Group #36459481 — subgroups

Subgroups — #36459481

Order 22784.

442 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 165 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	#26021213	C₂	1	2	yes	maximal
2	#26021054	C₂	1	1	yes	maximal, characteristic
3	#26020412	C₂	1	2	yes	maximal
4	#26020412	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	1	yes	characteristic
7	C₂×#15667946	—	2	2	no	—
8	C₂×#15667946	—	2	4	no	—
9	C₂×#15667937	—	2	2	no	—
10	#19151468	C₂²	1	2	yes	—
11	#19151468	C₂²	1	1	yes	characteristic
12	#19151361	—	2	4	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₇₈	—	2	1	no	—
16	#15668012	—	2	2	no	—
17	#15668012	—	2	1	no	—
18	#15668012	Q₈	1	2	yes	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	2	no	—
21	C₂×C₄⋊Dic₈₉	—	2	2	no	—
22	C₂×C₄⋊Dic₈₉	—	2	2	no	—
23	C₂×C₄⋊Dic₈₉	—	2	2	no	—
24	C₂×C₄⋊Dic₈₉	—	2	1	no	—
25	#15667946	—	4	8	no	—
26	#15667946	—	4	4	no	—
27	#15667937	—	4	4	no	—
28	C₂²×C₈×C₈₉	D₄	1	2	yes	—
29	C₂²×C₈×C₈₉	D₄	1	4	yes	—
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₈₉	—	2	2	no	—
32	C₄×C₈×C₈₉	—	2	2	no	—
33	C₄×C₈×C₈₉	—	2	2	no	—
34	C₂×Dic₁₇₈	—	2	2	no	—
35	C₂×Dic₁₇₈	—	4	2	no	—
36	C₂×Dic₁₇₈	—	4	2	no	—
37	C₂×Dic₁₇₈	—	4	1	no	—
38	C₂²×Dic₈₉	—	4	1	no	—
39	C₂²×Dic₈₉	—	4	2	no	—
40	C₂²×Dic₈₉	—	4	1	no	—
41	C₄⋊Dic₈₉	—	2	4	no	—
42	C₄⋊Dic₈₉	—	4	2	no	—
43	C₄⋊Dic₈₉	—	4	2	no	—
44	C₄⋊Dic₈₉	—	4	4	no	—
45	C₄⋊Dic₈₉	—	4	4	no	—
46	C₄⋊Dic₈₉	—	4	4	no	—
47	C₄⋊Dic₈₉	—	4	4	no	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	—	2	4	no	—
52	C₂×C₈×C₈₉	—	2	4	no	—
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	D₈	1	4	yes	—
55	C₂×C₈×C₈₉	Q₁₆	1	8	yes	—
56	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₈₉	#41	1	1	yes	characteristic
58	C₄²×C₈₉	#41	1	2	yes	—
59	C₂³×C₈₉	#127	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#123	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
66	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
67	C₂×C₄×C₈₉	#137	1	2	yes	—
68	C₂×Dic₈₉	—	4	2	no	—
69	C₂×Dic₈₉	—	8	1	no	—
70	C₂×Dic₈₉	—	8	1	no	—
71	C₂×Dic₈₉	—	8	1	no	—
72	C₂×Dic₈₉	—	8	2	no	—
73	C₂×Dic₈₉	—	8	2	no	—
74	C₂×Dic₈₉	—	8	2	no	—
75	C₂×Dic₈₉	—	8	2	no	—
76	Dic₁₇₈	—	4	4	no	—
77	Dic₁₇₈	—	8	2	no	—
78	C₈×C₈₉	—	2	4	no	—
79	C₈×C₈₉	—	2	8	no	—
80	C₂²×C₈₉	#399	1	1	yes	characteristic
81	C₂²×C₈₉	#493	1	1	yes	characteristic
82	C₂²×C₈₉	#494	1	2	yes	—
83	C₂²×C₈₉	#496	1	2	yes	—
84	C₂²×C₈₉	#497	1	1	yes	characteristic
85	C₄×C₈₉	#474	1	2	yes	—
86	C₄×C₈₉	#477	1	4	yes	—
87	C₄×C₈₉	#480	1	2	yes	—
88	C₄×C₈₉	#484	1	4	yes	—
89	Dic₈₉	—	8	2	no	—
90	Dic₈₉	—	16	1	no	—
91	Dic₈₉	—	16	2	no	—
92	#10674	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	#1713	1	1	yes	characteristic
94	C₂×C₈₉	#1721	1	1	yes	characteristic
95	C₂×C₈₉	#1723	1	2	yes	—
