Group #36442702 — subgroups

Subgroups — #36442702

Order 22784.

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

normal subgroups only

This group has 468 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	#26020001	C₂	1	2	yes	maximal
2	#26018078	C₂	1	4	yes	maximal
3	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667923	—	2	2	no	—
5	#19151353	—	2	2	no	—
6	#19151353	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₈₉	C₄	1	2	yes	—
9	#391×C₈₉	C₂²	1	2	yes	—
10	#389×C₈₉	—	2	2	no	—
11	#384×C₈₉	C₄	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
13	C₄³×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×#10623494	—	2	4	no	—
15	#15667923	—	2	8	no	—
16	C₂²×Q₈×C₈₉	C₂×C₄	1	2	yes	—
17	C₄⋊Q₈×C₈₉	—	2	4	no	—
18	C₄×Q₈×C₈₉	—	4	4	no	—
19	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
20	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₈₉	—	2	2	no	—
24	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	D₄	1	2	yes	—
27	C₂×C₄²×C₈₉	D₄	1	2	yes	—
28	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
29	#95×C₈₉	—	2	4	no	—
30	C₂×#10623494	—	2	8	no	—
31	C₂×#10623494	—	4	4	no	—
32	C₂×#10623494	—	4	4	no	—
33	C₂×Q₈×C₈₉	—	2	2	no	—
34	C₂×Q₈×C₈₉	—	4	2	no	—
35	C₂×Q₈×C₈₉	—	4	2	no	—
36	C₂×Q₈×C₈₉	M₁₆	1	4	yes	—
37	C₂²×C₄×C₈₉	—	2	2	no	—
38	C₂²×C₄×C₈₉	—	2	2	no	—
39	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
44	C₄⋊C₄×C₈₉	—	2	8	no	—
45	C₄⋊C₄×C₈₉	—	4	4	no	—
46	C₄⋊C₄×C₈₉	—	4	2	no	—
47	C₄⋊C₄×C₈₉	—	4	4	no	—
48	C₄²×C₈₉	—	2	2	no	—
49	C₄²×C₈₉	—	2	4	no	—
50	C₄²×C₈₉	—	2	2	no	—
51	C₄²×C₈₉	—	2	4	no	—
52	C₄²×C₈₉	—	4	1	no	—
53	C₄²×C₈₉	—	4	4	no	—
54	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	Q₈×C₈₉	—	4	4	no	—
56	Q₈×C₈₉	—	4	4	no	—
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	2	1	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	1	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	4	no	—
65	C₂×C₄×C₈₉	—	2	1	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	4	2	no	—
68	C₂×C₄×C₈₉	—	4	2	no	—
69	C₂×C₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	—	4	4	no	—
71	C₂×C₄×C₈₉	#100	1	2	yes	—
72	C₂×C₄×C₈₉	#104	1	4	yes	—
73	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₈₉	#137	1	4	yes	—
75	#10623494	—	4	8	no	—
76	C₂²×C₈₉	#407	1	1	yes	characteristic
77	C₂²×C₈₉	#410	1	1	yes	characteristic
78	C₂²×C₈₉	C₂×#100	1	1	yes	characteristic
79	C₂²×C₈₉	#419	1	2	yes	—
80	C₂²×C₈₉	C₂×#104	1	2	yes	—
81	C₄×C₈₉	—	2	2	no	—
82	C₄×C₈₉	—	2	2	no	—
83	C₄×C₈₉	—	2	4	no	—
84	C₄×C₈₉	—	2	4	no	—
85	C₄×C₈₉	—	4	4	no	—
86	C₄×C₈₉	—	4	1	no	—
87	C₄×C₈₉	—	8	2	no	—
88	#7639	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	#1466	1	1	yes	characteristic
90	C₂×C₈₉	#1491	1	2	yes	—
91	C₂×C₈₉	#1506	1	2	yes	—
92	C₂×C₈₉	#1525	1	1	yes	characteristic
93	C₂×C₈₉	#1558	1	1	yes	characteristic
94	C₄×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
95	#1745	—	89	2	no	—
96	#1272	—	89	4	no	—
97	C₈₉	#7639	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
99	C₂×C₄×Q₈	Dic₈₉	1	2	yes	—
100	C₂×#97	—	178	2	no	—
101	#391	C₂×D₈₉	1	2	yes	—
102	#389	—	2	2	no	—
103	#384	Dic₈₉	1	1	yes	characteristic
104	C₄×C₄⋊C₄	—	2	2	no	—
105	C₄³	C₂×D₈₉	1	1	yes	characteristic
106	#336	—	89	4	no	—
107	#336	—	178	2	no	—
108	C₂²×Q₈	C₂×Dic₈₉	1	2	yes	—
109	C₂²×C₈	—	178	4	no	—
110	C₄⋊Q₈	—	2	4	no	—
111	C₄×Q₈	—	4	4	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
116	C₂×C₄²	—	2	2	no	—
117	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
119	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
120	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
121	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
122	#97	—	178	8	no	—
123	#95	—	2	4	no	—
124	C₂×Q₈	—	2	2	no	—
125	C₂×Q₈	—	4	2	no	—
126	C₂×Q₈	—	4	2	no	—
127	C₂×Q₈	#13083558	1	4	yes	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
131	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
133	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
135	C₂×C₈	—	356	4	no	—
136	C₂×C₈	—	356	4	no	—
137	C₂×C₈	—	178	8	no	—
138	C₄⋊C₄	—	2	8	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	4	no	—
146	C₄²	—	4	1	no	—
147	C₄²	—	4	4	no	—
148	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
149	Q₈	—	4	4	no	—
150	Q₈	—	4	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	#15667929	1	2	yes	—
166	C₂×C₄	#15667939	1	4	yes	—
167	C₂×C₄	C₂×#13083558	1	2	yes	—
168	C₂×C₄	#15668043	1	4	yes	—
169	C₈	—	356	8	no	—
170	C₂²	#19151562	1	1	yes	characteristic
171	C₂²	#19151572	1	1	yes	characteristic
172	C₂²	C₂×#15667929	1	1	yes	characteristic
173	C₂²	#19151607	1	2	yes	—
174	C₂²	C₂×#15667939	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	4	no	—
179	C₄	—	4	1	no	—
180	C₄	—	4	4	no	—
181	C₄	—	8	2	no	—
182	C₂	#26018593	1	1	yes	characteristic
183	C₂	#26018699	1	2	yes	—
184	C₂	#26018764	1	2	yes	—
185	C₂	#26018844	1	1	yes	characteristic
186	C₂	#26018978	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020001

