Group #24995992 — subgroups

Subgroups — #24995992

Order 11008.

478 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 111 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	#19145023	C₂	1	1	yes	maximal, characteristic
2	#19133154	C₂	1	4	yes	maximal, complemented
3	#19132887	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#1835	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063347	—	2	2	no	—
6	#14583449	C₂²	1	2	yes	—
7	#14583283	—	2	2	no	—
8	#14583279	C₄	1	1	yes	characteristic, complemented
9	#14583276	C₄	1	1	yes	characteristic, complemented
10	#14583272	C₄	1	2	yes	complemented
11	#14582952	C₂²	1	2	yes	—
12	#14582632	—	2	8	no	—
13	#14582621	—	2	2	no	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#382	C₂²	1	2	yes	—
16	C₄₃×#340	—	2	2	no	—
17	C₂×#10621898	—	2	2	no	—
18	#13063347	—	4	8	no	—
19	C₂×Dic₁₇₂	—	2	2	no	—
20	C₂×#10621873	—	2	4	no	—
21	C₂×D₃₄₄	—	2	2	no	—
22	#13063325	C₂×C₄	1	2	yes	—
23	#13063322	C₂×C₄	1	2	yes	—
24	#13063313	—	2	2	no	—
25	#13063299	—	2	4	no	—
26	C₄×D₁₇₂	—	2	4	no	—
27	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
29	C₄×C₄₃⋊C₈	—	2	2	no	—
30	C₂²×C₈×C₄₃	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₄₃	D₄	1	2	yes	—
36	#95×C₄₃	—	2	4	no	—
37	C₂²×C₄×C₄₃	—	4	2	no	—
38	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₂²⋊D₄₃	—	4	2	no	—
41	#10621898	—	4	8	no	—
42	C₂×D₁₇₂	—	2	2	no	—
43	C₂×D₁₇₂	—	2	4	no	—
44	C₂×C₄×D₄₃	—	4	2	no	—
45	C₂×Dic₈₆	—	2	2	no	—
46	C₂×C₈×C₄₃	—	2	4	no	—
47	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
49	C₄⋊C₄×C₄₃	—	4	4	no	—
50	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₄₃	#41	1	2	yes	—
53	(C₂×D₄₃)⋊C₄	—	4	4	no	—
54	C₄⋊Dic₄₃	—	2	4	no	—
55	C₂×C₄₃⋊C₈	—	2	4	no	—
56	Dic₁₇₂	—	4	4	no	—
57	D₃₄₄	—	4	4	no	—
58	#10621873	—	4	8	no	—
59	C₂³×C₄₃	#127	1	1	yes	characteristic
60	C₂²×D₄₃	—	4	2	no	—
61	C₂×C₄×C₄₃	—	4	2	no	—
62	C₂×C₄×C₄₃	—	8	4	no	—
63	C₂×C₄×C₄₃	—	8	2	no	—
64	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
66	C₂×C₄×C₄₃	#121	1	2	yes	—
67	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
68	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
69	C₂²⋊D₄₃	—	8	4	no	—
70	C₂×Dic₄₃	—	4	2	no	—
71	D₁₇₂	—	4	4	no	—
72	D₁₇₂	—	8	4	no	—
73	C₄×D₄₃	—	8	4	no	—
74	Dic₈₆	—	4	4	no	—
75	C₈×C₄₃	—	2	2	no	—
76	C₈×C₄₃	—	2	2	no	—
77	C₄₃⋊C₈	—	4	4	no	—
78	#9760	—	43	1	no	maximal, Sylow
79	C₂²×C₄₃	—	2	2	no	—
80	C₂²×C₄₃	#390	1	1	yes	characteristic
81	C₂²×C₄₃	#496	1	2	yes	—
82	C₂×D₄₃	—	8	2	no	—
83	C₂×D₄₃	—	8	4	no	—
84	C₄×C₄₃	—	2	2	no	—
85	C₄×C₄₃	—	2	2	no	—
86	C₄×C₄₃	—	16	2	no	—
87	C₄×C₄₃	#418	1	2	yes	—
88	C₄×C₄₃	#490	1	2	yes	—
89	Dic₄₃	—	8	2	no	—
90	#3147	—	43	1	no	—
91	#1835	D₄₃	1	1	yes	characteristic, complemented
