Group #36469149 — subgroups

Subgroups — #36469149

Order 22784.

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

normal subgroups only

This group has 330 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	#26022509	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022507	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022504	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022501	C₂	1	1	yes	maximal, characteristic
5	#26019858	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1709	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15667958	—	2	2	no	—
9	C₂×#15667956	—	2	2	no	—
10	C₂×#15667955	—	2	2	no	—
11	C₂×#15667953	—	2	2	no	—
12	#19151978	C₂²	1	1	yes	characteristic
13	#19151975	C₂²	1	1	yes	characteristic
14	#19151945	—	2	2	no	—
15	C₄×#13083549	C₂²	1	1	yes	characteristic
16	C₄×#13083549	C₂²	1	1	yes	characteristic
17	#494×C₈₉	C₂²	1	1	yes	characteristic
18	#493×C₈₉	C₂²	1	1	yes	characteristic
19	#486×C₈₉	—	2	2	no	—
20	C₈²×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₅₆	—	2	1	no	—
22	C₂×Dic₃₅₆	—	2	1	no	—
23	C₂×D₇₁₂	—	2	1	no	—
24	C₂×D₇₁₂	—	2	1	no	—
25	#15668012	—	2	1	no	—
26	#15668009	—	2	1	no	—
27	#15667958	—	4	4	no	—
28	#15667956	—	4	4	no	—
29	#15667955	—	4	4	no	—
30	#15667953	—	4	4	no	—
31	C₂×#13083549	D₄	1	2	yes	—
32	C₂×#13083549	D₄	1	2	yes	—
33	#15667934	—	4	4	no	—
34	C₂×Q₁₆×C₈₉	—	2	1	no	—
35	C₂×Q₁₆×C₈₉	—	2	1	no	—
36	C₂×D₈×C₈₉	—	2	1	no	—
37	C₂×D₈×C₈₉	—	2	1	no	—
38	C₄⋊Q₈×C₈₉	—	2	1	no	—
39	#127×C₈₉	—	2	1	no	—
40	#102×C₈₉	—	4	4	no	—
41	C₄×C₈×C₈₉	D₄	1	2	yes	—
42	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×Dic₁₇₈	—	4	1	no	—
44	C₂×D₃₅₆	—	4	1	no	—
45	C₂×D₃₅₆	—	4	2	no	—
46	Dic₃₅₆	—	4	2	no	—
47	Dic₃₅₆	—	4	2	no	—
48	D₇₁₂	—	4	2	no	—
49	D₇₁₂	—	4	2	no	—
50	C₄⋊Dic₈₉	—	4	2	no	—
51	#13083549	D₈	1	4	yes	—
52	#13083549	D₈	1	4	yes	—
53	C₂×Q₈×C₈₉	—	4	1	no	—
54	C₂×D₄×C₈₉	—	4	1	no	—
55	C₂×D₄×C₈₉	—	4	2	no	—
56	Q₁₆×C₈₉	—	4	2	no	—
57	Q₁₆×C₈₉	—	4	2	no	—
58	D₈×C₈₉	—	4	2	no	—
59	D₈×C₈₉	—	4	2	no	—
60	C₂×C₈×C₈₉	D₈	1	4	yes	—
61	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₈₉	—	4	2	no	—
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂³×C₈₉	—	8	1	no	—
66	C₂²×D₈₉	—	8	1	no	—
67	Q₈×C₈₉	—	8	1	no	—
68	Q₈×C₈₉	—	8	1	no	—
69	D₄×C₈₉	—	8	1	no	—
70	D₄×C₈₉	—	8	1	no	—
71	D₄×C₈₉	—	8	4	no	—
72	C₂×C₄×C₈₉	—	8	1	no	—
73	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
75	C₂×Dic₈₉	—	8	1	no	—
76	D₃₅₆	—	8	1	no	—
77	D₃₅₆	—	8	1	no	—
78	D₃₅₆	—	8	4	no	—
79	Dic₁₇₈	—	8	1	no	—
80	Dic₁₇₈	—	8	1	no	—
81	C₈×C₈₉	—	2	4	no	—
82	C₈×C₈₉	C₂×D₈	1	2	yes	—
83	C₈×C₈₉	C₂×D₈	1	2	yes	—
84	C₂²×C₈₉	—	16	1	no	—
85	C₂²×C₈₉	—	16	2	no	—
86	C₂²×C₈₉	#493	1	1	yes	characteristic
87	C₂×D₈₉	—	16	1	no	—
88	C₂×D₈₉	—	16	2	no	—
89	C₄×C₈₉	—	16	1	no	—
90	C₄×C₈₉	#493	1	1	yes	characteristic
91	C₄×C₈₉	#493	1	1	yes	characteristic
