Group #36454400 — subgroups

Subgroups — #36454400

Order 22784.

514 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 155 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	#26030680	C₂	1	1	yes	maximal, characteristic
2	#26020424	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018704	C₂	1	4	yes	maximal, complemented
4	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15668034	—	2	2	no	—
6	#19152148	C₂²	1	2	yes	—
7	#19151982	—	2	2	no	—
8	#19151978	C₄	1	1	yes	characteristic
9	#19151975	C₄	1	1	yes	characteristic
10	#19151971	C₄	1	2	yes	—
11	C₂×#15667948	—	2	2	no	—
12	#19151622	C₂²	1	2	yes	—
13	#19151327	—	2	8	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#322×C₈₉	—	2	2	no	—
17	C₂×#13083574	—	2	2	no	—
18	#15668034	—	4	4	no	—
19	#15668034	—	4	4	no	—
20	C₂×Dic₃₅₆	—	2	2	no	—
21	C₂×#13083563	—	2	4	no	—
22	C₂×D₇₁₂	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	#15668012	C₂×C₄	1	2	yes	—
25	#15668009	C₂×C₄	1	2	yes	—
26	#15668000	—	2	2	no	—
27	#15667986	—	2	4	no	—
28	C₄×D₃₅₆	—	2	4	no	—
29	#15667948	—	4	4	no	—
30	C₄⋊#10623494	—	2	4	no	—
31	C₂²×C₈×C₈₉	—	2	2	no	—
32	C₂²×C₈×C₈₉	D₄	1	2	yes	—
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₈₉	—	2	2	no	—
35	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₉	D₄	1	4	yes	—
38	C₄×C₈×C₈₉	D₄	1	2	yes	—
39	#13083574	—	4	4	no	—
40	#13083574	—	4	4	no	—
41	C₂×Dic₁₇₈	—	2	2	no	—
42	C₂×D₃₅₆	—	2	2	no	—
43	C₂×D₃₅₆	—	2	4	no	—
44	C₂×C₂²⋊D₈₉	—	4	2	no	—
45	C₂×C₄×D₈₉	—	4	2	no	—
46	Dic₃₅₆	—	4	4	no	—
47	#13083563	—	4	8	no	—
48	D₇₁₂	—	4	4	no	—
49	#13083558	—	4	4	no	—
50	C₂×#10623494	—	4	2	no	—
51	C₄⋊Dic₈₉	—	2	4	no	—
52	#13083551	—	4	4	no	—
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	#41	1	2	yes	—
55	C₂×C₈×C₈₉	—	2	4	no	—
56	C₂×C₈×C₈₉	—	4	4	no	—
57	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₈₉	D₈	1	4	yes	—
60	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
61	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂×D₄	1	2	yes	—
64	C₂³×C₈₉	#124	1	1	yes	characteristic
65	C₂²×D₈₉	—	4	2	no	—
66	C₂×C₄×C₈₉	—	2	4	no	—
67	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
69	C₂×C₄×C₈₉	#121	1	2	yes	—
70	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
73	C₂²⋊D₈₉	—	8	4	no	—
74	C₂×Dic₈₉	—	4	2	no	—
75	D₃₅₆	—	4	4	no	—
76	D₃₅₆	—	4	8	no	—
77	C₄×D₈₉	—	8	2	no	—
78	C₄×D₈₉	—	8	2	no	—
79	Dic₁₇₈	—	4	4	no	—
80	C₈×C₈₉	—	4	4	no	—
81	C₈×C₈₉	#102	1	4	yes	—
82	C₈×C₈₉	#102	1	4	yes	—
83	#10623494	—	8	2	no	—
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	#390	1	1	yes	characteristic
86	C₂²×C₈₉	#486	1	2	yes	—
87	C₂×D₈₉	—	8	2	no	—
88	C₂×D₈₉	—	8	4	no	—
89	C₄×C₈₉	—	2	2	no	—
90	C₄×C₈₉	C₂×#102	1	2	yes	—
91	C₄×C₈₉	#444	1	4	yes	—
92	C₄×C₈₉	#492	1	1	yes	characteristic
93	C₄×C₈₉	#493	1	1	yes	characteristic
94	Dic₈₉	—	8	2	no	—
95	#9862	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	—	2	2	no	—
97	C₂×C₈₉	#1957	1	1	yes	characteristic
98	C₂×C₈₉	#1975	1	2	yes	—
99	D₈₉	—	16	2	no	—
100	#3147	—	89	1	no	—
