Group #36454455 — subgroups

Subgroups — #36454455

Order 22784.

434 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₄×#15668004	C₂	1	1	yes	maximal, characteristic
2	#26020369	C₂	1	1	yes	maximal, characteristic
3	#26018092	C₂	1	4	yes	maximal
4	C₈₉×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	C₄	1	1	yes	characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	#19151496	C₄	1	2	yes	—
8	#19151482	—	2	2	no	—
9	#19151464	C₄	1	1	yes	characteristic
10	C₄×#13083553	—	2	2	no	—
11	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
12	#19151353	—	2	2	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
14	C₄³×C₈₉	C₂²	1	1	yes	characteristic
15	#336×C₈₉	—	2	2	no	—
16	C₂²×#10623494	—	2	2	no	—
17	#15668004	—	2	4	no	—
18	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
20	C₂×#13083553	—	2	2	no	—
21	C₂×#13083553	—	2	2	no	—
22	C₂×C₄×Dic₈₉	—	2	2	no	—
23	C₄⋊#10623494	—	2	4	no	—
24	#15667919	—	2	2	no	—
25	#15667919	—	2	2	no	—
26	#15667919	—	2	2	no	—
27	C₂²×C₈×C₈₉	—	2	2	no	—
28	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	D₄	1	2	yes	—
31	C₂×C₄²×C₈₉	D₄	1	2	yes	—
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₈₉	—	2	2	no	—
34	C₄×C₈×C₈₉	Q₈	1	4	yes	—
35	C₂²×Dic₈₉	—	2	2	no	—
36	C₂²×Dic₈₉	—	2	2	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₂×#10623494	—	4	4	no	—
39	C₄⋊Dic₈₉	—	2	2	no	—
40	C₄⋊Dic₈₉	—	2	4	no	—
41	C₄⋊Dic₈₉	M₁₆	1	4	yes	—
42	#13083553	—	2	8	no	—
43	#13083553	—	4	4	no	—
44	C₄×Dic₈₉	—	4	4	no	—
45	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	—	2	4	no	—
51	C₂×C₈×C₈₉	—	4	4	no	—
52	C₄²×C₈₉	—	2	2	no	—
53	C₄²×C₈₉	—	2	2	no	—
54	C₄²×C₈₉	—	2	4	no	—
55	C₄²×C₈₉	—	4	2	no	—
56	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
57	C₄²×C₈₉	#41	1	2	yes	—
58	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
67	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₈₉	#131	1	2	yes	—
72	C₂×C₄×C₈₉	#135	1	4	yes	—
73	C₂×Dic₈₉	—	2	4	no	—
74	C₂×Dic₈₉	—	4	2	no	—
75	C₂×Dic₈₉	—	4	2	no	—
76	C₂×Dic₈₉	—	4	2	no	—
77	C₂×Dic₈₉	—	4	2	no	—
78	C₂×Dic₈₉	—	4	2	no	—
79	C₈×C₈₉	—	4	4	no	—
80	#10623494	—	8	2	no	—
81	C₂²×C₈₉	#391	1	1	yes	characteristic
82	C₂²×C₈₉	#407	1	1	yes	characteristic
83	C₂²×C₈₉	#408	1	1	yes	characteristic
84	C₂²×C₈₉	#416	1	2	yes	—
85	C₂²×C₈₉	#421	1	2	yes	—
86	C₄×C₈₉	—	2	2	no	—
87	C₄×C₈₉	—	4	2	no	—
88	C₄×C₈₉	—	4	2	no	—
89	C₄×C₈₉	#446	1	4	yes	—
90	C₄×C₈₉	#488	1	2	yes	—
91	C₄×C₈₉	#499	1	2	yes	—
92	Dic₈₉	—	4	4	no	—
93	Dic₈₉	—	8	2	no	—
94	#9875	—	89	1	no	maximal, Sylow
95	C₂×C₈₉	#1482	1	2	yes	—
96	C₂×C₈₉	#1488	1	2	yes	—
97	C₂×C₈₉	#1985	1	1	yes	characteristic
98	C₂×C₈₉	#1991	1	1	yes	characteristic
99	C₂×C₈₉	#1995	1	1	yes	characteristic
