Group #26030544 — subgroups

Subgroups — #26030544

Order 11392.

382 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 121 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	#19152203	C₂	1	1	yes	maximal, characteristic
2	#19152075	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151923	C₂	1	2	yes	maximal, complemented
4	#19151906	C₂	1	2	yes	maximal, complemented
5	#19151799	C₂	1	2	yes	maximal, complemented
6	#19151739	C₂	1	2	yes	maximal, complemented
7	#19151700	C₂	1	2	yes	maximal, complemented
8	C₄×#13083563	C₂	1	2	yes	maximal, complemented
9	#433×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
11	C₂×#13083563	C₂²	1	2	yes	—
12	C₂×#13083563	C₂²	1	2	yes	—
13	#15668009	C₂²	1	1	yes	characteristic
14	#15668004	C₂²	1	1	yes	characteristic
15	#15667997	—	2	2	no	—
16	#15667986	C₂²	1	2	yes	—
17	#15667985	—	2	2	no	—
18	C₄×Dic₁₇₈	C₂²	1	2	yes	—
19	C₄×Dic₁₇₈	C₂²	1	2	yes	—
20	#15667977	—	2	2	no	—
21	C₄×D₃₅₆	C₂²	1	2	yes	—
22	#15667971	C₂²	1	1	yes	characteristic
23	#15667968	—	2	2	no	—
24	#15667946	C₂²	1	2	yes	—
25	#15667944	C₂²	1	2	yes	—
26	#15667937	C₂²	1	2	yes	—
27	#15667937	C₂²	1	2	yes	—
28	#15667934	C₂²	1	2	yes	—
29	#15667934	C₂²	1	2	yes	—
30	#117×C₈₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
33	#97×C₈₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₇₈	—	2	1	no	—
36	C₂×Dic₁₇₈	—	2	4	no	—
37	C₂×Dic₁₇₈	C₂³	1	2	yes	—
38	C₂×D₃₅₆	—	2	2	no	—
39	C₂×D₃₅₆	C₂³	1	2	yes	—
40	C₂×C₂²⋊D₈₉	—	2	2	no	—
41	C₂²×Dic₈₉	—	2	1	no	—
42	C₂×C₄×D₈₉	—	2	2	no	—
43	#13083563	—	2	4	no	—
44	#13083563	—	2	4	no	—
45	C₄⋊Dic₈₉	C₂³	1	2	yes	—
46	C₄⋊Dic₈₉	C₂³	1	2	yes	—
47	#13083553	—	2	2	no	—
48	#13083553	—	2	2	no	—
49	#13083553	—	2	2	no	—
50	C₂²⋊Dic₈₉	—	2	1	no	—
51	#13083551	—	2	2	no	—
52	#13083551	—	2	2	no	—
53	C₄×Dic₈₉	—	2	2	no	—
54	C₄×Dic₈₉	—	2	2	no	—
55	C₄×Dic₈₉	—	2	1	no	—
56	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₂³	1	2	yes	—
58	C₂×C₈×C₈₉	C₂³	1	2	yes	—
59	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
60	C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
61	C₄²×C₈₉	C₂³	1	1	yes	characteristic
62	C₄²×C₈₉	C₂³	1	1	yes	characteristic
63	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×D₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
70	C₂²⋊D₈₉	—	8	2	no	—
71	C₂×Dic₈₉	—	2	4	no	—
72	C₂×Dic₈₉	—	2	2	no	—
73	C₂×Dic₈₉	—	2	2	no	—
74	C₂×Dic₈₉	—	2	2	no	—
75	D₃₅₆	—	2	2	no	—
76	D₃₅₆	—	2	2	no	—
77	D₃₅₆	—	8	2	no	—
78	C₄×D₈₉	—	8	2	no	—
79	Dic₁₇₈	—	2	4	no	—
80	Dic₁₇₈	—	2	2	no	—
81	Dic₁₇₈	#41	1	4	yes	—
82	C₈×C₈₉	—	2	2	no	—
83	C₈×C₈₉	—	2	2	no	—
84	C₂²×C₈₉	—	2	1	no	—
85	C₂²×C₈₉	—	2	2	no	—
86	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
87	C₂×D₈₉	—	4	2	no	—
88	C₂×D₈₉	—	4	2	no	—
89	C₂×D₈₉	—	4	2	no	—
90	C₄×C₈₉	—	2	1	no	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	—	2	2	no	—
93	C₄×C₈₉	C₂×#41	1	2	yes	—
94	Dic₈₉	—	2	4	no	—
95	Dic₈₉	—	4	2	no	—
