Group #36454403 — subgroups

Subgroups — #36454403

Order 22784.

412 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 119 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	#26030575	C₂	1	1	yes	maximal, characteristic
2	#26018706	C₂	1	2	yes	maximal, complemented
3	#26018702	C₂	1	2	yes	maximal
4	C₂×#19151319	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152155	C₂²	1	1	yes	characteristic
7	#19152148	C₂²	1	1	yes	characteristic
8	#19151953	—	2	2	no	—
9	#19151949	C₄	1	2	yes	—
10	#19151945	C₄	1	2	yes	—
11	#19151595	—	2	2	no	—
12	C₂×C₄×#10623494	C₂²	1	2	yes	—
13	#19151328	—	2	4	no	—
14	#19151325	—	2	4	no	—
15	#19151319	—	2	2	no	—
16	#423×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	#341×C₈₉	—	2	2	no	—
19	C₂×#13083574	—	2	1	no	—
20	C₂²×#10623494	—	2	2	no	—
21	#15668012	C₂×C₄	1	1	yes	characteristic
22	#15668012	C₂×C₄	1	2	yes	—
23	#15668009	C₂×C₄	1	1	yes	characteristic
24	#15668004	—	2	1	no	—
25	#15668000	—	2	1	no	—
26	#15667991	—	2	2	no	—
27	#15667986	—	2	2	no	—
28	C₄×Dic₁₇₈	—	2	2	no	—
29	C₄×D₃₅₆	—	2	2	no	—
30	#15667971	—	2	1	no	—
31	#15667937	—	2	4	no	—
32	#15667934	—	2	4	no	—
33	C₄×#10623494	—	2	2	no	—
34	C₄×#10623494	D₄	1	4	yes	—
35	C₂×M₁₆×C₈₉	—	2	2	no	—
36	C₂²×C₈×C₈₉	D₄	1	2	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₈₉	—	2	2	no	—
39	C₄⋊C₈×C₈₉	—	2	2	no	—
40	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₈₉	D₄	1	2	yes	—
42	#13083574	—	4	4	no	—
43	C₂×Dic₁₇₈	—	2	1	no	—
44	C₂×Dic₁₇₈	—	2	2	no	—
45	C₂×D₃₅₆	—	2	1	no	—
46	C₂×D₃₅₆	—	2	2	no	—
47	C₂×C₂²⋊D₈₉	—	4	1	no	—
48	C₂×C₄×D₈₉	—	4	1	no	—
49	C₂×#10623494	—	2	4	no	—
50	C₂×#10623494	—	4	4	no	—
51	C₄⋊Dic₈₉	—	2	2	no	—
52	C₄⋊Dic₈₉	—	2	2	no	—
53	C₄⋊Dic₈₉	—	2	2	no	—
54	#13083553	—	4	2	no	—
55	C₂²⋊Dic₈₉	—	4	1	no	—
56	#13083551	—	4	2	no	—
57	C₄×Dic₈₉	—	4	1	no	—
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	#41	1	2	yes	—
60	M₁₆×C₈₉	—	4	4	no	—
61	C₂×C₈×C₈₉	—	2	4	no	—
62	C₂×C₈×C₈₉	—	4	2	no	—
63	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
64	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×D₄	1	2	yes	—
67	C₂³×C₈₉	#124	1	1	yes	characteristic
68	C₂²×D₈₉	—	4	1	no	—
69	C₂×C₄×C₈₉	—	2	4	no	—
70	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₈₉	#121	1	2	yes	—
73	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#135	1	4	yes	—
75	C₂²⋊D₈₉	—	8	2	no	—
76	C₂×Dic₈₉	—	4	2	no	—
77	C₂×Dic₈₉	—	4	1	no	—
78	D₃₅₆	—	4	2	no	—
79	D₃₅₆	—	4	4	no	—
80	C₄×D₈₉	—	8	2	no	—
81	Dic₁₇₈	—	4	2	no	—
82	Dic₁₇₈	—	4	4	no	—
83	C₈×C₈₉	—	2	4	no	—
84	C₈×C₈₉	—	8	2	no	—
85	#10623494	—	4	4	no	—
86	C₂²×C₈₉	—	2	2	no	—
87	C₂²×C₈₉	#390	1	1	yes	characteristic
88	C₂²×C₈₉	#488	1	2	yes	—
89	C₂×D₈₉	—	8	1	no	—
90	C₂×D₈₉	—	8	2	no	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	#416	1	2	yes	—
93	C₄×C₈₉	#444	1	4	yes	—
94	C₄×C₈₉	#495	1	2	yes	—
95	Dic₈₉	—	8	1	no	—
96	Dic₈₉	—	8	2	no	—
97	#9863	—	89	1	no	maximal, Sylow
98	C₂×C₈₉	—	2	2	no	—
