Group #25015081 — subgroups

Subgroups — #25015081

Order 11008.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₂×#14583279	C₂	1	1	yes	maximal, characteristic
2	#19136351	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582714	C₂	1	4	yes	maximal
4	C₄×#108×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₇₂	—	2	1	no	—
6	C₂²×Dic₁₇₂	C₂²	1	2	yes	—
7	C₂×#13063325	—	2	1	no	—
8	C₂×#13063265	C₂²	1	2	yes	—
9	#14583279	—	2	2	no	—
10	#14583279	C₄	1	4	yes	—
11	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
12	#14582714	—	2	8	no	—
13	C₂×#108×C₄₃	C₂²	1	2	yes	—
14	C₄×M₁₆×C₄₃	—	2	1	no	—
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#341	—	2	1	no	—
17	C₂²×Dic₈₆	—	2	2	no	—
18	C₂×Dic₁₇₂	—	2	4	no	—
19	C₂×Dic₁₇₂	—	2	4	no	—
20	C₂×Dic₁₇₂	—	2	4	no	—
21	C₂×Dic₁₇₂	—	2	2	no	—
22	C₂×Dic₁₇₂	—	4	4	no	—
23	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
24	#13063325	—	2	4	no	—
25	#13063325	—	4	1	no	—
26	#13063325	—	4	1	no	—
27	C₂×C₄⋊Dic₄₃	—	4	2	no	—
28	#13063265	—	2	4	no	—
29	C₂×C₄₃⋊C₁₆	—	2	2	no	—
30	C₂×M₁₆×C₄₃	—	2	2	no	—
31	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
34	#108×C₄₃	—	2	2	no	—
35	#108×C₄₃	D₄	1	4	yes	—
36	#97×C₄₃	—	4	1	no	—
37	C₄×C₈×C₄₃	—	4	1	no	—
38	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₃	D₄	1	2	yes	—
40	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₄₃	—	4	2	no	—
43	C₂×Dic₈₆	—	2	4	no	—
44	C₂×Dic₈₆	—	4	4	no	—
45	C₂×Dic₈₆	—	4	4	no	—
46	C₂×Dic₈₆	—	4	2	no	—
47	M₁₆×C₄₃	—	2	4	no	—
48	M₁₆×C₄₃	—	4	2	no	—
49	C₂×C₈×C₄₃	—	2	2	no	—
50	C₂×C₈×C₄₃	—	4	2	no	—
51	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₄₃	#41	1	2	yes	—
56	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₄₃	D₈	1	1	yes	characteristic
58	C₄²×C₄₃	D₈	1	1	yes	characteristic
59	C₄⋊Dic₄₃	—	4	4	no	—
60	C₄⋊Dic₄₃	—	4	4	no	—
61	Dic₁₇₂	—	2	8	no	—
62	Dic₁₇₂	—	4	8	no	—
63	Dic₁₇₂	—	4	2	no	—
64	Dic₁₇₂	—	4	2	no	—
65	C₄₃⋊C₁₆	—	4	2	no	—
66	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₄₃	—	2	4	no	—
68	C₂×C₄×C₄₃	#102	1	2	yes	—
69	C₂×C₄×C₄₃	#102	1	2	yes	—
70	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₃	#121	1	2	yes	—
72	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×Dic₄₃	—	4	4	no	—
75	C₂×Dic₄₃	—	8	4	no	—
76	Dic₈₆	—	4	4	no	—
77	Dic₈₆	—	4	4	no	—
78	Dic₈₆	—	8	4	no	—
79	C₈×C₄₃	—	2	2	no	—
80	C₈×C₄₃	—	4	4	no	—
81	C₈×C₄₃	C₄×D₄	1	2	yes	—
82	C₈×C₄₃	#124	1	1	yes	characteristic
83	C₈×C₄₃	#127	1	1	yes	characteristic
84	#13256	—	43	1	no	maximal, Sylow
85	C₂²×C₄₃	—	2	2	no	—
86	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
87	C₂²×C₄₃	#460	1	2	yes	—
88	C₄×C₄₃	—	2	2	no	—
89	C₄×C₄₃	#362	1	4	yes	complemented
90	C₄×C₄₃	#390	1	1	yes	characteristic
91	C₄×C₄₃	#418	1	1	yes	characteristic
92	C₄×C₄₃	#459	1	2	yes	—
93	Dic₄₃	—	8	4	no	—
94	C₂×#494	—	43	1	no	—
95	#2150	—	43	1	no	—
96	C₂×#362	—	43	4	no	—
