Group #36459202 — subgroups

Subgroups — #36459202

Order 22784.

292 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 93 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	#26021233	C₂	1	1	yes	maximal, characteristic
2	#26018910	C₂	1	2	yes	maximal
3	#26018149	C₂	1	2	yes	maximal
4	#26018081	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1843	C₂	1	1	yes	maximal, characteristic
6	#19152155	C₂²	1	1	yes	characteristic
7	#19151651	C₂²	1	2	yes	—
8	#19151622	C₂²	1	1	yes	characteristic
9	#19151358	—	2	1	no	—
10	#19151358	—	2	1	no	—
11	#19151346	—	2	2	no	—
12	#19151343	—	2	2	no	—
13	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
14	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	#376×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083558	—	2	1	no	—
18	#15668012	D₄	1	2	yes	—
19	#15668004	—	2	1	no	—
20	#15667991	—	2	2	no	—
21	C₄×Dic₁₇₈	—	2	2	no	—
22	#15667971	—	2	1	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₂²⋊#10623494	—	2	2	no	—
26	C₄×#10623494	—	2	2	no	—
27	C₂²×C₈×C₈₉	—	2	2	no	—
28	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
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	—	4	2	no	—
34	C₄⋊C₈×C₈₉	—	4	2	no	—
35	#95×C₈₉	—	2	2	no	—
36	C₂×Dic₁₇₈	—	2	2	no	—
37	#13083558	—	4	1	no	—
38	#13083558	—	4	1	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₂×#10623494	—	4	1	no	—
41	C₄⋊Dic₈₉	—	2	2	no	—
42	C₄⋊Dic₈₉	—	2	2	no	—
43	#13083553	—	4	2	no	—
44	C₂²⋊Dic₈₉	—	4	1	no	—
45	C₄×Dic₈₉	—	4	1	no	—
46	C₂²×C₄×C₈₉	—	2	1	no	—
47	C₂²×C₄×C₈₉	—	2	1	no	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	#41	1	2	yes	—
50	C₂×C₈×C₈₉	—	4	2	no	—
51	C₂×C₈×C₈₉	—	4	2	no	—
52	C₂×C₈×C₈₉	—	4	2	no	—
53	C₄²×C₈₉	—	2	1	no	—
54	C₄²×C₈₉	—	2	1	no	—
55	C₄²×C₈₉	SD₁₆	1	2	yes	—
56	C₄²×C₈₉	Q₁₆	1	2	yes	—
57	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₉	#41	1	1	yes	characteristic
59	C₄²×C₈₉	#41	1	2	yes	—
60	C₂³×C₈₉	#123	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	4	2	no	—
64	C₂×C₄×C₈₉	—	4	2	no	—
65	C₂×C₄×C₈₉	—	4	2	no	—
66	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#123	1	2	yes	—
68	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#126	1	2	yes	—
70	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
74	C₂×Dic₈₉	—	4	2	no	—
75	Dic₁₇₈	—	8	2	no	—
76	C₈×C₈₉	—	8	2	no	—
77	#10623494	—	8	2	no	—
78	#10623494	—	8	1	no	—
79	C₂²×C₈₉	—	2	2	no	—
80	C₂²×C₈₉	#397	1	1	yes	characteristic
81	C₂²×C₈₉	#483	1	1	yes	characteristic
82	C₂²×C₈₉	#484	1	1	yes	characteristic
83	C₄×C₈₉	—	2	2	no	—
84	C₄×C₈₉	—	2	2	no	—
85	C₄×C₈₉	—	4	2	no	—
86	C₄×C₈₉	—	4	2	no	—
87	C₄×C₈₉	#460	1	1	yes	characteristic
88	C₄×C₈₉	#462	1	1	yes	characteristic
89	C₄×C₈₉	#487	1	1	yes	characteristic