92	#1517	—	43	4	no	—
93	#1452	—	43	1	no	—
94	C₂×C₄₃	—	2	2	no	—
95	C₂×C₄₃	#1961	1	1	yes	characteristic
96	C₂×C₄₃	#1976	1	2	yes	—
97	D₄₃	—	16	2	no	—
98	C₂×#135	—	86	2	no	—
99	#532	—	43	2	no	—
100	#495	—	86	2	no	—
101	#494	—	43	1	no	—
102	#493	—	43	1	no	—
103	#492	—	43	2	no	—
104	#431	—	43	2	no	—
105	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
106	#382	C₂×D₄₃	1	2	yes	—
107	#340	—	2	2	no	—
108	#327	—	86	8	no	—
109	#322	—	86	2	no	—
110	C₄₃	#9760	1	1	yes	characteristic, complemented, Sylow
111	C₂×#41	—	86	2	no	—
112	#135	—	172	8	no	—
113	C₂×Q₁₆	—	86	2	no	—
114	C₂×SD₁₆	—	86	4	no	—
115	C₂×D₈	—	86	2	no	—
116	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
117	C₄⋊Q₈	—	43	2	no	—
118	#127	—	43	2	no	—
119	#124	—	86	2	no	—
120	#121	—	86	4	no	—
121	C₄×D₄	—	86	4	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
124	C₄⋊C₈	—	86	2	no	—
125	C₂²⋊C₈	—	43	4	no	—
126	C₄×C₈	—	86	2	no	—
127	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
128	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
129	C₄×C₈	D₁₇₂	1	2	yes	—
130	#95	—	2	4	no	—
131	#41	—	172	8	no	—
132	C₂×Q₈	—	86	2	no	—
133	C₂×D₄	—	86	2	no	—
134	C₂×D₄	—	86	4	no	—
135	C₂×D₄	—	172	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	172	2	no	—
138	C₂²×C₄	D₄×D₄₃	1	2	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
140	Q₁₆	—	172	4	no	—
141	SD₁₆	—	172	8	no	—
142	D₈	—	172	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	86	4	no	—
145	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
146	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	86	4	no	—
149	C₂²⋊C₄	—	172	4	no	—
150	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
151	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
152	C₄²	#10621902	1	2	yes	—
153	C₂³	—	172	2	no	—
154	C₂³	#13063321	1	1	yes	characteristic
155	Q₈	—	172	4	no	—
156	D₄	—	344	4	no	—
157	D₄	—	344	4	no	—
158	D₄	—	172	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	344	4	no	—
163	C₂×C₄	—	172	2	no	—
164	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
165	C₂×C₄	#13063287	1	2	yes	—
166	C₂×C₄	#13063300	1	2	yes	—
167	C₂×C₄	#13063314	1	1	yes	characteristic
168	C₂×C₄	#13063350	1	4	yes	—
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	172	4	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	344	4	no	—
174	C₂²	—	344	2	no	—
175	C₂²	#14582800	1	1	yes	characteristic
176	C₂²	#14583289	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	16	2	no	—
180	C₄	—	344	2	no	—
181	C₄	#14582903	1	2	yes	—
182	C₄	#14583265	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	688	2	no	—
185	C₂	#19135418	1	1	yes	characteristic
186	C₂	#19135491	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145023

