Group #36454548 — subgroups

Subgroups — #36454548

Order 22784.

430 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 159 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	#26028615	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	#26018136	C₂	1	4	yes	maximal
4	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	—	2	1	no	—
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	—	2	1	no	—
8	#19151496	—	2	2	no	—
9	#19151496	C₄	1	4	yes	—
10	#19151468	C₂²	1	2	yes	—
11	#19151464	—	2	1	no	—
12	#19151353	—	2	1	no	—
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
14	C₄³×C₈₉	C₂²	1	1	yes	characteristic
15	#336×C₈₉	—	2	2	no	—
16	C₂²×#10623494	—	2	2	no	—
17	#15668004	—	4	4	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
21	C₂×#13083553	—	2	4	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	Q₈	1	4	yes	—
24	C₄×#10623494	—	4	2	no	—
25	#15667919	—	2	4	no	—
26	C₂²×C₈×C₈₉	—	2	2	no	—
27	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	D₄	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₈₉	—	2	4	no	—
33	C₂²×Dic₈₉	—	2	4	no	—
34	C₂×#10623494	—	2	4	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₂×#10623494	—	4	2	no	—
37	C₄⋊Dic₈₉	—	4	4	no	—
38	C₄⋊Dic₈₉	—	4	4	no	—
39	C₄⋊Dic₈₉	—	4	4	no	—
40	#13083553	—	4	8	no	—
41	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	—	4	2	no	—
47	C₂×C₈×C₈₉	—	4	4	no	—
48	C₄²×C₈₉	—	2	4	no	—
49	C₄²×C₈₉	—	2	2	no	—
50	C₄²×C₈₉	—	4	2	no	—
51	C₄²×C₈₉	SD₁₆	1	2	yes	—
52	C₄²×C₈₉	Q₁₆	1	2	yes	—
53	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
54	C₄²×C₈₉	#41	1	2	yes	—
55	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	#103	1	4	yes	—
63	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#135	1	4	yes	—
70	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₈₉	—	4	4	no	—
72	C₂×Dic₈₉	—	4	4	no	—
73	C₂×Dic₈₉	—	4	4	no	—
74	C₈×C₈₉	—	8	2	no	—
75	#10623494	—	4	4	no	—
76	C₂²×C₈₉	#391	1	1	yes	characteristic
77	C₂²×C₈₉	#413	1	2	yes	—
78	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
79	C₂²×C₈₉	#416	1	2	yes	—
80	C₂²×C₈₉	#418	1	1	yes	characteristic
81	C₄×C₈₉	—	2	4	no	—
82	C₄×C₈₉	—	2	2	no	—
83	C₄×C₈₉	—	4	2	no	—
84	C₄×C₈₉	#476	1	2	yes	—
85	C₄×C₈₉	#477	1	2	yes	—
86	C₄×C₈₉	#488	1	2	yes	—
87	C₄×C₈₉	#499	1	2	yes	—
88	Dic₈₉	—	8	4	no	—
89	#9897	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	#1542	1	2	yes	—
91	C₂×C₈₉	#1553	1	2	yes	—
92	C₂×C₈₉	#1987	1	1	yes	characteristic
93	C₂×C₈₉	#1991	1	1	yes	characteristic
94	C₂×C₈₉	#1998	1	1	yes	characteristic