99	C₂×C₈₉	#1958	1	1	yes	characteristic
100	C₂×C₈₉	#1975	1	2	yes	—
101	D₈₉	—	16	1	no	—
102	#3131	—	89	1	no	—
103	#1837	D₈₉	1	1	yes	characteristic, complemented
104	#1493	—	89	2	no	—
105	#1492	—	89	2	no	—
106	C₂×#322	—	89	1	no	—
107	C₈₉	#9863	1	1	yes	characteristic, complemented, Sylow
108	#533	—	89	1	no	—
109	#532	—	89	1	no	—
110	#488	—	178	2	no	—
111	#487	—	89	2	no	—
112	#486	—	89	2	no	—
113	#423	C₂×D₈₉	1	2	yes	—
114	#416	—	178	2	no	—
115	C₂×C₄×C₈	—	89	2	no	—
116	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
117	#341	—	2	2	no	—
118	Q₈⋊C₈	—	178	4	no	—
119	D₄⋊C₈	—	178	4	no	—
120	#322	—	178	2	no	—
121	C₂×#41	—	178	1	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂²×C₈	—	178	2	no	—
124	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
125	C₄⋊Q₈	—	89	1	no	—
126	C₄⋊Q₈	—	89	2	no	—
127	#127	—	89	1	no	—
128	#125	—	178	1	no	—
129	#124	—	178	1	no	—
130	C₂²⋊Q₈	—	178	2	no	—
131	#121	—	178	2	no	—
132	C₄×Q₈	—	178	2	no	—
133	C₄×D₄	—	178	2	no	—
134	#117	—	178	1	no	—
135	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	#103	—	178	4	no	—
139	#102	—	178	4	no	—
140	C₄×C₈	—	89	4	no	—
141	C₄×C₈	—	178	2	no	—
142	C₄×C₈	C₄×D₈₉	1	2	yes	—
143	C₄×C₈	D₃₅₆	1	2	yes	—
144	#41	—	356	4	no	—
145	C₂×Q₈	—	178	2	no	—
146	C₂×Q₈	—	178	1	no	—
147	C₂×D₄	—	356	1	no	—
148	C₂×D₄	—	178	2	no	—
149	C₂×D₄	—	178	1	no	—
150	C₂²×C₄	—	356	1	no	—
151	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
152	C₂²×C₄	#13083575	1	2	yes	—
153	M₁₆	—	4	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	356	4	no	—
157	C₂×C₈	—	178	4	no	—
158	C₂×C₈	#13083551	1	4	yes	—
159	C₄⋊C₄	—	356	2	no	—
160	C₄⋊C₄	—	178	2	no	—
161	C₄⋊C₄	—	178	2	no	—
162	C₄⋊C₄	—	178	2	no	—
163	C₂²⋊C₄	—	356	1	no	—
164	C₂²⋊C₄	—	356	2	no	—
165	C₄²	—	356	1	no	—
166	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
167	C₄²	D₄×D₈₉	1	2	yes	—
168	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
169	C₂³	—	356	1	no	—
170	C₂³	#15668001	1	1	yes	characteristic
171	Q₈	—	356	4	no	—
172	Q₈	—	356	2	no	—
173	D₄	—	356	4	no	—
174	D₄	—	356	2	no	—
175	D₄	—	712	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	356	2	no	—
178	C₂×C₄	—	356	1	no	—
179	C₂×C₄	—	712	2	no	—
180	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
181	C₂×C₄	#15667974	1	2	yes	—
182	C₂×C₄	#15667987	1	2	yes	—
183	C₂×C₄	#15668008	1	1	yes	characteristic
184	C₂×C₄	#15668032	1	4	yes	—
185	C₈	—	2	4	no	—
186	C₈	—	8	2	no	—
187	C₈	—	356	4	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	712	2	no	—
190	C₂²	—	712	1	no	—
191	C₂²	#19151499	1	1	yes	characteristic
192	C₂²	#19151955	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	712	2	no	—
195	C₄	—	712	1	no	—
196	C₄	#19151592	1	2	yes	—
197	C₄	#19151719	1	4	yes	—
198	C₄	#19151980	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	1424	1	no	—
201	C₂	#26021061	1	1	yes	characteristic
202	C₂	#26021143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26030575

