Group #36454459 — subgroups

Subgroups — #36454459

Order 22784.

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

normal subgroups only

This group has 486 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	C₄×#15668004	C₂	1	1	yes	maximal, characteristic
2	#26020404	C₂	1	1	yes	maximal, characteristic
3	#26018092	C₂	1	4	yes	maximal
4	C₄×#97×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668004	C₄	1	1	yes	characteristic
6	C₂×C₄⋊#10623494	—	2	2	no	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	#19151496	C₄	1	2	yes	—
9	#19151482	—	2	2	no	—
10	#19151464	C₄	1	1	yes	characteristic
11	C₄×#13083553	—	2	2	no	—
12	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	C₂×#97×C₈₉	—	2	2	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×#10623494	—	2	2	no	—
17	#15668004	—	2	4	no	—
18	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
20	C₂×#13083553	—	2	2	no	—
21	C₂×#13083553	—	2	2	no	—
22	C₂×C₄×Dic₈₉	—	2	2	no	—
23	C₄⋊#10623494	—	2	4	no	—
24	C₄⋊#10623494	—	4	4	no	—
25	#15667919	—	2	2	no	—
26	#15667919	—	2	2	no	—
27	#15667919	—	2	2	no	—
28	C₂²×C₈×C₈₉	—	2	2	no	—
29	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	D₄	1	2	yes	—
32	C₂×C₄²×C₈₉	D₄	1	2	yes	—
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	#97×C₈₉	—	2	8	no	—
35	C₄×C₈×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	Q₈	1	4	yes	—
37	C₂²×Dic₈₉	—	2	2	no	—
38	C₂²×Dic₈₉	—	2	2	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₂×#10623494	—	4	4	no	—
41	C₄⋊Dic₈₉	—	2	2	no	—
42	C₄⋊Dic₈₉	—	2	4	no	—
43	C₄⋊Dic₈₉	M₁₆	1	4	yes	—
44	#13083553	—	2	8	no	—
45	#13083553	—	4	4	no	—
46	C₄×Dic₈₉	—	4	2	no	—
47	C₄×Dic₈₉	—	4	2	no	—
48	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	—	2	8	no	—
54	C₂×C₈×C₈₉	—	2	4	no	—
55	C₄²×C₈₉	—	2	4	no	—
56	C₄²×C₈₉	—	2	2	no	—
57	C₄²×C₈₉	—	2	2	no	—
58	C₄²×C₈₉	—	2	2	no	—
59	C₄²×C₈₉	—	2	2	no	—
60	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
61	C₄²×C₈₉	#41	1	2	yes	—
62	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	2	2	no	—
69	C₂×C₄×C₈₉	#104	1	4	yes	—
70	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
72	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₈₉	#131	1	2	yes	—
76	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₈₉	#137	1	2	yes	—
78	C₂×Dic₈₉	—	2	4	no	—
79	C₂×Dic₈₉	—	4	2	no	—
80	C₂×Dic₈₉	—	4	2	no	—
81	C₂×Dic₈₉	—	4	2	no	—
82	C₂×Dic₈₉	—	4	2	no	—
83	C₂×Dic₈₉	—	4	2	no	—
84	C₈×C₈₉	—	2	8	no	—
85	#10623494	—	8	2	no	—
86	C₂²×C₈₉	#391	1	1	yes	characteristic
87	C₂²×C₈₉	#407	1	1	yes	characteristic
88	C₂²×C₈₉	#408	1	1	yes	characteristic
89	C₂²×C₈₉	#417	1	2	yes	—
90	C₂²×C₈₉	C₂×#104	1	2	yes	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	—	2	4	no	—
93	C₄×C₈₉	—	4	2	no	—
94	C₄×C₈₉	#446	1	4	yes	—
95	C₄×C₈₉	#490	1	1	yes	characteristic
96	C₄×C₈₉	#491	1	1	yes	characteristic
97	C₄×C₈₉	#501	1	2	yes	—
98	Dic₈₉	—	4	4	no	—
99	Dic₈₉	—	8	2	no	—
100	#9876	—	89	1	no	maximal, Sylow
101	C₂×C₈₉	#1483	1	2	yes	—
