Group #36542135 — subgroups

Subgroups — #36542135

Order 22784.

706 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 187 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	#26030837	C₂	1	4	yes	maximal
2	#26030763	C₂	1	2	yes	maximal
3	#26019869	C₂	1	2	yes	maximal, complemented
4	#26019862	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019848	C₂	1	4	yes	maximal, complemented
6	C₈²×D₈₉	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1710	C₂	1	1	yes	maximal, characteristic, complemented
8	#19152153	C₂²	1	4	yes	—
9	#19152003	C₂²	1	2	yes	—
10	#19152001	C₂²	1	4	yes	—
11	#19151999	—	2	4	no	—
12	#19151994	C₂²	1	4	yes	—
13	#19151977	C₂²	1	4	yes	—
14	#19151949	C₂²	1	4	yes	—
15	#19151838	—	2	8	no	—
16	#19151690	—	2	8	no	—
17	#19151635	—	2	4	no	—
18	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₈₉	C₂²	1	2	yes	—
20	C₈×#10623494	C₂²	1	2	yes	—
21	C₈×#10623494	C₂²	1	1	yes	characteristic
22	#500×C₈₉	C₂²	1	2	yes	—
23	#487×C₈₉	C₂²	1	4	yes	—
24	C₈²×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#13083564	—	2	8	no	—
26	C₂×C₈×D₈₉	D₄	1	4	yes	—
27	C₂×C₈×D₈₉	D₄	1	2	yes	—
28	#15668012	C₂³	1	4	yes	—
29	#15668003	—	2	4	no	—
30	#15667992	—	2	4	no	—
31	#15667988	—	2	4	no	—
32	Q₈×Dic₈₉	—	2	4	no	—
33	#15667977	—	2	4	no	—
34	#15667970	—	2	4	no	—
35	C₄²×D₈₉	C₂³	1	1	yes	characteristic
36	#15667946	—	2	4	no	—
37	#15667945	—	2	8	no	—
38	#15667943	—	2	8	no	—
39	#15667937	—	2	8	no	—
40	C₄×#10623494	C₂³	1	2	yes	—
41	C₄×#10623494	C₂³	1	1	yes	characteristic
42	C₈×Dic₈₉	D₄	1	4	yes	—
43	C₈×Dic₈₉	D₄	1	2	yes	—
44	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
45	#107×C₈₉	—	2	4	no	—
46	#103×C₈₉	—	2	8	no	—
47	C₄×C₈×C₈₉	C₂³	1	2	yes	—
48	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₇₈	—	2	4	no	—
50	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
52	#13083564	—	4	16	no	—
53	C₂×#10623494	C₂×D₄	1	2	yes	—
54	C₂×#10623494	C₂×D₄	1	4	yes	—
55	C₈×D₈₉	—	2	8	no	—
56	C₈×D₈₉	Q₁₆	1	8	yes	complemented
57	C₄⋊Dic₈₉	—	2	4	no	—
58	C₄⋊Dic₈₉	—	2	4	no	—
59	C₄⋊Dic₈₉	—	4	4	no	—
60	#13083553	—	4	4	no	—
61	#13083551	—	4	4	no	—
62	C₄×Dic₈₉	—	4	4	no	—
63	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
64	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₈₉	—	2	4	no	—
66	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
67	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
68	C₄⋊C₄×C₈₉	—	2	4	no	—
69	C₄⋊C₄×C₈₉	—	2	4	no	—
70	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
71	C₂²×D₈₉	#127	1	1	yes	characteristic
72	Q₈×C₈₉	—	4	8	no	—
73	C₂×C₄×C₈₉	—	4	4	no	—
74	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
76	C₂×Dic₈₉	—	4	4	no	—
77	C₂×Dic₈₉	#127	1	2	yes	—
78	C₂×Dic₈₉	#127	1	1	yes	characteristic
79	C₄×D₈₉	—	2	4	no	—
80	C₄×D₈₉	C₂×Q₁₆	1	4	yes	—
81	Dic₁₇₈	—	4	8	no	—
82	C₈×C₈₉	—	2	4	no	—
83	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
84	#10623494	C₂×Q₁₆	1	4	yes	—