97	C₄×#108	D₄₃	1	1	yes	characteristic
98	C₂×C₄₃	—	2	2	no	—
99	C₂×C₄₃	#1982	1	1	yes	characteristic
100	C₂×C₄₃	C₂×#362	1	2	yes	—
101	C₂²×Q₁₆	—	43	2	no	—
102	C₂²×Q₁₆	—	86	1	no	—
103	C₂×C₄⋊Q₈	—	86	1	no	—
104	C₂×M₃₂	—	43	2	no	—
105	#494	—	43	4	no	—
106	#494	—	86	2	no	—
107	C₂×#108	C₂×D₄₃	1	2	yes	—
108	C₄×M₁₆	—	2	1	no	—
109	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
110	C₄⋊C₁₆	—	86	2	no	—
111	#362	—	86	8	no	—
112	#341	—	2	1	no	—
113	C₄₃	#13256	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	86	2	no	—
115	C₂×Q₁₆	—	43	4	no	—
116	C₂×Q₁₆	—	86	4	no	—
117	C₂×Q₁₆	—	86	4	no	—
118	C₂×Q₁₆	—	86	2	no	—
119	C₂×Q₁₆	—	86	4	no	—
120	C₂×Q₁₆	—	172	4	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	86	4	no	—
125	C₄⋊Q₈	—	172	1	no	—
126	C₄⋊Q₈	—	172	1	no	—
127	C₂×C₄⋊C₄	—	172	2	no	—
128	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
129	M₃₂	—	86	4	no	—
130	C₂×C₁₆	—	86	2	no	—
131	#108	—	2	2	no	—
132	#108	D₁₇₂	1	4	yes	—
133	#97	—	4	1	no	—
134	C₄×C₈	—	4	1	no	—
135	C₄×C₈	C₄×D₄₃	1	2	yes	—
136	C₄×C₈	D₁₇₂	1	2	yes	—
137	C₂×Q₈	—	86	4	no	—
138	C₂×Q₈	—	172	4	no	—
139	C₂×Q₈	—	172	4	no	—
140	C₂×Q₈	—	172	2	no	—
141	C₂²×C₄	—	172	2	no	—
142	C₂²×C₄	#10621884	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
144	Q₁₆	—	86	8	no	—
145	Q₁₆	—	172	8	no	—
146	Q₁₆	—	172	2	no	—
147	Q₁₆	—	172	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	4	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
154	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
155	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
156	C₂×C₈	#10621898	1	2	yes	—
157	C₄⋊C₄	—	172	4	no	—
158	C₄⋊C₄	—	172	4	no	—
159	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
160	C₄²	#10621882	1	1	yes	characteristic
161	C₄²	#10621882	1	1	yes	characteristic
162	C₁₆	—	172	2	no	—
163	C₂³	C₂×#10621884	1	1	yes	characteristic
164	Q₈	—	344	4	no	—
165	Q₈	—	172	4	no	—
166	Q₈	—	172	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	344	4	no	—
169	C₂×C₄	—	172	4	no	—
170	C₂×C₄	#13063245	1	2	yes	—
171	C₂×C₄	#13063245	1	2	yes	—
172	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#13063299	1	2	yes	—
174	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
175	C₂×C₄	#13063350	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	4	4	no	—
178	C₈	C₄×D₁₇₂	1	2	yes	—
179	C₈	#13063313	1	1	yes	characteristic
180	C₈	#13063322	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₂×#13063245	1	1	yes	characteristic
183	C₂²	#14583099	1	2	yes	—
184	C₄	—	2	2	no	—
185	C₄	—	344	4	no	—
186	C₄	#14582714	1	4	yes	complemented
187	C₄	#14582801	1	1	yes	characteristic
188	C₄	#14582903	1	1	yes	characteristic
189	C₄	#14583094	1	2	yes	—
190	C₂	—	2	2	no	—
191	C₂	#19135522	1	1	yes	characteristic
192	C₂	C₂×#14582714	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583279