96	Dic₈₉	—	4	2	no	—
97	C₂×C₈₉	—	4	1	no	—
98	C₂×C₈₉	#525	1	1	yes	characteristic
99	C₂×C₈₉	#577	1	2	yes	—
100	D₈₉	—	8	2	no	—
101	#3126	—	89	1	no	maximal, Sylow
102	C₈₉	#3126	1	1	yes	characteristic, complemented, Sylow
103	#540	—	89	1	no	—
104	#519	—	89	1	no	—
105	#482	—	89	2	no	—
106	#479	—	89	2	no	—
107	#460	—	89	2	no	—
108	#448	—	89	2	no	—
109	#440	—	89	2	no	—
110	C₄×SD₁₆	—	89	2	no	—
111	#433	D₈₉	1	1	yes	characteristic, complemented
112	C₂²×Q₈	—	89	1	no	—
113	C₂×SD₁₆	—	89	2	no	—
114	C₂×SD₁₆	—	89	2	no	—
115	#127	—	89	1	no	—
116	#125	—	89	1	no	—
117	#123	—	178	2	no	—
118	#121	—	89	2	no	—
119	#121	—	178	2	no	—
120	C₄×Q₈	—	89	2	no	—
121	C₄×Q₈	—	89	2	no	—
122	C₄×Q₈	—	178	2	no	—
123	C₄×D₄	—	89	2	no	—
124	#117	—	89	1	no	—
125	#117	—	178	2	no	—
126	#117	C₂×D₈₉	1	1	yes	characteristic
127	#107	—	89	2	no	—
128	#106	—	89	2	no	—
129	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
130	#103	—	89	2	no	—
131	#103	—	89	2	no	—
132	#102	—	89	2	no	—
133	#102	—	89	2	no	—
134	C₂²⋊C₈	C₂×D₈₉	1	2	yes	—
135	#97	C₂×D₈₉	1	1	yes	characteristic
136	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
137	C₂×Q₈	—	89	2	no	—
138	C₂×Q₈	—	178	4	no	—
139	C₂×Q₈	—	178	1	no	—
140	C₂×D₄	—	89	2	no	—
141	C₂×D₄	—	178	2	no	—
142	C₂×D₄	—	178	2	no	—
143	C₂²×C₄	—	178	1	no	—
144	C₂²×C₄	—	178	2	no	—
145	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
146	SD₁₆	—	178	4	no	—
147	SD₁₆	—	178	4	no	—
148	C₂×C₈	C₂²×D₈₉	1	2	yes	—
149	C₂×C₈	C₂²×D₈₉	1	2	yes	—
150	C₄⋊C₄	—	89	2	no	—
151	C₄⋊C₄	—	89	2	no	—
152	C₄⋊C₄	—	178	2	no	—
153	C₄⋊C₄	—	178	2	no	—
154	C₄⋊C₄	—	178	2	no	—
155	C₄⋊C₄	D₃₅₆	1	2	yes	—
156	C₂²⋊C₄	—	178	1	no	—
157	C₂²⋊C₄	—	178	2	no	—
158	C₂²⋊C₄	—	178	2	no	—
159	C₂²⋊C₄	D₃₅₆	1	2	yes	—
160	C₄²	—	178	1	no	—
161	C₄²	—	178	2	no	—
162	C₄²	—	178	2	no	—
163	C₄²	C₂²×D₈₉	1	1	yes	characteristic
164	C₄²	C₂²×D₈₉	1	1	yes	characteristic
165	C₂³	—	178	2	no	—
166	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
167	Q₈	—	89	4	no	—
168	Q₈	—	178	4	no	—
169	Q₈	—	178	2	no	—
170	D₄	—	178	2	no	—
171	D₄	—	178	2	no	—
172	D₄	—	712	2	no	—
173	D₄	—	712	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	178	2	no	—
176	C₂×C₄	—	178	2	no	—
177	C₂×C₄	—	178	4	no	—
178	C₂×C₄	—	178	2	no	—
179	C₂×C₄	—	712	2	no	—
180	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
181	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
182	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
183	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	356	2	no	—
189	C₂²	—	356	2	no	—
190	C₂²	—	356	2	no	—
191	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	1	no	—
195	C₄	—	356	2	no	—
196	C₄	—	356	2	no	—
197	C₄	—	178	4	no	—
198	C₄	#41×D₈₉	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	712	2	no	—
201	C₂	#19152110	1	1	yes	characteristic
202	C₂	#19152410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152203