92	C₄×C₈₉	#508	1	4	yes	—
93	Dic₈₉	—	16	1	no	—
94	#12518	—	89	1	no	maximal, Sylow
95	C₂×C₈₉	—	32	1	no	—
96	C₂×C₈₉	#1712	1	1	yes	characteristic
97	C₂×C₈₉	#2249	1	2	yes	—
98	D₈₉	—	32	1	no	—
99	#2248	—	89	1	no	—
100	#2248	—	89	1	no	—
101	#2247	—	89	1	no	—
102	#2246	—	89	1	no	—
103	#1709	—	89	1	no	—
104	#1709	D₈₉	1	1	yes	characteristic, complemented
105	C₈×C₁₆	—	89	1	no	—
106	C₈₉	#12518	1	1	yes	characteristic, complemented, Sylow
107	C₂×Q₃₂	—	178	2	no	—
108	C₂×SD₃₂	—	178	2	no	—
109	C₂×SD₃₂	—	178	2	no	—
110	C₂×D₁₆	—	178	2	no	—
111	#494	—	89	1	no	—
112	#494	C₂×D₈₉	1	1	yes	characteristic
113	#493	—	89	1	no	—
114	#493	C₂×D₈₉	1	1	yes	characteristic
115	#486	—	2	2	no	—
116	#486	—	178	2	no	—
117	C₄×C₁₆	—	89	1	no	—
118	C₄×C₁₆	—	89	1	no	—
119	C₈²	C₂×D₈₉	1	1	yes	characteristic
120	C₂×Q₁₆	—	2	1	no	—
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×Q₁₆	—	178	1	no	—
123	C₂×Q₁₆	—	178	1	no	—
124	C₂×D₈	—	2	1	no	—
125	C₂×D₈	—	2	1	no	—
126	C₂×D₈	—	178	1	no	—
127	C₂×D₈	—	178	1	no	—
128	C₄⋊Q₈	—	2	1	no	—
129	C₄⋊Q₈	—	178	1	no	—
130	#127	—	2	1	no	—
131	#127	—	178	1	no	—
132	Q₃₂	—	356	4	no	—
133	SD₃₂	—	356	4	no	—
134	SD₃₂	—	356	4	no	—
135	D₁₆	—	356	4	no	—
136	C₂×C₁₆	—	89	2	no	—
137	C₂×C₁₆	—	89	2	no	—
138	#102	—	4	4	no	—
139	#102	—	356	4	no	—
140	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
141	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	356	1	no	—
144	C₂×D₄	—	4	1	no	—
145	C₂×D₄	—	4	2	no	—
146	C₂×D₄	—	356	1	no	—
147	C₂×D₄	—	356	2	no	—
148	Q₁₆	—	4	2	no	—
149	Q₁₆	—	4	2	no	—
150	Q₁₆	—	356	2	no	—
151	Q₁₆	—	356	2	no	—
152	D₈	—	4	2	no	—
153	D₈	—	4	2	no	—
154	D₈	—	356	2	no	—
155	D₈	—	356	2	no	—
156	C₂×C₈	#13083559	1	4	yes	—
157	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
158	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	356	2	no	—
161	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
162	C₁₆	—	89	4	no	—
163	C₁₆	—	89	4	no	—
164	C₂³	—	8	1	no	—
165	C₂³	—	712	1	no	—
166	Q₈	—	8	1	no	—
167	Q₈	—	8	1	no	—
168	Q₈	—	712	1	no	—
169	Q₈	—	712	1	no	—
170	D₄	—	8	1	no	—
171	D₄	—	8	1	no	—
172	D₄	—	8	4	no	—
173	D₄	—	712	1	no	—
174	D₄	—	712	1	no	—
175	D₄	—	712	4	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	712	1	no	—
178	C₂×C₄	#15668008	1	1	yes	characteristic
179	C₂×C₄	C₂×#13083559	1	2	yes	—
180	C₈	—	2	4	no	—
181	C₈	D₈×D₈₉	1	2	yes	—
182	C₈	D₈×D₈₉	1	2	yes	—
183	C₂²	—	16	1	no	—
184	C₂²	—	16	2	no	—
185	C₂²	—	1424	1	no	—
186	C₂²	—	1424	2	no	—
187	C₂²	#19151974	1	1	yes	characteristic
188	C₄	—	16	1	no	—
189	C₄	—	1424	1	no	—
190	C₄	#19151973	1	1	yes	characteristic
191	C₄	#19151973	1	1	yes	characteristic
192	C₄	#19152028	1	4	yes	—
193	C₂	—	32	1	no	—
194	C₂	—	2848	1	no	—
195	C₂	#26019866	1	1	yes	characteristic
196	C₂	#26022511	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022509