96	C₂×C₈₉	#2093	1	1	yes	characteristic
97	C₂×C₈₉	#2095	1	2	yes	—
98	#1990	—	89	2	no	—
99	#1956	—	89	1	no	—
100	#1845	—	89	2	no	—
101	#1845	—	89	1	no	—
102	#1273	D₈₉	1	1	yes	characteristic
103	C₈₉	#10674	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₄⋊Q₈	—	89	1	no	—
105	C₂×#107	—	178	4	no	—
106	C₂×#107	—	178	2	no	—
107	C₂×#103	—	178	2	no	—
108	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
110	#384	—	89	1	no	—
111	#384	—	89	2	no	—
112	#340	—	178	4	no	—
113	C₂²×Q₈	—	178	1	no	—
114	C₂²×C₈	D₃₅₆	1	2	yes	—
115	C₂²×C₈	D₃₅₆	1	4	yes	—
116	C₄⋊Q₈	—	89	2	no	—
117	C₄⋊Q₈	—	178	1	no	—
118	C₄⋊Q₈	—	178	2	no	—
119	C₂×C₄⋊C₄	—	178	2	no	—
120	C₂×C₄⋊C₄	—	178	1	no	—
121	C₂×C₄⋊C₄	—	178	2	no	—
122	C₂×C₄⋊C₄	—	178	2	no	—
123	C₂×C₄⋊C₄	—	178	2	no	—
124	C₂×C₄⋊C₄	—	178	2	no	—
125	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
126	#107	—	356	8	no	—
127	#107	—	356	4	no	—
128	#103	—	356	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₂×Q₈	—	356	1	no	—
133	C₂×Q₈	—	356	2	no	—
134	C₂×Q₈	—	356	2	no	—
135	C₂×Q₈	—	178	2	no	—
136	C₂²×C₄	—	356	1	no	—
137	C₂²×C₄	—	356	2	no	—
138	C₂²×C₄	—	356	1	no	—
139	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	D₇₁₂	1	4	yes	—
146	C₂×C₈	Dic₃₅₆	1	8	yes	—
147	C₄⋊C₄	—	356	2	no	—
148	C₄⋊C₄	—	356	2	no	—
149	C₄⋊C₄	—	356	4	no	—
150	C₄⋊C₄	—	356	4	no	—
151	C₄⋊C₄	—	356	4	no	—
152	C₄⋊C₄	—	356	4	no	—
153	C₄⋊C₄	—	178	4	no	—
154	C₄²	Q₈×D₈₉	1	1	yes	characteristic
155	C₄²	#13083573	1	1	yes	characteristic
156	C₄²	#13083573	1	2	yes	—
157	C₂³	#15668009	1	1	yes	characteristic
158	Q₈	—	356	4	no	—
159	Q₈	—	712	2	no	—
160	C₂×C₄	—	356	2	no	—
161	C₂×C₄	—	712	1	no	—
162	C₂×C₄	—	712	2	no	—
163	C₂×C₄	—	712	2	no	—
164	C₂×C₄	—	712	2	no	—
165	C₂×C₄	—	712	2	no	—
166	C₂×C₄	—	712	1	no	—
167	C₂×C₄	—	712	1	no	—
168	C₂×C₄	#15667992	1	1	yes	characteristic
169	C₂×C₄	#15667992	1	2	yes	—
170	C₂×C₄	#15667996	1	1	yes	characteristic
171	C₂×C₄	#15667996	1	2	yes	—
172	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
173	C₂×C₄	C₂×Dic₃₅₆	1	4	yes	—
174	C₂×C₄	#15668040	1	4	yes	—
175	C₂×C₄	#15668045	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	2	8	no	—
178	C₂²	#19151535	1	1	yes	characteristic
179	C₂²	#19151975	1	1	yes	characteristic
180	C₂²	#19151978	1	2	yes	—
181	C₂²	#19151987	1	2	yes	—
182	C₂²	#19151992	1	1	yes	characteristic
183	C₄	—	1424	1	no	—
184	C₄	—	1424	2	no	—
185	C₄	—	712	2	no	—
186	C₄	#19151879	1	2	yes	—
187	C₄	#19151894	1	4	yes	—
188	C₄	#19151912	1	2	yes	—
189	C₄	#19151934	1	4	yes	—
190	C₂	#26019870	1	1	yes	characteristic
191	C₂	#26019918	1	1	yes	characteristic
192	C₂	#26019929	1	2	yes	—
193	C₂	#26021779	1	1	yes	characteristic
194	C₂	#26021789	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021213

