Group #36451961 — subgroups

Subgroups — #36451961

Order 22784.

436 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 103 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	#26029358	C₂	1	1	yes	maximal, characteristic, complemented
2	#26018966	C₂	1	2	yes	maximal
3	#26018890	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018124	C₂	1	2	yes	maximal, complemented
5	C₈₉×#3138	C₂	1	1	yes	maximal, characteristic
6	#19151729	—	2	4	no	—
7	#19151659	—	2	2	no	—
8	#19151655	C₂²	1	2	yes	—
9	#19151655	C₂²	1	2	yes	—
10	#19151342	—	2	4	no	—
11	#19151340	—	2	2	no	—
12	#533×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
14	#490×C₈₉	—	2	2	no	—
15	#486×C₈₉	C₄	1	4	yes	—
16	#418×C₈₉	—	2	2	no	—
17	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊#10623494	—	2	4	no	—
19	C₄⋊#10623494	—	2	2	no	—
20	C₄⋊#10623494	D₄	1	4	yes	—
21	C₂²⋊#10623494	—	2	2	no	—
22	C₂²⋊#10623494	—	2	2	no	—
23	#15667923	—	2	2	no	—
24	#15667923	—	2	2	no	—
25	#15667923	—	2	2	no	—
26	C₄×#10623494	—	2	2	no	—
27	C₂²×D₄×C₈₉	—	2	1	no	—
28	C₂²×D₄×C₈₉	—	2	2	no	—
29	C₂×M₁₆×C₈₉	—	2	1	no	—
30	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
31	#127×C₈₉	—	2	1	no	—
32	#127×C₈₉	—	2	2	no	—
33	#127×C₈₉	C₂×C₄	1	2	yes	—
34	#125×C₈₉	D₄	1	2	yes	—
35	C₂²⋊Q₈×C₈₉	—	2	2	no	—
36	C₄×Q₈×C₈₉	—	2	2	no	—
37	#117×C₈₉	D₄	1	2	yes	—
38	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	#102×C₈₉	—	2	8	no	—
40	#97×C₈₉	—	2	1	no	—
41	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
42	C₂×#10623494	—	2	4	no	—
43	C₂×#10623494	—	4	2	no	—
44	C₂⁴×C₈₉	—	4	1	no	—
45	C₂×Q₈×C₈₉	—	2	2	no	—
46	C₂×D₄×C₈₉	—	2	2	no	—
47	C₂×D₄×C₈₉	—	2	4	no	—
48	C₂×D₄×C₈₉	—	4	4	no	—
49	C₂×D₄×C₈₉	—	4	1	no	—
50	C₂×D₄×C₈₉	—	4	2	no	—
51	C₂×D₄×C₈₉	—	8	4	no	—
52	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	#41	1	2	yes	—
54	M₁₆×C₈₉	—	4	2	no	—
55	C₂×C₈×C₈₉	—	2	2	no	—
56	C₄⋊C₄×C₈₉	—	2	2	no	—
57	C₄⋊C₄×C₈₉	—	2	4	no	—
58	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
59	C₂²⋊C₄×C₈₉	—	2	2	no	—
60	C₄²×C₈₉	—	2	2	no	—
61	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂×D₄	1	2	yes	—
64	C₂³×C₈₉	—	4	2	no	—
65	C₂³×C₈₉	—	8	2	no	—
66	C₂³×C₈₉	—	8	4	no	—
67	C₂³×C₈₉	#123	1	1	yes	characteristic
68	Q₈×C₈₉	—	8	2	no	—
69	D₄×C₈₉	—	4	4	no	—
70	D₄×C₈₉	—	8	4	no	—
71	D₄×C₈₉	—	8	4	no	—
72	D₄×C₈₉	—	8	2	no	—
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	2	4	no	—
75	C₂×C₄×C₈₉	—	4	2	no	—
76	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#121	1	2	yes	—
80	C₈×C₈₉	—	4	2	no	—
81	#10623494	—	4	4	no	—
82	#10623494	—	8	2	no	—
83	C₂²×C₈₉	—	2	1	no	—
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	—	8	2	no	—
86	C₂²×C₈₉	—	8	4	no	—
87	C₂²×C₈₉	—	16	4	no	—
88	C₂²×C₈₉	#386	1	1	yes	characteristic
89	C₄×C₈₉	—	2	1	no	—
90	C₄×C₈₉	—	2	2	no	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	—	4	4	no	—