90	C₄×C₈₉	#489	1	1	yes	characteristic
91	Dic₈₉	—	8	2	no	—
92	#10628	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	—	2	1	no	—
94	C₂×C₈₉	—	2	1	no	—
95	C₂×C₈₉	#2072	1	1	yes	characteristic
96	C₂×C₈₉	#2080	1	2	yes	—
97	#1995	—	89	1	no	—
98	#1843	D₈₉	1	1	yes	characteristic
99	#1542	—	89	2	no	—
100	#1301	—	89	2	no	—
101	#1274	—	89	1	no	—
102	C₈₉	#10628	1	1	yes	characteristic, complemented, Sylow
103	#533	—	89	1	no	—
104	#431	—	89	2	no	—
105	#423	—	89	1	no	—
106	C₂×C₄⋊C₈	—	2	1	no	—
107	C₂×C₄⋊C₈	—	2	1	no	—
108	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
109	#376	C₂×D₈₉	1	1	yes	characteristic
110	#339	—	178	1	no	—
111	#339	—	178	1	no	—
112	#333	—	178	2	no	—
113	#332	—	178	2	no	—
114	C₂×M₁₆	—	178	1	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₄⋊Q₈	—	89	2	no	—
117	#125	—	178	1	no	—
118	C₂²⋊Q₈	—	178	2	no	—
119	C₄×Q₈	—	178	2	no	—
120	#117	—	178	1	no	—
121	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
123	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	—	178	2	no	—
129	C₄⋊C₈	—	178	2	no	—
130	C₂²⋊C₈	—	178	2	no	—
131	C₄×C₈	—	178	2	no	—
132	#95	—	2	2	no	—
133	C₂×Q₈	—	178	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
137	C₂²×C₄	#13083574	1	2	yes	—
138	M₁₆	—	356	1	no	—
139	M₁₆	—	356	1	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	356	2	no	—
144	C₂×C₈	—	356	1	no	—
145	C₄⋊C₄	—	356	2	no	—
146	C₄⋊C₄	—	178	2	no	—
147	C₄⋊C₄	—	178	2	no	—
148	C₂²⋊C₄	—	356	1	no	—
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	356	1	no	—
152	C₄²	#13083562	1	2	yes	—
153	C₄²	#13083564	1	2	yes	—
154	C₄²	D₄×D₈₉	1	1	yes	characteristic
155	C₄²	#13083573	1	1	yes	characteristic
156	C₄²	#13083575	1	2	yes	—
157	C₂³	#15667995	1	1	yes	characteristic
158	Q₈	—	712	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	356	2	no	—
165	C₂×C₄	#15667983	1	1	yes	characteristic
166	C₂×C₄	#15667997	1	2	yes	—
167	C₂×C₄	#15668001	1	1	yes	characteristic
168	C₂×C₄	#15668007	1	2	yes	—
169	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
170	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
171	C₂×C₄	#15668038	1	1	yes	characteristic
172	C₂×C₄	#15668043	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	—	712	2	no	—
175	C₈	—	712	1	no	—
176	C₂²	—	2	2	no	—
177	C₂²	#19151524	1	1	yes	characteristic
178	C₂²	#19151928	1	1	yes	characteristic
179	C₂²	#19151933	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	712	2	no	—
185	C₄	#19151795	1	1	yes	characteristic
186	C₄	#19151805	1	1	yes	characteristic
187	C₄	#19151950	1	1	yes	characteristic
188	C₄	#19151960	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#26021661	1	1	yes	characteristic
192	C₂	#26021703	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021233

