Group #36453521 — subgroups

Subgroups — #36453521

Order 22784.

538 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 167 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	#26028650	C₂	1	1	yes	maximal, characteristic
2	#26020105	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151343	C₂	1	4	yes	maximal
4	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	—	2	1	no	—
6	C₂×#15668012	—	2	1	no	—
7	C₂×#15668012	C₂²	1	2	yes	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×#15667923	—	2	1	no	—
10	#19151468	—	2	2	no	—
11	#19151468	C₄	1	4	yes	—
12	#19151353	—	2	1	no	—
13	#19151343	—	2	8	no	—
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	#336×C₈₉	—	2	2	no	—
17	C₂²×Dic₁₇₈	—	2	2	no	—
18	C₂²×#10623494	—	2	2	no	—
19	#15668012	—	2	4	no	—
20	#15668012	—	2	4	no	—
21	#15668012	—	4	4	no	—
22	#15668012	—	4	2	no	—
23	#15668012	—	4	2	no	—
24	C₂×C₄⋊Dic₈₉	—	2	4	no	—
25	C₂×C₄⋊Dic₈₉	—	2	4	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
28	C₄⋊#10623494	—	2	2	no	—
29	C₄⋊#10623494	D₄	1	4	yes	—
30	#15667923	—	4	2	no	—
31	C₂²×C₈×C₈₉	—	2	2	no	—
32	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	D₄	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₈₉	—	2	4	no	—
38	C₂×Dic₁₇₈	—	4	4	no	—
39	C₂×Dic₁₇₈	—	4	2	no	—
40	C₂×Dic₁₇₈	—	4	2	no	—
41	C₂×Dic₁₇₈	—	4	4	no	—
42	C₂²×Dic₈₉	—	2	4	no	—
43	C₂×#10623494	—	2	4	no	—
44	C₂×#10623494	—	4	2	no	—
45	C₂×#10623494	—	4	2	no	—
46	C₄⋊Dic₈₉	—	2	8	no	—
47	C₄⋊Dic₈₉	—	4	8	no	—
48	C₄⋊Dic₈₉	—	4	8	no	—
49	C₄⋊Dic₈₉	—	4	2	no	—
50	C₄⋊Dic₈₉	—	4	2	no	—
51	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	—	4	2	no	—
57	C₂×C₈×C₈₉	—	4	4	no	—
58	C₄²×C₈₉	—	2	4	no	—
59	C₄²×C₈₉	—	2	2	no	—
60	C₄²×C₈₉	—	2	4	no	—
61	C₄²×C₈₉	D₈	1	2	yes	—
62	C₄²×C₈₉	SD₁₆	1	2	yes	—
63	C₄²×C₈₉	C₂×D₄	1	2	yes	—
64	C₄²×C₈₉	#41	1	2	yes	—
65	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
70	C₂×C₄×C₈₉	—	2	2	no	—
71	C₂×C₄×C₈₉	#101	1	4	yes	—
72	C₂×C₄×C₈₉	#102	1	4	yes	—
73	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₈₉	#121	1	2	yes	—
75	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₈₉	#137	1	4	yes	—
81	C₂×Dic₈₉	—	4	4	no	—
82	C₂×Dic₈₉	—	4	4	no	—
83	C₂×Dic₈₉	—	4	4	no	—
84	Dic₁₇₈	—	8	4	no	—
85	Dic₁₇₈	—	8	4	no	—
86	C₈×C₈₉	—	8	2	no	—
87	#10623494	—	4	4	no	—
88	C₂²×C₈₉	#390	1	1	yes	characteristic
89	C₂²×C₈₉	C₂×#101	1	2	yes	—
90	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
91	C₂²×C₈₉	#418	1	1	yes	characteristic
92	C₂²×C₈₉	#419	1	2	yes	—
93	C₄×C₈₉	—	2	4	no	—
94	C₄×C₈₉	—	2	4	no	—
95	C₄×C₈₉	—	2	2	no	—
96	C₄×C₈₉	#459	1	2	yes	—
97	C₄×C₈₉	#460	1	2	yes	—
98	C₄×C₈₉	#491	1	2	yes	—
99	C₄×C₈₉	#497	1	2	yes	—
100	Dic₈₉	—	8	4	no	—
101	#9705	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	#1546	1	2	yes	—
103	C₂×C₈₉	#1556	1	2	yes	—
104	C₂×C₈₉	#1960	1	1	yes	characteristic
105	C₂×C₈₉	#1966	1	1	yes	characteristic
106	C₂×C₈₉	#1982	1	1	yes	characteristic
