Group #36459492 — subgroups

Subgroups — #36459492

Order 22784.

410 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 147 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	#26021214	C₂	1	2	yes	maximal
2	#26020412	C₂	1	1	yes	maximal, characteristic
3	#26020379	C₂	1	2	yes	maximal
4	#26018079	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667946	—	2	2	no	—
7	C₂×C₄×#10623494	C₂²	1	2	yes	—
8	#19151468	C₂²	1	2	yes	—
9	#19151360	—	2	4	no	—
10	#19151360	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×#103×C₈₉	—	2	2	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#384×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×#10623494	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₈₉	—	2	2	no	—
17	C₂×C₄⋊Dic₈₉	—	2	2	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	#15667946	—	4	4	no	—
20	C₄×#10623494	—	2	2	no	—
21	C₄×#10623494	—	2	2	no	—
22	C₂²×Q₈×C₈₉	—	2	1	no	—
23	C₂²×C₈×C₈₉	D₄	1	2	yes	—
24	C₄⋊Q₈×C₈₉	—	2	2	no	—
25	C₄⋊Q₈×C₈₉	—	2	1	no	—
26	C₄⋊Q₈×C₈₉	Q₈	1	2	yes	—
27	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	#103×C₈₉	—	4	4	no	—
32	C₄×C₈×C₈₉	—	2	2	no	—
33	C₂²×Dic₈₉	—	4	2	no	—
34	C₂×#10623494	—	2	4	no	—
35	C₂×#10623494	—	2	4	no	—
36	C₂×#10623494	—	2	4	no	—
37	C₄⋊Dic₈₉	—	4	4	no	—
38	C₄⋊Dic₈₉	—	4	4	no	—
39	C₄⋊Dic₈₉	—	4	4	no	—
40	C₂×Q₈×C₈₉	—	2	2	no	—
41	C₂×Q₈×C₈₉	—	4	1	no	—
42	C₂×Q₈×C₈₉	—	4	2	no	—
43	C₂×Q₈×C₈₉	—	4	2	no	—
44	C₂²×C₄×C₈₉	—	4	1	no	—
45	C₂²×C₄×C₈₉	—	4	1	no	—
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₈₉	—	2	2	no	—
49	C₂×C₈×C₈₉	—	2	2	no	—
50	C₂×C₈×C₈₉	D₈	1	4	yes	—
51	C₄⋊C₄×C₈₉	—	2	4	no	—
52	C₄⋊C₄×C₈₉	—	4	2	no	—
53	C₄⋊C₄×C₈₉	—	4	2	no	—
54	C₄⋊C₄×C₈₉	—	4	4	no	—
55	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₈₉	#41	1	2	yes	—
57	C₄²×C₈₉	#41	1	1	yes	characteristic
58	C₂³×C₈₉	#127	1	1	yes	characteristic
59	Q₈×C₈₉	—	4	4	no	—
60	Q₈×C₈₉	—	8	2	no	—
61	C₂×C₄×C₈₉	—	4	2	no	—
62	C₂×C₄×C₈₉	—	8	1	no	—
63	C₂×C₄×C₈₉	—	8	1	no	—
64	C₂×C₄×C₈₉	—	8	1	no	—
65	C₂×C₄×C₈₉	—	8	2	no	—
66	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#123	1	2	yes	—
70	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
71	C₂×C₄×C₈₉	#135	1	4	yes	—
72	C₂×C₄×C₈₉	#137	1	4	yes	—
73	C₂×C₄×C₈₉	#137	1	2	yes	—
74	C₂×Dic₈₉	—	8	2	no	—
75	C₂×Dic₈₉	—	8	2	no	—
76	C₂×Dic₈₉	—	8	2	no	—
77	C₈×C₈₉	—	2	4	no	—
78	#10623494	—	4	4	no	—
79	C₂²×C₈₉	#399	1	1	yes	characteristic
80	C₂²×C₈₉	#493	1	1	yes	characteristic
81	C₂²×C₈₉	#495	1	2	yes	—
82	C₂²×C₈₉	#497	1	1	yes	characteristic
83	C₂²×C₈₉	#497	1	2	yes	—
84	C₄×C₈₉	—	8	2	no	—
85	C₄×C₈₉	—	16	1	no	—
86	C₄×C₈₉	#474	1	2	yes	—
87	C₄×C₈₉	#479	1	4	yes	—
88	C₄×C₈₉	#480	1	2	yes	—
89	C₄×C₈₉	#485	1	4	yes	—
90	Dic₈₉	—	16	2	no	—
