Group #36388382 — subgroups

Subgroups — C₈₉×#10534 #36388382

Order 22784.

358 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₈₉×#3255	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1561	C₂	1	2	yes	maximal
4	C₈₉×#1493	C₂	1	2	yes	maximal
5	C₈₉×#1309	C₂	1	1	yes	maximal, characteristic
6	#533×C₈₉	C₂²	1	2	yes	—
7	#475×C₈₉	—	2	2	no	—
8	#470×C₈₉	—	2	2	no	—
9	#462×C₈₉	—	2	2	no	—
10	C₂×#108×C₈₉	—	2	2	no	—
11	#423×C₈₉	C₂²	1	1	yes	characteristic
12	#423×C₈₉	C₂²	1	2	yes	—
13	#423×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	#332×C₈₉	—	2	2	no	—
16	#332×C₈₉	—	2	2	no	—
17	Q₈⋊C₈×C₈₉	—	2	4	no	—
18	C₂×Q₁₆×C₈₉	—	2	2	no	—
19	C₂×M₁₆×C₈₉	—	2	1	no	—
20	C₂×M₁₆×C₈₉	—	2	2	no	—
21	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
22	C₂²×C₈×C₈₉	—	2	1	no	—
23	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
24	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
25	#125×C₈₉	—	2	2	no	—
26	C₂²⋊Q₈×C₈₉	—	2	2	no	—
27	C₂²⋊Q₈×C₈₉	—	2	2	no	—
28	C₄×Q₈×C₈₉	—	2	2	no	—
29	C₄×Q₈×C₈₉	—	2	2	no	—
30	#117×C₈₉	—	2	2	no	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	#108×C₈₉	—	4	4	no	—
33	#106×C₈₉	—	2	2	no	—
34	C₄⋊C₈×C₈₉	—	2	2	no	—
35	C₄⋊C₈×C₈₉	—	2	2	no	—
36	C₄⋊C₈×C₈₉	—	2	2	no	—
37	C₄⋊C₈×C₈₉	—	2	2	no	—
38	#103×C₈₉	—	2	4	no	—
39	C₄×C₈×C₈₉	—	2	1	no	—
40	C₄×C₈×C₈₉	D₄	1	2	yes	—
41	C₂×Q₈×C₈₉	—	2	2	no	—
42	C₂×Q₈×C₈₉	—	2	2	no	—
43	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	#41	1	2	yes	—
45	Q₁₆×C₈₉	—	8	2	no	—
46	M₁₆×C₈₉	—	4	2	no	—
47	M₁₆×C₈₉	—	4	2	no	—
48	M₁₆×C₈₉	—	8	2	no	—
49	C₂×C₈×C₈₉	—	2	2	no	—
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	—	2	2	no	—
52	C₂×C₈×C₈₉	—	2	2	no	—
53	C₂×C₈×C₈₉	—	4	1	no	—
54	C₂×C₈×C₈₉	—	4	2	no	—
55	C₄⋊C₄×C₈₉	—	2	2	no	—
56	C₄⋊C₄×C₈₉	—	2	2	no	—
57	C₄⋊C₄×C₈₉	—	2	2	no	—
58	C₄⋊C₄×C₈₉	—	2	2	no	—
59	C₄⋊C₄×C₈₉	—	4	2	no	—
60	C₄⋊C₄×C₈₉	—	4	2	no	—
61	C₂²⋊C₄×C₈₉	—	4	2	no	—
62	C₄²×C₈₉	—	4	2	no	—
63	C₄²×C₈₉	C₂×D₄	1	2	yes	—
64	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₈₉	#41	1	1	yes	characteristic
66	C₂³×C₈₉	#124	1	1	yes	characteristic
67	Q₈×C₈₉	—	4	2	no	—
68	Q₈×C₈₉	—	4	2	no	—
69	Q₈×C₈₉	—	4	2	no	—
70	Q₈×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	—	2	1	no	—
72	C₂×C₄×C₈₉	—	2	1	no	—
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	2	2	no	—
75	C₂×C₄×C₈₉	—	4	2	no	—
76	C₂×C₄×C₈₉	—	4	2	no	—
77	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
78	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#121	1	2	yes	—
80	C₂×C₄×C₈₉	#123	1	2	yes	—
81	C₈×C₈₉	—	2	4	no	—
82	C₈×C₈₉	—	4	2	no	—
83	C₈×C₈₉	—	8	1	no	—
84	C₈×C₈₉	—	8	2	no	—
85	C₂²×C₈₉	—	2	2	no	—
86	C₂²×C₈₉	—	2	1	no	—
87	C₂²×C₈₉	#394	1	1	yes	characteristic