107	#2868	—	89	1	no	—
108	#1831	D₈₉	1	1	yes	characteristic
109	#1767	—	89	1	no	—
110	C₂×#332	—	89	4	no	—
111	C₈₉	#9705	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄⋊Q₈	—	89	2	no	—
113	C₂×C₄⋊Q₈	—	178	1	no	—
114	C₂×C₄⋊Q₈	—	178	1	no	—
115	C₂×C₄⋊C₈	—	89	2	no	—
116	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
117	C₂×#97	—	178	1	no	—
118	#384	—	89	4	no	—
119	#384	—	178	2	no	—
120	C₄³	C₂×D₈₉	1	1	yes	characteristic
121	#336	—	2	2	no	—
122	#336	—	178	1	no	—
123	#332	—	178	8	no	—
124	C₂²×Q₈	—	178	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	178	2	no	—
127	C₄⋊Q₈	—	356	4	no	—
128	C₄⋊Q₈	—	356	2	no	—
129	C₄⋊Q₈	—	356	2	no	—
130	C₄⋊Q₈	—	178	4	no	—
131	C₄⋊Q₈	—	178	4	no	—
132	C₂×C₄⋊C₄	—	89	4	no	—
133	C₂×C₄⋊C₄	—	178	4	no	—
134	C₂×C₄⋊C₄	—	178	4	no	—
135	C₂×C₄⋊C₄	—	178	2	no	—
136	C₂×C₄²	C₄×D₈₉	1	2	yes	—
137	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
138	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
140	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	89	4	no	—
143	C₄⋊C₈	—	178	2	no	—
144	#97	—	356	2	no	—
145	C₂×Q₈	—	356	4	no	—
146	C₂×Q₈	—	356	2	no	—
147	C₂×Q₈	—	356	2	no	—
148	C₂×Q₈	—	356	4	no	—
149	C₂²×C₄	—	178	4	no	—
150	C₂²×C₄	#13083551	1	2	yes	—
151	C₂²×C₄	#13083551	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	4	no	—
157	C₂×C₈	—	356	2	no	—
158	C₂×C₈	—	356	2	no	—
159	C₂×C₈	—	178	4	no	—
160	C₄⋊C₄	—	356	8	no	—
161	C₄⋊C₄	—	356	2	no	—
162	C₄⋊C₄	—	356	2	no	—
163	C₄⋊C₄	—	356	8	no	—
164	C₄⋊C₄	—	178	8	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	4	no	—
168	C₄²	D₇₁₂	1	2	yes	—
169	C₄²	#13083563	1	2	yes	—
170	C₄²	D₄×D₈₉	1	2	yes	—
171	C₄²	#13083575	1	2	yes	—
172	C₂³	C₂×#13083551	1	1	yes	characteristic
173	Q₈	—	712	4	no	—
174	Q₈	—	712	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	356	4	no	—
181	C₂×C₄	—	356	4	no	—
182	C₂×C₄	—	356	4	no	—
183	C₂×C₄	#15667930	1	4	yes	—
184	C₂×C₄	#15667934	1	4	yes	—
185	C₂×C₄	#15667974	1	2	yes	—
186	C₂×C₄	#15667987	1	2	yes	—
187	C₂×C₄	#15668001	1	1	yes	characteristic
188	C₂×C₄	#15668008	1	1	yes	characteristic
189	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
190	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
191	C₂×C₄	#15668040	1	2	yes	—
192	C₂×C₄	#15668042	1	4	yes	—
193	C₈	—	8	2	no	—
194	C₈	—	356	4	no	—
195	C₂²	#19151499	1	1	yes	characteristic
196	C₂²	C₂×#15667930	1	2	yes	—
197	C₂²	C₂×#15667934	1	1	yes	characteristic
198	C₂²	#19151604	1	1	yes	characteristic
199	C₂²	#19151606	1	2	yes	—
200	C₄	—	2	4	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	—	712	4	no	—
204	C₄	#19151794	1	2	yes	—
205	C₄	#19151799	1	2	yes	—
206	C₄	#19151968	1	2	yes	—
207	C₄	#19151993	1	2	yes	—
208	C₂	#26018923	1	2	yes	—
209	C₂	#26018968	1	2	yes	—
210	C₂	#26021071	1	1	yes	characteristic
211	C₂	#26021095	1	1	yes	characteristic
212	C₂	#26021178	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26028650

