Group #36472461 — subgroups

Subgroups — #36472461

Order 22784.

326 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 135 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×#19151426	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26020983	C₂	1	2	yes	maximal
4	#26018343	C₂	1	2	yes	maximal
5	C₂×#367×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	1	yes	characteristic
7	C₂×#15667946	C₂²	1	2	yes	—
8	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
9	#19151494	—	2	2	no	—
10	#19151426	C₄	1	4	yes	complemented
11	C₁₆×Dic₈₉	—	2	2	no	—
12	#19151360	—	2	2	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#107×C₈₉	C₂²	1	2	yes	—
15	#367×C₈₉	—	2	2	no	—
16	C₂²×#10623494	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₈₉	—	2	2	no	—
18	C₂×#13083553	—	4	2	no	—
19	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
20	C₂×#13083549	—	2	1	no	—
21	C₂×#13083549	—	2	1	no	—
22	C₂×#13083549	Q₈	1	2	yes	—
23	#15667946	—	2	2	no	—
24	#15667946	C₂×C₄	1	4	yes	—
25	C₈×Dic₈₉	—	2	1	no	—
26	C₈×Dic₈₉	—	2	1	no	—
27	#15667919	—	4	2	no	—
28	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
32	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
33	#107×C₈₉	—	2	2	no	—
34	#107×C₈₉	—	2	2	no	—
35	C₂²×Dic₈₉	—	4	2	no	—
36	C₂²×Dic₈₉	D₈	1	2	yes	—
37	C₂×#10623494	—	2	1	no	—
38	C₂×#10623494	—	2	1	no	—
39	C₂×#10623494	Q₁₆	1	2	yes	—
40	C₄⋊Dic₈₉	—	2	4	no	—
41	C₄⋊Dic₈₉	—	4	2	no	—
42	#13083553	—	8	4	no	—
43	C₄×Dic₈₉	—	2	1	no	—
44	C₄×Dic₈₉	—	2	1	no	—
45	#13083549	—	2	2	no	—
46	C₂²×C₄×C₈₉	—	4	2	no	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₈₉	—	4	2	no	—
55	C₄⋊C₄×C₈₉	—	4	2	no	—
56	C₁₆×C₈₉	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₈₉	—	8	2	no	—
59	C₂×C₄×C₈₉	—	8	2	no	—
60	C₂×C₄×C₈₉	—	8	2	no	—
61	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
67	C₂×Dic₈₉	—	2	2	no	—
68	C₂×Dic₈₉	—	2	2	no	—
69	C₂×Dic₈₉	—	4	4	no	—
70	C₂×Dic₈₉	—	8	2	no	—
71	C₂×Dic₈₉	—	8	2	no	—
72	C₂×Dic₈₉	SD₃₂	1	4	yes	—
73	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₈₉	#125	1	1	yes	characteristic
77	#10623494	—	2	2	no	—
78	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₈₉	#466	1	1	yes	characteristic
80	C₂²×C₈₉	#500	1	1	yes	characteristic
81	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
82	C₂²×C₈₉	#508	1	2	yes	—
83	C₄×C₈₉	—	16	2	no	—
84	C₄×C₈₉	#389	1	1	yes	characteristic
85	C₄×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
86	C₄×C₈₉	#467	1	1	yes	characteristic
87	C₄×C₈₉	#499	1	1	yes	characteristic
88	Dic₈₉	—	2	4	no	—
89	Dic₈₉	—	8	4	no	—
90	#13166	—	89	1	no	maximal, Sylow
91	C₂×C₈₉	#1942	1	1	yes	characteristic
92	C₂×C₈₉	C₄×SD₃₂	1	2	yes	—
93	C₂×C₈₉	#2217	1	2	yes	—
94	C₂×C₈₉	#2253	1	1	yes	characteristic
95	C₂×C₈₉	#2254	1	1	yes	characteristic
96	C₂×#367	—	89	1	no	—
97	C₂×#367	D₈₉	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	89	1	no	—
99	#1942	—	89	2	no	—
100	#1380	—	89	2	no	—
101	C₈₉	#13166	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	—	89	1	no	—
103	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
104	C₂×#107	—	89	2	no	—
105	C₂×#107	C₂×D₈₉	1	2	yes	—
106	C₂×C₄×C₈	—	89	1	no	—
107	#389	—	178	2	no	—
108	#367	—	2	2	no	—
109	#367	—	89	4	no	—
110	C₄×C₁₆	—	178	2	no	—
111	#340	—	178	2	no	—
112	C₂²×C₈	—	89	1	no	—
113	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	356	2	no	—
116	C₂×C₄⋊C₄	—	178	2	no	—
117	C₂×C₄²	—	89	1	no	—
118	C₂×C₁₆	—	89	2	no	—
119	C₂×C₁₆	—	178	1	no	—
120	C₂×C₁₆	—	178	1	no	—
121	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
122	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	89	4	no	—
127	#107	—	178	2	no	—
128	C₄×C₈	—	178	1	no	—
129	C₄×C₈	—	178	1	no	—
130	#95	—	356	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	89	2	no	—
133	C₂²×C₄	—	356	2	no	—
134	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
135	C₂×C₈	—	89	2	no	—
136	C₂×C₈	—	178	1	no	—
137	C₂×C₈	—	178	1	no	—
138	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
139	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
142	C₂×C₈	#13083573	1	1	yes	characteristic
143	C₂×C₈	#13083575	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	356	2	no	—
147	C₄⋊C₄	—	178	4	no	—
148	C₄⋊C₄	—	712	4	no	—
149	C₄²	—	178	1	no	—
150	C₄²	—	178	1	no	—
151	C₁₆	—	178	2	no	—
152	C₁₆	#13083553	1	4	yes	complemented
153	C₂³	D₈×D₈₉	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	89	4	no	—
158	C₂×C₄	—	356	4	no	—
159	C₂×C₄	—	178	2	no	—
160	C₂×C₄	—	178	2	no	—
161	C₂×C₄	—	712	2	no	—
162	C₂×C₄	—	712	2	no	—
163	C₂×C₄	#15667974	1	1	yes	characteristic
164	C₂×C₄	#15667983	1	1	yes	characteristic
165	C₂×C₄	#15668011	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
167	C₂×C₄	#15668031	1	1	yes	characteristic
168	C₂×C₄	#15668035	1	1	yes	characteristic
169	C₈	—	178	2	no	—
170	C₈	C₂×#13083553	1	1	yes	characteristic
171	C₈	#15667977	1	1	yes	characteristic
172	C₈	#15667988	1	1	yes	characteristic
173	C₈	#15668003	1	1	yes	characteristic
174	C₂²	#19151680	1	1	yes	characteristic
175	C₂²	#19151831	1	1	yes	characteristic
176	C₂²	#19152002	1	1	yes	characteristic
177	C₂²	SD₃₂×D₈₉	1	2	yes	—
178	C₂²	#19152029	1	2	yes	—
179	C₄	—	16	2	no	—
180	C₄	—	178	4	no	—
181	C₄	—	712	4	no	—
182	C₄	#19151494	1	1	yes	characteristic
183	C₄	#19151692	1	1	yes	characteristic
184	C₄	#19151836	1	1	yes	characteristic
185	C₄	#19151999	1	1	yes	characteristic
186	C₂	#26020983	1	1	yes	characteristic
187	C₂	#26022103	1	2	yes	—
188	C₂	#26022345	1	2	yes	—
189	C₂	#26022530	1	1	yes	characteristic
190	C₂	#26022533	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151426

