Group #36470582 — subgroups

Subgroups — #36470582

Order 22784.

358 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×#19152015	C₂	1	1	yes	maximal, characteristic
2	#26022024	C₂	1	2	yes	maximal, complemented
3	#26020302	C₂	1	1	yes	maximal, characteristic
4	#26018347	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1812	C₂	1	1	yes	maximal, characteristic
6	#19152015	—	2	4	no	—
7	C₂×#15667952	C₄	1	2	yes	complemented
8	C₂×#15667952	C₂²	1	1	yes	characteristic
9	C₂²×#13083549	C₄	1	2	yes	complemented
10	C₂²×#13083549	C₂²	1	1	yes	characteristic
11	C₂×#15667948	C₂²	1	2	yes	—
12	#19151582	—	2	1	no	—
13	#19151420	—	2	4	no	—
14	#19151356	—	2	1	no	—
15	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
16	#427×C₈₉	C₂²	1	2	yes	—
17	#410×C₈₉	—	2	1	no	—
18	H(3,ℤ/4)×C₈₉	—	2	1	no	—
19	C₂×#13083558	—	2	2	no	—
20	#15667952	—	2	4	no	—
21	#15667952	D₄	1	4	yes	—
22	C₂×#13083549	—	2	4	no	—
23	C₂×#13083549	—	2	2	no	—
24	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
25	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
26	C₂×#13083549	C₂×C₄	1	2	yes	—
27	#15667948	—	2	2	no	—
28	#15667948	—	2	2	no	—
29	#15667931	—	4	1	no	—
30	#15667929	—	4	1	no	—
31	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
32	#131×C₈₉	—	2	2	no	—
33	#131×C₈₉	—	2	2	no	—
34	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₈₉	Q₈	1	2	yes	—
36	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	#117×C₈₉	—	2	2	no	—
39	#107×C₈₉	—	2	2	no	—
40	#106×C₈₉	—	2	2	no	—
41	#99×C₈₉	—	4	2	no	—
42	#13083558	—	4	2	no	—
43	#13083558	—	4	2	no	—
44	C₂×#10623494	—	4	2	no	—
45	#13083549	—	2	1	no	—
46	#13083549	—	2	1	no	—
47	#13083549	C₂²⋊C₄	1	4	yes	—
48	#41×C₈₉	—	2	2	no	—
49	#41×C₈₉	—	2	2	no	—
50	C₂×Q₈×C₈₉	SD₁₆	1	1	yes	characteristic
51	C₂×D₄×C₈₉	SD₁₆	1	2	yes	—
52	C₂×D₄×C₈₉	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₈₉	—	2	2	no	—
56	M₁₆×C₈₉	—	2	4	no	—
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	—	2	2	no	—
59	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₈₉	#41	1	2	yes	—
64	C₄⋊C₄×C₈₉	—	4	2	no	—
65	C₂²⋊C₄×C₈₉	—	4	2	no	—
66	C₄²×C₈₉	—	4	2	no	—
67	C₂³×C₈₉	#106	1	2	yes	—
68	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
69	Q₈×C₈₉	—	2	2	no	—
70	D₄×C₈₉	—	2	4	no	—
71	D₄×C₈₉	—	2	2	no	—
72	C₂×C₄×C₈₉	—	2	2	no	—
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	8	2	no	—
75	C₂×C₄×C₈₉	#106	1	2	yes	—
76	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#121	1	2	yes	—
78	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#135	1	2	yes	—
80	C₈×C₈₉	—	2	2	no	—
81	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₈×C₈₉	#117	1	1	yes	characteristic
83	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
84	#10623494	—	8	2	no	—
85	C₂²×C₈₉	—	2	2	no	—
86	C₂²×C₈₉	—	4	1	no	—
87	C₂²×C₈₉	—	4	2	no	—
88	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
89	C₂²×C₈₉	#465	1	2	yes	—
90	C₄×C₈₉	—	2	2	no	—
91	C₄×C₈₉	—	16	2	no	—
