Group #25000756 — subgroups

Subgroups — #25000756

Order 11008.

342 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 77 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	#19134769	C₂	1	1	yes	maximal, characteristic
2	#19133325	C₂	1	2	yes	maximal
3	#19133210	C₂	1	2	yes	maximal
4	#19132506	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#3255	C₂	1	1	yes	maximal, characteristic
6	#14582956	C₂²	1	2	yes	—
7	C₂×#13063260	—	2	2	no	—
8	#14582923	C₂²	1	1	yes	characteristic
9	C₂×#13063236	C₂²	1	1	yes	characteristic
10	#14582648	—	2	2	no	—
11	#14582645	—	2	2	no	—
12	#14582641	—	2	4	no	—
13	C₄₃×#533	C₂²	1	2	yes	—
14	C₄₃×#475	—	2	2	no	—
15	C₄₃×#470	—	2	2	no	—
16	C₄₃×#462	—	2	2	no	—
17	C₄₃×#423	C₂²	1	1	yes	characteristic
18	C₂×#10621877	—	2	1	no	—
19	C₂²×C₄₃⋊C₈	—	2	1	no	—
20	#13063260	—	4	4	no	—
21	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
24	#13063236	—	2	1	no	—
25	#13063236	—	2	2	no	—
26	#13063236	D₄	1	2	yes	—
27	C₂×Q₁₆×C₄₃	—	2	2	no	—
28	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
29	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
30	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
31	#125×C₄₃	—	2	2	no	—
32	C₂²⋊Q₈×C₄₃	—	2	2	no	—
33	C₂²⋊Q₈×C₄₃	—	2	2	no	—
34	C₄×Q₈×C₄₃	—	2	2	no	—
35	C₄×Q₈×C₄₃	—	2	2	no	—
36	#117×C₄₃	—	2	2	no	—
37	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
38	#106×C₄₃	—	2	2	no	—
39	C₄⋊C₈×C₄₃	—	2	2	no	—
40	#103×C₄₃	—	2	4	no	—
41	C₂×Q₈×C₄₃	—	2	2	no	—
42	C₂×Q₈×C₄₃	—	2	2	no	—
43	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	#41	1	2	yes	—
45	Q₁₆×C₄₃	—	8	2	no	—
46	M₁₆×C₄₃	—	4	2	no	—
47	C₂×C₈×C₄₃	—	2	2	no	—
48	C₄⋊C₄×C₄₃	—	2	2	no	—
49	C₄⋊C₄×C₄₃	—	2	2	no	—
50	C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₄⋊C₄×C₄₃	—	4	2	no	—
53	C₄⋊C₄×C₄₃	—	4	2	no	—
54	C₂²⋊C₄×C₄₃	—	4	2	no	—
55	C₄²×C₄₃	—	4	2	no	—
56	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₃	C₂×D₄	1	2	yes	—
58	C₄²×C₄₃	#41	1	1	yes	characteristic
59	#10621877	—	4	2	no	—
60	C₂×C₄₃⋊C₈	—	2	2	no	—
61	C₂×C₄₃⋊C₈	—	2	2	no	—
62	C₂×C₄₃⋊C₈	—	2	2	no	—
63	C₂×C₄₃⋊C₈	—	4	1	no	—
64	C₂×C₄₃⋊C₈	—	4	2	no	—
65	C₂³×C₄₃	#124	1	1	yes	characteristic
66	Q₈×C₄₃	—	4	2	no	—
67	Q₈×C₄₃	—	4	2	no	—
68	Q₈×C₄₃	—	8	2	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	—	2	1	no	—
71	C₂×C₄×C₄₃	—	4	2	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	—	4	2	no	—
74	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
75	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₄₃	#121	1	2	yes	—
77	C₂×C₄×C₄₃	#123	1	2	yes	—
78	C₈×C₄₃	—	4	2	no	—
79	C₄₃⋊C₈	—	2	4	no	—
80	C₄₃⋊C₈	—	8	2	no	—
81	C₄₃⋊C₈	—	8	1	no	—
82	#10528	—	43	1	no	maximal, Sylow
83	C₂²×C₄₃	—	2	2	no	—
84	C₂²×C₄₃	—	2	1	no	—
85	C₂²×C₄₃	#394	1	1	yes	characteristic