C₂

yes

maximal, characteristic

#26018910

C₂

yes

maximal

#26018149

C₂

yes

maximal

#26018081

C₂

yes

maximal, characteristic

C₈₉×#1843

C₂

yes

maximal, characteristic

#19152155

C₂²

yes

characteristic

#19151651

C₂²

yes

—

#19151622

C₂²

yes

—

—

—

—

C₂×C₄⋊C₈×C₈₉

—

C₂×C₄⋊C₈×C₈₉

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

#376×C₈₉

C₂²

yes

characteristic

C₂×#13083558

—

#15668012

D₄

yes

—

#15668004

—

#15667991

—

C₄×Dic₁₇₈

—

—

—

—

—

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

#95×C₈₉

—

C₂×Dic₁₇₈

—

—

—

—

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

Q₁₆

yes

—

C₄²×C₈₉

C₂×D₄

yes

C₄²×C₈₉

yes

C₄²×C₈₉

yes

—

C₂³×C₈₉

#123

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

#123

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

#126

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#10628

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#2072

yes

characteristic

C₂×C₈₉

#2080

yes

—

#1995

—

#1843

D₈₉

yes

characteristic

#1542

—

100

#1301

—

101

#1274

—

102

C₈₉

#10628

yes

characteristic, complemented, Sylow

103

#533

—

104

#431

—

105

#423

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

109

#376

C₂×D₈₉

yes

characteristic

110

#339

—

178

—

111

#339

—

178

—

112

#333

—

178

—

113

#332

—

178

—

114

C₂×M₁₆

—

178

—

115

C₂²×C₈

—

116

C₄⋊Q₈

—

117

#125

—

178

—

118

C₂²⋊Q₈

—

178

—

119

C₄×Q₈

—

178

—

120

#117

—

178

—

121

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

122

C₂×C₄²

C₂²⋊D₈₉

yes

characteristic

123

C₂×C₄²

C₂²×D₈₉

yes

characteristic

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

—

178

—

129

C₄⋊C₈

—

178

—

130

C₂²⋊C₈

—

178

—

131

C₄×C₈

—

178

—

132

#95

—

133

C₂×Q₈

—

178

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

137

C₂²×C₄

#13083574

yes

—

138

M₁₆

—

356

—

139

M₁₆

—

356

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

356

—

144

C₂×C₈

—

356

—

145

C₄⋊C₄

—

356

—

146

C₄⋊C₄

—

178

—

147

C₄⋊C₄

—

178

—

148

C₂²⋊C₄

—

356

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

356

—

152

C₄²

#13083562

yes

—

153

C₄²

#13083564

yes

—

154

C₄²

D₄×D₈₉

yes

characteristic

155

C₄²

#13083573

yes

characteristic

156

C₄²

#13083575

yes

—

157

C₂³

#15667995

yes

characteristic

158

Q₈

—

712

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

356

—

165

C₂×C₄

#15667983

yes

characteristic

166

C₂×C₄

#15667997

yes

—

167

C₂×C₄

#15668001

yes

characteristic

168

C₂×C₄

#15668007

yes

—

169

C₂×C₄

C₂×#13083562

yes

characteristic

170

C₂×C₄

C₂×#13083564

yes

characteristic

171

C₂×C₄

#15668038

yes

characteristic

172

C₂×C₄

#15668043

yes

characteristic

173

C₈

—

174

C₈

—

712

—

175

C₈

—

712

—

176

C₂²

—

177

C₂²

#19151524

yes

characteristic

178

C₂²

#19151928

yes

characteristic

179

C₂²

#19151933

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

712

—

185

C₄

#19151795

yes

characteristic

186

C₄

#19151805

yes

characteristic

187

C₄

#19151950

yes

characteristic

188

C₄

#19151960

yes

characteristic

189

C₂

—

190

C₂

—

191

C₂

#26021661

yes

characteristic

192

C₂

#26021703

yes

—

	subgroup	G/H
center	C₂×C₄	#15668001
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1843	C₂
socle	C₂²×C₈₉	#397
prime radical	C₂³×C₈₉	#123
hypercenter	#1843	D₈₉
hypocenter	C₈₉	#10628
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

subgroup

G/H

center

C₂×C₄

#15668001

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1843

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1843

N	H ≅ G/N	type
102. C₈₉	#10628	semidirect

H ≅ G/N

type

102. C₈₉

#10628

semidirect

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

subgroup	index
#26021233	2
#26018910	2
#26018149	2
#26018081	2
C₈₉×#1843	2
#10628	89

subgroup

index

C₈₉×#1843

Subgroups — #36459202

Conjugacy classes of subgroups

Complements

Maximal subgroups