C₂

yes

maximal, characteristic

#19152075

C₂

yes

maximal, characteristic, complemented

#19151923

C₂

yes

maximal, complemented

#19151906

C₂

yes

maximal, complemented

#19151799

C₂

yes

maximal, complemented

#19151739

C₂

yes

maximal, complemented

#19151700

C₂

yes

maximal, complemented

C₄×#13083563

C₂

yes

maximal, complemented

#433×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂×#13083563

C₂²

yes

—

C₂×#13083563

C₂²

yes

—

#15668009

C₂²

yes

characteristic

#15668004

C₂²

yes

characteristic

#15667997

—

#15667986

C₂²

yes

—

#15667985

—

C₄×Dic₁₇₈

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

#15667977

—

C₄×D₃₅₆

C₂²

yes

—

#15667971

C₂²

yes

characteristic

#15667968

—

#15667946

C₂²

yes

—

#15667944

C₂²

yes

—

#15667937

C₂²

yes

—

#15667937

C₂²

yes

—

#15667934

C₂²

yes

—

#15667934

C₂²

yes

—

#117×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂²⋊C₈×C₈₉

C₂²

yes

—

#97×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

—

C₂×C₄×D₈₉

—

#13083563

—

#13083563

—

C₄⋊Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

#13083551

—

#13083551

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₂²⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

