Group #24990387 — subgroups

Subgroups — #24990387

Order 11008.

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

normal subgroups only

This group has 438 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	#19134754	C₂	1	2	yes	maximal
2	#19134752	C₂	1	1	yes	maximal, characteristic
3	#19134722	C₂	1	2	yes	maximal
4	C₂×#14582620	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063258	—	2	4	no	—
7	C₂×#13063257	—	2	2	no	—
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
9	#14582769	C₂²	1	1	yes	characteristic
10	#14582769	C₂²	1	1	yes	characteristic
11	#14582661	—	2	4	no	—
12	#14582620	C₄	1	4	yes	complemented
13	C₂×#107×C₄₃	—	2	2	no	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#384	C₂²	1	2	yes	—
16	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
17	C₂×C₄⋊Dic₄₃	—	2	2	no	—
18	C₂×C₄⋊Dic₄₃	—	2	1	no	—
19	C₂×C₄⋊Dic₄₃	—	2	1	no	—
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	#13063258	—	4	8	no	—
22	#13063257	—	4	4	no	—
23	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
24	C₄×C₄₃⋊C₈	Q₈	1	4	yes	—
25	C₂²×C₈×C₄₃	D₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
29	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
30	#107×C₄₃	—	4	4	no	—
31	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₃	Q₈	1	2	yes	—
33	C₂²×C₄×C₄₃	—	4	2	no	—
34	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₄₃	—	4	1	no	—
37	C₂²×Dic₄₃	—	4	1	no	—
38	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₄₃	D₈	1	4	yes	—
40	C₂×C₈×C₄₃	Q₁₆	1	4	yes	—
41	C₄⋊C₄×C₄₃	—	4	4	no	—
42	C₄⋊C₄×C₄₃	—	4	4	no	—
43	C₄⋊C₄×C₄₃	—	4	4	no	—
44	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
47	C₄⋊Dic₄₃	—	4	2	no	—
48	C₄⋊Dic₄₃	—	4	2	no	—
49	C₄⋊Dic₄₃	—	4	4	no	—
50	C₄⋊Dic₄₃	—	4	4	no	—
51	C₂×C₄₃⋊C₈	—	2	4	no	—
52	C₂×C₄₃⋊C₈	—	2	4	no	—
53	C₂×C₄₃⋊C₈	C₄⋊C₄	1	8	yes	—
54	C₂³×C₄₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₄₃	—	8	2	no	—
56	C₂×C₄×C₄₃	—	8	2	no	—
57	C₂×C₄×C₄₃	—	8	2	no	—
58	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
62	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
64	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
65	C₂×C₄×C₄₃	#137	1	4	yes	—
66	C₂×Dic₄₃	—	8	1	no	—
67	C₂×Dic₄₃	—	8	1	no	—
68	C₂×Dic₄₃	—	8	2	no	—
69	C₂×Dic₄₃	—	8	2	no	—
70	C₈×C₄₃	#107	1	8	yes	complemented
71	C₄₃⋊C₈	—	2	8	no	—
72	#8820	—	43	1	no	maximal, Sylow
73	C₂²×C₄₃	#384	1	1	yes	characteristic
74	C₂²×C₄₃	#493	1	1	yes	characteristic
75	C₂²×C₄₃	#494	1	1	yes	characteristic
76	C₂²×C₄₃	#496	1	2	yes	—
77	C₂²×C₄₃	#497	1	2	yes	—
78	C₄×C₄₃	—	16	2	no	—
79	C₄×C₄₃	C₂×#107	1	2	yes	—
80	C₄×C₄₃	#428	1	4	yes	—
81	C₄×C₄₃	#500	1	2	yes	—
82	C₄×C₄₃	#501	1	4	yes	—
83	Dic₄₃	—	16	1	no	—
84	Dic₄₃	—	16	1	no	—
85	#1845	—	43	2	no	—
86	#1845	D₄₃	1	1	yes	characteristic
87	#1844	—	43	1	no	—
88	#1834	—	43	2	no	—