92	C₄×C₈₉	#380	1	1	yes	characteristic
93	C₄×C₈₉	#427	1	1	yes	characteristic
94	C₄×C₈₉	#465	1	2	yes	—
95	#12832	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	—	2	2	no	—
97	C₂×C₈₉	—	2	1	no	—
98	C₂×C₈₉	—	4	2	no	—
99	C₂×C₈₉	#1807	1	1	yes	characteristic
100	C₂×#504	—	89	1	no	—
101	#2155	—	89	2	no	—
102	#1813	—	89	1	no	—
103	#1812	D₈₉	1	1	yes	characteristic
104	#1382	—	89	2	no	—
105	C₈₉	#12832	1	1	yes	characteristic, complemented, Sylow
106	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
107	#504	—	178	4	no	—
108	C₂×M₃₂	—	89	1	no	—
109	C₂×M₃₂	—	89	2	no	—
110	C₂²×C₁₆	—	89	2	no	—
111	C₂²×C₁₆	—	89	1	no	—
112	C₂×#108	—	89	2	no	—
113	#427	C₂×D₈₉	1	2	yes	—
114	#413	—	178	1	no	—
115	#410	—	2	1	no	—
116	#365	—	178	4	no	—
117	#338	—	178	1	no	—
118	H(3,ℤ/4)	—	2	1	no	—
119	C₂×#41	C₂²⋊D₈₉	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	178	2	no	—
123	C₂×M₁₆	Dic₁₇₈	1	2	yes	—
124	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
125	C₂²×C₈	C₄×D₈₉	1	2	yes	—
126	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
127	#117	—	2	2	no	—
128	M₃₂	—	89	4	no	—
129	M₃₂	—	178	4	no	—
130	C₂×C₁₆	—	89	2	no	—
131	C₂×C₁₆	—	89	1	no	—
132	C₂×C₁₆	—	89	1	no	—
133	C₂×C₁₆	—	178	4	no	—
134	C₂×C₁₆	—	178	2	no	—
135	#108	—	178	2	no	—
136	#108	—	178	2	no	—
137	#107	—	2	2	no	—
138	#106	—	2	2	no	—
139	#101	—	356	1	no	—
140	#100	—	356	1	no	—
141	#99	—	4	2	no	—
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	#13083561	1	1	yes	characteristic
145	C₂×D₄	#13083561	1	1	yes	characteristic
146	C₂×D₄	#13083562	1	2	yes	—
147	C₂²×C₄	#13083553	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	356	2	no	—
152	M₁₆	—	356	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	356	2	no	—
156	C₂×C₈	#13083553	1	2	yes	—
157	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
158	C₂×C₈	D₄×D₈₉	1	2	yes	—
159	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
160	C₂×C₈	#13083573	1	2	yes	—
161	C₄⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₄²	—	4	2	no	—
164	C₁₆	—	89	4	no	—
165	C₁₆	—	178	1	no	—
166	C₁₆	—	178	1	no	—
167	C₂³	#15667943	1	2	yes	—
168	C₂³	C₂×#13083562	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	#15667943	1	2	yes	—
176	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
177	C₂×C₄	#15667984	1	2	yes	—
178	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
179	C₂×C₄	#15668032	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	712	2	no	—
182	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
183	C₈	#15667968	1	1	yes	characteristic
184	C₈	#15667989	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#15667943	1	1	yes	characteristic
189	C₂²	#19151825	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	16	2	no	—
192	C₄	#19151456	1	1	yes	characteristic
193	C₄	#19151634	1	1	yes	characteristic
194	C₄	#19151829	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#26020275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152015

