Group #36466305 — subgroups

Subgroups — #36466305

Order 22784.

214 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 77 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	#26022121	C₂	1	1	yes	maximal, characteristic
2	#26019071	C₂	1	1	yes	maximal, characteristic
3	#26019070	C₂	1	1	yes	maximal, characteristic
4	#26018317	C₂	1	1	yes	maximal, characteristic
5	#26018223	C₂	1	1	yes	maximal, characteristic
6	#26018174	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1334	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667958	C₄	1	1	yes	characteristic
9	Q₈×#10623494	—	2	1	no	—
10	#19151722	—	2	1	no	—
11	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
12	#19151425	C₄	1	1	yes	characteristic
13	#19151402	C₄	1	1	yes	characteristic
14	#19151401	C₄	1	1	yes	characteristic
15	#19151374	C₂²	1	1	yes	characteristic
16	#19151349	C₂²	1	1	yes	characteristic
17	Q₈⋊#10623494	—	2	1	no	—
18	#19151328	—	2	1	no	—
19	#19151319	C₂²	1	1	yes	characteristic
20	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	#346×C₈₉	C₂²	1	1	yes	characteristic
22	#334×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₁₇₈	—	2	1	no	—
25	#15667958	C₈	1	4	yes	complemented
26	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
27	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
28	#15667946	C₂×C₄	1	1	yes	characteristic
29	C₄⋊#10623494	—	2	1	no	—
30	C₄⋊#10623494	—	4	1	no	—
31	#15667941	—	4	1	no	—
32	#15667937	—	2	1	no	—
33	C₄×#10623494	—	4	1	no	—
34	C₄×#10623494	D₄	1	1	yes	characteristic
35	C₄×#10623494	D₄	1	1	yes	characteristic
36	C₈×Dic₈₉	—	4	1	no	—
37	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₈₉	—	2	1	no	—
39	C₂×C₁₆×C₈₉	—	2	1	no	—
40	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₈₉	—	2	1	no	—
42	#103×C₈₉	—	2	1	no	—
43	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₇₈	—	2	1	no	—
45	Dic₃₅₆	—	2	1	no	—
46	Dic₃₅₆	C₂×C₈	1	2	yes	—
47	C₂×#10623494	—	2	1	no	—
48	C₂×#10623494	—	2	1	no	—
49	C₂×#10623494	—	4	1	no	—
50	C₄⋊Dic₈₉	—	2	1	no	—
51	#13083553	—	4	1	no	—
52	C₄×Dic₈₉	—	4	1	no	—
53	#13083549	—	2	1	no	—
54	#13083549	C₂×C₈	1	2	yes	—
55	C₂×Q₈×C₈₉	—	2	1	no	—
56	Q₁₆×C₈₉	—	2	1	no	—
57	Q₁₆×C₈₉	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₈₉	—	4	1	no	—
59	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₈₉	—	2	1	no	—
62	C₄⋊C₄×C₈₉	—	4	1	no	—
63	C₄²×C₈₉	—	4	1	no	—
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₁₆×C₈₉	—	4	1	no	—
66	Q₈×C₈₉	—	2	2	no	—
67	Q₈×C₈₉	—	4	1	no	—
68	C₂×C₄×C₈₉	—	4	1	no	—
69	C₂×C₄×C₈₉	—	4	1	no	—
70	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×Dic₈₉	—	4	1	no	—
74	C₂×Dic₈₉	—	4	1	no	—
75	Dic₁₇₈	—	2	2	no	—
76	Dic₁₇₈	—	4	1	no	—
77	C₈×C₈₉	—	2	1	no	—
78	C₈×C₈₉	—	8	1	no	—
79	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
80	C₈×C₈₉	#131	1	1	yes	characteristic
81	#10623494	—	4	1	no	—
82	#10623494	—	4	1	no	—
83	#10623494	—	8	1	no	—
84	C₂²×C₈₉	#442	1	1	yes	characteristic
85	C₄×C₈₉	—	2	1	no	—
86	C₄×C₈₉	—	2	1	no	—
87	C₄×C₈₉	—	4	2	no	—