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#26020983

C₂

yes

maximal

#26018343

C₂

yes

maximal

C₂×#367×C₈₉

C₂

yes

maximal, characteristic

C₂²×#13083549

C₂²

yes

characteristic

C₂×#15667946

C₂²

yes

—

C₂×C₈×Dic₈₉

C₂²

yes

characteristic

#19151494

—

#19151426

C₄

yes

complemented

C₁₆×Dic₈₉

—

#19151360

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#107×C₈₉

C₂²

yes

—

#367×C₈₉

—

C₂²×#10623494

D₄

yes

characteristic

C₂×C₄⋊Dic₈₉

—

C₂×#13083553

—

C₂×C₄×Dic₈₉

D₄

yes

—

—

Q₈

yes

—

#15667946

—

#15667946

C₂×C₄

yes

—

C₈×Dic₈₉

—

C₈×Dic₈₉

—

#15667919

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

Q₈

yes

—

#107×C₈₉

—

#107×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₈

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

Q₁₆

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

C₂×C₈×C₈₉

yes

C₂×C₈×C₈₉

yes

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

C₄⋊C₄

yes

complemented

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

SD₃₂

yes

—

C₈×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₄×Q₈

yes

characteristic

C₈×C₈₉

C₂²⋊Q₈

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₄×D₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×SD₃₂