95	#2859	—	89	1	no	—
96	#1831	D₈₉	1	1	yes	characteristic
97	C₄×C₄⋊C₈	—	89	1	no	—
98	#1296	—	89	4	no	—
99	C₈₉	#9897	1	1	yes	characteristic, complemented, Sylow
100	C₂×#125	—	178	1	no	—
101	C₂×C₄⋊C₈	—	89	2	no	—
102	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
103	C₂×C₄×C₈	—	178	1	no	—
104	#389	—	89	4	no	—
105	#389	—	178	2	no	—
106	#384	—	89	2	no	—
107	#382	—	178	1	no	—
108	C₄³	C₂×D₈₉	1	1	yes	characteristic
109	#336	—	2	2	no	—
110	#336	—	178	1	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	178	2	no	—
113	#125	—	356	4	no	—
114	C₂×C₄⋊C₄	—	89	4	no	—
115	C₂×C₄⋊C₄	—	178	4	no	—
116	C₂×C₄⋊C₄	—	178	2	no	—
117	C₂×C₄⋊C₄	—	178	2	no	—
118	C₂×C₄²	C₄×D₈₉	1	2	yes	—
119	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
120	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
122	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
123	C₄⋊C₈	—	2	4	no	—
124	C₄⋊C₈	—	89	4	no	—
125	C₄⋊C₈	—	178	2	no	—
126	C₄×C₈	—	356	2	no	—
127	#95	—	178	4	no	—
128	C₂²×C₄	—	178	4	no	—
129	C₂²×C₄	#13083551	1	2	yes	—
130	C₂²×C₄	#13083551	1	2	yes	—
131	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	356	2	no	—
137	C₂×C₈	—	356	2	no	—
138	C₂×C₈	—	178	4	no	—
139	C₄⋊C₄	—	356	4	no	—
140	C₄⋊C₄	—	356	8	no	—
141	C₄⋊C₄	—	356	4	no	—
142	C₄⋊C₄	—	356	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#13083563	1	2	yes	—
147	C₄²	Dic₃₅₆	1	2	yes	—
148	C₄²	Q₈×D₈₉	1	2	yes	—
149	C₄²	#13083573	1	2	yes	—
150	C₂³	C₂×#13083551	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	356	4	no	—
158	C₂×C₄	—	356	4	no	—
159	C₂×C₄	—	356	4	no	—
160	C₂×C₄	#15667937	1	4	yes	—
161	C₂×C₄	#15667977	1	2	yes	—
162	C₂×C₄	#15667992	1	2	yes	—
163	C₂×C₄	#15667999	1	1	yes	characteristic
164	C₂×C₄	#15668010	1	1	yes	characteristic
165	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
167	C₂×C₄	#15668032	1	4	yes	—
168	C₂×C₄	#15668040	1	2	yes	—
169	C₈	—	8	2	no	—
170	C₈	—	356	4	no	—
171	C₂²	#19151503	1	1	yes	characteristic
172	C₂²	#19151580	1	2	yes	—
173	C₂²	C₂×#15667937	1	1	yes	characteristic
174	C₂²	#19151592	1	2	yes	—
175	C₂²	#19151604	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	712	4	no	—
180	C₄	#19151889	1	2	yes	—
181	C₄	#19151894	1	2	yes	—
182	C₄	#19151952	1	2	yes	—
183	C₄	#19151997	1	2	yes	—
184	C₂	#26018907	1	2	yes	—
185	C₂	#26018958	1	2	yes	—
186	C₂	#26021199	1	1	yes	characteristic
187	C₂	#26021216	1	1	yes	characteristic
188	C₂	#26021244	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26028615

