Group #26030710 — subgroups

Subgroups — #26030710

Order 11392.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#15668044	C₂	1	4	yes	maximal, complemented
2	#19152149	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151991	C₂	1	4	yes	maximal, complemented
5	#19151977	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151972	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151971	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13083557	C₂	1	1	yes	maximal, characteristic, complemented
9	#494×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13083575	C₂²	1	2	yes	—
11	C₂×Q₈×D₈₉	C₂²	1	2	yes	—
12	#15668044	—	2	16	no	—
13	C₂×#13083564	C₂²	1	4	yes	—
14	C₂×#13083563	C₂²	1	4	yes	—
15	C₂×#13083561	C₂²	1	4	yes	—
16	C₂×#13083557	C₂²	1	2	yes	—
17	#15668012	C₂²	1	1	yes	characteristic
18	#15668011	—	2	2	no	—
19	#15668010	C₂²	1	2	yes	—
20	#15668009	C₂²	1	1	yes	characteristic
21	#15668001	C₂²	1	2	yes	—
22	#15667987	—	2	2	no	—
23	#15667974	—	2	2	no	—
24	C₄⋊C₄×D₈₉	—	2	2	no	—
25	C₄²×D₈₉	C₂²	1	1	yes	characteristic
26	#15667922	C₂²	1	2	yes	—
27	C₄×#10623494	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₈₉	C₂²	1	4	yes	—
29	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#13083575	—	2	8	no	—
33	C₂×Dic₁₇₈	—	2	2	no	—
34	C₂×Dic₁₇₈	C₂³	1	2	yes	—
35	Q₈×D₈₉	—	2	8	no	—
36	C₂×D₃₅₆	—	2	2	no	—
37	C₂×D₃₅₆	—	2	2	no	—
38	C₂×D₃₅₆	C₂³	1	2	yes	—
39	C₂×C₄×D₈₉	—	2	2	no	—
40	C₂×C₄×D₈₉	—	2	2	no	—
41	C₂×C₄×D₈₉	D₄	1	2	yes	—
42	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
43	#13083564	—	2	8	no	—
44	#13083563	—	2	8	no	—
45	#13083561	—	2	8	no	—
46	#13083557	D₄	1	8	yes	complemented
47	C₂×#10623494	C₂³	1	2	yes	—
48	C₄⋊Dic₈₉	—	2	2	no	—
49	#13083553	—	2	4	no	—
50	#13083551	—	2	4	no	—
51	C₄×Dic₈₉	D₄	1	2	yes	—
52	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
54	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
55	Q₁₆×C₈₉	—	2	8	no	—
56	C₂×C₈×C₈₉	C₂³	1	2	yes	—
57	C₄⋊C₄×C₈₉	—	2	2	no	—
58	C₄⋊C₄×C₈₉	—	2	2	no	—
59	C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₂²×D₈₉	—	2	2	no	—
61	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₈₉	—	2	4	no	—
63	Q₈×C₈₉	—	2	4	no	—
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₈₉	—	2	2	no	—
69	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
70	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
71	D₃₅₆	—	2	4	no	—
72	D₃₅₆	—	4	4	no	—
73	D₃₅₆	—	4	4	no	—
74	C₄×D₈₉	—	2	4	no	—
75	C₄×D₈₉	—	4	4	no	—
76	C₄×D₈₉	—	4	4	no	—
77	C₄×D₈₉	C₂×D₄	1	4	yes	—
78	Dic₁₇₈	—	2	4	no	—
79	Dic₁₇₈	—	4	4	no	—
80	C₈×C₈₉	C₂×D₄	1	4	yes	—
81	#10623494	C₂×D₄	1	4	yes	—
82	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
83	C₂×D₈₉	—	2	2	no	—
84	C₂×D₈₉	—	4	2	no	—
85	C₂×D₈₉	—	4	4	no	—
86	C₂×D₈₉	#127	1	2	yes	—
87	C₄×C₈₉	—	4	2	no	—
88	C₄×C₈₉	—	4	2	no	—
89	C₄×C₈₉	#137	1	4	yes	complemented
