1	yes	characteristic
83	#1386	—	43	2	no	—
84	#1383	—	43	2	no	—
85	#1376	—	43	1	no	—
86	C₂×C₄₃	—	2	1	no	—
87	C₂×C₄₃	—	2	1	no	—
88	C₂×C₄₃	#2100	1	1	yes	characteristic
89	C₂×C₄₃	#2231	1	2	yes	—
90	C₂×M₃₂	—	43	2	no	—
91	C₂×#108	—	43	1	no	—
92	#427	—	43	1	no	—
93	C₂×#107	—	2	1	no	—
94	C₂×#106	C₂×D₄₃	1	2	yes	—
95	#424	—	86	1	no	—
96	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
97	#382	—	2	1	no	—
98	#339	—	86	1	no	—
99	C₄₃	#13542	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	86	1	no	—
101	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
103	#117	—	86	1	no	—
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
106	M₃₂	—	86	2	no	—
107	M₃₂	—	86	2	no	—
108	C₂×C₁₆	—	86	2	no	—
109	#108	—	86	2	no	—
110	#107	—	4	2	no	—
111	#107	—	43	2	no	—
112	#106	—	2	4	no	—
113	#106	—	86	1	no	—
114	C₄⋊C₈	—	172	2	no	—
115	C₄×C₈	—	2	2	no	—
116	#95	—	4	2	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
120	M₁₆	—	172	1	no	—
121	M₁₆	—	172	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	172	1	no	—
125	C₂×C₈	#10621884	1	2	yes	—
126	C₂×C₈	#10621898	1	2	yes	—
127	C₂×C₈	#10621900	1	2	yes	—
128	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
129	C₂×C₈	#10621902	1	1	yes	characteristic
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	86	2	no	—
133	C₂²⋊C₄	—	172	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	172	1	no	—
136	C₄²	#10621885	1	2	yes	—
137	C₁₆	—	172	2	no	—
138	C₂³	#13063351	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	—	172	2	no	—
144	C₂×C₄	#13063301	1	1	yes	characteristic
145	C₂×C₄	#13063308	1	1	yes	characteristic
146	C₂×C₄	#13063312	1	1	yes	characteristic
147	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
148	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
149	C₂×C₄	#13063351	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	344	1	no	—
152	C₈	#13063315	1	2	yes	—
153	C₈	#13063320	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	#14583186	1	1	yes	characteristic
156	C₂²	#14583234	1	1	yes	characteristic
157	C₂²	#14583263	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	16	2	no	—
160	C₄	—	344	2	no	—
161	C₄	#14582837	1	1	yes	characteristic
162	C₄	#14583191	1	1	yes	characteristic
163	C₄	#14583217	1	1	yes	characteristic
164	C₄	#14583249	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#19136157	1	1	yes	characteristic
168	C₂	#19136768	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135343

C₂

yes

maximal, characteristic

#19132702

C₂

yes

maximal

#19132693

C₂

yes

maximal

#19132677

C₂

yes

maximal, characteristic

C₄₃×#1846

C₂

yes

maximal, characteristic

C₂×#13063265

C₂²

yes

—

C₂×#13063261

C₂²

yes

characteristic

#14582937

C₂²

yes

characteristic

#14582925

—

#14582659

—

C₂×#107×C₄₃

—

C₂×#106×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#382

—

—

—

—

—

C₂×C₄₃⋊C₁₆

—

#13063261

—

#13063259

Q₈

yes

—

#13063257

—

Dic₄₃⋊C₈

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

#107×C₄₃

—

#106×C₄₃

—

C₄×C₈×C₄₃

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

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

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

Q₁₆

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

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

—

C₈×C₄₃

—

C₈×C₄₃

#125

yes

—

C₈×C₄₃

#126

yes

—

C₄₃⋊C₈

—

#13542

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

#1945

—

#1846

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#2100

yes

characteristic

C₂×C₄₃

#2231

yes

—

C₂×M₃₂

—

—

—

—

C₂×D₄₃

yes

—

#424

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#382

—

#339

—

C₄₃

#13542

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

102

C₂²×C₈

C₂²×D₄₃

yes

characteristic

103

#117

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

#108

—

110

#107

—

111

#107

—

112

#106

—

113

#106

—

114

C₄⋊C₈

—

172

—

115

C₄×C₈

—

116

#95

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

120

M₁₆

—

172

—

121

M₁₆

—

172

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

172

—

125

C₂×C₈

#10621884

yes

—

126

C₂×C₈

#10621898

yes

—

127

C₂×C₈

#10621900

yes

—

128

C₂×C₈

Q₈×D₄₃

yes

characteristic

129

C₂×C₈

#10621902

yes

characteristic

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

172

—

134

C₄²

—

135

C₄²

—

172

—

136

C₄²

#10621885

yes

—

137

C₁₆

—

172

—

138

C₂³

#13063351

yes

characteristic

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

172

—

144

C₂×C₄

#13063301

yes

characteristic

145

C₂×C₄

#13063308

yes

characteristic

146

C₂×C₄

#13063312

yes

characteristic

147

C₂×C₄

C₂×#10621884

yes

characteristic

148

C₂×C₄

C₂×#10621885

yes

characteristic

149

C₂×C₄

#13063351

yes

characteristic

150

C₈

—

151

C₈

—

344

—

152

C₈

#13063315

yes

—

153

C₈

#13063320

yes

—

154

C₂²

—

155

C₂²

#14583186

yes

characteristic

156

C₂²

#14583234

yes

characteristic

157

C₂²

#14583263

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

344

—

161

C₄

#14582837

yes

characteristic

162

C₄

#14583191

yes

characteristic

163

C₄

#14583217

yes

characteristic

164

C₄

#14583249

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

#19136157

yes

characteristic

168

C₂

#19136768

yes

—

	subgroup	G/H
center	C₂²	#14583186
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1846	C₂
socle	C₂²×C₄₃	#476
prime radical	C₂³×C₄₃	AGL(1,ℤ/8)
hypercenter	#1846	D₄₃
hypocenter	C₄₃	#13542
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂²

#14583186

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1846

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
99. C₄₃	#13542	semidirect

H ≅ G/N

type

99. C₄₃

#13542

semidirect

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

subgroup	index
#19135343	2
#19132702	2
#19132693	2
#19132677	2
C₄₃×#1846	2
#13542	43

subgroup

index

C₄₃×#1846

Subgroups — #25016741

Conjugacy classes of subgroups

Complements

Maximal subgroups