85	#10623494	#135	1	8	yes	—
86	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
87	C₂×D₈₉	—	2	2	no	—
88	C₂×D₈₉	#494	1	2	yes	—
89	C₄×C₈₉	—	8	4	no	—
90	C₄×C₈₉	C₂²×Q₁₆	1	2	yes	—
91	C₄×C₈₉	C₂×#135	1	4	yes	—
92	Dic₈₉	—	2	2	no	—
93	Dic₈₉	—	8	4	no	—
94	Dic₈₉	#494	1	2	yes	—
95	#19604	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
97	C₂×C₈₉	#3147	1	2	yes	—
98	D₈₉	—	2	2	no	—
99	#3170	—	89	4	no	—
100	#3160	—	89	2	no	—
101	#1713	—	89	2	no	—
102	#1710	—	89	1	no	—
103	#1710	D₈₉	1	1	yes	characteristic, complemented
104	#1707	—	89	4	no	—
105	C₂×C₈²	—	89	1	no	—
106	C₈₉	#19604	1	1	yes	characteristic, complemented, Sylow
107	#533	—	89	4	no	—
108	#500	—	89	4	no	—
109	#500	—	89	2	no	—
110	#500	C₂×D₈₉	1	2	yes	—
111	#499	—	178	4	no	—
112	#498	—	89	4	no	—
113	#494	—	89	4	no	—
114	#487	—	89	4	no	—
115	#487	C₂×D₈₉	1	4	yes	—
116	#467	—	178	8	no	—
117	C₄×Q₁₆	—	178	8	no	—
118	#427	—	178	4	no	—
119	C₂×C₄×C₈	—	89	1	no	—
120	C₂×C₄×C₈	—	89	2	no	—
121	C₈²	—	89	2	no	—
122	C₈²	—	89	1	no	—
123	C₈²	C₂×D₈₉	1	1	yes	characteristic
124	C₂×Q₁₆	—	178	8	no	—
125	C₂²×C₈	—	89	2	no	—
126	C₂²×C₈	—	89	4	no	—
127	C₄⋊Q₈	—	89	4	no	—
128	C₄⋊Q₈	C₂²×D₈₉	1	4	yes	—
129	#125	—	178	4	no	—
130	C₂²⋊Q₈	—	178	4	no	—
131	C₂²⋊Q₈	—	178	4	no	—
132	C₄×Q₈	—	178	4	no	—
133	C₄×Q₈	—	178	4	no	—
134	#117	—	178	4	no	—
135	C₂×C₄²	—	89	1	no	—
136	#107	—	2	4	no	—
137	#107	—	178	8	no	—
138	#107	—	178	4	no	—
139	#106	—	178	8	no	—
140	#103	—	2	8	no	—
141	#103	—	178	8	no	—
142	C₄×C₈	—	89	4	no	—
143	C₄×C₈	—	89	2	no	—
144	C₄×C₈	—	89	2	no	—
145	C₄×C₈	—	89	1	no	—
146	C₄×C₈	C₂²×D₈₉	1	2	yes	—
147	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	178	4	no	—
150	C₂²×C₄	—	89	1	no	—
151	C₂²×C₄	—	89	2	no	—
152	Q₁₆	—	356	16	no	—
153	C₂×C₈	—	89	2	no	—
154	C₂×C₈	—	89	4	no	—
155	C₂×C₈	—	89	8	no	—
156	C₂×C₈	—	178	8	no	—
157	C₂×C₈	D₄×D₈₉	1	4	yes	—
158	C₂×C₈	D₄×D₈₉	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	356	4	no	—
162	C₄⋊C₄	—	356	4	no	—
163	C₄⋊C₄	—	178	4	no	—
164	C₄⋊C₄	—	178	4	no	—
165	C₂²⋊C₄	—	356	4	no	—
166	C₄²	—	89	2	no	—
167	C₄²	—	89	1	no	—
168	C₄²	—	356	4	no	—
169	C₄²	C₂³×D₈₉	1	1	yes	characteristic
170	C₂³	—	89	1	no	—
171	Q₈	—	4	8	no	—
172	Q₈	—	356	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	89	1	no	—
175	C₂×C₄	—	89	4	no	—
176	C₂×C₄	—	89	2	no	—
177	C₂×C₄	—	356	4	no	—
178	C₂×C₄	—	178	4	no	—
179	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
180	C₂×C₄	C₂×D₄×D₈₉	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	89	4	no	—
183	C₈	—	89	8	no	—
184	C₈	Q₁₆×D₈₉	1	4	yes	—
185	C₂²	—	89	2	no	—
186	C₂²	—	178	2	no	—
187	C₂²	#127×D₈₉	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	89	2	no	—
190	C₄	—	178	2	no	—
191	C₄	—	712	4	no	—
192	C₄	C₂×Q₁₆×D₈₉	1	2	yes	—
193	C₄	#135×D₈₉	1	4	yes	—
194	C₂	—	178	2	no	—
195	C₂	#494×D₈₉	1	1	yes	characteristic
196	C₂	#26030677	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26030837

