Group #36466343 — subgroups

Subgroups — #36466343

Order 22784.

210 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#26019148	C₂	1	1	yes	maximal, characteristic
3	#26019058	C₂	1	1	yes	maximal, characteristic
4	#26018215	C₂	1	1	yes	maximal, characteristic
5	#26018215	C₂	1	1	yes	maximal, characteristic
6	#26018171	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1371	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₁₂	C₄	1	1	yes	characteristic
9	#19151725	—	2	1	no	—
10	#19151722	—	2	1	no	—
11	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
13	#19151424	C₄	1	1	yes	characteristic
14	#19151403	C₄	1	1	yes	characteristic
15	#19151403	C₄	1	1	yes	characteristic
16	C₄×#13083549	C₂²	1	1	yes	characteristic
17	#19151328	—	2	1	no	—
18	#19151328	—	2	1	no	—
19	C₈×#10623494	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	#335×C₈₉	C₂²	1	1	yes	characteristic
22	#335×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₁₇₈	—	2	1	no	—
26	C₄×Dic₁₇₈	—	2	1	no	—
27	Dic₇₁₂	—	2	2	no	—
28	C₂×#13083549	—	2	1	no	—
29	#15667946	C₂×C₄	1	1	yes	characteristic
30	#15667946	C₂×C₄	1	1	yes	characteristic
31	#15667941	—	4	1	no	—
32	#15667941	—	4	1	no	—
33	#15667937	—	2	1	no	—
34	#15667937	—	2	1	no	—
35	#15667922	—	4	1	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×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₈₉	—	2	1	no	—
42	C₄⋊C₈×C₈₉	—	2	1	no	—
43	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₁₇₈	—	2	1	no	—
45	C₂×Dic₁₇₈	—	2	1	no	—
46	Dic₃₅₆	—	2	1	no	—
47	Dic₃₅₆	—	2	1	no	—
48	Dic₃₅₆	—	2	1	no	—
49	Dic₃₅₆	M₁₆	1	2	yes	—
50	C₂×#10623494	—	2	1	no	—
51	C₂×#10623494	D₈	1	1	yes	characteristic
52	C₂×#10623494	D₈	1	1	yes	characteristic
53	C₄⋊Dic₈₉	—	2	1	no	—
54	C₄⋊Dic₈₉	—	2	1	no	—
55	#13083553	—	4	1	no	—
56	#13083553	—	4	1	no	—
57	C₄×Dic₈₉	—	4	1	no	—
58	C₄×Dic₈₉	—	4	1	no	—
59	#13083549	—	4	1	no	—
60	C₂×C₈×C₈₉	—	4	1	no	—
61	C₂×C₈×C₈₉	—	4	1	no	—
62	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₁₆×C₈₉	—	2	1	no	—
66	C₁₆×C₈₉	—	2	1	no	—
67	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
70	C₂×Dic₈₉	—	4	1	no	—
71	C₂×Dic₈₉	—	4	1	no	—
72	C₂×Dic₈₉	—	4	1	no	—
73	C₂×Dic₈₉	—	4	1	no	—
74	Dic₁₇₈	—	2	2	no	—
75	Dic₁₇₈	—	4	1	no	—
76	Dic₁₇₈	—	4	1	no	—
77	Dic₁₇₈	—	4	1	no	—
78	C₈×C₈₉	—	2	1	no	—
79	C₈×C₈₉	—	8	1	no	—
80	C₈×C₈₉	—	8	1	no	—
81	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
82	C₈×C₈₉	#131	1	1	yes	characteristic
83	#10623494	—	2	1	no	—
84	#10623494	—	4	1	no	—
85	#10623494	Q₃₂	1	2	yes	complemented
86	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
87	C₄×C₈₉	—	2	1	no	—
88	C₄×C₈₉	#435	1	1	yes	characteristic
89	C₄×C₈₉	#444	1	1	yes	characteristic
90	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
91	C₄×C₈₉	#508	1	1	yes	characteristic
92	Dic₈₉	—	4	2	no	—
93	Dic₈₉	—	8	1	no	—
94	Dic₈₉	—	8	1	no	—
95	Dic₈₉	—	8	1	no	—
96	#12072	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	#1581	1	1	yes	characteristic