101	#1848	—	89	1	no	—
102	#1492	—	89	4	no	—
103	C₂×#322	D₈₉	1	1	yes	characteristic, complemented
104	C₈₉	#9862	1	1	yes	characteristic, complemented, Sylow
105	C₂×#135	—	178	2	no	—
106	#532	—	89	2	no	—
107	#495	—	178	2	no	—
108	#494	—	89	1	no	—
109	#493	—	89	1	no	—
110	#492	—	89	2	no	—
111	C₂×#108	—	178	2	no	—
112	#423	—	89	2	no	—
113	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
115	D₄⋊C₈	—	178	8	no	—
116	#322	—	2	2	no	—
117	C₂×#41	—	178	2	no	—
118	#135	—	356	4	no	—
119	#135	—	356	4	no	—
120	C₂×Q₁₆	—	178	2	no	—
121	C₂×SD₁₆	—	178	4	no	—
122	C₂×D₈	—	178	2	no	—
123	C₂×M₁₆	—	178	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	D₃₅₆	1	2	yes	—
126	C₄⋊Q₈	—	89	2	no	—
127	#127	—	89	2	no	—
128	#124	—	178	2	no	—
129	#121	—	178	4	no	—
130	C₄×D₄	—	178	4	no	—
131	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
132	#108	—	356	4	no	—
133	C₄⋊C₈	—	178	4	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
136	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
137	C₄×C₈	D₃₅₆	1	4	yes	—
138	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
139	#41	—	356	4	no	—
140	#41	—	356	4	no	—
141	C₂×Q₈	—	178	2	no	—
142	C₂×D₄	—	356	2	no	—
143	C₂×D₄	—	178	4	no	—
144	C₂×D₄	—	178	2	no	—
145	C₂²×C₄	—	356	2	no	—
146	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
147	C₂²×C₄	#13083574	1	2	yes	—
148	Q₁₆	—	356	4	no	—
149	SD₁₆	—	356	8	no	—
150	D₈	—	356	4	no	—
151	M₁₆	—	356	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	356	2	no	—
155	C₂×C₈	#13083551	1	2	yes	—
156	C₂×C₈	#13083551	1	2	yes	—
157	C₂×C₈	D₇₁₂	1	4	yes	—
158	C₂×C₈	#13083563	1	4	yes	—
159	C₄⋊C₄	—	178	4	no	—
160	C₂²⋊C₄	—	356	4	no	—
161	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
162	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
163	C₄²	C₂×D₃₅₆	1	2	yes	—
164	C₂³	—	356	2	no	—
165	C₂³	#15668000	1	1	yes	characteristic
166	Q₈	—	356	4	no	—
167	D₄	—	356	8	no	—
168	D₄	—	356	4	no	—
169	D₄	—	712	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	356	2	no	—
172	C₂×C₄	—	712	2	no	—
173	C₂×C₄	—	712	2	no	—
174	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
175	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
176	C₂×C₄	#15667986	1	2	yes	—
177	C₂×C₄	#15668009	1	1	yes	characteristic
178	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
179	C₂×C₄	C₂×#13083563	1	2	yes	—
180	C₈	—	4	4	no	—
181	C₈	—	712	2	no	—
182	C₈	#15667934	1	4	yes	—
183	C₈	#15667934	1	4	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	712	4	no	—
186	C₂²	—	712	2	no	—
187	C₂²	#19151500	1	1	yes	characteristic
188	C₂²	#19151945	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	712	2	no	—
191	C₄	C₂×#15667934	1	2	yes	—
192	C₄	#19151721	1	4	yes	—
193	C₄	#19151971	1	1	yes	characteristic
194	C₄	#19151975	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	1424	2	no	—
197	C₂	#26021058	1	1	yes	characteristic
198	C₂	#26021145	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26030680

