Group #36470766 — subgroups

Subgroups — #36470766

Order 22784.

366 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 127 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	#26021866	C₂	1	1	yes	maximal, characteristic
2	#26020298	C₂	1	1	yes	maximal, characteristic
3	#26018366	C₂	1	2	yes	maximal, complemented
4	#26018349	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2916	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	1	yes	characteristic
7	C₂²×#13083549	C₂²	1	1	yes	characteristic
8	#19151636	C₂²	1	2	yes	—
9	#19151572	—	2	1	no	—
10	C₂²⋊#13083549	—	2	2	no	—
11	#19151355	—	2	1	no	—
12	#568×C₈₉	—	2	1	no	—
13	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
14	#430×C₈₉	—	2	1	no	—
15	#430×C₈₉	C₄	1	4	yes	complemented
16	C₂×#108×C₈₉	—	2	2	no	—
17	C₂×#108×C₈₉	C₂²	1	2	yes	—
18	#15667971	—	2	2	no	—
19	#15667952	—	2	2	no	—
20	C₂×#13083549	—	2	1	no	—
21	C₂×#13083549	—	2	1	no	—
22	C₂×#13083549	—	2	2	no	—
23	#15667946	—	2	2	no	—
24	#15667944	—	2	2	no	—
25	#15667927	—	4	2	no	—
26	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
27	#131×C₈₉	—	2	2	no	—
28	#131×C₈₉	—	2	2	no	—
29	#131×C₈₉	—	4	4	no	—
30	C₂×M₁₆×C₈₉	—	2	4	no	—
31	C₂×M₁₆×C₈₉	—	2	2	no	—
32	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	D₄	1	2	yes	—
35	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
36	#108×C₈₉	—	2	4	no	—
37	#108×C₈₉	—	2	2	no	—
38	#108×C₈₉	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₈₉	—	4	2	no	—
40	C₂²⋊Dic₈₉	—	4	2	no	—
41	C₄×Dic₈₉	—	4	2	no	—
42	#13083549	—	4	1	no	—
43	#13083549	—	4	1	no	—
44	#41×C₈₉	—	2	2	no	—
45	#41×C₈₉	—	2	2	no	—
46	C₂×Q₈×C₈₉	Q₁₆	1	1	yes	characteristic
47	C₂×D₄×C₈₉	SD₁₆	1	2	yes	—
48	C₂×D₄×C₈₉	Q₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₈₉	—	2	2	no	—
52	M₁₆×C₈₉	—	2	4	no	—
53	M₁₆×C₈₉	—	2	4	no	—
54	M₁₆×C₈₉	—	4	4	no	—
55	M₁₆×C₈₉	—	4	2	no	—
56	C₂×C₈×C₈₉	—	2	2	no	—
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	—	4	4	no	—
59	C₂×C₈×C₈₉	—	4	2	no	—
60	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	#41	1	2	yes	—
64	C₂×C₈×C₈₉	#41	1	2	yes	—
65	C₂³×C₈₉	#103	1	2	yes	—
66	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈×C₈₉	—	2	2	no	—
68	D₄×C₈₉	—	2	4	no	—
69	D₄×C₈₉	—	2	2	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	#103	1	2	yes	—
73	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#123	1	2	yes	—
75	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×Dic₈₉	—	8	2	no	—
78	C₈×C₈₉	—	2	2	no	—
79	C₈×C₈₉	—	4	4	no	—
80	C₈×C₈₉	#117	1	2	yes	—
81	C₈×C₈₉	#123	1	1	yes	characteristic
82	C₈×C₈₉	#123	1	1	yes	characteristic
83	C₂²×C₈₉	—	2	2	no	—
84	C₂²×C₈₉	—	4	1	no	—
85	C₂²×C₈₉	—	4	2	no	—
86	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
87	C₂²×C₈₉	#481	1	2	yes	—
88	C₄×C₈₉	—	2	2	no	—