93	C₄×C₈₉	#454	1	2	yes	—
94	#9424	—	89	1	no	maximal, Sylow
95	C₂×C₈₉	—	4	1	no	—
96	C₂×C₈₉	—	16	2	no	—
97	C₂×C₈₉	#1888	1	1	yes	characteristic
98	C₂×C₈₉	#1903	1	2	yes	—
99	#3138	D₈₉	1	1	yes	characteristic
100	#2987	—	89	1	no	—
101	#1555	—	89	2	no	—
102	#1537	—	89	1	no	—
103	#1291	—	89	2	no	—
104	C₈₉	#9424	1	1	yes	characteristic, complemented, Sylow
105	#533	C₂×D₈₉	1	1	yes	characteristic
106	C₂×#127	C₂×D₈₉	1	1	yes	characteristic
107	#490	—	2	2	no	—
108	#486	Dic₈₉	1	4	yes	—
109	#446	—	178	4	no	—
110	#433	—	178	2	no	—
111	#432	—	89	2	no	—
112	#432	—	89	2	no	—
113	#418	—	2	2	no	—
114	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
115	#331	—	178	4	no	—
116	#330	—	178	2	no	—
117	C₂²×D₄	—	2	1	no	—
118	C₂²×D₄	—	2	2	no	—
119	C₂×M₁₆	—	2	1	no	—
120	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
121	#127	—	2	2	no	—
122	#127	—	2	1	no	—
123	#127	C₂×Dic₈₉	1	2	yes	—
124	#125	C₂²⋊D₈₉	1	2	yes	—
125	C₂²⋊Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	#117	C₂²⋊D₈₉	1	2	yes	—
128	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
129	C₄⋊C₈	—	89	4	no	—
130	C₄⋊C₈	—	178	4	no	—
131	C₄⋊C₈	—	178	2	no	—
132	#102	—	2	8	no	—
133	C₂²⋊C₈	—	178	2	no	—
134	C₂²⋊C₈	—	178	2	no	—
135	#97	—	2	1	no	—
136	#97	—	178	2	no	—
137	#97	—	178	2	no	—
138	#97	—	178	2	no	—
139	C₄×C₈	—	178	2	no	—
140	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
141	C₂⁴	—	4	1	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	4	4	no	—
146	C₂×D₄	—	4	1	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	8	4	no	—
149	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
150	C₂²×C₄	#13083573	1	2	yes	—
151	M₁₆	—	4	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	356	2	no	—
154	C₂×C₈	—	178	4	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	C₂²⋊Dic₈₉	1	4	yes	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
161	C₄²	D₄×D₈₉	1	2	yes	—
162	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
163	C₂³	—	4	2	no	—
164	C₂³	—	8	2	no	—
165	C₂³	—	8	4	no	—
166	C₂³	#15667993	1	1	yes	characteristic
167	Q₈	—	8	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	8	4	no	—
170	D₄	—	8	4	no	—
171	D₄	—	8	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
176	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
177	C₂×C₄	#15667980	1	1	yes	characteristic
178	C₂×C₄	#15667984	1	2	yes	—
179	C₈	—	4	2	no	—
180	C₈	—	356	4	no	—
181	C₈	—	712	2	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	8	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	—	16	4	no	—
187	C₂²	#19151478	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	#19151770	1	2	yes	—
193	C₂	—	4	1	no	—
194	C₂	—	16	2	no	—
195	C₂	#26020656	1	1	yes	characteristic
196	C₂	#26020734	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26029358

