Group #36457537 — subgroups

Subgroups — #36457537

Order 22784.

402 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 89 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	#26028639	C₂	1	1	yes	maximal, characteristic, complemented
2	#26021080	C₂	1	1	yes	maximal, characteristic
3	#26018813	C₂	1	2	yes	maximal
4	#26018111	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1276	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152148	C₂²	1	1	yes	characteristic
7	#19151622	C₂²	1	1	yes	characteristic
8	C₂×#15667940	—	2	2	no	—
9	#19151496	—	2	2	no	—
10	#19151482	—	2	4	no	—
11	#19151464	C₂²	1	2	yes	—
12	#19151333	—	2	4	no	—
13	#431×C₈₉	C₂²	1	2	yes	—
14	C₄³×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083574	—	2	1	no	—
16	C₂×#13083558	—	2	1	no	—
17	#15668012	Q₈	1	1	yes	characteristic
18	#15668009	Q₈	1	1	yes	characteristic
19	#15668000	—	2	1	no	—
20	#15667986	—	2	2	no	—
21	C₄×D₃₅₆	—	2	2	no	—
22	C₂×C₄⋊Dic₈₉	—	2	2	no	—
23	C₂×#13083553	—	2	4	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	#15667940	—	4	4	no	—
26	#15667919	—	2	4	no	—
27	#15667919	—	2	2	no	—
28	#15667919	D₄	1	4	yes	—
29	C₂×C₄²×C₈₉	—	2	2	no	—
30	C₂×C₄²×C₈₉	D₄	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₈₉	—	2	2	no	—
33	C₂²⋊C₈×C₈₉	—	2	2	no	—
34	C₄×C₈×C₈₉	—	2	2	no	—
35	#13083574	—	4	2	no	—
36	#13083574	—	4	2	no	—
37	C₂×Dic₁₇₈	—	2	1	no	—
38	C₂×D₃₅₆	—	2	1	no	—
39	C₂×D₃₅₆	—	2	2	no	—
40	C₂×C₂²⋊D₈₉	—	4	1	no	—
41	C₂²×Dic₈₉	—	2	4	no	—
42	C₂×C₄×D₈₉	—	4	1	no	—
43	#13083558	—	4	2	no	—
44	C₂×#10623494	—	4	1	no	—
45	C₄⋊Dic₈₉	—	2	2	no	—
46	C₄⋊Dic₈₉	—	4	2	no	—
47	C₄⋊Dic₈₉	—	4	2	no	—
48	#13083553	—	4	8	no	—
49	#13083551	—	4	2	no	—
50	C₂²×C₄×C₈₉	—	2	2	no	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	—	4	2	no	—
54	C₄²×C₈₉	—	2	2	no	—
55	C₄²×C₈₉	—	2	2	no	—
56	C₄²×C₈₉	—	4	4	no	—
57	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₈₉	#41	1	1	yes	characteristic
59	C₄²×C₈₉	#41	1	2	yes	—
60	C₂³×C₈₉	#120	1	1	yes	characteristic
61	C₂²×D₈₉	—	4	1	no	—
62	C₂×C₄×C₈₉	—	2	4	no	—
63	C₂×C₄×C₈₉	—	4	2	no	—
64	C₂×C₄×C₈₉	—	4	2	no	—
65	C₂×C₄×C₈₉	—	4	2	no	—
66	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	#124	1	2	yes	—
69	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
70	C₂×C₄×C₈₉	#135	1	2	yes	—
71	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
72	C₂²⋊D₈₉	—	8	2	no	—
73	C₂×Dic₈₉	—	4	8	no	—
74	C₂×Dic₈₉	—	4	1	no	—
75	C₂×Dic₈₉	—	4	4	no	—
76	D₃₅₆	—	4	2	no	—
77	D₃₅₆	—	8	2	no	—
78	C₄×D₈₉	—	8	1	no	—
79	C₄×D₈₉	—	8	1	no	—
80	Dic₁₇₈	—	4	2	no	—
81	C₈×C₈₉	—	8	2	no	—
82	#10623494	—	8	1	no	—
83	C₂²×C₈₉	—	2	2	no	—
84	C₂²×C₈₉	#393	1	1	yes	characteristic
85	C₂²×C₈₉	#449	1	2	yes	—
86	C₂×D₈₉	—	8	1	no	—
87	C₂×D₈₉	—	8	2	no	—
88	C₄×C₈₉	—	2	2	no	—
89	C₄×C₈₉	—	2	2	no	—
90	C₄×C₈₉	—	4	4	no	—
91	C₄×C₈₉	#478	1	2	yes	—
92	C₄×C₈₉	#488	1	1	yes	characteristic
93	C₄×C₈₉	#490	1	1	yes	characteristic