98	C₂×C₈₉	C₄×Q₃₂	1	1	yes	characteristic
99	C₂×C₈₉	#2179	1	1	yes	characteristic
100	C₄×Q₃₂	—	89	1	no	—
101	#1591	—	89	1	no	—
102	C₈×Q₁₆	—	89	1	no	—
103	#1371	D₈₉	1	1	yes	characteristic
104	#1332	—	89	1	no	—
105	#1332	—	89	1	no	—
106	#1312	—	89	1	no	—
107	C₈₉	#12072	1	1	yes	characteristic, complemented, Sylow
108	C₂×Q₃₂	—	89	1	no	—
109	#446	—	178	1	no	—
110	C₈×Q₈	—	178	1	no	—
111	C₄×Q₁₆	—	89	1	no	—
112	C₄×Q₁₆	—	89	1	no	—
113	#366	—	89	1	no	—
114	#358	—	89	1	no	—
115	#358	—	89	1	no	—
116	C₄×C₁₆	—	89	1	no	—
117	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
118	#335	C₂×D₈₉	1	1	yes	characteristic
119	#335	C₂×D₈₉	1	1	yes	characteristic
120	Q₈⋊C₈	—	178	1	no	—
121	Q₈⋊C₈	—	178	1	no	—
122	C₈²	—	89	1	no	—
123	C₂×Q₁₆	—	89	1	no	—
124	C₂×Q₁₆	—	89	1	no	—
125	C₄×Q₈	—	178	1	no	—
126	C₄×Q₈	—	178	1	no	—
127	Q₃₂	—	178	2	no	—
128	C₂×C₁₆	—	178	1	no	—
129	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
130	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
131	#107	—	89	1	no	—
132	#107	—	89	1	no	—
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	—	2	1	no	—
135	C₄⋊C₈	—	356	1	no	—
136	C₄⋊C₈	—	356	1	no	—
137	#103	—	178	1	no	—
138	#103	—	178	1	no	—
139	#97	—	356	1	no	—
140	C₄×C₈	—	89	1	no	—
141	C₄×C₈	—	89	1	no	—
142	C₄×C₈	—	356	1	no	—
143	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
144	C₂×Q₈	—	178	1	no	—
145	C₂×Q₈	—	178	1	no	—
146	Q₁₆	—	89	2	no	—
147	Q₁₆	—	178	1	no	—
148	Q₁₆	—	178	1	no	—
149	Q₁₆	—	178	1	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	89	1	no	—
153	C₂×C₈	—	89	1	no	—
154	C₂×C₈	—	178	1	no	—
155	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
156	C₂×C₈	#13083575	1	1	yes	characteristic
157	C₄⋊C₄	—	356	1	no	—
158	C₄⋊C₄	—	356	1	no	—
159	C₄⋊C₄	—	178	1	no	—
160	C₄⋊C₄	—	178	1	no	—
161	C₄²	—	356	1	no	—
162	C₄²	—	356	1	no	—
163	C₄²	D₄×D₈₉	1	1	yes	characteristic
164	C₁₆	—	2	1	no	—
165	C₁₆	—	2	1	no	—
166	C₁₆	—	356	1	no	—
167	Q₈	—	356	1	no	—
168	Q₈	—	356	1	no	—
169	Q₈	—	356	1	no	—
170	Q₈	—	178	2	no	—
171	C₂×C₄	—	356	1	no	—
172	C₂×C₄	—	356	1	no	—
173	C₂×C₄	—	356	1	no	—
174	C₂×C₄	—	356	1	no	—
175	C₂×C₄	#15667974	1	1	yes	characteristic
176	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
177	C₂×C₄	#15668035	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	8	1	no	—
180	C₈	—	8	1	no	—
181	C₈	—	89	2	no	—
182	C₈	—	356	1	no	—
183	C₈	—	178	1	no	—
184	C₈	M₁₆×D₈₉	1	1	yes	characteristic
185	C₈	#15668018	1	1	yes	characteristic
186	C₂²	#19151680	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	356	2	no	—
189	C₄	—	712	1	no	—
190	C₄	—	712	1	no	—
191	C₄	—	712	1	no	—
192	C₄	#19151668	1	1	yes	characteristic
193	C₄	#19151719	1	1	yes	characteristic
194	C₄	Q₃₂×D₈₉	1	1	yes	characteristic
195	C₄	#19152027	1	1	yes	characteristic
196	C₂	#26019080	1	1	yes	characteristic
197	C₂	#26022110	1	1	yes	characteristic
198	C₂	#26022137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₇₁₂