100	C₄×#125	—	89	1	no	—
101	#1831	—	89	1	no	—
102	#1744	D₈₉	1	1	yes	characteristic
103	#1279	—	89	4	no	—
104	C₈₉	#9875	1	1	yes	characteristic, complemented, Sylow
105	C₂×#125	—	89	1	no	—
106	C₂×C₄⋊C₈	—	89	2	no	—
107	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
108	#389	—	89	2	no	—
109	#387	—	178	2	no	—
110	#382	—	89	1	no	—
111	C₄×C₄⋊C₄	—	89	2	no	—
112	C₄×C₄⋊C₄	—	178	2	no	—
113	C₄³	C₂×D₈₉	1	1	yes	characteristic
114	#336	—	2	2	no	—
115	#336	—	178	2	no	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	178	2	no	—
118	#125	—	178	4	no	—
119	C₂×C₄⋊C₄	—	89	2	no	—
120	C₂×C₄⋊C₄	—	89	2	no	—
121	C₂×C₄⋊C₄	—	178	2	no	—
122	C₂×C₄⋊C₄	—	178	2	no	—
123	C₂×C₄²	—	178	2	no	—
124	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
126	C₂×C₄²	D₃₅₆	1	2	yes	—
127	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
128	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
129	C₄⋊C₈	—	178	4	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	Dic₁₇₈	1	4	yes	—
132	#95	—	178	2	no	—
133	#95	—	178	2	no	—
134	#95	—	178	2	no	—
135	C₂²×C₄	—	178	2	no	—
136	C₂²×C₄	—	178	2	no	—
137	C₂²×C₄	#13083551	1	2	yes	—
138	C₂²×C₄	#13083551	1	2	yes	—
139	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	4	no	—
144	C₂×C₈	—	356	2	no	—
145	C₂×C₈	—	356	4	no	—
146	C₄⋊C₄	—	89	4	no	—
147	C₄⋊C₄	—	356	4	no	—
148	C₄⋊C₄	—	178	8	no	—
149	C₄⋊C₄	—	178	4	no	—
150	C₄⋊C₄	—	178	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	2	no	—
155	C₄²	—	356	4	no	—
156	C₄²	C₂×Dic₁₇₈	1	2	yes	—
157	C₄²	#13083574	1	2	yes	—
158	C₂³	C₂×#13083551	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	356	2	no	—
167	C₂×C₄	—	356	2	no	—
168	C₂×C₄	—	356	2	no	—
169	C₂×C₄	—	356	2	no	—
170	C₂×C₄	—	356	2	no	—
171	C₂×C₄	—	178	4	no	—
172	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
173	C₂×C₄	#15667991	1	2	yes	—
174	C₂×C₄	#15668000	1	1	yes	characteristic
175	C₂×C₄	#15668012	1	1	yes	characteristic
176	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
177	C₂×C₄	#15668020	1	2	yes	—
178	C₂×C₄	#15668034	1	4	yes	—
179	C₈	—	4	4	no	—
180	C₈	—	712	2	no	—
181	C₂²	#19151504	1	1	yes	characteristic
182	C₂²	#19151561	1	1	yes	characteristic
183	C₂²	#19151566	1	1	yes	characteristic
184	C₂²	#19151595	1	2	yes	—
185	C₂²	#19151616	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	356	4	no	—
190	C₄	—	712	2	no	—
191	C₄	#19151726	1	4	yes	—
192	C₄	#19151953	1	2	yes	—
193	C₄	#19152000	1	2	yes	—
194	C₂	#26018661	1	2	yes	—
195	C₂	#26018687	1	2	yes	—
196	C₂	#26021192	1	1	yes	characteristic
197	C₂	#26021217	1	1	yes	characteristic
198	C₂	#26021233	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#15668004