86	C₄×C₄₃	—	2	2	no	—
87	C₄×C₄₃	—	2	2	no	—
88	C₄×C₄₃	—	2	1	no	—
89	C₄×C₄₃	—	8	2	no	—
90	C₄×C₄₃	—	8	2	no	—
91	C₄×C₄₃	#453	1	1	yes	characteristic
92	C₄×C₄₃	#456	1	1	yes	characteristic
93	#3255	D₄₃	1	1	yes	characteristic
94	#1849	—	43	1	no	—
95	#1559	—	43	2	no	—
96	#1531	—	43	2	no	—
97	#1307	—	43	1	no	—
98	C₂×C₄₃	—	4	1	no	—
99	C₂×C₄₃	#2039	1	1	yes	characteristic
100	C₂×C₄₃	#2049	1	2	yes	—
101	#533	C₂×D₄₃	1	2	yes	—
102	#475	—	2	2	no	—
103	#470	—	2	2	no	—
104	#462	—	2	2	no	—
105	#432	—	43	2	no	—
106	C₂×#108	—	86	2	no	—
107	#423	—	43	1	no	—
108	#423	C₂×D₄₃	1	1	yes	characteristic
109	C₂×#97	—	43	1	no	—
110	#333	—	86	2	no	—
111	#332	—	86	2	no	—
112	#330	—	86	4	no	—
113	C₄₃	#10528	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×M₁₆	—	86	1	no	—
116	C₂×M₁₆	C₂²⋊D₄₃	1	2	yes	—
117	C₂²×C₈	—	86	1	no	—
118	C₄⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
119	C₄⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
120	#125	—	2	2	no	—
121	C₂²⋊Q₈	—	2	2	no	—
122	C₂²⋊Q₈	—	2	2	no	—
123	C₄×Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	#117	—	2	2	no	—
126	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
127	#108	—	172	4	no	—
128	#106	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	86	2	no	—
131	C₄⋊C₈	—	86	2	no	—
132	#103	—	2	4	no	—
133	C₂²⋊C₈	—	86	2	no	—
134	#97	—	43	2	no	—
135	#97	—	86	2	no	—
136	#97	—	86	1	no	—
137	C₂×Q₈	—	2	2	no	—
138	C₂×Q₈	—	2	2	no	—
139	C₂²×C₄	#10621900	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
141	Q₁₆	—	8	2	no	—
142	M₁₆	—	4	2	no	—
143	M₁₆	—	172	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	86	2	no	—
146	C₂×C₈	—	86	2	no	—
147	C₂×C₈	—	86	2	no	—
148	C₂×C₈	—	172	1	no	—
149	C₂×C₈	—	172	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₂²⋊C₄	—	4	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	D₄×D₄₃	1	1	yes	characteristic
159	C₄²	#10621900	1	1	yes	characteristic
160	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
161	C₂³	#13063312	1	1	yes	characteristic
162	Q₈	—	4	2	no	—
163	Q₈	—	4	2	no	—
164	Q₈	—	8	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	#13063294	1	1	yes	characteristic
171	C₂×C₄	#13063296	1	1	yes	characteristic
172	C₂×C₄	#13063297	1	2	yes	—
173	C₂×C₄	#13063306	1	2	yes	—
174	C₈	—	4	2	no	—
175	C₈	—	86	4	no	—
176	C₈	—	344	2	no	—
177	C₈	—	344	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	2	no	—
180	C₂²	#14582813	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	2	no	—
185	C₄	—	8	2	no	—
186	C₄	#14583067	1	1	yes	characteristic
187	C₄	#14583082	1	1	yes	characteristic
188	C₂	—	4	1	no	—
189	C₂	#19135816	1	1	yes	characteristic
190	C₂	#19135872	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19134769

