Group #24996065 — subgroups

Subgroups — #24996065

Order 11008.

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

normal subgroups only

This group has 582 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	#19134767	C₂	1	1	yes	maximal, characteristic, complemented
2	#19133263	C₂	1	4	yes	maximal, complemented
3	#19132878	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic
5	#14582952	C₂²	1	2	yes	—
6	C₂×#13063261	—	2	2	no	—
7	#14582923	C₂²	1	2	yes	—
8	#14582643	—	2	8	no	—
9	#14582620	—	2	1	no	—
10	C₈×C₄₃⋊C₈	—	2	1	no	—
11	C₂²×D₈×C₄₃	—	2	2	no	—
12	C₂×#127×C₄₃	C₂²	1	2	yes	—
13	C₄₃×#493	—	2	1	no	—
14	C₄₃×#493	—	2	1	no	—
15	C₄₃×#493	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#10621877	—	2	2	no	—
18	#13063261	—	4	4	no	—
19	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
20	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
21	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
22	C₄×C₄₃⋊C₈	—	2	2	no	—
23	C₂²×D₄×C₄₃	—	2	2	no	—
24	C₂²×D₄×C₄₃	—	2	4	no	—
25	C₂×D₈×C₄₃	—	2	4	no	—
26	C₂×D₈×C₄₃	—	2	4	no	—
27	C₂×D₈×C₄₃	—	4	4	no	—
28	C₂×D₈×C₄₃	—	4	4	no	—
29	C₂²×C₈×C₄₃	D₄	1	2	yes	—
30	#127×C₄₃	—	2	4	no	—
31	#127×C₄₃	—	2	2	no	—
32	#127×C₄₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
36	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
37	C₂⁴×C₄₃	—	4	2	no	—
38	C₂×D₄×C₄₃	—	2	4	no	—
39	C₂×D₄×C₄₃	—	2	8	no	—
40	C₂×D₄×C₄₃	—	4	4	no	—
41	C₂×D₄×C₄₃	—	4	2	no	—
42	C₂×D₄×C₄₃	—	4	4	no	—
43	C₂×D₄×C₄₃	—	4	4	no	—
44	C₂×D₄×C₄₃	—	8	8	no	—
45	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	#41	1	2	yes	—
47	D₈×C₄₃	—	4	8	no	—
48	D₈×C₄₃	—	4	8	no	—
49	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₄₃	D₈	1	4	yes	—
52	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
53	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₄₃	C₂×D₄	1	2	yes	—
55	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
56	#10621877	—	4	4	no	—
57	C₂×C₄₃⋊C₈	—	2	4	no	—
58	C₂×C₄₃⋊C₈	—	4	2	no	—
59	C₂³×C₄₃	—	4	4	no	—
60	C₂³×C₄₃	—	8	8	no	—
61	C₂³×C₄₃	—	8	4	no	—
62	C₂³×C₄₃	#124	1	1	yes	characteristic
63	D₄×C₄₃	—	4	8	no	—
64	D₄×C₄₃	—	8	8	no	—
65	D₄×C₄₃	—	8	8	no	—
66	D₄×C₄₃	—	8	2	no	—
67	D₄×C₄₃	—	8	2	no	—
68	C₂×C₄×C₄₃	—	2	4	no	—
69	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₄₃	#121	1	2	yes	—
72	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
75	C₈×C₄₃	#102	1	4	yes	—
76	C₈×C₄₃	#102	1	4	yes	—
77	C₄₃⋊C₈	—	4	4	no	—
78	C₄₃⋊C₈	—	8	2	no	—
79	#9773	—	43	1	no	maximal, Sylow
80	C₂²×C₄₃	—	2	2	no	—
81	C₂²×C₄₃	—	8	4	no	—
82	C₂²×C₄₃	—	8	8	no	—
83	C₂²×C₄₃	—	16	8	no	—
84	C₂²×C₄₃	#390	1	1	yes	characteristic
85	C₂²×C₄₃	#486	1	2	yes	—
86	C₄×C₄₃	—	2	2	no	—
87	C₄×C₄₃	C₂×#102	1	2	yes	—
88	C₄×C₄₃	#472	1	4	yes	—
89	C₄×C₄₃	#492	1	1	yes	characteristic
90	C₄×C₄₃	#493	1	1	yes	characteristic
91	C₂×#493	D₄₃	1	1	yes	characteristic
