Group #36463776 — subgroups

Subgroups — #36463776

Order 22784.

288 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26029185	C₂	1	1	yes	maximal, characteristic
2	#26022052	C₂	1	2	yes	maximal
3	#26018194	C₂	1	1	yes	maximal, characteristic
4	#26018193	C₂	1	2	yes	maximal
5	C₈₉×#1323	C₂	1	1	yes	maximal, characteristic
6	#19152151	C₄	1	1	yes	characteristic
7	#19151727	—	2	2	no	—
8	#19151722	—	2	2	no	—
9	#19151624	C₂²	1	1	yes	characteristic
10	#19151621	C₄	1	1	yes	characteristic
11	#19151556	C₄	1	1	yes	characteristic
12	#19151554	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
14	#19151416	—	2	2	no	—
15	C₂²⋊#13083549	C₂²	1	2	yes	—
16	#19151373	—	2	2	no	—
17	#424×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
19	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
21	#15668011	C₈	1	2	yes	—
22	#15668003	C₈	1	2	yes	—
23	#15667989	—	2	2	no	—
24	C₄×Dic₁₇₈	—	2	2	no	—
25	#15667971	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×#13083549	—	2	2	no	—
28	#15667941	—	2	2	no	—
29	#15667941	—	2	2	no	—
30	#15667923	—	2	1	no	—
31	#15667922	—	2	1	no	—
32	C₄×#10623494	—	2	1	no	—
33	C₈×Dic₈₉	—	2	1	no	—
34	C₈×Dic₈₉	D₄	1	2	yes	—
35	C₈×Dic₈₉	D₄	1	2	yes	—
36	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₈₉	—	2	2	no	—
40	C₄⋊C₈×C₈₉	—	2	2	no	—
41	C₂×Dic₁₇₈	—	2	2	no	—
42	C₂²×Dic₈₉	M₁₆	1	2	yes	—
43	#13083558	—	4	1	no	—
44	C₂×#10623494	—	2	2	no	—
45	C₂×#10623494	—	2	1	no	—
46	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
48	C₄⋊Dic₈₉	—	2	2	no	—
49	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
50	#13083553	—	2	2	no	—
51	#13083553	—	2	2	no	—
52	C₂²⋊Dic₈₉	—	2	1	no	—
53	C₄×Dic₈₉	—	2	1	no	—
54	C₄×Dic₈₉	C₂²⋊C₄	1	2	yes	—
55	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
56	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
57	#13083549	—	4	2	no	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₈₉	—	4	1	no	—
60	C₂×C₈×C₈₉	—	2	1	no	—
61	C₂×C₈×C₈₉	—	4	1	no	—
62	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
63	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
64	C₂²⋊C₄×C₈₉	—	2	1	no	—
65	C₄²×C₈₉	—	2	1	no	—
66	C₁₆×C₈₉	—	4	2	no	—
67	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
73	C₂×Dic₈₉	—	2	2	no	—
74	C₂×Dic₈₉	—	4	2	no	—
75	C₂×Dic₈₉	C₂²⋊C₈	1	2	yes	—
76	C₂×Dic₈₉	C₂²⋊C₈	1	2	yes	—
77	Dic₁₇₈	—	4	4	no	—
78	C₈×C₈₉	—	2	2	no	—
79	C₈×C₈₉	—	4	1	no	—
80	#10623494	—	2	1	no	—
81	#10623494	—	4	1	no	—
82	#10623494	#101	1	2	yes	—
83	C₂²×C₈₉	—	2	2	no	—
84	C₂²×C₈₉	—	2	1	no	—
85	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
86	C₄×C₈₉	—	2	1	no	—
87	C₄×C₈₉	—	4	2	no	—
88	C₄×C₈₉	#410	1	1	yes	characteristic
89	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
90	Dic₈₉	—	2	2	no	—
91	Dic₈₉	—	2	2	no	—
92	Dic₈₉	—	4	2	no	—