C₂

yes

maximal, characteristic

#26020369

C₂

yes

maximal, characteristic

#26018092

C₂

yes

maximal

C₈₉×#1744

C₂

yes

maximal, characteristic

C₂×#15668004

C₄

yes

characteristic

C₂×C₄⋊#10623494

C₂²

yes

—

#19151496

C₄

yes

—

#19151482

—

#19151464

C₄

yes

characteristic

C₄×#13083553

—

C₄×C₄⋊Dic₈₉

C₂²

yes

—

#19151353

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

C₂²×#10623494

—

#15668004

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂×#13083553

—

C₂×#13083553

—

C₂×C₄×Dic₈₉

—

—

—

—

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

Q₈

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

M₁₆

yes

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#421

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#446

yes

—

C₄×C₈₉

#488

yes

—

C₄×C₈₉

#499

yes

—

Dic₈₉

—

Dic₈₉

—

#9875

—

maximal, Sylow

C₂×C₈₉

#1482

yes

—

C₂×C₈₉

#1488

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

100

—

101

#1831

—

102

#1744

D₈₉

yes

characteristic

103

#1279

—

104

C₈₉

#9875

yes

characteristic, complemented, Sylow

105

C₂×#125

—

106

C₂×C₄⋊C₈

—

107

C₂×C₄×C₈

C₂×D₈₉

yes

—

108

#389

—

109

#387

—

178

—

110

#382

—

111

C₄×C₄⋊C₄

—

112

C₄×C₄⋊C₄

—

178

—

113

C₄³

C₂×D₈₉

yes

characteristic

114

#336

—

115

#336

—

178

—

116

C₂²×C₈

—

117

C₂²×C₈

—

178

—

118

#125

—

178

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

178

—

122

C₂×C₄⋊C₄

—

178

—

123

C₂×C₄²

—

178

—

124

C₂×C₄²

C₄×D₈₉

yes

characteristic

125

C₂×C₄²

C₄×D₈₉

yes

characteristic

126

C₂×C₄²

D₃₅₆

yes

—

127

C₂×C₄²

C₂²⋊D₈₉

yes

—

128

C₂×C₄²

C₂²×D₈₉

yes

characteristic

129

C₄⋊C₈

—

178

—

130

C₄×C₈

—

131

C₄×C₈

Dic₁₇₈

yes

—

132

#95

—

178

—

133

#95

—

178

—

134

#95

—

178

—

135

C₂²×C₄

—

178

—

136

C₂²×C₄

—

178

—

137

C₂²×C₄

#13083551

yes

—

138

C₂²×C₄

#13083551

yes

—

139

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

141

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

356

—

145

C₂×C₈

—

356

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

356

—

148

C₄⋊C₄

—

178

—

149

C₄⋊C₄

—

178

—

150

C₄⋊C₄

—

178

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

356

—

156

C₄²

C₂×Dic₁₇₈

yes

—

157

C₄²

#13083574

yes

—

158

C₂³

C₂×#13083551

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

356

—

167

C₂×C₄

—

356

—

168

C₂×C₄

—

356

—

169

C₂×C₄

—

356

—

170

C₂×C₄

—

356

—

171

C₂×C₄

—

178

—

172

C₂×C₄

C₄×Dic₁₇₈

yes

—

173

C₂×C₄

#15667991

yes

—

174

C₂×C₄

#15668000

yes

characteristic

175

C₂×C₄

#15668012

yes

characteristic

176

C₂×C₄

M₁₆×D₈₉

yes

—

177

C₂×C₄

#15668020

yes

—

178

C₂×C₄

#15668034

yes

—

179

C₈

—

180

C₈

—

712

—

181

C₂²

#19151504

yes

characteristic

182

C₂²

#19151561

yes

characteristic

183

C₂²

#19151566

yes

characteristic

184

C₂²

#19151595

yes

—

185

C₂²

#19151616

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

356

—

190

C₄

—

712

—

191

C₄

#19151726

yes

—

192

C₄

#19151953

yes

—

193

C₄

#19152000

yes

—

194

C₂

#26018661

yes

—

195

C₂

#26018687

yes

—

196

C₂

#26021192

yes

characteristic

197

C₂

#26021217

yes

characteristic

198

C₂

#26021233

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1744	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1744	D₈₉
hypocenter	C₈₉	#9875
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₃₅₆

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1744

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
104. C₈₉	#9875	semidirect

H ≅ G/N

type

104. C₈₉

#9875

semidirect

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

subgroup	index
C₄×#15668004	2
#26020369	2
#26018092	2
C₈₉×#1744	2
#9875	89

subgroup

index

C₈₉×#1744

Subgroups — #36454455

Conjugacy classes of subgroups

Complements

Maximal subgroups