C₂

yes

maximal, characteristic

#19133325

C₂

yes

maximal

#19133210

C₂

yes

maximal

#19132506

C₂

yes

maximal, characteristic

C₄₃×#3255

C₂

yes

maximal, characteristic

#14582956

C₂²

yes

—

C₂×#13063260

—

#14582923

C₂²

yes

characteristic

C₂×#13063236

C₂²

yes

—

—

—

C₄₃×#533

C₂²

yes

—

C₄₃×#475

—

C₄₃×#470

—

C₄₃×#462

—

C₄₃×#423

C₂²

yes

characteristic

C₂×#10621877

—

C₂²×C₄₃⋊C₈

—

#13063260

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

#13063236

—

#13063236

D₄

yes

—

C₂×Q₁₆×C₄₃

—

C₂×M₁₆×C₄₃

D₄

yes

—

C₄⋊Q₈×C₄₃

D₄

yes

—

C₄⋊Q₈×C₄₃

D₄

yes

—

#125×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

#117×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#106×C₄₃

—

C₄⋊C₈×C₄₃

—

#103×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

Q₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

characteristic

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#124

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

#123

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#10528

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#394

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

D₄₃

yes

—

—

—

—

C₂×C₄₃

—

C₂×C₄₃

#2039

yes

characteristic

100

C₂×C₄₃

#2049

yes

—

101

#533

C₂×D₄₃

yes

—

102

#475

—

103

#470

—

104

#462

—

105

#432

—

106

C₂×#108

—

107

#423

—

108

#423

C₂×D₄₃

yes

characteristic

109

C₂×#97

—

110

#333

—

111

#332

—

112

#330

—

113

C₄₃

#10528

yes

characteristic, complemented, Sylow

114

C₂×Q₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

C₂²⋊D₄₃

yes

—

117

C₂²×C₈

—

118

C₄⋊Q₈

C₂²⋊D₄₃

yes

—

119

C₄⋊Q₈

C₂²⋊D₄₃

yes

—

120

#125

—

121

C₂²⋊Q₈

—

122

C₂²⋊Q₈

—

123

C₄×Q₈

—

124

C₄×Q₈

—

125

#117

—

126

C₂×C₄²

C₂²×D₄₃

yes

characteristic

127

#108

—

172

—

128

#106

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

#103

—

133

C₂²⋊C₈

—

134

#97

—

135

#97

—

136

#97

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

#10621900

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

141

Q₁₆

—

142

M₁₆

—

143

M₁₆

—

172

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

172

—

149

C₂×C₈

—

172

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₂²⋊C₄

—

157

C₄²

—

158

C₄²

D₄×D₄₃

yes

characteristic

159

C₄²

#10621900

yes

characteristic

160

C₄²

C₂×C₂²⋊D₄₃

yes

—

161

C₂³

#13063312

yes

characteristic

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

#13063294

yes

characteristic

171

C₂×C₄

#13063296

yes

characteristic

172

C₂×C₄

#13063297

yes

—

173

C₂×C₄

#13063306

yes

—

174

C₈

—

175

C₈

—

176

C₈

—

344

—

177

C₈

—

344

—

178

C₂²

—

179

C₂²

—

180

C₂²

#14582813

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

#14583067

yes

characteristic

187

C₄

#14583082

yes

characteristic

188

C₂

—

189

C₂

#19135816

yes

characteristic

190

C₂

#19135872

yes

—

	subgroup	G/H
center	C₂²	#14582813
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#3255	C₂
socle	C₂²×C₄₃	#394
prime radical	C₂³×C₄₃	#124
hypercenter	#3255	D₄₃
hypocenter	C₄₃	#10528
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3255

subgroup

G/H

center

C₂²

#14582813

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#3255

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3255

N	H ≅ G/N	type
113. C₄₃	#10528	semidirect

H ≅ G/N

type

113. C₄₃

#10528

semidirect

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

subgroup	index
#19134769	2
#19133325	2
#19133210	2
#19132506	2
C₄₃×#3255	2
#10528	43

subgroup

index

C₄₃×#3255

Subgroups — #25000756

Conjugacy classes of subgroups

Complements

Maximal subgroups