Group #36468578 — subgroups

Subgroups — #36468578

Order 22784.

218 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 65 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	#26022405	C₂	1	1	yes	maximal, characteristic
2	#26022299	C₂	1	1	yes	maximal, characteristic
3	#26019846	C₂	1	1	yes	maximal, characteristic
4	#26019058	C₂	1	1	yes	maximal, characteristic
5	#26018315	C₂	1	1	yes	maximal, characteristic
6	#26018311	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1362	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667958	—	2	1	no	—
9	#19152003	C₂²	1	1	yes	characteristic
10	#19151994	—	2	1	no	—
11	#19151950	—	2	1	no	—
12	#19151722	—	2	1	no	—
13	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
14	#19151424	—	2	1	no	—
15	C₄⋊#13083549	C₂²	1	1	yes	characteristic
16	#19151403	—	2	1	no	—
17	#19151402	—	2	1	no	—
18	#19151328	—	2	1	no	—
19	C₈×#10623494	C₂²	1	1	yes	characteristic
20	#494×C₈₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	#335×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₅₆	D₄	1	1	yes	characteristic
24	#15668012	—	2	1	no	—
25	C₄×Dic₁₇₈	—	2	1	no	—
26	#15667958	—	4	2	no	—
27	C₂×#13083549	—	2	1	no	—
28	#15667946	—	2	1	no	—
29	#15667946	—	2	1	no	—
30	#15667946	D₄	1	1	yes	characteristic
31	#15667943	—	4	1	no	—
32	#15667943	—	4	1	no	—
33	#15667941	—	4	1	no	—
34	#15667937	—	2	1	no	—
35	#15667935	—	4	2	no	—
36	C₄×#10623494	D₄	1	1	yes	characteristic
37	C₄×#10623494	D₄	1	1	yes	characteristic
38	C₈×Dic₈₉	—	4	1	no	—
39	C₂×Q₁₆×C₈₉	—	2	1	no	—
40	C₂×Q₁₆×C₈₉	—	2	1	no	—
41	C₄⋊Q₈×C₈₉	—	2	1	no	—
42	C₂×C₁₆×C₈₉	—	2	1	no	—
43	C₄⋊C₈×C₈₉	—	2	1	no	—
44	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₇₈	—	2	1	no	—
46	C₂×Dic₁₇₈	—	4	1	no	—
47	Dic₃₅₆	—	2	1	no	—
48	Dic₃₅₆	SD₁₆	1	2	yes	—
49	C₂×#10623494	—	2	1	no	—
50	C₂×#10623494	D₈	1	1	yes	characteristic
51	C₂×#10623494	D₈	1	1	yes	characteristic
52	C₄⋊Dic₈₉	—	2	1	no	—
53	C₄⋊Dic₈₉	—	4	1	no	—
54	C₄⋊Dic₈₉	—	4	1	no	—
55	#13083553	—	4	1	no	—
56	C₄×Dic₈₉	—	4	1	no	—
57	#13083549	—	4	1	no	—
58	C₂×Q₈×C₈₉	—	4	1	no	—
59	Q₁₆×C₈₉	—	4	1	no	—
60	Q₁₆×C₈₉	—	4	1	no	—
61	Q₁₆×C₈₉	—	8	1	no	—
62	C₂×C₈×C₈₉	—	4	1	no	—
63	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
65	C₄⋊C₄×C₈₉	—	4	1	no	—
66	C₄⋊C₄×C₈₉	—	4	1	no	—
67	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₁₆×C₈₉	—	4	1	no	—
69	Q₈×C₈₉	—	8	1	no	—
70	Q₈×C₈₉	—	8	1	no	—
71	C₂×C₄×C₈₉	—	8	1	no	—
72	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×Dic₈₉	—	4	1	no	—
76	C₂×Dic₈₉	—	4	1	no	—
77	C₂×Dic₈₉	—	8	1	no	—
78	Dic₁₇₈	—	2	2	no	—
79	Dic₁₇₈	—	4	1	no	—
80	Dic₁₇₈	—	8	2	no	—
81	C₈×C₈₉	—	2	1	no	—
82	C₈×C₈₉	—	8	1	no	—
83	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
84	C₈×C₈₉	#135	1	1	yes	characteristic
85	#10623494	—	2	1	no	—
86	#10623494	—	4	1	no	—
87	#10623494	Q₃₂	1	2	yes	complemented
88	C₂²×C₈₉	#459	1	1	yes	characteristic
89	C₄×C₈₉	—	2	1	no	—
90	C₄×C₈₉	—	16	1	no	—
91	C₄×C₈₉	#465	1	1	yes	characteristic
92	C₄×C₈₉	#471	1	1	yes	characteristic
93	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
94	C₄×C₈₉	#510	1	1	yes	characteristic