94	Dic₈₉	—	8	4	no	—
95	Dic₈₉	—	8	1	no	—
96	#10366	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	—	2	2	no	—
98	C₂×C₈₉	#2025	1	1	yes	characteristic
99	C₂×C₈₉	#2031	1	2	yes	—
100	D₈₉	—	16	1	no	—
101	#2864	—	89	1	no	—
102	#1962	—	89	1	no	—
103	#1517	—	89	2	no	—
104	#1286	—	89	2	no	—
105	#1276	D₈₉	1	1	yes	characteristic, complemented
106	C₈₉	#10366	1	1	yes	characteristic, complemented, Sylow
107	#532	—	89	1	no	—
108	#431	C₂×D₈₉	1	2	yes	—
109	#423	—	89	1	no	—
110	C₂×#104	—	178	2	no	—
111	#389	—	178	2	no	—
112	#387	—	178	4	no	—
113	#382	—	89	2	no	—
114	C₄³	C₂×D₈₉	1	1	yes	characteristic
115	#327	—	178	4	no	—
116	C₂×#41	—	178	1	no	—
117	C₂×M₁₆	—	178	1	no	—
118	C₄⋊Q₈	—	89	1	no	—
119	#127	—	89	1	no	—
120	#124	—	178	1	no	—
121	#121	—	178	2	no	—
122	C₄×D₄	—	178	2	no	—
123	C₂×C₄⋊C₄	—	178	4	no	—
124	C₂×C₄⋊C₄	—	178	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	D₃₅₆	1	2	yes	—
127	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	178	2	no	—
130	#104	—	356	4	no	—
131	C₂²⋊C₈	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	#95	—	89	4	no	—
134	#95	—	178	4	no	—
135	#95	—	178	2	no	—
136	#41	—	356	2	no	—
137	#41	—	356	2	no	—
138	C₂×Q₈	—	178	1	no	—
139	C₂×D₄	—	356	1	no	—
140	C₂×D₄	—	178	2	no	—
141	C₂×D₄	—	178	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	356	1	no	—
144	C₂²×C₄	—	178	4	no	—
145	C₂²×C₄	D₄×D₈₉	1	2	yes	—
146	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
147	M₁₆	—	356	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	356	1	no	—
150	C₄⋊C₄	—	356	8	no	—
151	C₄⋊C₄	—	356	2	no	—
152	C₄⋊C₄	—	356	2	no	—
153	C₄⋊C₄	—	178	2	no	—
154	C₂²⋊C₄	—	356	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	4	no	—
158	C₄²	Q₈×D₈₉	1	1	yes	characteristic
159	C₄²	#13083573	1	1	yes	characteristic
160	C₄²	#13083574	1	2	yes	—
161	C₂³	—	356	1	no	—
162	C₂³	#15667982	1	1	yes	characteristic
163	Q₈	—	356	2	no	—
164	D₄	—	356	2	no	—
165	D₄	—	712	2	no	—
166	D₄	—	712	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	356	8	no	—
172	C₂×C₄	—	356	4	no	—
173	C₂×C₄	—	356	1	no	—
174	C₂×C₄	—	712	1	no	—
175	C₂×C₄	—	712	1	no	—
176	C₂×C₄	#15667990	1	2	yes	—
177	C₂×C₄	#15667992	1	1	yes	characteristic
178	C₂×C₄	#15667998	1	2	yes	—
179	C₂×C₄	#15668000	1	1	yes	characteristic
180	C₂×C₄	#15668034	1	2	yes	—
181	C₂×C₄	#15668040	1	2	yes	—
182	C₈	—	8	2	no	—
183	C₈	—	712	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	712	1	no	—
186	C₂²	—	712	2	no	—
187	C₂²	#19151511	1	1	yes	characteristic
188	C₂²	#19151745	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	712	4	no	—
193	C₄	—	712	1	no	—
194	C₄	#19151899	1	2	yes	—
195	C₄	#19151953	1	1	yes	characteristic
196	C₄	#19151963	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	—	1424	1	no	—
199	C₂	#26021399	1	1	yes	characteristic
200	C₂	#26021425	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26028639