C₂

yes

maximal, characteristic

#19136351

C₂

yes

maximal, characteristic

C₂×#14582714

C₂

yes

maximal

C₄×#108×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₂

—

C₂²×Dic₁₇₂

C₂²

yes

—

C₂×#13063325

—

C₂×#13063265

C₂²

yes

—

#14583279

—

#14583279

C₄

yes

—

C₄⋊(C₄₃⋊C₁₆)

—

#14582714

—

C₂×#108×C₄₃

C₂²

yes

—

C₄×M₁₆×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#341

—

C₂²×Dic₈₆

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

C₂×C₄

yes

—

#13063325

—

#13063325

—

#13063325

—

C₂×C₄⋊Dic₄₃

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

#108×C₄₃

—

#108×C₄₃

D₄

yes

—

#97×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

SD₁₆

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

D₈

yes

characteristic

C₄²×C₄₃

D₈

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₄×D₄

yes

—

C₈×C₄₃

yes

C₈×C₄₃

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×#102

yes

characteristic

C₂²×C₄₃

#460

yes

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

—

—

—

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

#1982

yes

characteristic

100

C₂×C₄₃

C₂×#362

yes

—

101

C₂²×Q₁₆

—

102

C₂²×Q₁₆

—

103

C₂×C₄⋊Q₈

—

104

C₂×M₃₂

—

105

#494

—

106

#494

—

107

C₂×#108

C₂×D₄₃

yes

—

108

C₄×M₁₆

—

109

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

110

C₄⋊C₁₆

—

111

#362

—

112

#341

—

113

C₄₃

#13256

yes

characteristic, complemented, Sylow

114

C₂²×Q₈

—

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂×Q₁₆

—

119

C₂×Q₁₆

—

120

C₂×Q₁₆

—

172

—

121

C₂×M₁₆

—

122

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

123

C₂²×C₈

C₂²×D₄₃

yes

characteristic

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

172

—

126

C₄⋊Q₈

—

172

—

127

C₂×C₄⋊C₄

—

172

—

128

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

129

M₃₂

—

130

C₂×C₁₆

—

131

#108

—

132

#108

D₁₇₂

yes

—

133

#97

—

134

C₄×C₈

—

135

C₄×C₈

C₄×D₄₃

yes

—

136

C₄×C₈

D₁₇₂

yes

—

137

C₂×Q₈

—

138

C₂×Q₈

—

172

—

139

C₂×Q₈

—

172

—

140

C₂×Q₈

—

172

—

141

C₂²×C₄

—

172

—

142

C₂²×C₄

#10621884

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

144

Q₁₆

—

145

Q₁₆

—

172

—

146

Q₁₆

—

172

—

147

Q₁₆

—

172

—

148

M₁₆

—

149

M₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

153

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

154

C₂×C₈

C₂×D₁₇₂

yes

—

155

C₂×C₈

C₂×D₁₇₂

yes

characteristic

156

C₂×C₈

#10621898

yes

—

157

C₄⋊C₄

—

172

—

158

C₄⋊C₄

—

172

—

159

C₄²

(C₂×D₄₃)⋊C₄

yes

—

160

C₄²

#10621882

yes

characteristic

161

C₄²

#10621882

yes

characteristic

162

C₁₆

—

172

—

163

C₂³

C₂×#10621884

yes

characteristic

164

Q₈

—

344

—

165

Q₈

—

172

—

166

Q₈

—

172

—

167

C₂×C₄

—

168

C₂×C₄

—

344

—

169

C₂×C₄

—

172

—

170

C₂×C₄

#13063245

yes

—

171

C₂×C₄

#13063245

yes

—

172

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

173

C₂×C₄

#13063299

yes

—

174

C₂×C₄

C₂×#10621882

yes

characteristic

175

C₂×C₄

#13063350

yes

—

176

C₈

—

177

C₈

—

178

C₈

C₄×D₁₇₂

yes

—

179

C₈

#13063313

yes

characteristic

180

C₈

#13063322

yes

characteristic

181

C₂²

—

182

C₂²

C₂×#13063245

yes

characteristic

183

C₂²

#14583099

yes

—

184

C₄

—

185

C₄

—

344

—

186

C₄

#14582714

yes

complemented

187

C₄

#14582801

yes

characteristic

188

C₄

#14582903

yes

characteristic

189

C₄

#14583094

yes

—

190

C₂

—

191

C₂

#19135522

yes

characteristic

192

C₂

C₂×#14582714

yes

—

	subgroup	G/H
center	C₂²	C₂×#13063245
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄×#108×C₄₃	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×C₄₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₄×#108	D₄₃
hypocenter	C₄₃	#13256
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂²

C₂×#13063245

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄×#108×C₄₃

C₂

socle

C₂²×C₄₃

C₂×#102

prime radical

C₂³×C₄₃

C₂×SD₁₆

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
89. C₄×C₄₃	#362	semidirect
113. C₄₃	#13256	semidirect
186. C₄	#14582714	semidirect

H ≅ G/N

type

89. C₄×C₄₃

#362

semidirect

113. C₄₃

#13256

semidirect

186. C₄

#14582714

semidirect

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

subgroup	index
C₂×#14583279	2
#19136351	2
C₂×#14582714	2
C₄×#108×C₄₃	2
#13256	43

subgroup

index

C₄×#108×C₄₃

Subgroups — #25015081

Conjugacy classes of subgroups

Complements

Maximal subgroups