C₂

yes

maximal

#26021054

C₂

yes

maximal, characteristic

#26020412

C₂

yes

maximal

#26020412

C₂

yes

maximal, characteristic

C₈₉×#1273

C₂

yes

maximal, characteristic

C₂×#15668012

C₂²

yes

—

—

—

C₂²

yes

—

#19151468

C₂²

yes

characteristic

#19151361

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂²×Dic₁₇₈

—

#15668012

—

#15668012

—

#15668012

Q₈

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667946

—

#15667946

—

#15667937

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₄²×C₈₉

C₂×Q₈

yes

C₄²×C₈₉

yes

C₄²×C₈₉

yes

—

C₂³×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#123

yes

characteristic

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#496

yes

—

C₂²×C₈₉

#497

yes

characteristic

C₄×C₈₉

#474

yes

—

C₄×C₈₉

#477

yes

—

C₄×C₈₉

#480

yes

—

C₄×C₈₉

#484

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#2093

yes

characteristic

C₂×C₈₉

#2095

yes

—

#1990

—

#1956

—

100

#1845

—

101

#1845

—

102

#1273

D₈₉

yes

characteristic

103

C₈₉

#10674

yes

characteristic, complemented, Sylow

104

C₂×C₄⋊Q₈

—

105

C₂×#107

—

178

—

106

C₂×#107

—

178

—

107

C₂×#103

—

178

—

108

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

109

C₂×C₄×C₈

C₂×D₈₉

yes

—

110

#384

—

111

#384

—

112

#340

—

178

—

113

C₂²×Q₈

—

178

—

114

C₂²×C₈

D₃₅₆

yes

—

115

C₂²×C₈

D₃₅₆

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

178

—

118

C₄⋊Q₈

—

178

—

119

C₂×C₄⋊C₄

—

178

—

120

C₂×C₄⋊C₄

—

178

—

121

C₂×C₄⋊C₄

—

178

—

122

C₂×C₄⋊C₄

—

178

—

123

C₂×C₄⋊C₄

—

178

—

124

C₂×C₄⋊C₄

—

178

—

125

C₂×C₄²

C₂²×D₈₉

yes

characteristic

126

#107

—

356

—

127

#107

—

356

—

128

#103

—

356

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

356

—

133

C₂×Q₈

—

356

—

134

C₂×Q₈

—

356

—

135

C₂×Q₈

—

178

—

136

C₂²×C₄

—

356

—

137

C₂²×C₄

—

356

—

138

C₂²×C₄

—

356

—

139

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

140

C₂²×C₄

C₂×D₃₅₆

yes

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

D₇₁₂

yes

—

146

C₂×C₈

Dic₃₅₆

yes

—

147

C₄⋊C₄

—

356

—

148

C₄⋊C₄

—

356

—

149

C₄⋊C₄

—

356

—

150

C₄⋊C₄

—

356

—

151

C₄⋊C₄

—

356

—

152

C₄⋊C₄

—

356

—

153

C₄⋊C₄

—

178

—

154

C₄²

Q₈×D₈₉

yes

characteristic

155

C₄²

#13083573

yes

characteristic

156

C₄²

#13083573

yes

—

157

C₂³

#15668009

yes

characteristic

158

Q₈

—

356

—

159

Q₈

—

712

—

160

C₂×C₄

—

356

—

161

C₂×C₄

—

712

—

162

C₂×C₄

—

712

—

163

C₂×C₄

—

712

—

164

C₂×C₄

—

712

—

165

C₂×C₄

—

712

—

166

C₂×C₄

—

712

—

167

C₂×C₄

—

712

—

168

C₂×C₄

#15667992

yes

characteristic

169

C₂×C₄

#15667992

yes

—

170

C₂×C₄

#15667996

yes

characteristic

171

C₂×C₄

#15667996

yes

—

172

C₂×C₄

C₂×D₇₁₂

yes

—

173

C₂×C₄

C₂×Dic₃₅₆

yes

—

174

C₂×C₄

#15668040

yes

—

175

C₂×C₄

#15668045

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#19151535

yes

characteristic

179

C₂²

#19151975

yes

characteristic

180

C₂²

#19151978

yes

—

181

C₂²

#19151987

yes

—

182

C₂²

#19151992

yes

characteristic

183

C₄

—

1424

—

184

C₄

—

1424

—

185

C₄

—

712

—

186

C₄

#19151879

yes

—

187

C₄

#19151894

yes

—

188

C₄

#19151912

yes

—

189

C₄

#19151934

yes

—

190

C₂

#26019870

yes

characteristic

191

C₂

#26019918

yes

characteristic

192

C₂

#26019929

yes

—

193

C₂

#26021779

yes

characteristic

194

C₂

#26021789

yes

—

	subgroup	G/H
center	C₂³	#15668009
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1273	C₂
socle, prime radical	C₂³×C₈₉	#127
hypercenter	#1273	D₈₉
hypocenter	C₈₉	#10674
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#15668009

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1273

C₂

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
103. C₈₉	#10674	semidirect

H ≅ G/N

type

103. C₈₉

#10674

semidirect

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

subgroup	index
#26021213	2
#26021054	2
#26020412	2
C₈₉×#1273	2
#10674	89

subgroup

index

C₈₉×#1273

Subgroups — #36459481

Conjugacy classes of subgroups

Complements

Maximal subgroups