90	C₄×C₈₉	C₂²×D₄	1	2	yes	—
91	Dic₈₉	—	2	2	no	—
92	Dic₈₉	—	4	2	no	—
93	Dic₈₉	#127	1	2	yes	—
94	C₂×C₈₉	C₂×#127	1	1	yes	characteristic
95	C₂×C₈₉	C₂×#137	1	2	yes	—
96	D₈₉	—	2	2	no	—
97	D₈₉	—	8	2	no	—
98	#3152	—	89	1	no	maximal, Sylow
99	C₈₉	#3152	1	1	yes	characteristic, complemented, Sylow
100	C₂×#137	—	89	4	no	—
101	#532	—	89	1	no	—
102	C₂×C₄⋊Q₈	—	89	1	no	—
103	#497	—	89	4	no	—
104	#494	—	89	1	no	—
105	#494	D₈₉	1	1	yes	characteristic, complemented
106	#492	—	89	1	no	—
107	#492	—	89	1	no	—
108	C₄×M₁₆	—	89	1	no	—
109	C₂×#41	—	89	2	no	—
110	C₂²×Q₈	—	89	2	no	—
111	#137	—	178	16	no	—
112	C₂×Q₁₆	—	89	4	no	—
113	C₂×Q₁₆	C₂×D₈₉	1	4	yes	—
114	C₂×SD₁₆	—	89	4	no	—
115	C₂×SD₁₆	—	89	4	no	—
116	C₂×M₁₆	—	89	2	no	—
117	C₄⋊Q₈	—	89	2	no	—
118	C₄⋊Q₈	—	89	1	no	—
119	C₄⋊Q₈	—	178	2	no	—
120	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
121	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
122	#127	—	89	1	no	—
123	#124	—	89	2	no	—
124	#121	—	178	2	no	—
125	C₄×D₄	—	178	2	no	—
126	C₂×C₄⋊C₄	—	178	2	no	—
127	C₂×C₄²	—	89	1	no	—
128	#97	—	89	2	no	—
129	C₄×C₈	—	89	1	no	—
130	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
131	#41	—	178	8	no	—
132	C₂×Q₈	—	89	2	no	—
133	C₂×Q₈	—	178	2	no	—
134	C₂×Q₈	—	178	8	no	—
135	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
136	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
137	C₂×D₄	—	89	2	no	—
138	C₂×D₄	—	178	2	no	—
139	C₂×D₄	—	178	2	no	—
140	C₂²×C₄	—	89	1	no	—
141	C₂²×C₄	—	89	2	no	—
142	C₂²×C₄	—	178	2	no	—
143	C₂²×C₄	—	178	2	no	—
144	Q₁₆	—	2	8	no	—
145	Q₁₆	—	178	8	no	—
146	SD₁₆	—	178	8	no	—
147	SD₁₆	—	178	8	no	—
148	M₁₆	—	89	8	no	—
149	C₂×C₈	—	89	2	no	—
150	C₂×C₈	C₂²×D₈₉	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	178	4	no	—
154	C₄⋊C₄	—	178	2	no	—
155	C₂²⋊C₄	—	178	4	no	—
156	C₄²	—	89	2	no	—
157	C₄²	—	89	1	no	—
158	C₄²	C₂²×D₈₉	1	1	yes	characteristic
159	C₂³	—	89	1	no	—
160	C₂³	—	178	2	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	2	4	no	—
163	Q₈	—	356	4	no	—
164	Q₈	—	178	4	no	—
165	D₄	—	356	4	no	—
166	D₄	—	356	4	no	—
167	D₄	—	178	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	89	1	no	—
171	C₂×C₄	—	89	4	no	—
172	C₂×C₄	—	89	2	no	—
173	C₂×C₄	—	356	4	no	—
174	C₂×C₄	—	356	4	no	—
175	C₂×C₄	—	178	2	no	—
176	C₂×C₄	—	178	4	no	—
177	C₂×C₄	D₄×D₈₉	1	2	yes	—
178	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
179	C₈	—	89	4	no	—
180	C₈	D₄×D₈₉	1	4	yes	—
181	C₂²	—	89	2	no	—
182	C₂²	—	356	2	no	—
183	C₂²	—	356	4	no	—
184	C₂²	—	178	2	no	—
185	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	89	2	no	—
189	C₄	—	356	2	no	—
190	C₄	—	178	2	no	—
191	C₄	#15668044	1	4	yes	complemented
192	C₄	C₂×D₄×D₈₉	1	2	yes	—
193	C₂	—	178	2	no	—
194	C₂	—	712	2	no	—
195	C₂	#127×D₈₉	1	1	yes	characteristic
196	C₂	C₂×#15668044	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668044