89	C₂×#322	—	43	1	no	—
90	C₂×C₄₃	#1721	1	1	yes	characteristic
91	C₂×C₄₃	#1722	1	1	yes	characteristic
92	C₂×C₄₃	#1723	1	2	yes	—
93	C₂×C₄₃	#1845	1	1	yes	characteristic
94	C₂×C₄₃	#1847	1	2	yes	—
95	C₂×#107	—	2	2	no	—
96	C₂×#107	—	86	4	no	—
97	C₂×#106	—	86	2	no	—
98	C₂×C₄×C₈	—	43	2	no	—
99	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
100	#384	—	43	1	no	—
101	#384	—	43	1	no	—
102	#384	C₂×D₄₃	1	2	yes	—
103	#340	—	86	4	no	—
104	#322	—	43	4	no	—
105	C₄₃	#8820	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	43	4	no	—
107	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	86	2	no	—
112	C₂×C₄⋊C₄	—	86	2	no	—
113	C₂×C₄⋊C₄	—	86	1	no	—
114	C₂×C₄⋊C₄	—	86	1	no	—
115	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
116	#107	—	4	4	no	—
117	#107	—	172	8	no	—
118	#106	—	172	4	no	—
119	C₄×C₈	—	43	4	no	—
120	C₄×C₈	—	43	4	no	—
121	C₄×C₈	Dic₈₆	1	2	yes	—
122	C₄×C₈	C₄×D₄₃	1	2	yes	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	172	1	no	—
125	C₂²×C₄	—	172	1	no	—
126	C₂²×C₄	D₄×D₄₃	1	2	yes	—
127	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
128	C₂×C₈	—	43	8	no	—
129	C₂×C₈	—	86	4	no	—
130	C₂×C₈	—	86	4	no	—
131	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
132	C₂×C₈	#10621882	1	4	yes	—
133	C₂×C₈	#10621885	1	4	yes	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	4	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	172	2	no	—
138	C₄⋊C₄	—	172	2	no	—
139	C₄⋊C₄	—	172	4	no	—
140	C₄⋊C₄	—	172	4	no	—
141	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
142	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
143	C₄²	Q₈×D₄₃	1	2	yes	—
144	C₂³	#13063321	1	1	yes	characteristic
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	—	344	1	no	—
149	C₂×C₄	—	344	1	no	—
150	C₂×C₄	—	344	2	no	—
151	C₂×C₄	—	344	2	no	—
152	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
153	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
154	C₂×C₄	#13063323	1	1	yes	characteristic
155	C₂×C₄	#13063324	1	2	yes	—
156	C₂×C₄	C₂×#10621882	1	2	yes	—
157	C₂×C₄	C₂×#10621885	1	2	yes	—
158	C₂×C₄	#13063349	1	4	yes	—
159	C₂×C₄	#13063357	1	4	yes	—
160	C₈	—	86	8	no	—
161	C₈	#13063258	1	8	yes	complemented
162	C₂²	#14582768	1	1	yes	characteristic
163	C₂²	#14583275	1	1	yes	characteristic
164	C₂²	#14583278	1	1	yes	characteristic
165	C₂²	#14583287	1	2	yes	—
166	C₂²	#14583292	1	2	yes	—
167	C₄	—	16	2	no	—
168	C₄	—	688	1	no	—
169	C₄	—	688	1	no	—
170	C₄	C₂×#13063258	1	2	yes	—
171	C₄	#14582940	1	4	yes	—
172	C₄	#14583302	1	2	yes	—
173	C₄	#14583307	1	4	yes	—
174	C₂	#19134262	1	1	yes	characteristic
175	C₂	#19134267	1	1	yes	characteristic
176	C₂	#19134273	1	2	yes	—
177	C₂	#19134754	1	1	yes	characteristic
178	C₂	#19134762	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19134754