C₂

yes

maximal, characteristic, complemented

#26021080

C₂

yes

maximal, characteristic

#26018813

C₂

yes

maximal

#26018111

C₂

yes

maximal, complemented

C₈₉×#1276

C₂

yes

maximal, characteristic, complemented

#19152148

C₂²

yes

characteristic

#19151622

C₂²

yes

—

—

—

C₂²

yes

—

#19151333

—

#431×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

—

—

Q₈

yes

characteristic

#15668009

Q₈

yes

characteristic

#15668000

—

#15667986

—

C₄×D₃₅₆

—

C₂×C₄⋊Dic₈₉

—

—

—

—

—

—

D₄

yes

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

#13083574

—

#13083574

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

—

C₂×C₄×D₈₉

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083551

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

C₄²×C₈₉

yes

C₄²×C₈₉

yes

—

C₂³×C₈₉

#120

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#393

yes

characteristic

C₂²×C₈₉

#449

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#478

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

#10366

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2025

yes

characteristic

C₂×C₈₉

#2031

yes

—

100

D₈₉

—

101

#2864

—

102

#1962

—

103

#1517

—

104

#1286

—

105

#1276

D₈₉

yes

characteristic, complemented

106

C₈₉

#10366

yes

characteristic, complemented, Sylow

107

#532

—

108

#431

C₂×D₈₉

yes

—

109

#423

—

110

C₂×#104

—

178

—

111

#389

—

178

—

112

#387

—

178

—

113

#382

—

114

C₄³

C₂×D₈₉

yes

characteristic

115

#327

—

178

—

116

C₂×#41

—

178

—

117

C₂×M₁₆

—

178

—

118

C₄⋊Q₈

—

119

#127

—

120

#124

—

178

—

121

#121

—

178

—

122

C₄×D₄

—

178

—

123

C₂×C₄⋊C₄

—

178

—

124

C₂×C₄⋊C₄

—

178

—

125

C₂×C₄²

—

126

C₂×C₄²

D₃₅₆

yes

—

127

C₂×C₄²

C₂²×D₈₉

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

—

178

—

130

#104

—

356

—

131

C₂²⋊C₈

—

132

C₄×C₈

—

133

#95

—

134

#95

—

178

—

135

#95

—

178

—

136

#41

—

356

—

137

#41

—

356

—

138

C₂×Q₈

—

178

—

139

C₂×D₄

—

356

—

140

C₂×D₄

—

178

—

141

C₂×D₄

—

178

—

142

C₂²×C₄

—

143

C₂²×C₄

—

356

—

144

C₂²×C₄

—

178

—

145

C₂²×C₄

D₄×D₈₉

yes

—

146

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

147

M₁₆

—

356

—

148

C₂×C₈

—

149

C₂×C₈

—

356

—

150

C₄⋊C₄

—

356

—

151

C₄⋊C₄

—

356

—

152

C₄⋊C₄

—

356

—

153

C₄⋊C₄

—

178

—

154

C₂²⋊C₄

—

356

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

Q₈×D₈₉

yes

characteristic

159

C₄²

#13083573

yes

characteristic

160

C₄²

#13083574

yes

—

161

C₂³

—

356

—

162

C₂³

#15667982

yes

characteristic

163

Q₈

—

356

—

164

D₄

—

356

—

165

D₄

—

712

—

166

D₄

—

712

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

356

—

172

C₂×C₄

—

356

—

173

C₂×C₄

—

356

—

174

C₂×C₄

—

712

—

175

C₂×C₄

—

712

—

176

C₂×C₄

#15667990

yes

—

177

C₂×C₄

#15667992

yes

characteristic

178

C₂×C₄

#15667998

yes

—

179

C₂×C₄

#15668000

yes

characteristic

180

C₂×C₄

#15668034

yes

—

181

C₂×C₄

#15668040

yes

—

182

C₈

—

183

C₈

—

712

—

184

C₂²

—

185

C₂²

—

712

—

186

C₂²

—

712

—

187

C₂²

#19151511

yes

characteristic

188

C₂²

#19151745

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

712

—

193

C₄

—

712

—

194

C₄

#19151899

yes

—

195

C₄

#19151953

yes

characteristic

196

C₄

#19151963

yes

characteristic

197

C₂

—

198

C₂

—

1424

—

199

C₂

#26021399

yes

characteristic

200

C₂

#26021425

yes

—

	subgroup	G/H
center	C₂²	#19151511
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1276	C₂
socle	C₂²×C₈₉	#393
prime radical	#19152148	C₂²
hypercenter	#1276	D₈₉
hypocenter	C₈₉	#10366
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#19151511

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1276

C₂

C₂²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
1. #26028639	C₂	semidirect
4. #26018111	C₂	semidirect
5. C₈₉×#1276	C₂	semidirect
105. #1276	D₈₉	semidirect
106. C₈₉	#10366	semidirect

H ≅ G/N

type

1. #26028639

C₂

semidirect

4. #26018111

C₂

semidirect

5. C₈₉×#1276

C₂

semidirect

105. #1276

D₈₉

semidirect

106. C₈₉

#10366

semidirect

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

subgroup	index
#26028639	2
#26021080	2
#26018813	2
#26018111	2
C₈₉×#1276	2
#10366	89

subgroup

index

C₈₉×#1276

Subgroups — #36457537

Conjugacy classes of subgroups

Complements

Maximal subgroups