C₂

yes

maximal, characteristic

#26019148

C₂

yes

maximal, characteristic

#26019058

C₂

yes

maximal, characteristic

#26018215

C₂

yes

maximal, characteristic

#26018215

C₂

yes

maximal, characteristic

#26018171

C₂

yes

maximal, characteristic

C₈₉×#1371

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

C₄

yes

characteristic

#19151725

—

#19151722

—

C₄×Dic₃₅₆

C₂²

yes

characteristic

C₄×Dic₃₅₆

C₂²

yes

characteristic

#19151424

C₄

yes

characteristic

#19151403

C₄

yes

characteristic

#19151403

C₄

yes

characteristic

C₄×#13083549

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

#335×C₈₉

C₂²

yes

characteristic

#335×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

C₂×C₄

yes

characteristic

C₂×Dic₃₅₆

C₂×C₄

yes

characteristic

C₄×Dic₁₇₈

—

C₄×Dic₁₇₈

—

Dic₇₁₂

—

C₂×#13083549

—

#15667946

C₂×C₄

yes

characteristic

#15667946

C₂×C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#10623494

D₄

yes

characteristic

C₈×Dic₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

M₁₆

yes

—

C₂×#10623494

—

C₂×#10623494

D₈

yes

characteristic

C₂×#10623494

D₈

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×M₁₆

yes

C₈×C₈₉

yes

—

—

Q₃₂

yes

complemented

C₂²×C₈₉

C₄×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×Q₃₂

yes

characteristic

C₄×C₈₉

#508

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

C₄×Q₃₂

yes

characteristic

C₂×C₈₉

#2179

yes

characteristic

100

C₄×Q₃₂

—

101

#1591

—

102

C₈×Q₁₆

—

103

#1371

D₈₉

yes

characteristic

104

#1332

—

105

#1332

—

106

#1312

—

107

C₈₉

#12072

yes

characteristic, complemented, Sylow

108

C₂×Q₃₂

—

109

#446

—

178

—

110

C₈×Q₈

—

178

—

111

C₄×Q₁₆

—

112

C₄×Q₁₆

—

113

#366

—

114

#358

—

115

#358

—

116

C₄×C₁₆

—

117

C₄×C₁₆

C₂×D₈₉

yes

characteristic

118

#335

C₂×D₈₉

yes

characteristic

119

#335

C₂×D₈₉

yes

characteristic

120

Q₈⋊C₈

—

178

—

121

Q₈⋊C₈

—

178

—

122

C₈²

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

125

C₄×Q₈

—

178

—

126

C₄×Q₈

—

178

—

127

Q₃₂

—

178

—

128

C₂×C₁₆

—

178

—

129

C₂×C₁₆

C₄×D₈₉

yes

characteristic

130

C₂×C₁₆

C₄×D₈₉

yes

characteristic

131

#107

—

132

#107

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

356

—

136

C₄⋊C₈

—

356

—

137

#103

—

178

—

138

#103

—

178

—

139

#97

—

356

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

356

—

143

C₄×C₈

C₂²×D₈₉

yes

characteristic

144

C₂×Q₈

—

178

—

145

C₂×Q₈

—

178

—

146

Q₁₆

—

147

Q₁₆

—

178

—

148

Q₁₆

—

178

—

149

Q₁₆

—

178

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

178

—

155

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

156

C₂×C₈

#13083575

yes

characteristic

157

C₄⋊C₄

—

356

—

158

C₄⋊C₄

—

356

—

159

C₄⋊C₄

—

178

—

160

C₄⋊C₄

—

178

—

161

C₄²

—

356

—

162

C₄²

—

356

—

163

C₄²

D₄×D₈₉

yes

characteristic

164

C₁₆

—

165

C₁₆

—

166

C₁₆

—

356

—

167

Q₈

—

356

—

168

Q₈

—

356

—

169

Q₈

—

356

—

170

Q₈

—

178

—

171

C₂×C₄

—

356

—

172

C₂×C₄

—

356

—

173

C₂×C₄

—

356

—

174

C₂×C₄

—

356

—

175

C₂×C₄

#15667974

yes

characteristic

176

C₂×C₄

D₈×D₈₉

yes

characteristic

177

C₂×C₄

#15668035

yes

characteristic

178

C₈

—

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

356

—

183

C₈

—

178

—

184

C₈

M₁₆×D₈₉

yes

characteristic

185

C₈

#15668018

yes

characteristic

186

C₂²

#19151680

yes

characteristic

187

C₄

—

188

C₄

—

356

—

189

C₄

—

712

—

190

C₄

—

712

—

191

C₄

—

712

—

192

C₄

#19151668

yes

characteristic

193

C₄

#19151719

yes

characteristic

194

C₄

Q₃₂×D₈₉

yes

characteristic

195

C₄

#19152027

yes

characteristic

196

C₂

#26019080

yes

characteristic

197

C₂

#26022110

yes

characteristic

198

C₂

#26022137

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1371	C₂
socle, prime radical	C₂²×C₈₉	C₄×D₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1371	D₈₉
hypocenter	C₈₉	#12072
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂×C₄

D₈×D₈₉

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1371

C₂

socle, prime radical

C₂²×C₈₉

C₄×D₈

cosocle

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
85. #10623494	Q₃₂	semidirect
107. C₈₉	#12072	semidirect

H ≅ G/N

type

85. #10623494

Q₃₂

semidirect

107. C₈₉

#12072

semidirect

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

subgroup	index
C₄×Dic₇₁₂	2
#26019148	2
#26019058	2
#26018215	2
#26018171	2
C₈₉×#1371	2
#12072	89

subgroup

index

C₄×Dic₇₁₂

C₈₉×#1371

Subgroups — #36466343

Conjugacy classes of subgroups

Complements

Maximal subgroups