95	Dic₈₉	—	4	2	no	—
96	Dic₈₉	—	8	1	no	—
97	Dic₈₉	—	16	1	no	—
98	#12421	—	89	1	no	maximal, Sylow
99	C₂×C₈₉	#1665	1	1	yes	characteristic
100	C₂×C₈₉	#2209	1	1	yes	characteristic
101	C₂×C₈₉	#2212	1	1	yes	characteristic
102	#2227	—	89	1	no	—
103	#2209	—	89	1	no	—
104	#1707	—	89	1	no	—
105	C₈×Q₁₆	—	89	1	no	—
106	#1367	—	89	1	no	—
107	#1365	—	89	1	no	—
108	#1362	D₈₉	1	1	yes	characteristic
109	C₈₉	#12421	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₃₂	—	178	1	no	—
111	#500	—	89	1	no	—
112	#498	—	178	1	no	—
113	#494	C₂×D₈₉	1	1	yes	characteristic
114	#487	—	178	1	no	—
115	C₈×Q₈	—	178	1	no	—
116	C₄×Q₁₆	—	89	1	no	—
117	#366	—	178	1	no	—
118	C₄⋊C₁₆	—	89	1	no	—
119	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
120	#358	—	178	1	no	—
121	#358	—	178	1	no	—
122	#335	C₂×D₈₉	1	1	yes	characteristic
123	Q₈⋊C₈	—	178	1	no	—
124	C₈²	—	89	1	no	—
125	C₂×Q₁₆	—	2	1	no	—
126	C₂×Q₁₆	—	2	1	no	—
127	C₂×Q₁₆	—	89	1	no	—
128	C₄⋊Q₈	—	2	1	no	—
129	C₄⋊Q₈	—	178	1	no	—
130	C₄×Q₈	—	178	1	no	—
131	Q₃₂	—	356	2	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	178	1	no	—
134	#107	—	89	1	no	—
135	#107	—	178	1	no	—
136	#107	—	178	1	no	—
137	#106	—	356	1	no	—
138	#106	—	356	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	—	356	1	no	—
141	#103	—	356	2	no	—
142	#103	—	178	1	no	—
143	C₄×C₈	—	89	1	no	—
144	C₄×C₈	—	89	1	no	—
145	C₄×C₈	—	356	1	no	—
146	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
147	C₂×Q₈	—	4	1	no	—
148	C₂×Q₈	—	356	1	no	—
149	C₂×Q₈	—	178	1	no	—
150	Q₁₆	—	4	1	no	—
151	Q₁₆	—	4	1	no	—
152	Q₁₆	—	8	1	no	—
153	Q₁₆	—	89	2	no	—
154	Q₁₆	—	178	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	—	89	1	no	—
157	C₂×C₈	—	89	1	no	—
158	C₂×C₈	—	178	1	no	—
159	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
160	C₂×C₈	#13083574	1	1	yes	characteristic
161	C₄⋊C₄	—	4	1	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₄⋊C₄	—	356	1	no	—
164	C₄⋊C₄	—	356	1	no	—
165	C₄⋊C₄	—	356	1	no	—
166	C₄⋊C₄	—	178	1	no	—
167	C₄²	—	356	1	no	—
168	C₄²	D₄×D₈₉	1	1	yes	characteristic
169	C₁₆	—	4	1	no	—
170	C₁₆	—	356	1	no	—
171	Q₈	—	8	1	no	—
172	Q₈	—	8	1	no	—
173	Q₈	—	356	1	no	—
174	Q₈	—	178	2	no	—
175	Q₈	—	712	2	no	—
176	C₂×C₄	—	8	1	no	—
177	C₂×C₄	—	356	1	no	—
178	C₂×C₄	—	356	1	no	—
179	C₂×C₄	—	712	1	no	—
180	C₂×C₄	#15667985	1	1	yes	characteristic
181	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
182	C₂×C₄	#15668039	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	8	1	no	—
185	C₈	—	89	2	no	—
186	C₈	—	356	1	no	—
187	C₈	—	178	1	no	—
188	C₈	SD₁₆×D₈₉	1	1	yes	characteristic
189	C₈	#15668031	1	1	yes	characteristic
190	C₂²	#19151792	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	16	1	no	—
193	C₄	—	356	2	no	—
194	C₄	—	1424	1	no	—
195	C₄	—	712	1	no	—
196	C₄	#19151826	1	1	yes	characteristic
197	C₄	#19151860	1	1	yes	characteristic
198	C₄	Q₃₂×D₈₉	1	1	yes	characteristic
199	C₄	#19152037	1	1	yes	characteristic
200	C₂	#26019592	1	1	yes	characteristic
201	C₂	#26022301	1	1	yes	characteristic
202	C₂	#26022324	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022405