yes

—

C₂²×C₈₉

#508

yes

—

C₄×C₈₉

—

C₄×C₈₉

#389

yes

characteristic

C₄×C₈₉

C₄×Q₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

C₄×SD₃₂

yes

—

C₂×C₈₉

#2217

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

C₂×C₄×C₁₆

—

#1942

—

100

#1380

—

101

C₈₉

#13166

yes

characteristic, complemented, Sylow

102

C₂²×C₁₆

—

103

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

104

C₂×#107

—

105

C₂×#107

C₂×D₈₉

yes

—

106

C₂×C₄×C₈

—

107

#389

—

178

—

108

#367

—

109

#367

—

110

C₄×C₁₆

—

178

—

111

#340

—

178

—

112

C₂²×C₈

—

113

C₂²×C₈

C₂²×D₈₉

yes

characteristic

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

356

—

116

C₂×C₄⋊C₄

—

178

—

117

C₂×C₄²

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

178

—

120

C₂×C₁₆

—

178

—

121

C₂×C₁₆

Dic₁₇₈

yes

—

122

C₂×C₁₆

C₄×D₈₉

yes

—

123

C₂×C₁₆

C₂²⋊D₈₉

yes

—

124

#107

—

125

#107

—

126

#107

—

127

#107

—

178

—

128

C₄×C₈

—

178

—

129

C₄×C₈

—

178

—

130

#95

—

356

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

356

—

134

C₂²×C₄

D₄×D₈₉

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

178

—

137

C₂×C₈

—

178

—

138

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

139

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

140

C₂×C₈

Q₈×D₈₉

yes

characteristic

141

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

142

C₂×C₈

#13083573

yes

characteristic

143

C₂×C₈

#13083575

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

356

—

147

C₄⋊C₄

—

178

—

148

C₄⋊C₄

—

712

—

149

C₄²

—

178

—

150

C₄²

—

178

—

151

C₁₆

—

178

—

152

C₁₆

#13083553

yes

complemented

153

C₂³

D₈×D₈₉

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

356

—

159

C₂×C₄

—

178

—

160

C₂×C₄

—

178

—

161

C₂×C₄

—

712

—

162

C₂×C₄

—

712

—

163

C₂×C₄

#15667974

yes

characteristic

164

C₂×C₄

#15667983

yes

characteristic

165

C₂×C₄

#15668011

yes

characteristic

166

C₂×C₄

Q₁₆×D₈₉

yes

characteristic

167

C₂×C₄

#15668031

yes

characteristic

168

C₂×C₄

#15668035

yes

characteristic

169

C₈

—

178

—

170

C₈

C₂×#13083553

yes

characteristic

171

C₈

#15667977

yes

characteristic

172

C₈

#15667988

yes

characteristic

173

C₈

#15668003

yes

characteristic

174

C₂²

#19151680

yes

characteristic

175

C₂²

#19151831

yes

characteristic

176

C₂²

#19152002

yes

characteristic

177

C₂²

SD₃₂×D₈₉

yes

—

178

C₂²

#19152029

yes

—

179

C₄

—

180

C₄

—

178

—

181

C₄

—

712

—

182

C₄

#19151494

yes

characteristic

183

C₄

#19151692

yes

characteristic

184

C₄

#19151836

yes

characteristic

185

C₄

#19151999

yes

characteristic

186

C₂

#26020983

yes

characteristic

187

C₂

#26022103

yes

—

188

C₂

#26022345

yes

—

189

C₂

#26022530

yes

characteristic

190

C₂

#26022533

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#367×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#367	D₈₉
hypocenter	C₈₉	#13166
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂³

D₈×D₈₉

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#367×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₈

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
10. #19151426	C₄	semidirect
56. C₁₆×C₈₉	C₄⋊C₄	semidirect
101. C₈₉	#13166	semidirect
152. C₁₆	#13083553	semidirect

H ≅ G/N

type

10. #19151426

C₄

semidirect

56. C₁₆×C₈₉

C₄⋊C₄

semidirect

101. C₈₉

#13166

semidirect

152. C₁₆

#13083553

semidirect

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

subgroup	index
C₂×#19151426	2
C₂×C₁₆×Dic₈₉	2
#26020983	2
#26018343	2
C₂×#367×C₈₉	2
#13166	89

subgroup

index

C₂×C₁₆×Dic₈₉

C₂×#367×C₈₉

Subgroups — #36472461

Conjugacy classes of subgroups

Complements

Maximal subgroups