88	C₄×C₈₉	—	8	1	no	—
89	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
90	C₄×C₈₉	#444	1	1	yes	characteristic
91	Dic₈₉	—	4	2	no	—
92	Dic₈₉	—	8	1	no	—
93	#12066	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	#1580	1	1	yes	characteristic
95	C₂×C₈₉	#2180	1	1	yes	characteristic
96	C₂×C₈₉	#2180	1	1	yes	characteristic
97	#2176	—	89	1	no	—
98	#1579	—	89	1	no	—
99	#1579	—	89	1	no	—
100	#1368	—	89	1	no	—
101	#1334	—	89	1	no	—
102	#1334	D₈₉	1	1	yes	characteristic
103	#1313	—	89	1	no	—
104	C₈₉	#12066	1	1	yes	characteristic, complemented, Sylow
105	C₂×Q₃₂	—	89	1	no	—
106	C₈×Q₈	—	178	1	no	—
107	C₈×Q₈	—	178	1	no	—
108	C₄×Q₁₆	—	89	1	no	—
109	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
110	#367	—	89	1	no	—
111	#358	—	89	1	no	—
112	#358	—	89	1	no	—
113	#346	—	89	1	no	—
114	#346	C₂×D₈₉	1	1	yes	characteristic
115	#334	—	89	1	no	—
116	#334	C₂×D₈₉	1	1	yes	characteristic
117	Q₈⋊C₈	—	178	1	no	—
118	Q₈⋊C₈	—	178	1	no	—
119	#322	—	89	1	no	—
120	C₂×Q₁₆	—	89	1	no	—
121	C₂×Q₁₆	C₄×D₈₉	1	1	yes	characteristic
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	—	178	1	no	—
124	Q₃₂	—	89	4	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	89	1	no	—
127	C₂×C₁₆	—	89	1	no	—
128	#107	—	89	1	no	—
129	#107	C₄×D₈₉	1	1	yes	characteristic
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	—	356	1	no	—
132	C₄⋊C₈	—	356	1	no	—
133	C₄⋊C₈	—	178	1	no	—
134	#103	—	2	1	no	—
135	#103	—	178	1	no	—
136	C₄×C₈	—	89	1	no	—
137	C₄×C₈	—	89	1	no	—
138	C₄×C₈	—	356	1	no	—
139	C₄×C₈	—	356	1	no	—
140	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	178	1	no	—
143	Q₁₆	—	2	1	no	—
144	Q₁₆	—	89	2	no	—
145	Q₁₆	—	178	1	no	—
146	Q₁₆	C₈×D₈₉	1	2	yes	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	356	1	no	—
149	C₂×C₈	—	178	1	no	—
150	C₂×C₈	—	178	1	no	—
151	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
152	C₂×C₈	#13083573	1	1	yes	characteristic
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	356	1	no	—
156	C₄⋊C₄	—	178	1	no	—
157	C₄²	—	4	1	no	—
158	C₄²	—	356	1	no	—
159	C₄²	D₄×D₈₉	1	1	yes	characteristic
160	C₁₆	—	4	1	no	—
161	C₁₆	—	89	2	no	—
162	C₁₆	—	178	1	no	—
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	356	1	no	—
166	Q₈	—	178	2	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	356	1	no	—
170	C₂×C₄	—	356	1	no	—
171	C₂×C₄	#15667972	1	1	yes	characteristic
172	C₂×C₄	#15668036	1	1	yes	characteristic
173	C₂×C₄	#15668039	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	8	1	no	—
176	C₈	—	356	1	no	—
177	C₈	—	356	1	no	—
178	C₈	—	712	1	no	—
179	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
180	C₈	#15668020	1	1	yes	characteristic
181	C₂²	#19151707	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	8	1	no	—
186	C₄	—	356	2	no	—
187	C₄	—	712	1	no	—
188	C₄	#19151661	1	1	yes	characteristic
189	C₄	#19151717	1	1	yes	characteristic
190	C₂	#26019073	1	1	yes	characteristic
191	C₂	#26022140	1	1	yes	characteristic
192	C₂	#26022146	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022121