C₂

yes

maximal, characteristic

#19133154

C₂

yes

maximal, complemented

#19132887

C₂

yes

maximal, characteristic, complemented

C₄₃×#1835

C₂

yes

maximal, characteristic, complemented

C₂×#13063347

—

#14583449

C₂²

yes

—

#14583283

—

#14583279

C₄

yes

characteristic, complemented

#14583276

C₄

yes

characteristic, complemented

#14583272

C₄

yes

complemented

#14582952

C₂²

yes

—

#14582632

—

#14582621

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#382

C₂²

yes

—

C₄₃×#340

—

C₂×#10621898

—

#13063347

—

C₂×Dic₁₇₂

—

C₂×#10621873

—

C₂×D₃₄₄

—

#13063325

C₂×C₄

yes

—

#13063322

C₂×C₄

yes

—

#13063313

—

#13063299

—

C₄×D₁₇₂

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

D₄

yes

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

#41

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

D₃₄₄

—

#10621873

—

C₂³×C₄₃

#127

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×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

#124

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#9760

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#390

yes

characteristic

C₂²×C₄₃

#496

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#418

yes

—

C₄×C₄₃

#490

yes

—

Dic₄₃

—

#3147

—

#1835

D₄₃

yes

characteristic, complemented

#1517

—

#1452

—

C₂×C₄₃

—

C₂×C₄₃

#1961

yes

characteristic

C₂×C₄₃

#1976

yes

—

D₄₃

—

C₂×#135

—

#532

—

100

#495

—

101

#494

—

102

#493

—

103

#492

—

104

#431

—

105

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

106

#382

C₂×D₄₃

yes

—

107

#340

—

108

#327

—

109

#322

—

110

C₄₃

#9760

yes

characteristic, complemented, Sylow

111

C₂×#41

—

112

#135

—

172

—

113

C₂×Q₁₆

—

114

C₂×SD₁₆

—

115

C₂×D₈

—

116

C₂²×C₈

C₂²⋊D₄₃

yes

—

117

C₄⋊Q₈

—

118

#127

—

119

#124

—

120

#121

—

121

C₄×D₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄²

C₂²×D₄₃

yes

characteristic

124

C₄⋊C₈

—

125

C₂²⋊C₈

—

126

C₄×C₈

—

127

C₄×C₈

C₄×D₄₃

yes

characteristic

128

C₄×C₈

C₄×D₄₃

yes

characteristic

129

C₄×C₈

D₁₇₂

yes

—

130

#95

—

131

#41

—

172

—

132

C₂×Q₈

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

172

—

136

C₂²×C₄

—

137

C₂²×C₄

—

172

—

138

C₂²×C₄

D₄×D₄₃

yes

—

139

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

140

Q₁₆

—

172

—

141

SD₁₆

—

172

—

142

D₈

—

172

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

146

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₂²⋊C₄

—

172

—

150

C₄²

C₂×C₄×D₄₃

yes

characteristic

151

C₄²

C₂×D₁₇₂

yes

characteristic

152

C₄²

#10621902

yes

—

153

C₂³

—

172

—

154

C₂³

#13063321

yes

characteristic

155

Q₈

—

172

—

156

D₄

—

344

—

157

D₄

—

344

—

158

D₄

—

172

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

344

—

163

C₂×C₄

—

172

—

164

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

165

C₂×C₄

#13063287

yes

—

166

C₂×C₄

#13063300

yes

—

167

C₂×C₄

#13063314

yes

characteristic

168

C₂×C₄

#13063350

yes

—

169

C₈

—

170

C₈

—

171

C₈

—

172

—

172

C₂²

—

173

C₂²

—

344

—

174

C₂²

—

344

—

175

C₂²

#14582800

yes

characteristic

176

C₂²

#14583289

yes

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

344

—

181

C₄

#14582903

yes

—

182

C₄

#14583265

yes

—

183

C₂

—

184

C₂

—

688

—

185

C₂

#19135418

yes

characteristic

186

C₂

#19135491

yes

—

	subgroup	G/H
center	C₂²	#14582800
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1835	C₂
socle	C₂²×C₄₃	#390
prime radical	#19145023	C₂
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1835	D₄₃
hypocenter	C₄₃	#9760
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#14582800

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1835

C₂

C₂²×C₄₃

C₂

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
2. #19133154	C₂	semidirect
3. #19132887	C₂	semidirect
4. C₄₃×#1835	C₂	semidirect
8. #14583279	C₄	semidirect
9. #14583276	C₄	semidirect
10. #14583272	C₄	semidirect
91. #1835	D₄₃	semidirect
110. C₄₃	#9760	semidirect

H ≅ G/N

type

2. #19133154

C₂

semidirect

3. #19132887

C₂

semidirect

4. C₄₃×#1835

C₂

semidirect

8. #14583279

C₄

semidirect

9. #14583276

C₄

semidirect

10. #14583272

C₄

semidirect

91. #1835

D₄₃

semidirect

110. C₄₃

#9760

semidirect

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

subgroup	index
#19145023	2
#19133154	2
#19132887	2
C₄₃×#1835	2
#9760	43

subgroup

index

C₄₃×#1835

Subgroups — #24995992

Conjugacy classes of subgroups

Complements

Maximal subgroups