91	#10676	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	#1709	1	1	yes	characteristic
93	C₂×C₈₉	#1718	1	2	yes	—
94	C₂×C₈₉	#1722	1	1	yes	characteristic
95	C₂×C₈₉	#2093	1	1	yes	characteristic
96	C₂×C₈₉	#2097	1	2	yes	—
97	#1990	—	89	2	no	—
98	#1956	D₈₉	1	1	yes	characteristic
99	#1845	—	89	1	no	—
100	#1834	—	89	2	no	—
101	#1273	—	89	1	no	—
102	C₈₉	#10676	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
104	C₂×#107	—	178	2	no	—
105	C₂×#103	—	2	2	no	—
106	C₂×C₄×C₈	—	89	2	no	—
107	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
108	#384	—	89	2	no	—
109	#384	C₂×D₈₉	1	1	yes	characteristic
110	#340	—	178	4	no	—
111	#340	—	178	4	no	—
112	C₂²×Q₈	—	2	1	no	—
113	C₂²×C₈	—	89	4	no	—
114	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
115	C₄⋊Q₈	—	2	2	no	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	Dic₁₇₈	1	2	yes	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	—	178	2	no	—
122	C₂×C₄⋊C₄	—	178	2	no	—
123	C₂×C₄⋊C₄	—	178	2	no	—
124	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
125	#107	—	356	4	no	—
126	#103	—	4	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	178	2	no	—
129	C₄×C₈	—	178	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	—	4	1	no	—
136	C₂²×C₄	—	356	2	no	—
137	C₂²×C₄	D₄×D₈₉	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	178	4	no	—
142	C₂×C₈	—	178	4	no	—
143	C₂×C₈	—	178	4	no	—
144	C₂×C₈	#13083559	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	356	4	no	—
150	C₄⋊C₄	—	356	4	no	—
151	C₄⋊C₄	—	356	4	no	—
152	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
153	C₄²	#13083573	1	2	yes	—
154	C₄²	#13083574	1	1	yes	characteristic
155	C₂³	#15668008	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	712	2	no	—
164	C₂×C₄	—	712	2	no	—
165	C₂×C₄	—	712	2	no	—
166	C₂×C₄	#15667989	1	2	yes	—
167	C₂×C₄	#15667991	1	1	yes	characteristic
168	C₂×C₄	#15667993	1	1	yes	characteristic
169	C₂×C₄	#15667994	1	2	yes	—
170	C₂×C₄	C₂×#13083559	1	2	yes	—
171	C₂×C₄	#15668035	1	4	yes	—
172	C₂×C₄	#15668041	1	4	yes	—
173	C₂×C₄	#15668042	1	2	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	356	4	no	—
176	C₂²	#19151534	1	1	yes	characteristic
177	C₂²	#19151974	1	1	yes	characteristic
178	C₂²	#19151983	1	2	yes	—
179	C₂²	#19151989	1	2	yes	—
180	C₂²	#19151993	1	1	yes	characteristic
181	C₄	—	8	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	1424	2	no	—
184	C₄	#19151878	1	2	yes	—
185	C₄	#19151907	1	4	yes	—
186	C₄	#19151909	1	2	yes	—
187	C₄	#19151938	1	4	yes	—
188	C₂	#26019859	1	1	yes	characteristic
189	C₂	#26019901	1	2	yes	—
190	C₂	#26019921	1	1	yes	characteristic
191	C₂	#26021780	1	1	yes	characteristic
192	C₂	#26021798	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26021214