C₂

yes

maximal, characteristic

#26018706

C₂

yes

maximal, complemented

#26018702

C₂

yes

maximal

C₂×#19151319

C₂

yes

maximal, characteristic, complemented

C₈₉×#1837

C₂

yes

maximal, characteristic, complemented

#19152155

C₂²

yes

characteristic

#19152148

C₂²

yes

characteristic

#19151953

—

#19151949

C₄

yes

—

#19151945

C₄

yes

—

#19151595

—

C₂×C₄×#10623494

C₂²

yes

—

#19151328

—

#19151325

—

#19151319

—

#423×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#341×C₈₉

—

C₂×#13083574

—

C₂²×#10623494

—

#15668012

C₂×C₄

yes

characteristic

#15668012

C₂×C₄

yes

—

#15668009

C₂×C₄

yes

—

—

—

—

C₄×Dic₁₇₈

—

C₄×D₃₅₆

—

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#13083574

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

#13083551

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

#124

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#488

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#416

yes

—

C₄×C₈₉

#444

yes

—

C₄×C₈₉

#495

yes

—

Dic₈₉

—

Dic₈₉

—

#9863

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1958

yes

characteristic

100

C₂×C₈₉

#1975

yes

—

101

D₈₉

—

102

#3131

—

103

#1837

D₈₉

yes

characteristic, complemented

104

#1493

—

105

#1492

—

106

C₂×#322

—

107

C₈₉

#9863

yes

characteristic, complemented, Sylow

108

#533

—

109

#532

—

110

#488

—

178

—

111

#487

—

112

#486

—

113

#423

C₂×D₈₉

yes

—

114

#416

—

178

—

115

C₂×C₄×C₈

—

116

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

117

#341

—

118

Q₈⋊C₈

—

178

—

119

D₄⋊C₈

—

178

—

120

#322

—

178

—

121

C₂×#41

—

178

—

122

C₂×M₁₆

—

123

C₂²×C₈

—

178

—

124

C₂²×C₈

C₂²⋊D₈₉

yes

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

#127

—

128

#125

—

178

—

129

#124

—

178

—

130

C₂²⋊Q₈

—

178

—

131

#121

—

178

—

132

C₄×Q₈

—

178

—

133

C₄×D₄

—

178

—

134

#117

—

178

—

135

C₂×C₄²

C₂²×D₈₉

yes

characteristic

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#103

—

178

—

139

#102

—

178

—

140

C₄×C₈

—

141

C₄×C₈

—

178

—

142

C₄×C₈

C₄×D₈₉

yes

—

143

C₄×C₈

D₃₅₆

yes

—

144

#41

—

356

—

145

C₂×Q₈

—

178

—

146

C₂×Q₈

—

178

—

147

C₂×D₄

—

356

—

148

C₂×D₄

—

178

—

149

C₂×D₄

—

178

—

150

C₂²×C₄

—

356

—

151

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

152

C₂²×C₄

#13083575

yes

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

356

—

157

C₂×C₈

—

178

—

158

C₂×C₈

#13083551

yes

—

159

C₄⋊C₄

—

356

—

160

C₄⋊C₄

—

178

—

161

C₄⋊C₄

—

178

—

162

C₄⋊C₄

—

178

—

163

C₂²⋊C₄

—

356

—

164

C₂²⋊C₄

—

356

—

165

C₄²

—

356

—

166

C₄²

C₂×C₄×D₈₉

yes

characteristic

167

C₄²

D₄×D₈₉

yes

—

168

C₄²

C₂×D₃₅₆

yes

characteristic

169

C₂³

—

356

—

170

C₂³

#15668001

yes

characteristic

171

Q₈

—

356

—

172

Q₈

—

356

—

173

D₄

—

356

—

174

D₄

—

356

—

175

D₄

—

712

—

176

C₂×C₄

—

177

C₂×C₄

—

356

—

178

C₂×C₄

—

356

—

179

C₂×C₄

—

712

—

180

C₂×C₄

C₂×#13083551

yes

characteristic

181

C₂×C₄

#15667974

yes

—

182

C₂×C₄

#15667987

yes

—

183

C₂×C₄

#15668008

yes

characteristic

184

C₂×C₄

#15668032

yes

—

185

C₈

—

186

C₈

—

187

C₈

—

356

—

188

C₂²

—

189

C₂²

—

712

—

190

C₂²

—

712

—

191

C₂²

#19151499

yes

characteristic

192

C₂²

#19151955

yes

—

193

C₄

—

194

C₄

—

712

—

195

C₄

—

712

—

196

C₄

#19151592

yes

—

197

C₄

#19151719

yes

—

198

C₄

#19151980

yes

—

199

C₂

—

200

C₂

—

1424

—

201

C₂

#26021061

yes

characteristic

202

C₂

#26021143

yes

—

	subgroup	G/H
center	C₂×C₄	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1837	C₂
socle	C₂²×C₈₉	#390
prime radical	#19152148	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1837	D₈₉
hypocenter	C₈₉	#9863
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂×C₄

#15668008

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1837

C₂

C₂²×C₈₉

C₂²

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
2. #26018706	C₂	semidirect
4. C₂×#19151319	C₂	semidirect
5. C₈₉×#1837	C₂	semidirect
103. #1837	D₈₉	semidirect
107. C₈₉	#9863	semidirect

H ≅ G/N

type

2. #26018706

C₂

semidirect

4. C₂×#19151319

C₂

semidirect

5. C₈₉×#1837

C₂

semidirect

103. #1837

D₈₉

semidirect

107. C₈₉

#9863

semidirect

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

subgroup	index
#26030575	2
#26018706	2
#26018702	2
C₂×#19151319	2
C₈₉×#1837	2
#9863	89

subgroup

index

C₈₉×#1837

Subgroups — #36454403

Conjugacy classes of subgroups

Complements

Maximal subgroups