C₂

yes

maximal

#19134752

C₂

yes

maximal, characteristic

#19134722

C₂

yes

maximal

C₂×#14582620

C₂

yes

maximal, characteristic

C₄₃×#1845

C₂

yes

maximal, characteristic

C₂×#13063258

—

C₂×#13063257

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582769

C₂²

yes

characteristic

#14582769

C₂²

yes

characteristic

#14582661

—

#14582620

C₄

yes

complemented

C₂×#107×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#384

C₂²

yes

—

C₂²×C₄₃⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063258

—

#13063257

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

Q₈

yes

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#107×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

Q₁₆

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

#107

yes

complemented

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#497

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#107

yes

—

C₄×C₄₃

#428

yes

—

C₄×C₄₃

#500

yes

—

C₄×C₄₃

#501

yes

—

Dic₄₃

—

Dic₄₃

—

#1845

—

#1845

D₄₃

yes

—

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#1845

yes

characteristic

C₂×C₄₃

#1847

yes

—

C₂×#107

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

100

#384

—

101

#384

—

102

#384

C₂×D₄₃

yes

—

103

#340

—

104

#322

—

105

C₄₃

#8820

yes

characteristic, complemented, Sylow

106

C₂²×C₈

—

107

C₂²×C₈

C₂²⋊D₄₃

yes

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₂²×D₄₃

yes

characteristic

116

#107

—

117

#107

—

172

—

118

#106

—

172

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

Dic₈₆

yes

—

122

C₄×C₈

C₄×D₄₃

yes

—

123

C₂²×C₄

—

124

C₂²×C₄

—

172

—

125

C₂²×C₄

—

172

—

126

C₂²×C₄

D₄×D₄₃

yes

—

127

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

Dic₄₃⋊C₄

yes

—

132

C₂×C₈

#10621882

yes

—

133

C₂×C₈

#10621885

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

172

—

138

C₄⋊C₄

—

172

—

139

C₄⋊C₄

—

172

—

140

C₄⋊C₄

—

172

—

141

C₄²

C₂×Dic₈₆

yes

characteristic

142

C₄²

C₂×C₄×D₄₃

yes

characteristic

143

C₄²

Q₈×D₄₃

yes

—

144

C₂³

#13063321

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

344

—

149

C₂×C₄

—

344

—

150

C₂×C₄

—

344

—

151

C₂×C₄

—

344

—

152

C₂×C₄

C₄⋊C₄×D₄₃

yes

—

153

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

154

C₂×C₄

#13063323

yes

characteristic

155

C₂×C₄

#13063324

yes

—

156

C₂×C₄

C₂×#10621882

yes

—

157

C₂×C₄

C₂×#10621885

yes

—

158

C₂×C₄

#13063349

yes

—

159

C₂×C₄

#13063357

yes

—

160

C₈

—

161

C₈

#13063258

yes

complemented

162

C₂²

#14582768

yes

characteristic

163

C₂²

#14583275

yes

characteristic

164

C₂²

#14583278

yes

characteristic

165

C₂²

#14583287

yes

—

166

C₂²

#14583292

yes

—

167

C₄

—

168

C₄

—

688

—

169

C₄

—

688

—

170

C₄

C₂×#13063258

yes

—

171

C₄

#14582940

yes

—

172

C₄

#14583302

yes

—

173

C₄

#14583307

yes

—

174

C₂

#19134262

yes

characteristic

175

C₂

#19134267

yes

characteristic

176

C₂

#19134273

yes

—

177

C₂

#19134754

yes

characteristic

178

C₂

#19134762

yes

—

	subgroup	G/H
center	C₂³	#13063321
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1845	C₂
socle, prime radical	C₂³×C₄₃	#127
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1845	D₄₃
hypocenter	C₄₃	#8820
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#13063321

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1845

C₂

socle, prime radical

C₂³×C₄₃

#127

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
12. #14582620	C₄	semidirect
70. C₈×C₄₃	#107	semidirect
105. C₄₃	#8820	semidirect
161. C₈	#13063258	semidirect

H ≅ G/N

type

12. #14582620

C₄

semidirect

70. C₈×C₄₃

#107

semidirect

105. C₄₃

#8820

semidirect

161. C₈

#13063258

semidirect

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

subgroup	index
#19134754	2
#19134752	2
#19134722	2
C₂×#14582620	2
C₄₃×#1845	2
#8820	43

subgroup

index

C₄₃×#1845

Subgroups — #24990387

Conjugacy classes of subgroups

Complements

Maximal subgroups