88	C₄×C₈₉	—	2	2	no	—
89	C₄×C₈₉	—	2	1	no	—
90	C₄×C₈₉	—	8	2	no	—
91	C₄×C₈₉	—	8	2	no	—
92	C₄×C₈₉	#453	1	1	yes	characteristic
93	C₄×C₈₉	#456	1	1	yes	characteristic
94	C₄×C₈₉	#471	1	2	yes	—
95	C₄×C₈₉	#473	1	2	yes	—
96	#10534	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂×C₈₉	—	4	1	no	—
98	C₂×C₈₉	#2039	1	1	yes	characteristic
99	C₂×C₈₉	#2052	1	2	yes	—
100	#3255	C₂×C₈₉	1	1	yes	characteristic
101	#1848	C₂×C₈₉	1	1	yes	characteristic
102	#1561	C₂×C₈₉	1	2	yes	—
103	#1493	C₂×C₈₉	1	2	yes	—
104	#1309	C₂×C₈₉	1	1	yes	characteristic
105	C₈₉	#10534	1	1	yes	characteristic, direct product factor, Sylow
106	#533	C₂²×C₈₉	1	2	yes	—
107	#475	—	2	2	no	—
108	#470	—	2	2	no	—
109	#462	—	2	2	no	—
110	C₂×#108	—	2	2	no	—
111	#423	C₂²×C₈₉	1	1	yes	characteristic
112	#423	C₂²×C₈₉	1	2	yes	—
113	#423	C₂²×C₈₉	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
115	#332	—	2	2	no	—
116	#332	—	2	2	no	—
117	Q₈⋊C₈	—	2	4	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×M₁₆	—	2	1	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	D₄×C₈₉	1	2	yes	—
122	C₂²×C₈	—	2	1	no	—
123	C₄⋊Q₈	D₄×C₈₉	1	2	yes	—
124	C₄⋊Q₈	D₄×C₈₉	1	2	yes	—
125	#125	—	2	2	no	—
126	C₂²⋊Q₈	—	2	2	no	—
127	C₂²⋊Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	C₄×Q₈	—	2	2	no	—
130	#117	—	2	2	no	—
131	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
132	#108	—	4	4	no	—
133	#106	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	#103	—	2	4	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	D₄×C₈₉	1	2	yes	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
144	C₂²×C₄	#41×C₈₉	1	2	yes	—
145	Q₁₆	—	8	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	8	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₂×D₄×C₈₉	1	2	yes	—
164	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
165	C₄²	#41×C₈₉	1	1	yes	characteristic
166	C₂³	#124×C₈₉	1	1	yes	characteristic
167	Q₈	—	4	2	no	—
168	Q₈	—	4	2	no	—
169	Q₈	—	4	2	no	—
170	Q₈	—	4	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
178	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
179	C₂×C₄	#121×C₈₉	1	2	yes	—
180	C₂×C₄	#123×C₈₉	1	2	yes	—
181	C₈	—	2	4	no	—
182	C₈	—	4	2	no	—
183	C₈	—	8	1	no	—
184	C₈	—	8	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	2	no	—
187	C₂²	#394×C₈₉	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	2	no	—
190	C₄	—	8	2	no	—
191	C₄	—	8	2	no	—
192	C₄	#453×C₈₉	1	1	yes	characteristic
193	C₄	#456×C₈₉	1	1	yes	characteristic
194	C₄	#471×C₈₉	1	2	yes	—
195	C₄	#473×C₈₉	1	2	yes	—
196	C₂	—	4	1	no	—
197	C₂	C₈₉×#2039	1	1	yes	characteristic
198	C₂	C₈₉×#2052	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#3255

