Group #36454265 — subgroups

Subgroups — #36454265

Order 22784.

710 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 191 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×#19151975	C₂	1	1	yes	maximal, characteristic
2	#26021057	C₂	1	4	yes	maximal, complemented
3	C₂×#19151319	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₇₁₂	—	2	2	no	—
6	C₂×#15668009	C₂²	1	2	yes	—
7	#19151975	—	2	2	no	—
8	#19151975	C₄	1	4	yes	complemented
9	C₂×#15667932	—	2	8	no	—
10	C₂×C₄×#10623494	C₂²	1	2	yes	—
11	#19151319	—	2	2	no	—
12	C₂×#106×C₈₉	—	2	2	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#384×C₈₉	C₂²	1	2	yes	—
15	C₂²×D₃₅₆	—	2	4	no	—
16	C₂²×D₃₅₆	—	2	2	no	—
17	C₂×D₇₁₂	—	2	8	no	—
18	C₂×D₇₁₂	—	4	4	no	—
19	C₂×D₇₁₂	—	4	4	no	—
20	C₂²×#10623494	D₄	1	4	yes	—
21	#15668009	—	2	2	no	—
22	#15668009	—	2	4	no	—
23	#15668009	C₂×C₄	1	4	yes	—
24	#15667932	—	4	16	no	—
25	C₄×#10623494	—	2	2	no	—
26	C₄×#10623494	—	2	2	no	—
27	C₂²×C₈×C₈₉	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	#106×C₈₉	—	4	4	no	—
32	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₈₉	D₄	1	2	yes	—
34	C₂³×D₈₉	—	4	2	no	—
35	C₂×D₃₅₆	—	2	4	no	—
36	C₂×D₃₅₆	—	2	8	no	—
37	C₂×D₃₅₆	—	4	4	no	—
38	C₂×D₃₅₆	—	4	4	no	—
39	C₂×D₃₅₆	—	4	8	no	—
40	C₂×D₃₅₆	—	4	8	no	—
41	C₂×D₃₅₆	—	4	4	no	—
42	C₂×D₃₅₆	—	4	2	no	—
43	D₇₁₂	—	4	16	no	—
44	C₂×#10623494	—	2	4	no	—
45	C₂×#10623494	—	2	8	no	—
46	C₂²×C₄×C₈₉	—	4	2	no	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
50	C₂×C₈×C₈₉	D₈	1	4	yes	—
51	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
52	C₄⋊C₄×C₈₉	—	4	4	no	—
53	C₄⋊C₄×C₈₉	—	4	8	no	—
54	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₈₉	#41	1	2	yes	—
57	C₂³×C₈₉	#127	1	1	yes	characteristic
58	C₂²×D₈₉	—	4	4	no	—
59	C₂²×D₈₉	—	8	2	no	—
60	C₂²×D₈₉	—	8	2	no	—
61	C₂²×D₈₉	—	8	4	no	—
62	C₂²×D₈₉	—	8	4	no	—
63	C₂×C₄×C₈₉	—	8	2	no	—
64	C₂×C₄×C₈₉	—	8	4	no	—
65	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₈₉	#121	1	2	yes	—
68	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	8	yes	—
72	D₃₅₆	—	4	8	no	—
73	D₃₅₆	—	4	16	no	—
74	D₃₅₆	—	8	8	no	—
75	D₃₅₆	—	8	4	no	—
76	C₈×C₈₉	#102	1	8	yes	complemented
77	#10623494	—	4	4	no	—
78	C₂²×C₈₉	#390	1	1	yes	characteristic
79	C₂²×C₈₉	#492	1	1	yes	characteristic
80	C₂²×C₈₉	#493	1	1	yes	characteristic
81	C₂²×C₈₉	#496	1	4	yes	—
82	C₂×D₈₉	—	8	4	no	—
83	C₂×D₈₉	—	8	8	no	—
84	C₂×D₈₉	—	16	2	no	—
85	C₂×D₈₉	—	16	2	no	—
86	C₂×D₈₉	—	16	4	no	—
87	C₄×C₈₉	—	16	2	no	—
88	C₄×C₈₉	C₂×#102	1	2	yes	—
89	C₄×C₈₉	#442	1	4	yes	—
90	C₄×C₈₉	#469	1	4	yes	—
91	C₄×C₈₉	#486	1	2	yes	—
92	#9837	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	#1715	1	2	yes	—
94	C₂×C₈₉	#1721	1	2	yes	—
95	C₂×C₈₉	#1957	1	1	yes	characteristic
96	C₂×C₈₉	#1972	1	2	yes	—