C₂

yes

maximal, characteristic, complemented

#26018966

C₂

yes

maximal

#26018890

C₂

yes

maximal, characteristic, complemented

#26018124

C₂

yes

maximal, complemented

C₈₉×#3138

C₂

yes

maximal, characteristic

#19151729

—

#19151659

—

#19151655

C₂²

yes

—

#19151655

C₂²

yes

—

#19151342

—

#19151340

—

#533×C₈₉

C₂²

yes

characteristic

C₂×#127×C₈₉

C₂²

yes

characteristic

#490×C₈₉

—

#486×C₈₉

C₄

yes

—

#418×C₈₉

—

C₄×M₁₆×C₈₉

C₂²

yes

—

—

D₄

yes

—

—

—

—

—

—

—

C₂²×D₄×C₈₉

—

C₂²×D₄×C₈₉

—

C₂×M₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

—

#127×C₈₉

—

#127×C₈₉

—

#127×C₈₉

C₂×C₄

yes

—

#125×C₈₉

D₄

yes

—

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#102×C₈₉

—

#97×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₂⁴×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#123

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂×C₄×C₈₉

#121

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#386

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#454

yes

—

#9424

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1888

yes

characteristic

C₂×C₈₉

#1903

yes

—

#3138

D₈₉

yes

characteristic

100

#2987

—

101

#1555

—

102

#1537

—

103

#1291

—

104

C₈₉

#9424

yes

characteristic, complemented, Sylow

105

#533

C₂×D₈₉

yes

characteristic

106

C₂×#127

C₂×D₈₉

yes

characteristic

107

#490

—

108

#486

Dic₈₉

yes

—

109

#446

—

178

—

110

#433

—

178

—

111

#432

—

112

#432

—

113

#418

—

114

C₄×M₁₆

C₂×D₈₉

yes

characteristic

115

#331

—

178

—

116

#330

—

178

—

117

C₂²×D₄

—

118

C₂²×D₄

—

119

C₂×M₁₆

—

120

C₄⋊Q₈

C₂×Dic₈₉

yes

—

121

#127

—

122

#127

—

123

#127

C₂×Dic₈₉

yes

—

124

#125

C₂²⋊D₈₉

yes

—

125

C₂²⋊Q₈

—

126

C₄×Q₈

—

127

#117

C₂²⋊D₈₉

yes

—

128

C₂×C₄²

C₂²×D₈₉

yes

characteristic

129

C₄⋊C₈

—

130

C₄⋊C₈

—

178

—

131

C₄⋊C₈

—

178

—

132

#102

—

133

C₂²⋊C₈

—

178

—

134

C₂²⋊C₈

—

178

—

135

#97

—

136

#97

—

178

—

137

#97

—

178

—

138

#97

—

178

—

139

C₄×C₈

—

178

—

140

C₄×C₈

C₂×Dic₈₉

yes

—

141

C₂⁴

—

142

C₂×Q₈

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

150

C₂²×C₄

#13083573

yes

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

356

—

154

C₂×C₈

—

178

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

C₂²⋊Dic₈₉

yes

—

158

C₂²⋊C₄

—

159

C₄²

—

160

C₄²

C₂²×Dic₈₉

yes

characteristic

161

C₄²

D₄×D₈₉

yes

—

162

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

163

C₂³

—

164

C₂³

—

165

C₂³

—

166

C₂³

#15667993

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

176

C₂×C₄

D₄×Dic₈₉

yes

—

177

C₂×C₄

#15667980

yes

characteristic

178

C₂×C₄

#15667984

yes

—

179

C₈

—

180

C₈

—

356

—

181

C₈

—

712

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#19151478

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#19151770

yes

—

193

C₂

—

194

C₂

—

195

C₂

#26020656

yes

characteristic

196

C₂

#26020734

yes

—

	subgroup	G/H
center	C₂²	#19151478
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#3138	C₂
socle	C₂²×C₈₉	#386
prime radical	C₂×#127×C₈₉	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#3138	D₈₉
hypocenter	C₈₉	#9424
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3138

subgroup

G/H

center

C₂²

#19151478

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#3138

C₂

socle

C₂²×C₈₉

#386

prime radical

C₂×#127×C₈₉

C₂²

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3138

N	H ≅ G/N	type
1. #26029358	C₂	semidirect
3. #26018890	C₂	semidirect
4. #26018124	C₂	semidirect
104. C₈₉	#9424	semidirect

H ≅ G/N

type

1. #26029358

C₂

semidirect

3. #26018890

C₂

semidirect

4. #26018124

C₂

semidirect

104. C₈₉

#9424

semidirect

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

subgroup	index
#26029358	2
#26018966	2
#26018890	2
#26018124	2
C₈₉×#3138	2
#9424	89

subgroup

index

C₈₉×#3138

Subgroups — #36451961

Conjugacy classes of subgroups

Complements

Maximal subgroups