102	C₂×C₈₉	#1489	1	2	yes	—
103	C₂×C₈₉	#1985	1	1	yes	characteristic
104	C₂×C₈₉	#1992	1	1	yes	characteristic
105	C₂×C₈₉	#1994	1	1	yes	characteristic
106	C₄×#125	—	89	1	no	—
107	#1842	—	89	1	no	—
108	C₄×#97	D₈₉	1	1	yes	characteristic
109	#1279	—	89	4	no	—
110	C₈₉	#9876	1	1	yes	characteristic, complemented, Sylow
111	C₂×#125	—	89	1	no	—
112	C₂×C₄⋊C₈	—	89	2	no	—
113	C₂×C₄⋊C₈	—	178	2	no	—
114	C₂×#97	—	2	2	no	—
115	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
116	#389	—	89	2	no	—
117	#387	—	178	2	no	—
118	#382	—	89	1	no	—
119	C₄×C₄⋊C₄	—	89	2	no	—
120	C₄×C₄⋊C₄	—	178	2	no	—
121	C₄³	C₂×D₈₉	1	1	yes	characteristic
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	178	2	no	—
124	#125	—	178	4	no	—
125	C₂×C₄⋊C₄	—	89	2	no	—
126	C₂×C₄⋊C₄	—	89	2	no	—
127	C₂×C₄⋊C₄	—	178	2	no	—
128	C₂×C₄⋊C₄	—	178	2	no	—
129	C₂×C₄²	—	178	2	no	—
130	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
132	C₂×C₄²	D₃₅₆	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
134	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
135	C₄⋊C₈	—	356	4	no	—
136	C₄⋊C₈	—	178	4	no	—
137	#97	—	2	8	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	Dic₁₇₈	1	4	yes	—
140	#95	—	178	2	no	—
141	#95	—	178	2	no	—
142	#95	—	178	2	no	—
143	C₂²×C₄	—	178	2	no	—
144	C₂²×C₄	—	178	2	no	—
145	C₂²×C₄	#13083551	1	2	yes	—
146	C₂²×C₄	#13083551	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	356	2	no	—
153	C₂×C₈	—	356	4	no	—
154	C₄⋊C₄	—	89	4	no	—
155	C₄⋊C₄	—	356	4	no	—
156	C₄⋊C₄	—	178	8	no	—
157	C₄⋊C₄	—	178	4	no	—
158	C₄⋊C₄	—	178	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	356	2	no	—
165	C₄²	—	356	2	no	—
166	C₄²	C₂×Dic₁₇₈	1	2	yes	—
167	C₄²	#13083574	1	2	yes	—
168	C₂³	C₂×#13083551	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	356	2	no	—
176	C₂×C₄	—	356	2	no	—
177	C₂×C₄	—	356	2	no	—
178	C₂×C₄	—	356	2	no	—
179	C₂×C₄	—	356	2	no	—
180	C₂×C₄	—	178	4	no	—
181	C₂×C₄	#15667940	1	4	yes	—
182	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
183	C₂×C₄	#15667991	1	2	yes	—
184	C₂×C₄	#15668000	1	1	yes	characteristic
185	C₂×C₄	#15668012	1	1	yes	characteristic
186	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
187	C₂×C₄	#15668020	1	2	yes	—
188	C₂×C₄	#15668040	1	2	yes	—
189	C₂×C₄	#15668045	1	2	yes	—
190	C₈	—	2	8	no	—
191	C₈	—	712	2	no	—
192	C₂²	#19151504	1	1	yes	characteristic
193	C₂²	#19151561	1	1	yes	characteristic
194	C₂²	#19151566	1	1	yes	characteristic
195	C₂²	#19151600	1	2	yes	—
196	C₂²	C₂×#15667940	1	2	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	356	4	no	—
201	C₄	—	712	2	no	—
202	C₄	#19151726	1	4	yes	—
203	C₄	#19151963	1	1	yes	characteristic
204	C₄	#19151967	1	1	yes	characteristic
205	C₄	#19152007	1	2	yes	—
206	C₂	#26018666	1	2	yes	—
207	C₂	#26018692	1	2	yes	—
208	C₂	#26021192	1	1	yes	characteristic
209	C₂	#26021221	1	1	yes	characteristic
210	C₂	#26021229	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#15668004