89	C₄×C₈₉	#381	1	1	yes	characteristic
90	C₄×C₈₉	#418	1	1	yes	characteristic
91	C₄×C₈₉	#484	1	2	yes	—
92	Dic₈₉	—	16	2	no	—
93	#12877	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	—	2	2	no	—
95	C₂×C₈₉	—	2	1	no	—
96	C₂×C₈₉	—	4	2	no	—
97	C₂×C₈₉	#1824	1	1	yes	characteristic
98	#2916	D₈₉	1	1	yes	characteristic
99	#2113	—	89	1	no	—
100	#1812	—	89	1	no	—
101	#1389	—	89	2	no	—
102	#1382	—	89	2	no	—
103	C₈₉	#12877	1	1	yes	characteristic, complemented, Sylow
104	#568	—	2	1	no	—
105	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
106	C₂×M₃₂	—	89	1	no	—
107	C₂²×C₁₆	—	89	1	no	—
108	#430	—	2	1	no	—
109	#430	Dic₈₉	1	4	yes	complemented
110	C₂×#108	—	2	2	no	—
111	C₂×#108	C₂×D₈₉	1	2	yes	—
112	#427	—	89	2	no	—
113	#410	—	178	1	no	—
114	C₂²⋊C₁₆	—	178	2	no	—
115	H(3,ℤ/4)	—	178	1	no	—
116	C₂×#41	C₂²⋊D₈₉	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	#131	—	4	4	no	—
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
123	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
125	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
126	#117	—	178	2	no	—
127	M₃₂	—	178	2	no	—
128	C₂×C₁₆	—	178	1	no	—
129	C₂×C₁₆	—	178	1	no	—
130	C₂×C₁₆	—	178	2	no	—
131	#108	—	2	4	no	—
132	#108	—	2	2	no	—
133	#108	C₂×Dic₈₉	1	4	yes	—
134	#107	—	178	2	no	—
135	#106	—	178	2	no	—
136	#99	—	356	2	no	—
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	#13083564	1	1	yes	characteristic
140	C₂×D₄	#13083561	1	2	yes	—
141	C₂×D₄	#13083564	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	2	4	no	—
147	M₁₆	—	4	4	no	—
148	M₁₆	—	4	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
154	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
155	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
156	C₂×C₈	#13083574	1	2	yes	—
157	C₂×C₈	#13083574	1	2	yes	—
158	C₄⋊C₄	—	356	2	no	—
159	C₂²⋊C₄	—	356	2	no	—
160	C₄²	—	356	2	no	—
161	C₁₆	—	356	1	no	—
162	C₁₆	—	356	1	no	—
163	C₂³	#15667936	1	2	yes	—
164	C₂³	C₂×#13083561	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	712	2	no	—
171	C₂×C₄	#15667936	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
173	C₂×C₄	#15667995	1	2	yes	—
174	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
175	C₂×C₄	#15668038	1	2	yes	—
176	C₈	—	2	2	no	—
177	C₈	—	4	4	no	—
178	C₈	#15667971	1	2	yes	—
179	C₈	#15667995	1	1	yes	characteristic
180	C₈	#15667995	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂×#15667936	1	1	yes	characteristic
185	C₂²	#19151916	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	1424	2	no	—
188	C₄	#19151461	1	1	yes	characteristic
189	C₄	#19151603	1	1	yes	characteristic
190	C₄	#19151933	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#26020342	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021866

