Group #36458869 — subgroups

Subgroups — #36458869

Order 22784.

494 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 137 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	#26025281	C₂	1	1	yes	maximal, characteristic
2	#26021016	C₂	1	2	yes	maximal
3	#26020408	C₂	1	1	yes	maximal, characteristic
4	#26018091	C₂	1	2	yes	maximal
5	C₈₉×#1296	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668011	—	2	2	no	—
7	C₂×#15667944	—	2	2	no	—
8	C₂×#15667943	—	2	2	no	—
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	#19151518	—	2	2	no	—
11	#19151494	—	2	2	no	—
12	#19151468	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
14	#19151361	—	2	2	no	—
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	#384×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₇₈	—	2	2	no	—
18	C₂²×#10623494	—	2	1	no	—
19	#15668011	—	4	4	no	—
20	#15668011	—	4	4	no	—
21	C₂×C₄⋊Dic₈₉	—	2	2	no	—
22	C₂×C₄⋊Dic₈₉	—	2	1	no	—
23	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
24	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
25	C₂×#13083553	—	2	4	no	—
26	C₂×C₄×Dic₈₉	—	2	2	no	—
27	#15667944	—	4	4	no	—
28	#15667943	—	4	4	no	—
29	C₄×#10623494	—	2	1	no	—
30	C₄×#10623494	Q₈	1	2	yes	complemented
31	#15667919	—	2	2	no	—
32	#15667919	—	2	2	no	—
33	C₂²×C₈×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₈₉	—	2	2	no	—
38	C₄⋊C₈×C₈₉	Q₈	1	4	yes	complemented
39	C₂×Dic₁₇₈	—	4	8	no	—
40	C₂×Dic₁₇₈	—	8	2	no	—
41	C₂²×Dic₈₉	—	2	2	no	—
42	C₂²×Dic₈₉	—	2	2	no	—
43	C₂²×Dic₈₉	—	4	1	no	—
44	C₂×#10623494	—	2	2	no	—
45	C₂×#10623494	—	2	2	no	—
46	C₂×#10623494	—	2	2	no	—
47	C₄⋊Dic₈₉	—	2	2	no	—
48	C₄⋊Dic₈₉	—	2	2	no	—
49	C₄⋊Dic₈₉	—	4	2	no	—
50	C₄⋊Dic₈₉	—	4	2	no	—
51	C₄⋊Dic₈₉	—	4	2	no	—
52	C₄⋊Dic₈₉	SD₁₆	1	4	yes	—
53	C₄⋊Dic₈₉	SD₁₆	1	4	yes	—
54	#13083553	—	4	8	no	—
55	C₄×Dic₈₉	—	4	2	no	—
56	C₄×Dic₈₉	—	4	2	no	—
57	C₂²×C₄×C₈₉	—	2	2	no	—
58	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	—	4	2	no	—
62	C₂×C₈×C₈₉	—	4	2	no	—
63	C₄⋊C₄×C₈₉	—	4	2	no	—
64	C₄⋊C₄×C₈₉	—	4	2	no	—
65	C₄⋊C₄×C₈₉	—	4	2	no	—
66	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
68	C₄²×C₈₉	#41	1	1	yes	characteristic
69	C₂³×C₈₉	#120	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	1	no	—
71	C₂×C₄×C₈₉	—	2	1	no	—
72	C₂×C₄×C₈₉	—	4	4	no	—
73	C₂×C₄×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
77	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×C₄×C₈₉	#137	1	2	yes	—
82	C₂×Dic₈₉	—	2	4	no	—
83	C₂×Dic₈₉	—	2	4	no	—
84	C₂×Dic₈₉	—	4	2	no	—
85	C₂×Dic₈₉	—	4	2	no	—
86	C₂×Dic₈₉	—	4	2	no	—
87	C₂×Dic₈₉	—	4	2	no	—
88	C₂×Dic₈₉	—	8	2	no	—
89	C₂×Dic₈₉	—	8	1	no	—
90	Dic₁₇₈	—	8	8	no	—
91	C₈×C₈₉	—	4	4	no	—
92	#10623494	—	2	4	no	—
93	C₂²×C₈₉	#395	1	1	yes	characteristic
94	C₂²×C₈₉	#448	1	2	yes	—
95	C₂²×C₈₉	#450	1	2	yes	—
96	C₂²×C₈₉	#453	1	1	yes	characteristic
97	C₂²×C₈₉	#456	1	1	yes	characteristic
98	C₄×C₈₉	—	2	1	no	—
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	—	8	2	no	—
101	C₄×C₈₉	#475	1	2	yes	—
102	C₄×C₈₉	#476	1	2	yes	—
103	C₄×C₈₉	#498	1	2	yes	—
104	C₄×C₈₉	#501	1	2	yes	—
105	Dic₈₉	—	4	4	no	—
106	Dic₈₉	—	4	4	no	—
107	Dic₈₉	—	16	1	no	—
108	#10574	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	#1623	1	1	yes	characteristic
110	C₂×C₈₉	#1629	1	1	yes	characteristic