C₂

yes

maximal, characteristic

#26020105

C₂

yes

maximal, characteristic

C₂×#19151343

C₂

yes

maximal

C₈₉×#1831

C₂

yes

maximal, characteristic

C₂×#15668012

—

C₂×#15668012

—

C₂×#15668012

C₂²

yes

—

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×#15667923

—

#19151468

—

#19151468

C₄

yes

—

#19151353

—

#19151343

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

C₂²×Dic₁₇₈

—

—

—

—

—

—

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄⋊#10623494

—

C₄⋊#10623494

D₄

yes

—

#15667923

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

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

D₈

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

C₂×D₄

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

#101

yes

—

C₂×C₄×C₈₉

#102

yes

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

C₂×#101

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#459

yes

—

C₄×C₈₉

#460

yes

—

C₄×C₈₉

#491

yes

—

C₄×C₈₉

#497

yes

—

100

Dic₈₉

—

101

#9705

—

maximal, Sylow

102

C₂×C₈₉

#1546

yes

—

103

C₂×C₈₉

#1556

yes

—

104

C₂×C₈₉

#1960

yes

characteristic

105

C₂×C₈₉

#1966

yes

characteristic

106

C₂×C₈₉

#1982

yes

characteristic

107

#2868

—

108

#1831

D₈₉

yes

characteristic

109

#1767

—

110

C₂×#332

—

111

C₈₉

#9705

yes

characteristic, complemented, Sylow

112

C₂×C₄⋊Q₈

—

113

C₂×C₄⋊Q₈

—

178

—

114

C₂×C₄⋊Q₈

—

178

—

115

C₂×C₄⋊C₈

—

116

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

117

C₂×#97

—

178

—

118

#384

—

119

#384

—

178

—

120

C₄³

C₂×D₈₉

yes

characteristic

121

#336

—

122

#336

—

178

—

123

#332

—

178

—

124

C₂²×Q₈

—

178

—

125

C₂²×C₈

—

126

C₂²×C₈

—

178

—

127

C₄⋊Q₈

—

356

—

128

C₄⋊Q₈

—

356

—

129

C₄⋊Q₈

—

356

—

130

C₄⋊Q₈

—

178

—

131

C₄⋊Q₈

—

178

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

178

—

134

C₂×C₄⋊C₄

—

178

—

135

C₂×C₄⋊C₄

—

178

—

136

C₂×C₄²

C₄×D₈₉

yes

—

137

C₂×C₄²

D₃₅₆

yes

characteristic

138

C₂×C₄²

D₃₅₆

yes

characteristic

139

C₂×C₄²

C₂²⋊D₈₉

yes

—

140

C₂×C₄²

C₂²×D₈₉

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

178

—

144

#97

—

356

—

145

C₂×Q₈

—

356

—

146

C₂×Q₈

—

356

—

147

C₂×Q₈

—

356

—

148

C₂×Q₈

—

356

—

149

C₂²×C₄

—

178

—

150

C₂²×C₄

#13083551

yes

—

151

C₂²×C₄

#13083551

yes

—

152

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

153

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

154

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

356

—

158

C₂×C₈

—

356

—

159

C₂×C₈

—

178

—

160

C₄⋊C₄

—

356

—

161

C₄⋊C₄

—

356

—

162

C₄⋊C₄

—

356

—

163

C₄⋊C₄

—

356

—

164

C₄⋊C₄

—

178

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

D₇₁₂

yes

—

169

C₄²

#13083563

yes

—

170

C₄²

D₄×D₈₉

yes

—

171

C₄²

#13083575

yes

—

172

C₂³

C₂×#13083551

yes

characteristic

173

Q₈

—

712

—

174

Q₈

—

712

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

356

—

181

C₂×C₄

—

356

—

182

C₂×C₄

—

356

—

183

C₂×C₄

#15667930

yes

—

184

C₂×C₄

#15667934

yes

—

185

C₂×C₄

#15667974

yes

—

186

C₂×C₄

#15667987

yes

—

187

C₂×C₄

#15668001

yes

characteristic

188

C₂×C₄

#15668008

yes

characteristic

189

C₂×C₄

C₂×D₇₁₂

yes

characteristic

190

C₂×C₄

C₂×#13083563

yes

characteristic

191

C₂×C₄

#15668040

yes

—

192

C₂×C₄

#15668042

yes

—

193

C₈

—

194

C₈

—

356

—

195

C₂²

#19151499

yes

characteristic

196

C₂²

C₂×#15667930

yes

—

197

C₂²

C₂×#15667934

yes

characteristic

198

C₂²

#19151604

yes

characteristic

199

C₂²

#19151606

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

712

—

204

C₄

#19151794

yes

—

205

C₄

#19151799

yes

—

206

C₄

#19151968

yes

—

207

C₄

#19151993

yes

—

208

C₂

#26018923

yes

—

209

C₂

#26018968

yes

—

210

C₂

#26021071

yes

characteristic

211

C₂

#26021095

yes

characteristic

212

C₂

#26021178

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1831	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#9705
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×#13083551

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1831

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
111. C₈₉	#9705	semidirect

H ≅ G/N

type

111. C₈₉

#9705

semidirect

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

subgroup	index
#26028650	2
#26020105	2
C₂×#19151343	2
C₈₉×#1831	2
#9705	89

subgroup

index

C₈₉×#1831

Subgroups — #36453521

Conjugacy classes of subgroups

Complements

Maximal subgroups