C₂

yes

maximal, characteristic

#26020298

C₂

yes

maximal, characteristic

#26018366

C₂

yes

maximal, complemented

#26018349

C₂

yes

maximal, complemented

C₈₉×#2916

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

characteristic

C₂²×#13083549

C₂²

yes

characteristic

#19151636

C₂²

yes

—

#19151572

—

C₂²⋊#13083549

—

#19151355

—

#568×C₈₉

—

C₂×#131×C₈₉

C₂²

yes

characteristic

#430×C₈₉

—

#430×C₈₉

C₄

yes

complemented

C₂×#108×C₈₉

—

C₂×#108×C₈₉

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂×#41×C₈₉

D₄

yes

characteristic

#131×C₈₉

—

#131×C₈₉

—

#131×C₈₉

—

C₂×M₁₆×C₈₉

—

C₂×M₁₆×C₈₉

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

—

C₂×M₁₆×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

#108×C₈₉

—

#108×C₈₉

—

#108×C₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

#13083549

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

Q₁₆

yes

characteristic

C₂×D₄×C₈₉

SD₁₆

yes

—

C₂×D₄×C₈₉

Q₁₆

yes

characteristic

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₂³×C₈₉

#103

yes

—

C₂³×C₈₉

C₂×SD₁₆

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#103

yes

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#117

yes

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#103

yes

characteristic

C₂²×C₈₉

#481

yes

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#12877

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1824

yes

characteristic

#2916

D₈₉

yes

characteristic

#2113

—

100

#1812

—

101

#1389

—

102

#1382

—

103

C₈₉

#12877

yes

characteristic, complemented, Sylow

104

#568

—

105

C₂×#131

C₂×D₈₉

yes

characteristic

106

C₂×M₃₂

—

107

C₂²×C₁₆

—

108

#430

—

109

#430

Dic₈₉

yes

complemented

110

C₂×#108

—

111

C₂×#108

C₂×D₈₉

yes

—

112

#427

—

113

#410

—

178

—

114

C₂²⋊C₁₆

—

178

—

115

H(3,ℤ/4)

—

178

—

116

C₂×#41

C₂²⋊D₈₉

yes

characteristic

117

#131

—

118

#131

—

119

#131

—

120

C₂×M₁₆

—

121

C₂×M₁₆

—

122

C₂×M₁₆

C₂×Dic₈₉

yes

—

123

C₂×M₁₆

C₂²⋊D₈₉

yes

characteristic

124

C₂²×C₈

C₂²⋊D₈₉

yes

—

125

C₂²×C₈

C₂²×D₈₉

yes

characteristic

126

#117

—

178

—

127

M₃₂

—

178

—

128

C₂×C₁₆

—

178

—

129

C₂×C₁₆

—

178

—

130

C₂×C₁₆

—

178

—

131

#108

—

132

#108

—

133

#108

C₂×Dic₈₉

yes

—

134

#107

—

178

—

135

#106

—

178

—

136

#99

—

356

—

137

#41

—

138

#41

—

139

C₂×Q₈

#13083564

yes

characteristic

140

C₂×D₄

#13083561

yes

—

141

C₂×D₄

#13083564

yes

characteristic

142

C₂²×C₄

C₂²⋊Dic₈₉

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²⋊Dic₈₉

yes

—

154

C₂×C₈

C₂²×Dic₈₉

yes

characteristic

155

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

156

C₂×C₈

#13083574

yes

—

157

C₂×C₈

#13083574

yes

—

158

C₄⋊C₄

—

356

—

159

C₂²⋊C₄

—

356

—

160

C₄²

—

356

—

161

C₁₆

—

356

—

162

C₁₆

—

356

—

163

C₂³

#15667936

yes

—

164

C₂³

C₂×#13083561

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

712

—

171

C₂×C₄

#15667936

yes

—

172

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

173

C₂×C₄

#15667995

yes

—

174

C₂×C₄

C₂×#13083564

yes

characteristic

175

C₂×C₄

#15668038

yes

—

176

C₈

—

177

C₈

—

178

C₈

#15667971

yes

—

179

C₈

#15667995

yes

characteristic

180

C₈

#15667995

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

C₂×#15667936

yes

characteristic

185

C₂²

#19151916

yes

—

186

C₄

—

187

C₄

—

1424

—

188

C₄

#19151461

yes

characteristic

189

C₄

#19151603

yes

characteristic

190

C₄

#19151933

yes

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#26020342

yes

characteristic

	subgroup	G/H
center	C₄	#19151603
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2916	C₂
socle	C₂×C₈₉	#1824
prime radical	C₂×#41×C₈₉	D₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2916	D₈₉
hypocenter	C₈₉	#12877
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

subgroup

G/H

center

C₄

#19151603

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#2916

C₂

socle

C₂×C₈₉

#1824

prime radical

C₂×#41×C₈₉

D₄

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2916

N	H ≅ G/N	type
3. #26018366	C₂	semidirect
4. #26018349	C₂	semidirect
15. #430×C₈₉	C₄	semidirect
103. C₈₉	#12877	semidirect
109. #430	Dic₈₉	semidirect

H ≅ G/N

type

3. #26018366

C₂

semidirect

4. #26018349

C₂

semidirect

15. #430×C₈₉

C₄

semidirect

103. C₈₉

#12877

semidirect

109. #430

Dic₈₉

semidirect

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

subgroup	index
#26021866	2
#26020298	2
#26018366	2
#26018349	2
C₈₉×#2916	2
#12877	89

subgroup

index

C₈₉×#2916

Subgroups — #36470766

Conjugacy classes of subgroups

Complements

Maximal subgroups