93	#11497	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	—	4	1	no	—
95	C₂×C₈₉	#1461	1	1	yes	characteristic
96	C₂×C₈₉	#2117	1	2	yes	—
97	#2960	—	89	1	no	—
98	#2163	—	89	2	no	—
99	#1323	—	89	1	no	—
100	#1323	—	89	2	no	—
101	#1323	D₈₉	1	1	yes	characteristic
102	C₈₉	#11497	1	1	yes	characteristic, complemented, Sylow
103	#533	—	89	1	no	—
104	#446	—	178	2	no	—
105	C₈×Q₈	—	178	2	no	—
106	#424	—	89	1	no	—
107	#424	C₂×D₈₉	1	1	yes	characteristic
108	#423	—	89	1	no	—
109	#405	—	89	1	no	—
110	#405	—	89	1	no	—
111	C₂×C₄×C₈	—	89	1	no	—
112	C₄⋊C₁₆	—	178	2	no	—
113	C₂²⋊C₁₆	—	89	2	no	—
114	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
115	#346	—	178	2	no	—
116	C₂×M₁₆	—	89	1	no	—
117	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
118	C₂²×C₈	—	89	1	no	—
119	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
120	C₄⋊Q₈	—	89	2	no	—
121	#125	—	89	2	no	—
122	C₂²⋊Q₈	—	178	2	no	—
123	C₄×Q₈	—	178	2	no	—
124	#117	—	89	1	no	—
125	#117	C₄×D₈₉	1	1	yes	characteristic
126	C₂×C₄²	—	89	1	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	178	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	178	2	no	—
131	C₄⋊C₈	—	178	2	no	—
132	#97	—	178	1	no	—
133	#97	—	178	1	no	—
134	C₄×C₈	—	89	2	no	—
135	C₄×C₈	—	89	2	no	—
136	C₄×C₈	—	178	1	no	—
137	C₄×C₈	—	178	1	no	—
138	C₂×Q₈	—	178	2	no	—
139	C₂²×C₄	—	89	2	no	—
140	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
141	M₁₆	—	4	1	no	—
142	M₁₆	—	356	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	—	89	1	no	—
146	C₂×C₈	—	89	1	no	—
147	C₂×C₈	—	178	2	no	—
148	C₂×C₈	—	178	1	no	—
149	C₂×C₈	D₄×D₈₉	1	2	yes	—
150	C₄⋊C₄	—	89	2	no	—
151	C₄⋊C₄	—	178	2	no	—
152	C₄⋊C₄	—	178	2	no	—
153	C₄⋊C₄	—	178	2	no	—
154	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
155	C₂²⋊C₄	—	2	1	no	—
156	C₂²⋊C₄	—	178	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	89	1	no	—
159	C₄²	—	89	1	no	—
160	C₄²	—	89	2	no	—
161	C₄²	—	178	1	no	—
162	C₁₆	—	4	2	no	—
163	C₁₆	—	356	2	no	—
164	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
165	Q₈	—	356	4	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	89	2	no	—
170	C₂×C₄	—	89	2	no	—
171	C₂×C₄	—	356	2	no	—
172	C₂×C₄	—	178	2	no	—
173	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
174	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	4	1	no	—
177	C₈	—	89	2	no	—
178	C₈	—	356	1	no	—
179	C₈	—	178	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	2	no	—
182	C₂²	C₂²⋊C₈×D₈₉	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	356	2	no	—
186	C₄	—	178	2	no	—
187	C₄	—	178	2	no	—
188	C₄	#19151570	1	1	yes	characteristic
189	C₄	#101×D₈₉	1	1	yes	characteristic
190	C₂	—	4	1	no	—
191	C₂	#26018574	1	1	yes	characteristic
192	C₂	#26021878	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26029185