C₂

yes

maximal

#26018078

C₂

yes

maximal

C₄×C₄⋊Q₈×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667923

—

#19151353

—

#19151353

C₂²

yes

—

C₂×C₄⋊Q₈×C₈₉

C₄

yes

characteristic

C₂×C₄×Q₈×C₈₉

C₄

yes

—

#391×C₈₉

C₂²

yes

—

#389×C₈₉

—

#384×C₈₉

C₄

yes

characteristic

C₄×C₄⋊C₄×C₈₉

—

C₄³×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

—

#15667923

—

C₂²×Q₈×C₈₉

C₂×C₄

yes

—

C₄⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#95×C₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

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

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#100

yes

—

C₂×C₄×C₈₉

#104

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

#137

yes

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

C₂×#104

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#1506

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₄×C₄⋊Q₈

D₈₉

yes

characteristic

#1745

—

#1272

—

C₈₉

#7639

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

Dic₈₉

yes

characteristic

C₂×C₄×Q₈

Dic₈₉

yes

—

100

C₂×#97

—

178

—

101

#391

C₂×D₈₉

yes

—

102

#389

—

103

#384

Dic₈₉

yes

characteristic

104

C₄×C₄⋊C₄

—

105

C₄³

C₂×D₈₉

yes

characteristic

106

#336

—

107

#336

—

178

—

108

C₂²×Q₈

C₂×Dic₈₉

yes

—

109

C₂²×C₈

—

178

—

110

C₄⋊Q₈

—

111

C₄×Q₈

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

C₂×Dic₈₉

yes

—

116

C₂×C₄²

—

117

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

118

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

119

C₂×C₄²

C₂²⋊D₈₉

yes

—

120

C₂×C₄²

C₂²⋊D₈₉

yes

—

121

C₂×C₄²

C₂²×D₈₉

yes

characteristic

122

#97

—

178

—

123

#95

—

124

C₂×Q₈

—

125

C₂×Q₈

—

126

C₂×Q₈

—

127

C₂×Q₈

#13083558

yes

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂²⋊Dic₈₉

yes

characteristic

131

C₂²×C₄

C₂²⋊Dic₈₉

yes

characteristic

132

C₂²×C₄

C₂²⋊Dic₈₉

yes

—

133

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

134

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

—

135

C₂×C₈

—

356

—

136

C₂×C₈

—

356

—

137

C₂×C₈

—

178

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₂³

C₂×C₂²⋊Dic₈₉

yes

characteristic

149

Q₈

—

150

Q₈

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#15667929

yes

—

166

C₂×C₄

#15667939

yes

—

167

C₂×C₄

C₂×#13083558

yes

—

168

C₂×C₄

#15668043

yes

—

169

C₈

—

356

—

170

C₂²

#19151562

yes

characteristic

171

C₂²

#19151572

yes

characteristic

172

C₂²

C₂×#15667929

yes

characteristic

173

C₂²

#19151607

yes

—

174

C₂²

C₂×#15667939

yes

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₂

#26018593

yes

characteristic

183

C₂

#26018699

yes

—

184

C₂

#26018764

yes

—

185

C₂

#26018844

yes

characteristic

186

C₂

#26018978

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#7639
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₈₉

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₄×C₄⋊Q₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

hypercenter

C₄×C₄⋊Q₈

D₈₉

hypocenter

C₈₉

#7639

LCS

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

N	H ≅ G/N	type
97. C₈₉	#7639	semidirect

H ≅ G/N

type

97. C₈₉

#7639

semidirect

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

subgroup	index
#26020001	2
#26018078	2
C₄×C₄⋊Q₈×C₈₉	2
#7639	89

subgroup

index

#26020001

#26018078

C₄×C₄⋊Q₈×C₈₉

#7639

Subgroups — #36442702

Conjugacy classes of subgroups

Complements

Maximal subgroups