C₂

yes

maximal, characteristic, complemented

#26022507

C₂

yes

maximal, characteristic, complemented

#26022504

C₂

yes

maximal, characteristic, complemented

#26022501

C₂

yes

maximal, characteristic

#26019858

C₂

yes

maximal, characteristic, complemented

C₈×#13083549

C₂

yes

maximal, characteristic, complemented

C₈₉×#1709

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

#19151975

C₂²

yes

characteristic

#19151945

—

C₄×#13083549

C₂²

yes

characteristic

C₄×#13083549

C₂²

yes

characteristic

#494×C₈₉

C₂²

yes

characteristic

#493×C₈₉

C₂²

yes

characteristic

#486×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

—

—

—

—

—

—

D₄

yes

—

C₂×#13083549

D₄

yes

—

#15667934

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

#127×C₈₉

—

#102×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

#13083549

D₈

yes

—

#13083549

D₈

yes

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

C₂×D₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#12518

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1712

yes

characteristic

C₂×C₈₉

#2249

yes

—

D₈₉

—

#2248

—

100

#2248

—

101

#2247

—

102

#2246

—

103

#1709

—

104

#1709

D₈₉

yes

characteristic, complemented

105

C₈×C₁₆

—

106

C₈₉

#12518

yes

characteristic, complemented, Sylow

107

C₂×Q₃₂

—

178

—

108

C₂×SD₃₂

—

178

—

109

C₂×SD₃₂

—

178

—

110

C₂×D₁₆

—

178

—

111

#494

—

112

#494

C₂×D₈₉

yes

characteristic

113

#493

—

114

#493

C₂×D₈₉

yes

characteristic

115

#486

—

116

#486

—

178

—

117

C₄×C₁₆

—

118

C₄×C₁₆

—

119

C₈²

C₂×D₈₉

yes

characteristic

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

178

—

123

C₂×Q₁₆

—

178

—

124

C₂×D₈

—

125

C₂×D₈

—

126

C₂×D₈

—

178

—

127

C₂×D₈

—

178

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

178

—

130

#127

—

131

#127

—

178

—

132

Q₃₂

—

356

—

133

SD₃₂

—

356

—

134

SD₃₂

—

356

—

135

D₁₆

—

356

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

#102

—

139

#102

—

356

—

140

C₄×C₈

C₂²⋊D₈₉

yes

—

141

C₄×C₈

C₂²×D₈₉

yes

characteristic

142

C₂×Q₈

—

143

C₂×Q₈

—

356

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

356

—

147

C₂×D₄

—

356

—

148

Q₁₆

—

149

Q₁₆

—

150

Q₁₆

—

356

—

151

Q₁₆

—

356

—

152

D₈

—

153

D₈

—

154

D₈

—

356

—

155

D₈

—

356

—

156

C₂×C₈

#13083559

yes

—

157

C₂×C₈

D₄×D₈₉

yes

characteristic

158

C₂×C₈

D₄×D₈₉

yes

characteristic

159

C₄⋊C₄

—

160

C₄⋊C₄

—

356

—

161

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

162

C₁₆

—

163

C₁₆

—

164

C₂³

—

165

C₂³

—

712

—

166

Q₈

—

167

Q₈

—

168

Q₈

—

712

—

169

Q₈

—

712

—

170

D₄

—

171

D₄

—

172

D₄

—

173

D₄

—

712

—

174

D₄

—

712

—

175

D₄

—

712

—

176

C₂×C₄

—

177

C₂×C₄

—

712

—

178

C₂×C₄

#15668008

yes

characteristic

179

C₂×C₄

C₂×#13083559

yes

—

180

C₈

—

181

C₈

D₈×D₈₉

yes

—

182

C₈

D₈×D₈₉

yes

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

1424

—

186

C₂²

—

1424

—

187

C₂²

#19151974

yes

characteristic

188

C₄

—

189

C₄

—

1424

—

190

C₄

#19151973

yes

characteristic

191

C₄

#19151973

yes

characteristic

192

C₄

#19152028

yes

—

193

C₂

—

194

C₂

—

2848

—

195

C₂

#26019866

yes

characteristic

196

C₂

#26022511

yes

—

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1709	C₂
socle	C₂²×C₈₉	#493
prime radical	#26022501	C₂
hypercenter	#1709	D₈₉
hypocenter	C₈₉	#12518
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1709

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #26022509	C₂	semidirect
2. #26022507	C₂	semidirect
3. #26022504	C₂	semidirect
5. #26019858	C₂	semidirect
6. C₈×#13083549	C₂	semidirect
7. C₈₉×#1709	C₂	semidirect
104. #1709	D₈₉	semidirect
106. C₈₉	#12518	semidirect

H ≅ G/N

type

1. #26022509

C₂

semidirect

2. #26022507

C₂

semidirect

3. #26022504

C₂

semidirect

5. #26019858

C₂

semidirect

6. C₈×#13083549

C₂

semidirect

7. C₈₉×#1709

C₂

semidirect

104. #1709

D₈₉

semidirect

106. C₈₉

#12518

semidirect

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

subgroup	index
#26022509	2
#26022507	2
#26022504	2
#26022501	2
#26019858	2
C₈×#13083549	2
C₈₉×#1709	2
#12518	89

subgroup

index

C₈₉×#1709

Subgroups — #36469149

Conjugacy classes of subgroups

Complements

Maximal subgroups