C₂

yes

maximal, characteristic

#26022052

C₂

yes

maximal

#26018194

C₂

yes

maximal, characteristic

#26018193

C₂

yes

maximal

C₈₉×#1323

C₂

yes

maximal, characteristic

#19152151

C₄

yes

—

—

C₂²

yes

characteristic

#19151621

C₄

yes

characteristic

#19151556

C₄

yes

characteristic

#19151554

C₄

yes

characteristic

C₂×C₈×Dic₈₉

C₂²

yes

characteristic

#19151416

—

C₂²⋊#13083549

C₂²

yes

—

#19151373

—

#424×C₈₉

C₂²

yes

characteristic

C₂²⋊C₁₆×C₈₉

C₂²

yes

—

C₂×#13083558

C₂×C₄

yes

characteristic

C₂²×#10623494

C₂×C₄

yes

characteristic

#15668011

C₈

yes

—

#15668003

C₈

yes

—

#15667989

—

C₄×Dic₁₇₈

—

#15667971

C₂×C₄

yes

characteristic

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

—

—

—

—

—

C₈×Dic₈₉

—

C₈×Dic₈₉

D₄

yes

—

C₈×Dic₈₉

D₄

yes

—

C₂×M₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

#117×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₄⋊C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

M₁₆

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10623494

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂×C₈

yes

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂²⋊C₄

yes

—

C₄×Dic₈₉

C₂×C₈

yes

characteristic

C₄×Dic₈₉

C₂×C₈

yes

characteristic

#13083549

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

C₂×C₈

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂²⋊C₈

yes

—

C₂×Dic₈₉

C₂²⋊C₈

yes

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

—

—

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

#11497

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1461

yes

characteristic

C₂×C₈₉

#2117

yes

—

#2960

—

#2163

—

#1323

—

100

#1323

—

101

#1323

D₈₉

yes

characteristic

102

C₈₉

#11497

yes

characteristic, complemented, Sylow

103

#533

—

104

#446

—

178

—

105

C₈×Q₈

—

178

—

106

#424

—

107

#424

C₂×D₈₉

yes

characteristic

108

#423

—

109

#405

—

110

#405

—

111

C₂×C₄×C₈

—

112

C₄⋊C₁₆

—

178

—

113

C₂²⋊C₁₆

—

114

C₂²⋊C₁₆

C₂×D₈₉

yes

—

115

#346

—

178

—

116

C₂×M₁₆

—

117

C₂×M₁₆

C₄×D₈₉

yes

characteristic

118

C₂²×C₈

—

119

C₂²×C₈

C₂²×D₈₉

yes

characteristic

120

C₄⋊Q₈

—

121

#125

—

122

C₂²⋊Q₈

—

178

—

123

C₄×Q₈

—

178

—

124

#117

—

125

#117

C₄×D₈₉

yes

characteristic

126

C₂×C₄²

—

127

C₂×C₁₆

—

128

C₂×C₁₆

—

178

—

129

C₄⋊C₈

—

130

C₄⋊C₈

—

178

—

131

C₄⋊C₈

—

178

—

132

#97

—

178

—

133

#97

—

178

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

178

—

137

C₄×C₈

—

178

—

138

C₂×Q₈

—

178

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

141

M₁₆

—

142

M₁₆

—

356

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

178

—

148

C₂×C₈

—

178

—

149

C₂×C₈

D₄×D₈₉

yes

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

178

—

152

C₄⋊C₄

—

178

—

153

C₄⋊C₄

—

178

—

154

C₄⋊C₄

C₈×D₈₉

yes

—

155

C₂²⋊C₄

—

156

C₂²⋊C₄

—

178

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

178

—

162

C₁₆

—

163

C₁₆

—

356

—

164

C₂³

M₁₆×D₈₉

yes

characteristic

165

Q₈

—

356

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

356

—

172

C₂×C₄

—

178

—

173

C₂×C₄

C₂²⋊C₄×D₈₉

yes

characteristic

174

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

356

—

179

C₈

—

178

—

180

C₂²

—

181

C₂²

—

182

C₂²

C₂²⋊C₈×D₈₉

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

356

—

186

C₄

—

178

—

187

C₄

—

178

—

188

C₄

#19151570

yes

characteristic

189

C₄

#101×D₈₉

yes

characteristic

190

C₂

—

191

C₂

#26018574

yes

characteristic

192

C₂

#26021878

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1323	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1323	D₈₉
hypocenter	C₈₉	#11497
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1323

subgroup

G/H

center

C₂×C₄

C₂²⋊C₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1323

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₈

prime radical

C₂³×C₈₉

C₂×M₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1323

N	H ≅ G/N	type
102. C₈₉	#11497	semidirect

H ≅ G/N

type

102. C₈₉

#11497

semidirect

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

subgroup	index
#26029185	2
#26022052	2
#26018194	2
#26018193	2
C₈₉×#1323	2
#11497	89

subgroup

index

C₈₉×#1323

Subgroups — #36463776

Conjugacy classes of subgroups

Complements

Maximal subgroups