Group #25001614 — subgroups

Subgroups — #25001614

Order 11008.

354 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 97 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	#19135427	C₂	1	1	yes	maximal, characteristic
2	#19133369	C₂	1	2	yes	maximal, complemented
3	#19132993	C₂	1	2	yes	maximal
4	#19132424	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1843	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583449	C₂²	1	1	yes	characteristic
7	#14582952	C₂²	1	2	yes	—
8	#14582923	C₂²	1	1	yes	characteristic
9	C₂×#13063253	—	2	2	no	—
10	#14582651	—	2	4	no	—
11	#14582626	—	2	4	no	—
12	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
13	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
14	C₄₃×#376	C₂²	1	1	yes	characteristic
15	C₂×#10621898	—	2	1	no	—
16	C₂×#10621877	—	2	1	no	—
17	#13063325	Q₈	1	1	yes	characteristic
18	#13063322	Q₈	1	1	yes	characteristic
19	#13063313	—	2	1	no	—
20	#13063299	—	2	2	no	—
21	C₄×D₁₇₂	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
24	#13063253	—	4	4	no	—
25	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
26	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
27	C₂²×C₈×C₄₃	—	2	2	no	—
28	C₂×C₄²×C₄₃	D₄	1	2	yes	—
29	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₃	—	2	2	no	—
31	C₄⋊C₈×C₄₃	—	2	2	no	—
32	C₄⋊C₈×C₄₃	—	4	4	no	—
33	#95×C₄₃	—	2	2	no	—
34	C₂²×C₄×C₄₃	—	2	2	no	—
35	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	#41	1	2	yes	—
37	C₂×C₂²⋊D₄₃	—	4	1	no	—
38	#10621898	—	4	2	no	—
39	#10621898	—	4	2	no	—
40	C₂×D₁₇₂	—	2	2	no	—
41	C₂×D₁₇₂	—	2	1	no	—
42	C₂×C₄×D₄₃	—	4	1	no	—
43	C₂×Dic₈₆	—	2	1	no	—
44	C₂×C₈×C₄₃	—	4	2	no	—
45	C₂×C₈×C₄₃	—	4	4	no	—
46	C₄²×C₄₃	—	2	2	no	—
47	C₄²×C₄₃	D₈	1	4	yes	—
48	C₄²×C₄₃	C₂×D₄	1	2	yes	—
49	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₄₃	#41	1	1	yes	characteristic
51	(C₂×D₄₃)⋊C₄	—	4	2	no	—
52	C₄⋊Dic₄₃	—	2	2	no	—
53	#10621877	—	4	2	no	—
54	C₂×C₄₃⋊C₈	—	2	4	no	—
55	C₂×C₄₃⋊C₈	—	4	1	no	—
56	C₂³×C₄₃	#123	1	1	yes	characteristic
57	C₂²×D₄₃	—	4	1	no	—
58	C₂×C₄×C₄₃	—	2	4	no	—
59	C₂×C₄×C₄₃	—	4	2	no	—
60	C₂×C₄×C₄₃	—	4	4	no	—
61	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
63	C₂×C₄×C₄₃	#123	1	2	yes	—
64	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
65	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₄₃	#137	1	2	yes	—
67	C₂²⋊D₄₃	—	8	2	no	—
68	C₂×Dic₄₃	—	4	1	no	—
69	D₁₇₂	—	4	4	no	—
70	D₁₇₂	—	4	2	no	—
71	C₄×D₄₃	—	8	1	no	—
72	C₄×D₄₃	—	8	1	no	—
73	Dic₈₆	—	4	2	no	—
74	C₈×C₄₃	—	8	2	no	—
75	C₄₃⋊C₈	—	4	4	no	—
76	C₄₃⋊C₈	—	8	1	no	—
77	#10667	—	43	1	no	maximal, Sylow
78	C₂²×C₄₃	—	2	2	no	—
79	C₂²×C₄₃	#399	1	1	yes	characteristic
80	C₂²×C₄₃	#480	1	2	yes	—
81	C₂×D₄₃	—	8	1	no	—
82	C₂×D₄₃	—	8	2	no	—
83	C₄×C₄₃	—	2	2	no	—
84	C₄×C₄₃	—	2	2	no	—
85	C₄×C₄₃	—	4	4	no	—