C₂

yes

maximal, characteristic

#26019071

C₂

yes

maximal, characteristic

#26019070

C₂

yes

maximal, characteristic

#26018317

C₂

yes

maximal, characteristic

#26018223

C₂

yes

maximal, characteristic

#26018174

C₂

yes

maximal, characteristic

C₈₉×#1334

C₂

yes

maximal, characteristic

C₂×#15667958

C₄

yes

characteristic

Q₈×#10623494

—

#19151722

—

C₄×Dic₃₅₆

C₂²

yes

characteristic

#19151425

C₄

yes

characteristic

#19151402

C₄

yes

characteristic

#19151401

C₄

yes

characteristic

#19151374

C₂²

yes

characteristic

#19151349

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×Q₁₆×C₈₉

C₂²

yes

characteristic

#346×C₈₉

C₂²

yes

characteristic

#334×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

C₂×C₄

yes

characteristic

C₄×Dic₁₇₈

—

#15667958

C₈

yes

complemented

C₂×#13083549

C₂×C₄

yes

characteristic

C₂×#13083549

C₂×C₄

yes

characteristic

#15667946

C₂×C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#10623494

D₄

yes

characteristic

C₈×Dic₈₉

—

C₂×Q₁₆×C₈₉

C₂×C₄

yes

characteristic

C₄×Q₈×C₈₉

—

C₂×C₁₆×C₈₉

—

#107×C₈₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₈₉

—

#103×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

C₂×C₈

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

#13083549

—

#13083549

C₂×C₈

yes

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

C₂×C₈

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂²×C₈

yes

C₈×C₈₉

yes

—

—

—

C₂²×C₈₉

#442

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₈×D₄

yes

characteristic

C₄×C₈₉

#444

yes

characteristic

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

100

#1368

—

101

#1334

—

102

#1334

D₈₉

yes

characteristic

103

#1313

—

104

C₈₉

#12066

yes

characteristic, complemented, Sylow

105

C₂×Q₃₂

—

106

C₈×Q₈

—

178

—

107

C₈×Q₈

—

178

—

108

C₄×Q₁₆

—

109

C₄×Q₁₆

C₂×D₈₉

yes

characteristic

110

#367

—

111

#358

—

112

#358

—

113

#346

—

114

#346

C₂×D₈₉

yes

characteristic

115

#334

—

116

#334

C₂×D₈₉

yes

characteristic

117

Q₈⋊C₈

—

178

—

118

Q₈⋊C₈

—

178

—

119

#322

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

C₄×D₈₉

yes

characteristic

122

C₄×Q₈

—

123

C₄×Q₈

—

178

—

124

Q₃₂

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

#107

—

129

#107

C₄×D₈₉

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

356

—

132

C₄⋊C₈

—

356

—

133

C₄⋊C₈

—

178

—

134

#103

—

135

#103

—

178

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

356

—

139

C₄×C₈

—

356

—

140

C₄×C₈

C₂²×D₈₉

yes

characteristic

141

C₂×Q₈

—

142

C₂×Q₈

—

178

—

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

178

—

146

Q₁₆

C₈×D₈₉

yes

—

147

C₂×C₈

—

148

C₂×C₈

—

356

—

149

C₂×C₈

—

178

—

150

C₂×C₈

—

178

—

151

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

152

C₂×C₈

#13083573

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

356

—

156

C₄⋊C₄

—

178

—

157

C₄²

—

158

C₄²

—

356

—

159

C₄²

D₄×D₈₉

yes

characteristic

160

C₁₆

—

161

C₁₆

—

162

C₁₆

—

178

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

356

—

166

Q₈

—

178

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

356

—

170

C₂×C₄

—

356

—

171

C₂×C₄

#15667972

yes

characteristic

172

C₂×C₄

#15668036

yes

characteristic

173

C₂×C₄

#15668039

yes

characteristic

174

C₈

—

175

C₈

—

176

C₈

—

356

—

177

C₈

—

356

—

178

C₈

—

712

—

179

C₈

C₂×C₈×D₈₉

yes

characteristic

180

C₈

#15668020

yes

characteristic

181

C₂²

#19151707

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

356

—

187

C₄

—

712

—

188

C₄

#19151661

yes

characteristic

189

C₄

#19151717

yes

characteristic

190

C₂

#26019073

yes

characteristic

191

C₂

#26022140

yes

characteristic

192

C₂

#26022146

yes

characteristic

	subgroup	G/H
center	C₂²	#19151707
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1334	C₂
socle, prime radical	C₂²×C₈₉	#442
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1334	D₈₉
hypocenter	C₈₉	#12066
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1334

subgroup

G/H

center

C₂²

#19151707

commutator

C₈×C₈₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1334

C₂

socle, prime radical

C₂²×C₈₉

#442

cosocle

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1334

N	H ≅ G/N	type
25. #15667958	C₈	semidirect
104. C₈₉	#12066	semidirect

H ≅ G/N

type

25. #15667958

C₈

semidirect

104. C₈₉

#12066

semidirect

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

subgroup	index
#26022121	2
#26019071	2
#26019070	2
#26018317	2
#26018223	2
#26018174	2
C₈₉×#1334	2
#12066	89

subgroup

index

C₈₉×#1334

Subgroups — #36466305

Conjugacy classes of subgroups

Complements

Maximal subgroups