C₂

yes

maximal, characteristic

C₄×C₄⋊#10623494

C₂

yes

maximal, characteristic

#26018136

C₂

yes

maximal

C₈₉×#1831

C₂

yes

maximal, characteristic

C₂×#15668004

—

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×C₄×#10623494

—

#19151496

—

#19151496

C₄

yes

—

#19151468

C₂²

yes

—

#19151464

—

#19151353

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

C₂²×#10623494

—

#15668004

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂×#13083553

—

C₄⋊#10623494

—

C₄⋊#10623494

Q₈

yes

—

C₄×#10623494

—

#15667919

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

Q₁₆

yes

—

C₄²×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#103

yes

—

C₂×C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#391

yes

characteristic

C₂²×C₈₉

#413

yes

—

C₂²×C₈₉

C₂×#103

yes

characteristic

C₂²×C₈₉

#416

yes

—

C₂²×C₈₉

#418

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#476

yes

—

C₄×C₈₉

#477

yes

—

C₄×C₈₉

#488

yes

—

C₄×C₈₉

#499

yes

—

Dic₈₉

—

#9897

—

maximal, Sylow

C₂×C₈₉

#1542

yes

—

C₂×C₈₉

#1553

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

C₄×C₄⋊C₈

—

#1296

—

C₈₉

#9897

yes

characteristic, complemented, Sylow

100

C₂×#125

—

178

—

101

C₂×C₄⋊C₈

—

102

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

103

C₂×C₄×C₈

—

178

—

104

#389

—

105

#389

—

178

—

106

#384

—

107

#382

—

178

—

108

C₄³

C₂×D₈₉

yes

characteristic

109

#336

—

110

#336

—

178

—

111

C₂²×C₈

—

112

C₂²×C₈

—

178

—

113

#125

—

356

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

178

—

116

C₂×C₄⋊C₄

—

178

—

117

C₂×C₄⋊C₄

—

178

—

118

C₂×C₄²

C₄×D₈₉

yes

—

119

C₂×C₄²

D₃₅₆

yes

characteristic

120

C₂×C₄²

D₃₅₆

yes

characteristic

121

C₂×C₄²

C₂²⋊D₈₉

yes

—

122

C₂×C₄²

C₂²×D₈₉

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

178

—

126

C₄×C₈

—

356

—

127

#95

—

178

—

128

C₂²×C₄

—

178

—

129

C₂²×C₄

#13083551

yes

—

130

C₂²×C₄

#13083551

yes

—

131

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

132

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

133

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

356

—

137

C₂×C₈

—

356

—

138

C₂×C₈

—

178

—

139

C₄⋊C₄

—

356

—

140

C₄⋊C₄

—

356

—

141

C₄⋊C₄

—

356

—

142

C₄⋊C₄

—

356

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#13083563

yes

—

147

C₄²

Dic₃₅₆

yes

—

148

C₄²

Q₈×D₈₉

yes

—

149

C₄²

#13083573

yes

—

150

C₂³

C₂×#13083551

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

356

—

158

C₂×C₄

—

356

—

159

C₂×C₄

—

356

—

160

C₂×C₄

#15667937

yes

—

161

C₂×C₄

#15667977

yes

—

162

C₂×C₄

#15667992

yes

—

163

C₂×C₄

#15667999

yes

characteristic

164

C₂×C₄

#15668010

yes

characteristic

165

C₂×C₄

C₂×#13083563

yes

characteristic

166

C₂×C₄

C₂×Dic₃₅₆

yes

characteristic

167

C₂×C₄

#15668032

yes

—

168

C₂×C₄

#15668040

yes

—

169

C₈

—

170

C₈

—

356

—

171

C₂²

#19151503

yes

characteristic

172

C₂²

#19151580

yes

—

173

C₂²

C₂×#15667937

yes

characteristic

174

C₂²

#19151592

yes

—

175

C₂²

#19151604

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

712

—

180

C₄

#19151889

yes

—

181

C₄

#19151894

yes

—

182

C₄

#19151952

yes

—

183

C₄

#19151997

yes

—

184

C₂

#26018907

yes

—

185

C₂

#26018958

yes

—

186

C₂

#26021199

yes

characteristic

187

C₂

#26021216

yes

characteristic

188

C₂

#26021244

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1831	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#9897
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×#13083551

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1831

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
99. C₈₉	#9897	semidirect

H ≅ G/N

type

99. C₈₉

#9897

semidirect

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

subgroup	index
#26028615	2
C₄×C₄⋊#10623494	2
#26018136	2
C₈₉×#1831	2
#9897	89

subgroup

index

C₈₉×#1831

Subgroups — #36454548

Conjugacy classes of subgroups

Complements

Maximal subgroups