86	C₄×C₄₃	#474	1	2	yes	—
87	C₄×C₄₃	#493	1	1	yes	characteristic
88	C₄×C₄₃	#497	1	1	yes	characteristic
89	Dic₄₃	—	8	1	no	—
90	#1963	—	43	1	no	—
91	#1843	D₄₃	1	1	yes	characteristic, complemented
92	#1571	—	43	2	no	—
93	#1480	—	43	2	no	—
94	#1274	—	43	1	no	—
95	C₂×C₄₃	—	2	2	no	—
96	C₂×C₄₃	#2093	1	1	yes	characteristic
97	C₂×C₄₃	#2098	1	2	yes	—
98	D₄₃	—	16	1	no	—
99	#532	—	43	1	no	—
100	#431	—	43	2	no	—
101	#423	—	43	1	no	—
102	C₂×C₄⋊C₈	—	2	2	no	—
103	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
104	C₂×#104	—	86	2	no	—
105	#376	C₂×D₄₃	1	1	yes	characteristic
106	#335	—	86	4	no	—
107	D₄⋊C₈	—	86	4	no	—
108	C₄₃	#10667	1	1	yes	characteristic, complemented, Sylow
109	C₂×#41	—	86	1	no	—
110	C₂×M₁₆	—	86	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₄⋊Q₈	—	43	1	no	—
113	#127	—	43	1	no	—
114	#124	—	86	1	no	—
115	#121	—	86	2	no	—
116	C₄×D₄	—	86	2	no	—
117	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
118	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄⋊C₈	—	86	2	no	—
123	C₄⋊C₈	—	86	2	no	—
124	#104	—	172	4	no	—
125	C₂²⋊C₈	—	86	2	no	—
126	C₄×C₈	—	43	4	no	—
127	#95	—	2	2	no	—
128	#41	—	172	2	no	—
129	#41	—	172	2	no	—
130	C₂×Q₈	—	86	1	no	—
131	C₂×D₄	—	86	1	no	—
132	C₂×D₄	—	86	2	no	—
133	C₂×D₄	—	172	1	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	172	1	no	—
136	C₂²×C₄	#10621898	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
138	M₁₆	—	172	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	86	4	no	—
142	C₂×C₈	—	172	1	no	—
143	C₄⋊C₄	—	86	2	no	—
144	C₂²⋊C₄	—	172	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	#10621882	1	4	yes	—
147	C₄²	D₄×D₄₃	1	2	yes	—
148	C₄²	Q₈×D₄₃	1	1	yes	characteristic
149	C₄²	#10621902	1	1	yes	characteristic
150	C₂³	—	172	1	no	—
151	C₂³	#13063308	1	1	yes	characteristic
152	Q₈	—	172	2	no	—
153	D₄	—	344	2	no	—
154	D₄	—	172	4	no	—
155	D₄	—	172	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	344	1	no	—
160	C₂×C₄	—	344	1	no	—
161	C₂×C₄	—	172	1	no	—
162	C₂×C₄	#13063301	1	1	yes	characteristic
163	C₂×C₄	#13063303	1	2	yes	—
164	C₂×C₄	#13063310	1	2	yes	—
165	C₂×C₄	#13063321	1	1	yes	characteristic
166	C₂×C₄	C₂×#10621882	1	2	yes	—
167	C₂×C₄	#13063356	1	2	yes	—
168	C₈	—	8	2	no	—
169	C₈	—	344	1	no	—
170	C₈	—	172	4	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	344	1	no	—
173	C₂²	—	344	2	no	—
174	C₂²	#14582834	1	1	yes	characteristic
175	C₂²	#14583212	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	4	no	—
179	C₄	—	344	1	no	—
180	C₄	#14583176	1	2	yes	—
181	C₄	#14583275	1	1	yes	characteristic
182	C₄	#14583291	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	688	1	no	—
185	C₂	#19136124	1	1	yes	characteristic
186	C₂	#19136151	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135427