C₂

yes

maximal, complemented

#19152149

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₈₉

C₂

yes

maximal, characteristic, complemented

#19151991

C₂

yes

maximal, complemented

#19151977

C₂

yes

maximal, characteristic, complemented

#19151972

C₂

yes

maximal, characteristic, complemented

#19151971

C₂

yes

maximal, characteristic, complemented

C₄×#13083557

C₂

yes

maximal, characteristic, complemented

#494×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083575

C₂²

yes

—

C₂×Q₈×D₈₉

C₂²

yes

—

#15668044

—

C₂×#13083564

C₂²

yes

—

C₂×#13083563

C₂²

yes

—

C₂×#13083561

C₂²

yes

—

C₂×#13083557

C₂²

yes

—

#15668012

C₂²

yes

characteristic

#15668011

—

#15668010

C₂²

yes

—

#15668009

C₂²

yes

characteristic

#15668001

C₂²

yes

—

#15667987

—

#15667974

—

C₄⋊C₄×D₈₉

—

C₄²×D₈₉

C₂²

yes

characteristic

#15667922

C₂²

yes

—

C₄×#10623494

C₂²

yes

characteristic

C₂×Q₁₆×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

#13083575

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

Q₈×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂³

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

D₄

yes

—

C₂×C₄×D₈₉

C₂³

yes

—

—

—

D₄

yes

complemented

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083553

—

#13083551

—

C₄×Dic₈₉

D₄

yes

—

C₄×Dic₈₉

C₂³

yes

characteristic

C₂×Q₈×C₈₉

C₂³

yes

—

C₂×Q₈×C₈₉

C₂³

yes

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

characteristic

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

C₂×D₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₂×D₄

yes

—

#10623494

C₂×D₄

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

#127

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#137

yes

complemented

C₄×C₈₉

C₂²×D₄

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

#127

yes

—

C₂×C₈₉

C₂×#127

yes

characteristic

C₂×C₈₉

C₂×#137

yes

—

D₈₉

—

D₈₉

—

#3152

—

maximal, Sylow

C₈₉

#3152

yes

characteristic, complemented, Sylow

100

C₂×#137

—

101

#532

—

102

C₂×C₄⋊Q₈

—

103

#497

—

104

#494

—

105

#494

D₈₉

yes

characteristic, complemented

106

#492

—

107

#492

—

108

C₄×M₁₆

—

109

C₂×#41

—

110

C₂²×Q₈

—

111

#137

—

178

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

C₂×D₈₉

yes

—

114

C₂×SD₁₆

—

115

C₂×SD₁₆

—

116

C₂×M₁₆

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

178

—

120

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

121

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

122

#127

—

123

#124

—

124

#121

—

178

—

125

C₄×D₄

—

178

—

126

C₂×C₄⋊C₄

—

178

—

127

C₂×C₄²

—

128

#97

—

129

C₄×C₈

—

130

C₄×C₈

C₂×D₈₉

yes

characteristic

131

#41

—

178

—

132

C₂×Q₈

—

133

C₂×Q₈

—

178

—

134

C₂×Q₈

—

178

—

135

C₂×Q₈

C₂²×D₈₉

yes

—

136

C₂×Q₈

C₂²×D₈₉

yes

—

137

C₂×D₄

—

138

C₂×D₄

—

178

—

139

C₂×D₄

—

178

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

178

—

143

C₂²×C₄

—

178

—

144

Q₁₆

—

145

Q₁₆

—

178

—

146

SD₁₆

—

178

—

147

SD₁₆

—

178

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

C₂²×D₈₉

yes

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

178

—

154

C₄⋊C₄

—

178

—

155

C₂²⋊C₄

—

178

—

156

C₄²

—

157

C₄²

—

158

C₄²

C₂²×D₈₉

yes

characteristic

159

C₂³

—

160

C₂³

—

178

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

356

—

164

Q₈

—

178

—

165

D₄

—

356

—

166

D₄

—

356

—

167

D₄

—

178

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

356

—

174

C₂×C₄

—

356

—

175

C₂×C₄

—

178

—

176

C₂×C₄

—

178

—

177

C₂×C₄

D₄×D₈₉

yes

—

178

C₂×C₄

C₂³×D₈₉

yes

characteristic

179

C₈

—

180

C₈

D₄×D₈₉

yes

—

181

C₂²

—

182

C₂²

—

356

—

183

C₂²

—

356

—

184

C₂²

—

178

—

185

C₂²

C₂×D₄×D₈₉

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

356

—

190

C₄

—

178

—

191

C₄

#15668044

yes

complemented

192

C₄

C₂×D₄×D₈₉

yes

—

193

C₂

—

178

—

194

C₂

—

712

—

195

C₂

#127×D₈₉

yes

characteristic

196

C₂

C₂×#15668044

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#494×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152149	C₂
hypercenter	#494	D₈₉
hypocenter	C₈₉	#3152
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#494×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

C₂

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
1. C₂×#15668044	C₂	semidirect
3. C₄⋊Q₈×D₈₉	C₂	semidirect
4. #19151991	C₂	semidirect
5. #19151977	C₂	semidirect
6. #19151972	C₂	semidirect
7. #19151971	C₂	semidirect
8. C₄×#13083557	C₂	semidirect
9. #494×C₈₉	C₂	semidirect
46. #13083557	D₄	semidirect
89. C₄×C₈₉	#137	semidirect
99. C₈₉	#3152	semidirect
105. #494	D₈₉	semidirect
191. C₄	#15668044	semidirect

H ≅ G/N

type

1. C₂×#15668044

C₂

semidirect

3. C₄⋊Q₈×D₈₉

C₂

semidirect

4. #19151991

C₂

semidirect

5. #19151977

C₂

semidirect

6. #19151972

C₂

semidirect

7. #19151971

C₂

semidirect

8. C₄×#13083557

C₂

semidirect

9. #494×C₈₉

C₂

semidirect

46. #13083557

D₄

semidirect

89. C₄×C₈₉

#137

semidirect

99. C₈₉

#3152

semidirect

105. #494

D₈₉

semidirect

191. C₄

#15668044

semidirect

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

subgroup	index
C₂×#15668044	2
#19152149	2
C₄⋊Q₈×D₈₉	2
#19151991	2
#19151977	2
#19151972	2
#19151971	2
C₄×#13083557	2
#494×C₈₉	2
#3152	89

subgroup

index

C₄⋊Q₈×D₈₉

#494×C₈₉

Subgroups — #26030710

Conjugacy classes of subgroups

Complements

Maximal subgroups