C₂

yes

maximal

#26030763

C₂

yes

maximal

#26019869

C₂

yes

maximal, complemented

#26019862

C₂

yes

maximal, characteristic, complemented

#26019848

C₂

yes

maximal, complemented

C₈²×D₈₉

C₂

yes

maximal, characteristic

C₈₉×#1710

C₂

yes

maximal, characteristic, complemented

#19152153

C₂²

yes

—

#19152003

C₂²

yes

—

#19152001

C₂²

yes

—

#19151999

—

#19151994

C₂²

yes

—

#19151977

C₂²

yes

—

#19151949

C₂²

yes

—

#19151838

—

#19151690

—

#19151635

—

C₄×C₈×D₈₉

C₂²

yes

characteristic

C₄×C₈×D₈₉

C₂²

yes

—

C₈×#10623494

C₂²

yes

—

C₈×#10623494

C₂²

yes

characteristic

#500×C₈₉

C₂²

yes

—

#487×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×#13083564

—

C₂×C₈×D₈₉

D₄

yes

—

C₂×C₈×D₈₉

D₄

yes

—

#15668012

C₂³

yes

—

#15668003

—

#15667992

—

#15667988

—

Q₈×Dic₈₉

—

#15667977

—

#15667970

—

C₄²×D₈₉

C₂³

yes

—

—

—

—

C₂³

yes

—

C₄×#10623494

C₂³

yes

characteristic

C₈×Dic₈₉

D₄

yes

—

C₈×Dic₈₉

D₄

yes

—

C₄⋊Q₈×C₈₉

C₂³

yes

—

#107×C₈₉

—

#103×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×C₄×D₈₉

C₂×D₄

yes

—

C₂×C₄×D₈₉

C₂×D₄

yes

characteristic

#13083564

—

C₂×#10623494

C₂×D₄

yes

—

C₂×#10623494

C₂×D₄

yes

—

C₈×D₈₉

—

C₈×D₈₉

Q₁₆

yes

complemented

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083551

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂×D₄

yes

—

C₄×Dic₈₉

C₂×D₄

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

#127

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×D₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#127