C₂

yes

maximal, characteristic

#26020404

C₂

yes

maximal, characteristic

#26018092

C₂

yes

maximal

C₄×#97×C₈₉

C₂

yes

maximal, characteristic

C₂×#15668004

C₄

yes

characteristic

C₂×C₄⋊#10623494

—

C₂×C₄⋊#10623494

C₂²

yes

—

#19151496

C₄

yes

—

#19151482

—

#19151464

C₄

yes

characteristic

C₄×#13083553

—

C₄×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#97×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

—

#15668004

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂×#13083553

—

C₂×#13083553

—

C₂×C₄×Dic₈₉

—

—

—

—

—

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#97×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

Q₈

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

M₁₆

yes

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

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

—

C₄²×C₈₉

—

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

#104

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×M₁₆

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

C₂×#104

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#446

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

100

#9876

—

maximal, Sylow

101

C₂×C₈₉

#1483

yes

—

102

C₂×C₈₉

#1489

yes

—

103

C₂×C₈₉

#1985

yes

characteristic

104

C₂×C₈₉

#1992

yes

characteristic

105

C₂×C₈₉

#1994

yes

characteristic

106

C₄×#125

—

107

#1842

—

108

C₄×#97

D₈₉

yes

characteristic

109

#1279

—

110

C₈₉

#9876

yes

characteristic, complemented, Sylow

111

C₂×#125

—

112

C₂×C₄⋊C₈

—

113

C₂×C₄⋊C₈

—

178

—

114

C₂×#97

—

115

C₂×C₄×C₈

C₂×D₈₉

yes

—

116

#389

—

117

#387

—

178

—

118

#382

—

119

C₄×C₄⋊C₄

—

120

C₄×C₄⋊C₄

—

178

—

121

C₄³

C₂×D₈₉

yes

characteristic

122

C₂²×C₈

—

123

C₂²×C₈

—

178

—

124

#125

—

178

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

178

—

128

C₂×C₄⋊C₄

—

178

—

129

C₂×C₄²

—

178

—

130

C₂×C₄²

C₄×D₈₉

yes

characteristic

131

C₂×C₄²

C₄×D₈₉

yes

characteristic

132

C₂×C₄²

D₃₅₆

yes

—

133

C₂×C₄²

C₂²⋊D₈₉

yes

—

134

C₂×C₄²

C₂²×D₈₉

yes

characteristic

135

C₄⋊C₈

—

356

—

136

C₄⋊C₈

—

178

—

137

#97

—

138

C₄×C₈

—

139

C₄×C₈

Dic₁₇₈

yes

—

140

#95

—

178

—

141

#95

—

178

—

142

#95

—

178

—

143

C₂²×C₄

—

178

—

144

C₂²×C₄

—

178

—

145

C₂²×C₄

#13083551

yes

—

146

C₂²×C₄

#13083551

yes

—

147

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

149

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

356

—

153

C₂×C₈

—

356

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

356

—

156

C₄⋊C₄

—

178

—

157

C₄⋊C₄

—

178

—

158

C₄⋊C₄

—

178

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

356

—

165

C₄²

—

356

—

166

C₄²

C₂×Dic₁₇₈

yes

—

167

C₄²

#13083574

yes

—

168

C₂³

C₂×#13083551

yes

characteristic

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

356

—

176

C₂×C₄

—

356

—

177

C₂×C₄

—

356

—

178

C₂×C₄

—

356

—

179

C₂×C₄

—

356

—

180

C₂×C₄

—

178

—

181

C₂×C₄

#15667940

yes

—

182

C₂×C₄

C₄×Dic₁₇₈

yes

—

183

C₂×C₄

#15667991

yes

—

184

C₂×C₄

#15668000

yes

characteristic

185

C₂×C₄

#15668012

yes

characteristic

186

C₂×C₄

M₁₆×D₈₉

yes

—

187

C₂×C₄

#15668020

yes

—

188

C₂×C₄

#15668040

yes

—

189

C₂×C₄

#15668045

yes

—

190

C₈

—

191

C₈

—

712

—

192

C₂²

#19151504

yes

characteristic

193

C₂²

#19151561

yes

characteristic

194

C₂²

#19151566

yes

characteristic

195

C₂²

#19151600

yes

—

196

C₂²

C₂×#15667940

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

356

—

201

C₄

—

712

—

202

C₄

#19151726

yes

—

203

C₄

#19151963

yes

characteristic

204

C₄

#19151967

yes

characteristic

205

C₄

#19152007

yes

—

206

C₂

#26018666

yes

—

207

C₂

#26018692

yes

—

208

C₂

#26021192

yes

characteristic

209

C₂

#26021221

yes

characteristic

210

C₂

#26021229

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#97×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#97	D₈₉
hypocenter	C₈₉	#9876
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

subgroup

G/H

center

C₂²×C₄

C₂×D₃₅₆

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₄×#97×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

hypercenter

C₄×#97

D₈₉

hypocenter

C₈₉

#9876

LCS

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

N	H ≅ G/N	type
110. C₈₉	#9876	semidirect

H ≅ G/N

type

110. C₈₉

#9876

semidirect

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

subgroup	index
C₄×#15668004	2
#26020404	2
#26018092	2
C₄×#97×C₈₉	2
#9876	89

subgroup

index

C₄×#97×C₈₉

Subgroups — #36454459

Conjugacy classes of subgroups

Complements

Maximal subgroups