C₂

yes

maximal, characteristic

#19133369

C₂

yes

maximal, complemented

#19132993

C₂

yes

maximal

#19132424

C₂

yes

maximal, characteristic, complemented

C₄₃×#1843

C₂

yes

maximal, characteristic, complemented

#14583449

C₂²

yes

characteristic

#14582952

C₂²

yes

—

#14582923

C₂²

yes

—

—

—

C₂×C₄⋊C₈×C₄₃

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

—

C₄₃×#376

C₂²

yes

—

—

Q₈

yes

characteristic

#13063322

Q₈

yes

characteristic

#13063313

—

#13063299

—

C₄×D₁₇₂

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063253

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₈

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#123

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#10667

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#399

yes

characteristic

C₂²×C₄₃

#480

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#474

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

#1963

—

#1843

D₄₃

yes

characteristic, complemented

#1571

—

#1480

—

#1274

—

C₂×C₄₃

—

C₂×C₄₃

#2093

yes

characteristic

C₂×C₄₃

#2098

yes

—

D₄₃

—

#532

—

100

#431

—

101

#423

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄⋊C₈

C₂×D₄₃

yes

—

104

C₂×#104

—

105

#376

C₂×D₄₃

yes

characteristic

106

#335

—

107

D₄⋊C₈

—

108

C₄₃

#10667

yes

characteristic, complemented, Sylow

109

C₂×#41

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₄⋊Q₈

—

113

#127

—

114

#124

—

115

#121

—

116

C₄×D₄

—

117

C₂×C₄²

C₂²⋊D₄₃

yes

—

118

C₂×C₄²

C₂²×D₄₃

yes

characteristic

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

#104

—

172

—

125

C₂²⋊C₈

—

126

C₄×C₈

—

127

#95

—

128

#41

—

172

—

129

#41

—

172

—

130

C₂×Q₈

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

172

—

134

C₂²×C₄

—

135

C₂²×C₄

—

172

—

136

C₂²×C₄

#10621898

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

138

M₁₆

—

172

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

172

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

172

—

145

C₄²

—

146

C₄²

#10621882

yes

—

147

C₄²

D₄×D₄₃

yes

—

148

C₄²

Q₈×D₄₃

yes

characteristic

149

C₄²

#10621902

yes

characteristic

150

C₂³

—

172

—

151

C₂³

#13063308

yes

characteristic

152

Q₈

—

172

—

153

D₄

—

344

—

154

D₄

—

172

—

155

D₄

—

172

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

344

—

160

C₂×C₄

—

344

—

161

C₂×C₄

—

172

—

162

C₂×C₄

#13063301

yes

characteristic

163

C₂×C₄

#13063303

yes

—

164

C₂×C₄

#13063310

yes

—

165

C₂×C₄

#13063321

yes

characteristic

166

C₂×C₄

C₂×#10621882

yes

—

167

C₂×C₄

#13063356

yes

—

168

C₈

—

169

C₈

—

344

—

170

C₈

—

172

—

171

C₂²

—

172

C₂²

—

344

—

173

C₂²

—

344

—

174

C₂²

#14582834

yes

characteristic

175

C₂²

#14583212

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

344

—

180

C₄

#14583176

yes

—

181

C₄

#14583275

yes

characteristic

182

C₄

#14583291

yes

characteristic

183

C₂

—

184

C₂

—

688

—

185

C₂

#19136124

yes

characteristic

186

C₂

#19136151

yes

—

	subgroup	G/H
center	C₂×C₄	#13063321
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1843	C₂
socle	C₂²×C₄₃	#399
prime radical	#14583449	C₂²
hypercenter	#1843	D₄₃
hypocenter	C₄₃	#10667
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#13063321

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1843

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
2. #19133369	C₂	semidirect
4. #19132424	C₂	semidirect
5. C₄₃×#1843	C₂	semidirect
91. #1843	D₄₃	semidirect
108. C₄₃	#10667	semidirect

H ≅ G/N

type

2. #19133369

C₂

semidirect

4. #19132424

C₂

semidirect

5. C₄₃×#1843

C₂

semidirect

91. #1843

D₄₃

semidirect

108. C₄₃

#10667

semidirect

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

subgroup	index
#19135427	2
#19133369	2
#19132993	2
#19132424	2
C₄₃×#1843	2
#10667	43

subgroup

index

C₄₃×#1843

Subgroups — #25001614

Conjugacy classes of subgroups

Complements

Maximal subgroups