#41

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#41

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#525

yes

characteristic

C₂×C₈₉

#577

yes

—

100

D₈₉

—

101

#3126

—

maximal, Sylow

102

C₈₉

#3126

yes

characteristic, complemented, Sylow

103

#540

—

104

#519

—

105

#482

—

106

#479

—

107

#460

—

108

#448

—

109

#440

—

110

C₄×SD₁₆

—

111

#433

D₈₉

yes

characteristic, complemented

112

C₂²×Q₈

—

113

C₂×SD₁₆

—

114

C₂×SD₁₆

—

115

#127

—

116

#125

—

117

#123

—

178

—

118

#121

—

119

#121

—

178

—

120

C₄×Q₈

—

121

C₄×Q₈

—

122

C₄×Q₈

—

178

—

123

C₄×D₄

—

124

#117

—

125

#117

—

178

—

126

#117

C₂×D₈₉

yes

characteristic

127

#107

—

128

#106

—

129

C₄⋊C₈

C₂×D₈₉

yes

—

130

#103

—

131

#103

—

132

#102

—

133

#102

—

134

C₂²⋊C₈

C₂×D₈₉

yes

—

135

#97

C₂×D₈₉

yes

characteristic

136

C₄×C₈

C₂×D₈₉

yes

characteristic

137

C₂×Q₈

—

138

C₂×Q₈

—

178

—

139

C₂×Q₈

—

178

—

140

C₂×D₄

—

141

C₂×D₄

—

178

—

142

C₂×D₄

—

178

—

143

C₂²×C₄

—

178

—

144

C₂²×C₄

—

178

—

145

C₂²×C₄

C₂²×D₈₉

yes

characteristic

146

SD₁₆

—

178

—

147

SD₁₆

—

178

—

148

C₂×C₈

C₂²×D₈₉

yes

—

149

C₂×C₈

C₂²×D₈₉

yes

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

178

—

153

C₄⋊C₄

—

178

—

154

C₄⋊C₄

—

178

—

155

C₄⋊C₄

D₃₅₆

yes

—

156

C₂²⋊C₄

—

178

—

157

C₂²⋊C₄

—

178

—

158

C₂²⋊C₄

—

178

—

159

C₂²⋊C₄

D₃₅₆

yes

—

160

C₄²

—

178

—

161

C₄²

—

178

—

162

C₄²

—

178

—

163

C₄²

C₂²×D₈₉

yes

characteristic

164

C₄²

C₂²×D₈₉

yes

characteristic

165

C₂³

—

178

—

166

C₂³

C₂×D₃₅₆

yes

characteristic

167

Q₈

—

168

Q₈

—

178

—

169

Q₈

—

178

—

170

D₄

—

178

—

171

D₄

—

178

—

172

D₄

—

712

—

173

D₄

—

712

—

174

C₂×C₄

—

175

C₂×C₄

—

178

—

176

C₂×C₄

—

178

—

177

C₂×C₄

—

178

—

178

C₂×C₄

—

178

—

179

C₂×C₄

—

712

—

180

C₂×C₄

C₂×D₃₅₆

yes

—

181

C₂×C₄

C₂×D₃₅₆

yes

—

182

C₂×C₄

C₂×D₃₅₆

yes

characteristic

183

C₂×C₄

C₂³×D₈₉

yes

characteristic

184

C₈

—

185

C₈

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

356

—

189

C₂²

—

356

—

190

C₂²

—

356

—

191

C₂²

C₂²×D₃₅₆

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

356

—

196

C₄

—

356

—

197

C₄

—

178

—

198

C₄

#41×D₈₉

yes

—

199

C₂

—

200

C₂

—

712

—

201

C₂

#19152110

yes

characteristic

202

C₂

#19152410

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#433×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152203	C₂
hypercenter	#433	D₈₉
hypocenter	C₈₉	#3126
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#433×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

C₂

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #433

N	H ≅ G/N	type
2. #19152075	C₂	semidirect
3. #19151923	C₂	semidirect
4. #19151906	C₂	semidirect
5. #19151799	C₂	semidirect
6. #19151739	C₂	semidirect
7. #19151700	C₂	semidirect
8. C₄×#13083563	C₂	semidirect
9. #433×C₈₉	C₂	semidirect
102. C₈₉	#3126	semidirect
111. #433	D₈₉	semidirect

H ≅ G/N

type

2. #19152075

C₂

semidirect

3. #19151923

C₂

semidirect

4. #19151906

C₂

semidirect

5. #19151799

C₂

semidirect

6. #19151739

C₂

semidirect

7. #19151700

C₂

semidirect

8. C₄×#13083563

C₂

semidirect

9. #433×C₈₉

C₂

semidirect

102. C₈₉

#3126

semidirect

111. #433

D₈₉

semidirect

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

subgroup	index
#19152203	2
#19152075	2
#19151923	2
#19151906	2
#19151799	2
#19151739	2
#19151700	2
C₄×#13083563	2
#433×C₈₉	2
#3126	89

subgroup

index

#433×C₈₉

Subgroups — #26030544

Conjugacy classes of subgroups

Complements

Maximal subgroups