yes

—

C₂×Dic₈₉

#127

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

C₂×Q₁₆

yes

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₁₆

yes

—

#10623494

C₂×Q₁₆

yes

—

#10623494

#135

yes

—

C₂²×C₈₉

C₂×#127

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

#494

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂²×Q₁₆

yes

—

C₄×C₈₉

C₂×#135

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

#494

yes

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

#3170

—

100

#3160

—

101

#1713

—

102

#1710

—

103

#1710

D₈₉

yes

characteristic, complemented

104

#1707

—

105

C₂×C₈²

—

106

C₈₉

#19604

yes

characteristic, complemented, Sylow

107

#533

—

108

#500

—

109

#500

—

110

#500

C₂×D₈₉

yes

—

111

#499

—

178

—

112

#498

—

113

#494

—

114

#487

—

115

#487

C₂×D₈₉

yes

—

116

#467

—

178

—

117

C₄×Q₁₆

—

178

—

118

#427

—

178

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₈²

—

122

C₈²

—

123

C₈²

C₂×D₈₉

yes

characteristic

124

C₂×Q₁₆

—

178

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂²×D₈₉

yes

—

129

#125

—

178

—

130

C₂²⋊Q₈

—

178

—

131

C₂²⋊Q₈

—

178

—

132

C₄×Q₈

—

178

—

133

C₄×Q₈

—

178

—

134

#117

—

178

—

135

C₂×C₄²

—

136

#107

—

137

#107

—

178

—

138

#107

—

178

—

139

#106

—

178

—

140

#103

—

141

#103

—

178

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂²×D₈₉

yes

—

147

C₄×C₈

C₂²×D₈₉

yes

characteristic

148

C₂×Q₈

—

149

C₂×Q₈

—

178

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

Q₁₆

—

356

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

178

—

157

C₂×C₈

D₄×D₈₉

yes

—

158

C₂×C₈

D₄×D₈₉

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

356

—

162

C₄⋊C₄

—

356

—

163

C₄⋊C₄

—

178

—

164

C₄⋊C₄

—

178

—

165

C₂²⋊C₄

—

356

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

356

—

169

C₄²

C₂³×D₈₉

yes

characteristic

170

C₂³

—

171

Q₈

—

172

Q₈

—

356

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

356

—

178

C₂×C₄

—

178

—

179

C₂×C₄

C₂×D₄×D₈₉

yes

characteristic

180

C₂×C₄

C₂×D₄×D₈₉

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

Q₁₆×D₈₉

yes

—

185

C₂²

—

186

C₂²

—

178

—

187

C₂²

#127×D₈₉

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

178

—

191

C₄

—

712

—

192

C₄

C₂×Q₁₆×D₈₉

yes

—

193

C₄

#135×D₈₉

yes

—

194

C₂

—

178

—

195

C₂

#494×D₈₉

yes

characteristic

196

C₂

#26030677

yes

—

	subgroup	G/H
center	C₂²	#127×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1710	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	C₂²×D₈₉	#127
hypercenter	#1710	D₈₉
hypocenter	C₈₉	#19604
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1710

subgroup

G/H

center

C₂²

#127×D₈₉

commutator, cosocle

C₄²×C₈₉

C₂⁴

Frattini

C₄²

C₂³×D₈₉

Fitting

C₈₉×#1710

C₂

C₂²×C₈₉

C₂²×D₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #1710

N	H ≅ G/N	type
3. #26019869	C₂	semidirect
4. #26019862	C₂	semidirect
5. #26019848	C₂	semidirect
7. C₈₉×#1710	C₂	semidirect
56. C₈×D₈₉	Q₁₆	semidirect
103. #1710	D₈₉	semidirect
106. C₈₉	#19604	semidirect

H ≅ G/N

type

3. #26019869

C₂

semidirect

4. #26019862

C₂

semidirect

5. #26019848

C₂

semidirect

7. C₈₉×#1710

C₂

semidirect

56. C₈×D₈₉

Q₁₆

semidirect

103. #1710

D₈₉

semidirect

106. C₈₉

#19604

semidirect

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

subgroup	index
#26030837	2
#26030763	2
#26019869	2
#26019862	2
#26019848	2
C₈²×D₈₉	2
C₈₉×#1710	2
#19604	89

subgroup

index

C₈²×D₈₉

C₈₉×#1710

#19604

Subgroups — #36542135

Conjugacy classes of subgroups

Complements

Maximal subgroups