97	D₈₉	—	16	4	no	—
98	C₂×#493	—	89	1	no	—
99	#1957	—	89	4	no	—
100	#1844	D₈₉	1	1	yes	characteristic, complemented
101	C₂×#322	—	89	1	no	—
102	C₈₉	#9837	1	1	yes	characteristic, complemented, Sylow
103	C₂²×D₈	—	178	2	no	—
104	C₂×#127	—	89	2	no	—
105	#493	—	89	4	no	—
106	#493	—	178	2	no	—
107	C₂×#106	—	2	2	no	—
108	C₂×#102	—	178	8	no	—
109	C₂×C₄×C₈	—	89	2	no	—
110	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
111	#384	C₂×D₈₉	1	2	yes	—
112	#322	—	178	2	no	—
113	C₂²×D₄	—	178	4	no	—
114	C₂²×D₄	—	178	2	no	—
115	C₂×D₈	—	356	4	no	—
116	C₂×D₈	—	356	4	no	—
117	C₂×D₈	—	178	8	no	—
118	C₂²×C₈	—	89	4	no	—
119	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
120	#127	—	89	4	no	—
121	#127	—	178	2	no	—
122	#127	—	178	4	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
126	#106	—	4	4	no	—
127	#102	—	356	16	no	—
128	C₄×C₈	—	178	2	no	—
129	C₄×C₈	—	178	2	no	—
130	C₄×C₈	C₄×D₈₉	1	2	yes	—
131	C₄×C₈	D₃₅₆	1	2	yes	—
132	C₂⁴	—	356	2	no	—
133	C₂×D₄	—	356	4	no	—
134	C₂×D₄	—	356	2	no	—
135	C₂×D₄	—	356	4	no	—
136	C₂×D₄	—	356	4	no	—
137	C₂×D₄	—	356	8	no	—
138	C₂×D₄	—	356	8	no	—
139	C₂×D₄	—	178	8	no	—
140	C₂×D₄	—	178	4	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	D₄×D₈₉	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
144	D₈	—	356	16	no	—
145	C₂×C₈	—	178	4	no	—
146	C₂×C₈	—	178	8	no	—
147	C₂×C₈	#13083551	1	4	yes	—
148	C₂×C₈	#13083559	1	4	yes	—
149	C₂×C₈	#13083561	1	4	yes	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
153	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
154	C₄²	#13083575	1	2	yes	—
155	C₂³	—	356	4	no	—
156	C₂³	—	712	2	no	—
157	C₂³	—	712	2	no	—
158	C₂³	—	712	4	no	—
159	C₂³	—	712	4	no	—
160	C₂³	#15668008	1	1	yes	characteristic
161	D₄	—	356	16	no	—
162	D₄	—	356	8	no	—
163	D₄	—	712	4	no	—
164	D₄	—	712	8	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
168	C₂×C₄	#15667974	1	2	yes	—
169	C₂×C₄	#15667987	1	2	yes	—
170	C₂×C₄	#15668001	1	1	yes	characteristic
171	C₂×C₄	C₂×#13083559	1	2	yes	—
172	C₂×C₄	C₂×#13083561	1	2	yes	—
173	C₂×C₄	#15668039	1	8	yes	—
174	C₈	—	356	4	no	—
175	C₈	#15667932	1	8	yes	complemented
176	C₂²	—	1424	2	no	—
177	C₂²	—	1424	2	no	—
178	C₂²	—	1424	4	no	—
179	C₂²	—	712	8	no	—
180	C₂²	—	712	4	no	—
181	C₂²	#19151499	1	1	yes	characteristic
182	C₂²	#19151972	1	1	yes	characteristic
183	C₂²	#19151974	1	1	yes	characteristic
184	C₂²	#19151984	1	4	yes	—
185	C₄	—	16	2	no	—
186	C₄	C₂×#15667932	1	2	yes	—
187	C₄	#19151709	1	4	yes	—
188	C₄	#19151852	1	4	yes	—
189	C₄	#19151946	1	2	yes	—
190	C₂	—	1424	4	no	—
191	C₂	#26019881	1	2	yes	—
192	C₂	#26019916	1	2	yes	—
193	C₂	#26021057	1	1	yes	characteristic
194	C₂	#26021126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151975