92	#1848	—	43	1	no	—
93	#1545	—	43	4	no	—
94	#1450	—	43	1	no	—
95	C₂×C₄₃	—	2	2	no	—
96	C₂×C₄₃	—	16	4	no	—
97	C₂×C₄₃	#1957	1	1	yes	characteristic
98	C₂×C₄₃	#1978	1	2	yes	—
99	C₂²×D₈	—	2	2	no	—
100	C₂×#127	C₂×D₄₃	1	2	yes	—
101	#493	—	2	1	no	—
102	#493	—	2	1	no	—
103	#493	Dic₄₃	1	4	yes	—
104	#431	—	43	2	no	—
105	C₂×#108	—	86	2	no	—
106	#423	—	43	2	no	—
107	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
108	#331	—	86	8	no	—
109	#322	—	86	1	no	—
110	C₈²	—	86	1	no	—
111	C₄₃	#9773	1	1	yes	characteristic, complemented, Sylow
112	C₂²×D₄	—	2	4	no	—
113	C₂²×D₄	—	2	2	no	—
114	C₂×D₈	—	2	4	no	—
115	C₂×D₈	—	2	4	no	—
116	C₂×D₈	—	4	4	no	—
117	C₂×D₈	—	4	4	no	—
118	C₂×M₁₆	—	86	2	no	—
119	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
120	#127	—	2	4	no	—
121	#127	—	2	2	no	—
122	#127	C₂×Dic₄₃	1	4	yes	—
123	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
124	#108	—	172	4	no	—
125	C₄⋊C₈	—	43	4	no	—
126	C₄⋊C₈	—	86	4	no	—
127	C₂²⋊C₈	—	86	2	no	—
128	C₄×C₈	—	86	2	no	—
129	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
130	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
132	C₂⁴	—	4	2	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	2	8	no	—
135	C₂×D₄	—	4	4	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂×D₄	—	4	4	no	—
138	C₂×D₄	—	4	4	no	—
139	C₂×D₄	—	8	8	no	—
140	C₂²×C₄	#10621900	1	2	yes	—
141	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
142	D₈	—	4	8	no	—
143	D₈	—	4	8	no	—
144	M₁₆	—	172	4	no	—
145	C₂×C₈	—	86	4	no	—
146	C₂×C₈	—	172	2	no	—
147	C₂×C₈	#10621882	1	4	yes	—
148	C₂×C₈	#10621883	1	4	yes	—
149	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
150	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
151	C₄²	D₄×D₄₃	1	2	yes	—
152	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
154	C₂³	—	4	4	no	—
155	C₂³	—	8	8	no	—
156	C₂³	—	8	4	no	—
157	C₂³	#13063312	1	1	yes	characteristic
158	D₄	—	4	8	no	—
159	D₄	—	8	8	no	—
160	D₄	—	8	8	no	—
161	D₄	—	8	2	no	—
162	D₄	—	8	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
165	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
166	C₂×C₄	#13063296	1	2	yes	—
167	C₂×C₄	#13063321	1	1	yes	characteristic
168	C₂×C₄	C₂×#10621882	1	2	yes	—
169	C₂×C₄	C₂×#10621883	1	2	yes	—
170	C₈	—	344	2	no	—
171	C₈	—	172	4	no	—
172	C₈	#13063246	1	4	yes	—
173	C₈	#13063246	1	4	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	—	8	8	no	—
176	C₂²	—	8	4	no	—
177	C₂²	—	16	8	no	—
178	C₂²	#14582802	1	1	yes	characteristic
179	C₂²	#14583245	1	2	yes	—
180	C₄	—	2	2	no	—
181	C₄	C₂×#13063246	1	2	yes	—
182	C₄	#14583166	1	4	yes	—
183	C₄	#14583271	1	1	yes	characteristic
184	C₄	#14583275	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	—	16	4	no	—
187	C₂	#19135402	1	1	yes	characteristic
188	C₂	#19135506	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19134767