C₂

yes

maximal, characteristic

#26020424

C₂

yes

maximal, characteristic, complemented

#26018704

C₂

yes

maximal, complemented

C₂×#322×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668034

—

#19152148

C₂²

yes

—

#19151982

—

#19151978

C₄

yes

characteristic

#19151975

C₄

yes

characteristic

#19151971

C₄

yes

—

C₂×#15667948

—

#19151622

C₂²

yes

—

#19151327

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#322×C₈₉

—

C₂×#13083574

—

#15668034

—

#15668034

—

C₂×Dic₃₅₆

—

C₂×#13083563

—

C₂×D₇₁₂

—

C₂×#13083558

—

#15668012

C₂×C₄

yes

—

#15668009

C₂×C₄

yes

—

#15668000

—

#15667986

—

C₄×D₃₅₆

—

#15667948

—

C₄⋊#10623494

—

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

Dic₃₅₆

—

#13083563

—

D₇₁₂

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

SD₁₆

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

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

#102

yes

—

C₈×C₈₉

#102

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#486

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#102

yes

—

C₄×C₈₉

#444

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#9862

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1957

yes

characteristic

C₂×C₈₉

#1975

yes

—

D₈₉

—

100

#3147

—

101

#1848

—

102

#1492

—

103

C₂×#322

D₈₉

yes

characteristic, complemented

104

C₈₉

#9862

yes

characteristic, complemented, Sylow

105

C₂×#135

—

178

—

106

#532

—

107

#495

—

178

—

108

#494

—

109

#493

—

110

#492

—

111

C₂×#108

—

178

—

112

#423

—

113

C₂×C₄×C₈

C₂×D₈₉

yes

—

114

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

115

D₄⋊C₈

—

178

—

116

#322

—

117

C₂×#41

—

178

—

118

#135

—

356

—

119

#135

—

356

—

120

C₂×Q₁₆

—

178

—

121

C₂×SD₁₆

—

178

—

122

C₂×D₈

—

178

—

123

C₂×M₁₆

—

178

—

124

C₂²×C₈

—

125

C₂²×C₈

D₃₅₆

yes

—

126

C₄⋊Q₈

—

127

#127

—

128

#124

—

178

—

129

#121

—

178

—

130

C₄×D₄

—

178

—

131

C₂×C₄²

C₂²×D₈₉

yes

characteristic

132

#108

—

356

—

133

C₄⋊C₈

—

178

—

134

C₄×C₈

—

135

C₄×C₈

C₄×D₈₉

yes

characteristic

136

C₄×C₈

C₄×D₈₉

yes

characteristic

137

C₄×C₈

D₃₅₆

yes

—

138

C₄×C₈

C₂²⋊D₈₉

yes

—

139

#41

—

356

—

140

#41

—

356

—

141

C₂×Q₈

—

178

—

142

C₂×D₄

—

356

—

143

C₂×D₄

—

178

—

144

C₂×D₄

—

178

—

145

C₂²×C₄

—

356

—

146

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

147

C₂²×C₄

#13083574

yes

—

148

Q₁₆

—

356

—

149

SD₁₆

—

356

—

150

D₈

—

356

—

151

M₁₆

—

356

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

356

—

155

C₂×C₈

#13083551

yes

—

156

C₂×C₈

#13083551

yes

—

157

C₂×C₈

D₇₁₂

yes

—

158

C₂×C₈

#13083563

yes

—

159

C₄⋊C₄

—

178

—

160

C₂²⋊C₄

—

356

—

161

C₄²

C₂×C₄×D₈₉

yes

characteristic

162

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

163

C₄²

C₂×D₃₅₆

yes

—

164

C₂³

—

356

—

165

C₂³

#15668000

yes

characteristic

166

Q₈

—

356

—

167

D₄

—

356

—

168

D₄

—

356

—

169

D₄

—

712

—

170

C₂×C₄

—

171

C₂×C₄

—

356

—

172

C₂×C₄

—

712

—

173

C₂×C₄

—

712

—

174

C₂×C₄

C₂×#13083551

yes

characteristic

175

C₂×C₄

C₄×D₃₅₆

yes

—

176

C₂×C₄

#15667986

yes

—

177

C₂×C₄

#15668009

yes

characteristic

178

C₂×C₄

C₂×D₇₁₂

yes

—

179

C₂×C₄

C₂×#13083563

yes

—

180

C₈

—

181

C₈

—

712

—

182

C₈

#15667934

yes

—

183

C₈

#15667934

yes

—

184

C₂²

—

185

C₂²

—

712

—

186

C₂²

—

712

—

187

C₂²

#19151500

yes

characteristic

188

C₂²

#19151945

yes

—

189

C₄

—

190

C₄

—

712

—

191

C₄

C₂×#15667934

yes

—

192

C₄

#19151721

yes

—

193

C₄

#19151971

yes

characteristic

194

C₄

#19151975

yes

characteristic

195

C₂

—

196

C₂

—

1424

—

197

C₂

#26021058

yes

characteristic

198

C₂

#26021145

yes

—

	subgroup	G/H
center	C₂×C₄	#15668009
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂²×C₈₉	#390
prime radical	#26030680	C₂
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#9862
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#15668009

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₂×#322×C₈₉

C₂

C₂²×C₈₉

C₂

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
2. #26020424	C₂	semidirect
3. #26018704	C₂	semidirect
4. C₂×#322×C₈₉	C₂	semidirect
103. C₂×#322	D₈₉	semidirect
104. C₈₉	#9862	semidirect

H ≅ G/N

type

2. #26020424

C₂

semidirect

3. #26018704

C₂

semidirect

4. C₂×#322×C₈₉

C₂

semidirect

103. C₂×#322

D₈₉

semidirect

104. C₈₉

#9862

semidirect

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

subgroup	index
#26030680	2
#26020424	2
#26018704	2
C₂×#322×C₈₉	2
#9862	89

subgroup

index

C₂×#322×C₈₉

Subgroups — #36454400

Conjugacy classes of subgroups

Complements

Maximal subgroups