C₂

yes

maximal, characteristic

#26021057

C₂

yes

maximal, complemented

C₂×#19151319

C₂

yes

maximal, characteristic, complemented

C₈₉×#1844

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₁₂

—

C₂×#15668009

C₂²

yes

—

#19151975

—

#19151975

C₄

yes

complemented

C₂×#15667932

—

C₂×C₄×#10623494

C₂²

yes

—

#19151319

—

C₂×#106×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₂²

yes

—

C₂²×D₃₅₆

—

C₂²×D₃₅₆

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂²×#10623494

D₄

yes

—

#15668009

—

#15668009

—

#15668009

C₂×C₄

yes

—

#15667932

—

C₄×#10623494

—

C₄×#10623494

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#106×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

C₂×#10623494

—

C₂×#10623494

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

#127

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

#102

yes

complemented

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#102

yes

—

C₄×C₈₉

#442

yes

—

C₄×C₈₉

#469

yes

—

C₄×C₈₉

#486

yes

—

#9837

—

maximal, Sylow

C₂×C₈₉

#1715

yes

—

C₂×C₈₉

#1721

yes

—

C₂×C₈₉

#1957

yes

characteristic

C₂×C₈₉

#1972

yes

—

D₈₉

—

C₂×#493

—

#1957

—

100

#1844

D₈₉

yes

characteristic, complemented

101

C₂×#322

—

102

C₈₉

#9837

yes

characteristic, complemented, Sylow

103

C₂²×D₈

—

178

—

104

C₂×#127

—

105

#493

—

106

#493

—

178

—

107

C₂×#106

—

108

C₂×#102

—

178

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

111

#384

C₂×D₈₉

yes

—

112

#322

—

178

—

113

C₂²×D₄

—

178

—

114

C₂²×D₄

—

178

—

115

C₂×D₈

—

356

—

116

C₂×D₈

—

356

—

117

C₂×D₈

—

178

—

118

C₂²×C₈

—

119

C₂²×C₈

C₂²⋊D₈₉

yes

—

120

#127

—

121

#127

—

178

—

122

#127

—

178

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

C₂²×D₈₉

yes

characteristic

126

#106

—

127

#102

—

356

—

128

C₄×C₈

—

178

—

129

C₄×C₈

—

178

—

130

C₄×C₈

C₄×D₈₉

yes

—

131

C₄×C₈

D₃₅₆

yes

—

132

C₂⁴

—

356

—

133

C₂×D₄

—

356

—

134

C₂×D₄

—

356

—

135

C₂×D₄

—

356

—

136

C₂×D₄

—

356

—

137

C₂×D₄

—

356

—

138

C₂×D₄

—

356

—

139

C₂×D₄

—

178

—

140

C₂×D₄

—

178

—

141

C₂²×C₄

—

142

C₂²×C₄

D₄×D₈₉

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

144

D₈

—

356

—

145

C₂×C₈

—

178

—

146

C₂×C₈

—

178

—

147

C₂×C₈

#13083551

yes

—

148

C₂×C₈

#13083559

yes

—

149

C₂×C₈

#13083561

yes

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×C₄×D₈₉

yes

characteristic

153

C₄²

C₂×D₃₅₆

yes

characteristic

154

C₄²

#13083575

yes

—

155

C₂³

—

356

—

156

C₂³

—

712

—

157

C₂³

—

712

—

158

C₂³

—

712

—

159

C₂³

—

712

—

160

C₂³

#15668008

yes

characteristic

161

D₄

—

356

—

162

D₄

—

356

—

163

D₄

—

712

—

164

D₄

—

712

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

C₂×#13083551

yes

characteristic

168

C₂×C₄

#15667974

yes

—

169

C₂×C₄

#15667987

yes

—

170

C₂×C₄

#15668001

yes

characteristic

171

C₂×C₄

C₂×#13083559

yes

—

172

C₂×C₄

C₂×#13083561

yes

—

173

C₂×C₄

#15668039

yes

—

174

C₈

—

356

—

175

C₈

#15667932

yes

complemented

176

C₂²

—

1424

—

177

C₂²

—

1424

—

178

C₂²

—

1424

—

179

C₂²

—

712

—

180

C₂²

—

712

—

181

C₂²

#19151499

yes

characteristic

182

C₂²

#19151972

yes

characteristic

183

C₂²

#19151974

yes

characteristic

184

C₂²

#19151984

yes

—

185

C₄

—

186

C₄

C₂×#15667932

yes

—

187

C₄

#19151709

yes

—

188

C₄

#19151852

yes

—

189

C₄

#19151946

yes

—

190

C₂

—

1424

—

191

C₂

#26019881

yes

—

192

C₂

#26019916

yes

—

193

C₂

#26021057

yes

characteristic

194

C₂

#26021126

yes

—

	subgroup	G/H
center	C₂³	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1844	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#19151975	C₂
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1844	D₈₉
hypocenter	C₈₉	#9837
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#15668008

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1844

C₂

C₂³×C₈₉

C₂

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
2. #26021057	C₂	semidirect
3. C₂×#19151319	C₂	semidirect
4. C₈₉×#1844	C₂	semidirect
8. #19151975	C₄	semidirect
76. C₈×C₈₉	#102	semidirect
100. #1844	D₈₉	semidirect
102. C₈₉	#9837	semidirect
175. C₈	#15667932	semidirect

H ≅ G/N

type

2. #26021057

C₂

semidirect

3. C₂×#19151319

C₂

semidirect

4. C₈₉×#1844

C₂

semidirect

8. #19151975

C₄

semidirect

76. C₈×C₈₉

#102

semidirect

100. #1844

D₈₉

semidirect

102. C₈₉

#9837

semidirect

175. C₈

#15667932

semidirect

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

subgroup	index
C₂×#19151975	2
#26021057	2
C₂×#19151319	2
C₈₉×#1844	2
#9837	89

subgroup

index

C₈₉×#1844

Subgroups — #36454265

Conjugacy classes of subgroups

Complements

Maximal subgroups