C₂

yes

maximal

#26020412

C₂

yes

maximal, characteristic

#26020379

C₂

yes

maximal

#26018079

C₂

yes

maximal, characteristic

C₈₉×#1956

C₂

yes

maximal, characteristic

C₂×#15667946

—

C₂×C₄×#10623494

C₂²

yes

—

#19151468

C₂²

yes

—

#19151360

—

#19151360

—

C₂×C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₂×#103×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

D₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667946

—

C₄×#10623494

—

C₄×#10623494

—

C₂²×Q₈×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

Q₈

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#103×C₈₉

—

C₄×C₈×C₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₄²×C₈₉

yes

C₂³×C₈₉

yes

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#497

yes

characteristic

C₂²×C₈₉

#497

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#474

yes

—

C₄×C₈₉

#479

yes

—

C₄×C₈₉

#480

yes

—

C₄×C₈₉

#485

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#1990

—

#1956

D₈₉

yes

characteristic

#1845

—

100

#1834

—

101

#1273

—

102

C₈₉

#10676

yes

characteristic, complemented, Sylow

103

C₂×C₄⋊Q₈

C₂×D₈₉

yes

characteristic

104

C₂×#107

—

178

—

105

C₂×#103

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

108

#384

—

109

#384

C₂×D₈₉

yes

characteristic

110

#340

—

178

—

111

#340

—

178

—

112

C₂²×Q₈

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²⋊D₈₉

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

Dic₁₇₈

yes

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

178

—

122

C₂×C₄⋊C₄

—

178

—

123

C₂×C₄⋊C₄

—

178

—

124

C₂×C₄²

C₂²×D₈₉

yes

characteristic

125

#107

—

356

—

126

#103

—

127

C₄×C₈

—

128

C₄×C₈

—

178

—

129

C₄×C₈

—

178

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

—

136

C₂²×C₄

—

356

—

137

C₂²×C₄

D₄×D₈₉

yes

—

138

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

178

—

142

C₂×C₈

—

178

—

143

C₂×C₈

—

178

—

144

C₂×C₈

#13083559

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

356

—

150

C₄⋊C₄

—

356

—

151

C₄⋊C₄

—

356

—

152

C₄²

C₂×Dic₁₇₈

yes

characteristic

153

C₄²

#13083573

yes

—

154

C₄²

#13083574

yes

characteristic

155

C₂³

#15668008

yes

characteristic

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

712

—

164

C₂×C₄

—

712

—

165

C₂×C₄

—

712

—

166

C₂×C₄

#15667989

yes

—

167

C₂×C₄

#15667991

yes

characteristic

168

C₂×C₄

#15667993

yes

characteristic

169

C₂×C₄

#15667994

yes

—

170

C₂×C₄

C₂×#13083559

yes

—

171

C₂×C₄

#15668035

yes

—

172

C₂×C₄

#15668041

yes

—

173

C₂×C₄

#15668042

yes

—

174

C₈

—

175

C₈

—

356

—

176

C₂²

#19151534

yes

characteristic

177

C₂²

#19151974

yes

characteristic

178

C₂²

#19151983

yes

—

179

C₂²

#19151989

yes

—

180

C₂²

#19151993

yes

characteristic

181

C₄

—

182

C₄

—

183

C₄

—

1424

—

184

C₄

#19151878

yes

—

185

C₄

#19151907

yes

—

186

C₄

#19151909

yes

—

187

C₄

#19151938

yes

—

188

C₂

#26019859

yes

characteristic

189

C₂

#26019901

yes

—

190

C₂

#26019921

yes

characteristic

191

C₂

#26021780

yes

characteristic

192

C₂

#26021798

yes

—

	subgroup	G/H
center	C₂³	#15668008
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1956	C₂
socle, prime radical	C₂³×C₈₉	#127
hypercenter	#1956	D₈₉
hypocenter	C₈₉	#10676
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#15668008

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1956

C₂

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
102. C₈₉	#10676	semidirect

H ≅ G/N

type

102. C₈₉

#10676

semidirect

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

subgroup	index
#26021214	2
#26020412	2
#26020379	2
#26018079	2
C₈₉×#1956	2
#10676	89

subgroup

index

C₈₉×#1956

Subgroups — #36459492

Conjugacy classes of subgroups

Complements

Maximal subgroups