C₂

yes

maximal, characteristic

#26022024

C₂

yes

maximal, complemented

#26020302

C₂

yes

maximal, characteristic

#26018347

C₂

yes

maximal, complemented

C₈₉×#1812

C₂

yes

maximal, characteristic

#19152015

—

C₂×#15667952

C₄

yes

complemented

C₂×#15667952

C₂²

yes

characteristic

C₂²×#13083549

C₄

yes

complemented

C₂²×#13083549

C₂²

yes

characteristic

C₂×#15667948

C₂²

yes

—

#19151582

—

#19151420

—

#19151356

—

C₂×#131×C₈₉

C₂²

yes

characteristic

#427×C₈₉

C₂²

yes

—

#410×C₈₉

—

H(3,ℤ/4)×C₈₉

—

C₂×#13083558

—

#15667952

—

#15667952

D₄

yes

—

C₂×#13083549

—

C₂×#13083549

—

C₂×#13083549

C₂×C₄

yes

characteristic

C₂×#13083549

C₂×C₄

yes

characteristic

C₂×#13083549

C₂×C₄

yes

—

—

—

—

—

C₂×#41×C₈₉

D₄

yes

characteristic

#131×C₈₉

—

#131×C₈₉

—

C₂×M₁₆×C₈₉

D₄

yes

characteristic

C₂×M₁₆×C₈₉

Q₈

yes

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

#117×C₈₉

—

#107×C₈₉

—

#106×C₈₉

—

#99×C₈₉

—

—

—

—

—

—

C₂²⋊C₄

yes

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

SD₁₆

yes

characteristic

C₂×D₄×C₈₉

SD₁₆

yes

—

C₂×D₄×C₈₉

SD₁₆

yes

characteristic

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

#106

yes

—

C₂³×C₈₉

C₂×SD₁₆

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#106

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₈×C₈₉

—

C₈×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₈₉

#117

yes

characteristic

C₈×C₈₉

C₂²⋊Q₈

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#106

yes

characteristic

C₂²×C₈₉

#465

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#12832

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1807

yes

characteristic

100

C₂×#504

—

101

#2155

—

102

#1813

—

103

#1812

D₈₉

yes

characteristic

104

#1382

—

105

C₈₉

#12832

yes

characteristic, complemented, Sylow

106

C₂×#131

C₂×D₈₉

yes

characteristic

107

#504

—

178

—

108

C₂×M₃₂

—

109

C₂×M₃₂

—

110

C₂²×C₁₆

—

111

C₂²×C₁₆

—

112

C₂×#108

—

113

#427

C₂×D₈₉

yes

—

114

#413

—

178

—

115

#410

—

116

#365

—

178

—

117

#338

—

178

—

118

H(3,ℤ/4)

—

119

C₂×#41

C₂²⋊D₈₉

yes

characteristic

120

#131

—

121

#131

—

122

C₂×M₁₆

—

178

—

123

C₂×M₁₆

Dic₁₇₈

yes

—

124

C₂×M₁₆

C₂²⋊D₈₉

yes

characteristic

125

C₂²×C₈

C₄×D₈₉

yes

—

126

C₂²×C₈

C₂²×D₈₉

yes

characteristic

127

#117

—

128

M₃₂

—

129

M₃₂

—

178

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

178

—

134

C₂×C₁₆

—

178

—

135

#108

—

178

—

136

#108

—

178

—

137

#107

—

138

#106

—

139

#101

—

356

—

140

#100

—

356

—

141

#99

—

142

#41

—

143

#41

—

144

C₂×Q₈

#13083561

yes

characteristic

145

C₂×D₄

#13083561

yes

characteristic

146

C₂×D₄

#13083562

yes

—

147

C₂²×C₄

#13083553

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

149

M₁₆

—

150

M₁₆

—

151

M₁₆

—

356

—

152

M₁₆

—

356

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

356

—

156

C₂×C₈

#13083553

yes

—

157

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

158

C₂×C₈

D₄×D₈₉

yes

—

159

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

160

C₂×C₈

#13083573

yes

—

161

C₄⋊C₄

—

162

C₂²⋊C₄

—

163

C₄²

—

164

C₁₆

—

165

C₁₆

—

178

—

166

C₁₆

—

178

—

167

C₂³

#15667943

yes

—

168

C₂³

C₂×#13083562

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

#15667943

yes

—

176

C₂×C₄

C₂×#13083553

yes

characteristic

177

C₂×C₄

#15667984

yes

—

178

C₂×C₄

C₂×#13083561

yes

characteristic

179

C₂×C₄

#15668032

yes

—

180

C₈

—

181

C₈

—

712

—

182

C₈

C₂²⋊C₄×D₈₉

yes

characteristic

183

C₈

#15667968

yes

characteristic

184

C₈

#15667989

yes

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#15667943

yes

characteristic

189

C₂²

#19151825

yes

—

190

C₄

—

191

C₄

—

192

C₄

#19151456

yes

characteristic

193

C₄

#19151634

yes

characteristic

194

C₄

#19151829

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#26020275

yes

characteristic

	subgroup	G/H
center	C₄	#19151634
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1812	C₂
socle	C₂×C₈₉	#1807
prime radical	C₂×#41×C₈₉	D₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1812	D₈₉
hypocenter	C₈₉	#12832
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1812

subgroup

G/H

center

C₄

#19151634

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1812

C₂

socle

C₂×C₈₉

#1807

prime radical

C₂×#41×C₈₉

D₄

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1812

N	H ≅ G/N	type
2. #26022024	C₂	semidirect
4. #26018347	C₂	semidirect
7. C₂×#15667952	C₄	semidirect
9. C₂²×#13083549	C₄	semidirect
105. C₈₉	#12832	semidirect

H ≅ G/N

type

2. #26022024

C₂

semidirect

4. #26018347

C₂

semidirect

7. C₂×#15667952

C₄

semidirect

9. C₂²×#13083549

C₄

semidirect

105. C₈₉

#12832

semidirect

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

subgroup	index
C₂×#19152015	2
#26022024	2
#26020302	2
#26018347	2
C₈₉×#1812	2
#12832	89

subgroup

index

C₈₉×#1812

Subgroups — #36470582

Conjugacy classes of subgroups

Complements

Maximal subgroups