C₂

yes

maximal, characteristic, complemented

#19133263

C₂

yes

maximal, complemented

#19132878

C₂

yes

maximal, characteristic, complemented

C₂×C₄₃×#493

C₂

yes

maximal, characteristic

#14582952

C₂²

yes

—

C₂×#13063261

—

#14582923

C₂²

yes

—

#14582643

—

#14582620

—

C₈×C₄₃⋊C₈

—

C₂²×D₈×C₄₃

—

C₂×#127×C₄₃

C₂²

yes

—

C₄₃×#493

—

C₄₃×#493

—

C₄₃×#493

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂²×D₄×C₄₃

—

C₂²×D₄×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

#127×C₄₃

—

#127×C₄₃

—

#127×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₂⁴×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

D₈×C₄₃

—

D₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

#124

yes

characteristic

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₈×C₄₃

#102

yes

—

C₈×C₄₃

#102

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#9773

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#390

yes

characteristic

C₂²×C₄₃

#486

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#102

yes

—

C₄×C₄₃

#472

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#1957

yes

characteristic

C₂×C₄₃

#1978

yes

—

C₂²×D₈

—

100

C₂×#127

C₂×D₄₃

yes

—

101

#493

—

102

#493

—

103

#493

Dic₄₃

yes

—

104

#431

—

105

C₂×#108

—

106

#423

—

107

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

108

#331

—

109

#322

—

110

C₈²

—

111

C₄₃

#9773

yes

characteristic, complemented, Sylow

112

C₂²×D₄

—

113

C₂²×D₄

—

114

C₂×D₈

—

115

C₂×D₈

—

116

C₂×D₈

—

117

C₂×D₈

—

118

C₂×M₁₆

—

119

C₂²×C₈

C₂²⋊D₄₃

yes

—

120

#127

—

121

#127

—

122

#127

C₂×Dic₄₃

yes

—

123

C₂×C₄²

C₂²×D₄₃

yes

characteristic

124

#108

—

172

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₂²⋊C₈

—

128

C₄×C₈

—

129

C₄×C₈

C₂×Dic₄₃

yes

—

130

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

131

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

132

C₂⁴

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

#10621900

yes

—

141

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

142

D₈

—

143

D₈

—

144

M₁₆

—

172

—

145

C₂×C₈

—

146

C₂×C₈

—

172

—

147

C₂×C₈

#10621882

yes

—

148

C₂×C₈

#10621883

yes

—

149

C₂×C₈

C₂²⋊Dic₄₃

yes

—

150

C₂×C₈

C₂²⋊Dic₄₃

yes

—

151

C₄²

D₄×D₄₃

yes

—

152

C₄²

C₂²×Dic₄₃

yes

characteristic

153

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

#13063312

yes

characteristic

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

165

C₂×C₄

D₄×Dic₄₃

yes

—

166

C₂×C₄

#13063296

yes

—

167

C₂×C₄

#13063321

yes

characteristic

168

C₂×C₄

C₂×#10621882

yes

—

169

C₂×C₄

C₂×#10621883

yes

—

170

C₈

—

344

—

171

C₈

—

172

—

172

C₈

#13063246

yes

—

173

C₈

#13063246

yes

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

#14582802

yes

characteristic

179

C₂²

#14583245

yes

—

180

C₄

—

181

C₄

C₂×#13063246

yes

—

182

C₄

#14583166

yes

—

183

C₄

#14583271

yes

characteristic

184

C₄

#14583275

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#19135402

yes

characteristic

188

C₂

#19135506

yes

—

	subgroup	G/H
center	C₂²	#14582802
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting, prime radical	C₂×C₄₃×#493	C₂
socle	C₂²×C₄₃	#390
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#493	D₄₃
hypocenter	C₄₃	#9773
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂²

#14582802

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting, prime radical

C₂×C₄₃×#493

C₂

socle

C₂²×C₄₃

#390

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
1. #19134767	C₂	semidirect
2. #19133263	C₂	semidirect
3. #19132878	C₂	semidirect
111. C₄₃	#9773	semidirect

H ≅ G/N

type

1. #19134767

C₂

semidirect

2. #19133263

C₂

semidirect

3. #19132878

C₂

semidirect

111. C₄₃

#9773

semidirect

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

subgroup	index
#19134767	2
#19133263	2
#19132878	2
C₂×C₄₃×#493	2
#9773	43

subgroup

index

C₂×C₄₃×#493

Subgroups — #24996065

Conjugacy classes of subgroups

Complements

Maximal subgroups