C₂

yes

maximal, characteristic

C₈₉×#1848

C₂

yes

maximal, characteristic

C₈₉×#1561

C₂

yes

maximal

C₈₉×#1493

C₂

yes

maximal

C₈₉×#1309

C₂

yes

maximal, characteristic

#533×C₈₉

C₂²

yes

—

#475×C₈₉

—

#470×C₈₉

—

#462×C₈₉

—

C₂×#108×C₈₉

—

#423×C₈₉

C₂²

yes

characteristic

#423×C₈₉

C₂²

yes

—

#423×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#332×C₈₉

—

#332×C₈₉

—

Q₈⋊C₈×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×M₁₆×C₈₉

—

C₂×M₁₆×C₈₉

—

C₂×M₁₆×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

—

C₄⋊Q₈×C₈₉

D₄

yes

—

C₄⋊Q₈×C₈₉

D₄

yes

—

#125×C₈₉

—

C₂²⋊Q₈×C₈₉

—

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#108×C₈₉

—

#106×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

#103×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

Q₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

C₄²×C₈₉

yes

C₂³×C₈₉

yes

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

#123

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#394

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₄×C₈₉

#473

yes

—

#10534

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2039

yes

characteristic

C₂×C₈₉

#2052

yes

—

100

#3255

C₂×C₈₉

yes

characteristic

101

#1848

C₂×C₈₉

yes

characteristic

102

#1561

C₂×C₈₉

yes

—

103

#1493

C₂×C₈₉

yes

—

104

#1309

C₂×C₈₉

yes

characteristic

105

C₈₉

#10534

yes

characteristic, direct product factor, Sylow

106

#533

C₂²×C₈₉

yes

—

107

#475

—

108

#470

—

109

#462

—

110

C₂×#108

—

111

#423

C₂²×C₈₉

yes

characteristic

112

#423

C₂²×C₈₉

yes

—

113

#423

C₂²×C₈₉

yes

characteristic

114

C₂×C₄×C₈

C₂²×C₈₉

yes

characteristic

115

#332

—

116

#332

—

117

Q₈⋊C₈

—

118

C₂×Q₁₆

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂×M₁₆

D₄×C₈₉

yes

—

122

C₂²×C₈

—

123

C₄⋊Q₈

D₄×C₈₉

yes

—

124

C₄⋊Q₈

D₄×C₈₉

yes

—

125

#125

—

126

C₂²⋊Q₈

—

127

C₂²⋊Q₈

—

128

C₄×Q₈

—

129

C₄×Q₈

—

130

#117

—

131

C₂×C₄²

C₂³×C₈₉

yes

characteristic

132

#108

—

133

#106

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#103

—

139

C₄×C₈

—

140

C₄×C₈

D₄×C₈₉

yes

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂²×C₄

C₂×D₄×C₈₉

yes

characteristic

144

C₂²×C₄

#41×C₈₉

yes

—

145

Q₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₄²

C₂×D₄×C₈₉

yes

—

164

C₄²

C₂×D₄×C₈₉

yes

characteristic

165

C₄²

#41×C₈₉

yes

characteristic

166

C₂³

#124×C₈₉

yes

characteristic

167

Q₈

—

168

Q₈

—

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

#120×C₈₉

yes

characteristic

178

C₂×C₄

#121×C₈₉

yes

characteristic

179

C₂×C₄

#121×C₈₉

yes

—

180

C₂×C₄

#123×C₈₉

yes

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₂²

—

186

C₂²

—

187

C₂²

#394×C₈₉

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#453×C₈₉

yes

characteristic

193

C₄

#456×C₈₉

yes

characteristic

194

C₄

#471×C₈₉

yes

—

195

C₄

#473×C₈₉

yes

—

196

C₂

—

197

C₂

C₈₉×#2039

yes

characteristic

198

C₂

C₈₉×#2052

yes

—

	subgroup	G/H
center, socle	C₂²×C₈₉	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₂³×C₈₉	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₉

#394

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₈₉

prime radical

C₂³×C₈₉

#124

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

N	H ≅ G/N	type
96. #10534	C₈₉	direct (N × H)
105. C₈₉	#10534	direct (N × H)

H ≅ G/N

type

96. #10534

C₈₉

direct (N × H)

105. C₈₉

#10534

direct (N × H)

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

subgroup	index
C₈₉×#3255	2
C₈₉×#1848	2
C₈₉×#1561	2
C₈₉×#1493	2
C₈₉×#1309	2
#10534	89

subgroup

index

C₈₉×#3255

C₈₉×#1848

C₈₉×#1561

C₈₉×#1493

C₈₉×#1309

#10534

Subgroups — C₈₉×#10534 #36388382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#10534 #36388382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#10534 #36388382