C₂

yes

maximal, characteristic

#26022299

C₂

yes

maximal, characteristic

#26019846

C₂

yes

maximal, characteristic

#26019058

C₂

yes

maximal, characteristic

#26018315

C₂

yes

maximal, characteristic

#26018311

C₂

yes

maximal, characteristic

C₈₉×#1362

C₂

yes

maximal, characteristic

C₂×#15667958

—

#19152003

C₂²

yes

—

—

—

C₄×Dic₃₅₆

C₂²

yes

characteristic

#19151424

—

C₄⋊#13083549

C₂²

yes

—

—

—

C₂²

yes

characteristic

#494×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

C₂²

yes

characteristic

#335×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

D₄

yes

characteristic

#15668012

—

C₄×Dic₁₇₈

—

—

—

—

—

D₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#10623494

D₄

yes

characteristic

C₈×Dic₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

SD₁₆

yes

—

C₂×#10623494

—

C₂×#10623494

D₈

yes

characteristic

C₂×#10623494

D₈

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×SD₁₆

yes

C₈×C₈₉

yes

—

—

Q₃₂

yes

complemented

C₂²×C₈₉

#459

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×Q₃₂

yes

characteristic

C₄×C₈₉

#510

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

100

C₂×C₈₉

#2209

yes

characteristic

101

C₂×C₈₉

#2212

yes

characteristic

102

#2227

—

103

#2209

—

104

#1707

—

105

C₈×Q₁₆

—

106

#1367

—

107

#1365

—

108

#1362

D₈₉

yes

characteristic

109

C₈₉

#12421

yes

characteristic, complemented, Sylow

110

C₂×Q₃₂

—

178

—

111

#500

—

112

#498

—

178

—

113

#494

C₂×D₈₉

yes

characteristic

114

#487

—

178

—

115

C₈×Q₈

—

178

—

116

C₄×Q₁₆

—

117

#366

—

178

—

118

C₄⋊C₁₆

—

119

C₄⋊C₁₆

C₂×D₈₉

yes

characteristic

120

#358

—

178

—

121

#358

—

178

—

122

#335

C₂×D₈₉

yes

characteristic

123

Q₈⋊C₈

—

178

—

124

C₈²

—

125

C₂×Q₁₆

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

178

—

130

C₄×Q₈

—

178

—

131

Q₃₂

—

356

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

178

—

134

#107

—

135

#107

—

178

—

136

#107

—

178

—

137

#106

—

356

—

138

#106

—

356

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

356

—

141

#103

—

356

—

142

#103

—

178

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

356

—

146

C₄×C₈

C₂²×D₈₉

yes

characteristic

147

C₂×Q₈

—

148

C₂×Q₈

—

356

—

149

C₂×Q₈

—

178

—

150

Q₁₆

—

151

Q₁₆

—

152

Q₁₆

—

153

Q₁₆

—

154

Q₁₆

—

178

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

178

—

159

C₂×C₈

D₄×D₈₉

yes

characteristic

160

C₂×C₈

#13083574

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

356

—

164

C₄⋊C₄

—

356

—

165

C₄⋊C₄

—

356

—

166

C₄⋊C₄

—

178

—

167

C₄²

—

356

—

168

C₄²

D₄×D₈₉

yes

characteristic

169

C₁₆

—

170

C₁₆

—

356

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

356

—

174

Q₈

—

178

—

175

Q₈

—

712

—

176

C₂×C₄

—

177

C₂×C₄

—

356

—

178

C₂×C₄

—

356

—

179

C₂×C₄

—

712

—

180

C₂×C₄

#15667985

yes

characteristic

181

C₂×C₄

D₈×D₈₉

yes

characteristic

182

C₂×C₄

#15668039

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

356

—

187

C₈

—

178

—

188

C₈

SD₁₆×D₈₉

yes

characteristic

189

C₈

#15668031

yes

characteristic

190

C₂²

#19151792

yes

characteristic

191

C₄

—

192

C₄

—

193

C₄

—

356

—

194

C₄

—

1424

—

195

C₄

—

712

—

196

C₄

#19151826

yes

characteristic

197

C₄

#19151860

yes

characteristic

198

C₄

Q₃₂×D₈₉

yes

characteristic

199

C₄

#19152037

yes

characteristic

200

C₂

#26019592

yes

characteristic

201

C₂

#26022301

yes

characteristic

202

C₂

#26022324

yes

characteristic

	subgroup	G/H
center	C₂²	#19151792
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1362	C₂
socle, prime radical	C₂²×C₈₉	#459
hypercenter	#1362	D₈₉
hypocenter	C₈₉	#12421
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1362

subgroup

G/H

center

C₂²

#19151792

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1362

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1362

N	H ≅ G/N	type
87. #10623494	Q₃₂	semidirect
109. C₈₉	#12421	semidirect

H ≅ G/N

type

87. #10623494

Q₃₂

semidirect

109. C₈₉

#12421

semidirect

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

subgroup	index
#26022405	2
#26022299	2
#26019846	2
#26019058	2
#26018315	2
#26018311	2
C₈₉×#1362	2
#12421	89

subgroup

index

C₈₉×#1362

Subgroups — #36468578

Conjugacy classes of subgroups

Complements

Maximal subgroups