111	C₂×C₈₉	#1651	1	2	yes	—
112	C₂×C₈₉	#2058	1	2	yes	—
113	C₂×C₈₉	#2060	1	1	yes	characteristic
114	#2551	—	89	1	no	—
115	#1948	—	89	2	no	—
116	#1844	—	89	1	no	—
117	#1296	D₈₉	1	1	yes	characteristic
118	#1279	—	89	2	no	—
119	C₈₉	#10574	1	1	yes	characteristic, complemented, Sylow
120	C₂×C₄⋊Q₈	—	178	2	no	—
121	C₂×#106	—	178	2	no	—
122	C₂×#106	—	178	2	no	—
123	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
124	C₂×C₄×C₈	—	89	1	no	—
125	#395	—	178	2	no	—
126	#389	—	178	2	no	—
127	#384	—	89	1	no	—
128	#384	C₂×D₈₉	1	1	yes	characteristic
129	C₄×C₄⋊C₄	—	89	2	no	—
130	#340	—	178	2	no	—
131	C₂²×Q₈	—	178	2	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	178	1	no	—
134	C₄⋊Q₈	—	356	4	no	—
135	C₄⋊Q₈	—	356	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	89	2	no	—
138	C₂×C₄⋊C₄	—	89	2	no	—
139	C₂×C₄⋊C₄	—	178	4	no	—
140	C₂×C₄⋊C₄	—	178	2	no	—
141	C₂×C₄⋊C₄	—	178	1	no	—
142	C₂×C₄⋊C₄	D₃₅₆	1	2	yes	—
143	C₂×C₄²	—	178	2	no	—
144	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
145	#106	—	356	4	no	—
146	#106	—	356	4	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	Dic₁₇₈	1	4	yes	complemented
149	C₄×C₈	—	89	2	no	—
150	C₄×C₈	—	178	1	no	—
151	#95	—	178	2	no	—
152	#95	—	178	2	no	—
153	C₂×Q₈	—	356	8	no	—
154	C₂×Q₈	—	712	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	356	1	no	—
157	C₂²×C₄	—	178	2	no	—
158	C₂²×C₄	—	178	2	no	—
159	C₂²×C₄	D₄×D₈₉	1	2	yes	—
160	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	178	2	no	—
165	C₂×C₈	—	178	2	no	—
166	C₂×C₈	—	178	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	89	4	no	—
171	C₄⋊C₄	—	89	4	no	—
172	C₄⋊C₄	—	356	2	no	—
173	C₄⋊C₄	—	356	8	no	—
174	C₄⋊C₄	—	356	2	no	—
175	C₄⋊C₄	—	356	2	no	—
176	C₄⋊C₄	—	178	2	no	—
177	C₄⋊C₄	—	178	2	no	—
178	C₄²	—	356	2	no	—
179	C₄²	—	356	2	no	—
180	C₄²	Q₈×D₈₉	1	1	yes	characteristic
181	C₄²	C₂×Dic₁₇₈	1	2	yes	—
182	C₄²	#13083573	1	1	yes	characteristic
183	C₂³	#15667982	1	1	yes	characteristic
184	Q₈	—	712	8	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	356	2	no	—
190	C₂×C₄	—	356	2	no	—
191	C₂×C₄	—	356	2	no	—
192	C₂×C₄	—	356	2	no	—
193	C₂×C₄	—	178	4	no	—
194	C₂×C₄	—	178	4	no	—
195	C₂×C₄	—	712	2	no	—
196	C₂×C₄	—	712	1	no	—
197	C₂×C₄	#15667989	1	2	yes	—
198	C₂×C₄	#15667992	1	1	yes	characteristic
199	C₂×C₄	#15668011	1	2	yes	—
200	C₂×C₄	#15668012	1	1	yes	characteristic
201	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
202	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
203	C₂×C₄	#15668036	1	2	yes	—
204	C₂×C₄	#15668044	1	2	yes	—
205	C₈	—	4	4	no	—
206	C₈	—	178	4	no	—
207	C₂²	#19151519	1	1	yes	characteristic
208	C₂²	#19151738	1	2	yes	—
209	C₂²	#19151750	1	2	yes	—
210	C₂²	#19151768	1	1	yes	characteristic
211	C₂²	#19151783	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	—	356	4	no	—
216	C₄	—	356	4	no	—
217	C₄	—	1424	1	no	—
218	C₄	#19151881	1	2	yes	—
219	C₄	#19151886	1	2	yes	—
220	C₄	#19151995	1	2	yes	—
221	C₄	#19152006	1	2	yes	—
222	C₂	#26019331	1	1	yes	characteristic
223	C₂	#26019369	1	1	yes	characteristic
224	C₂	#26019516	1	2	yes	—
225	C₂	#26021584	1	2	yes	—
226	C₂	#26021598	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26025281

C₂

yes

maximal, characteristic

#26021016

C₂

yes

maximal

#26020408

C₂

yes

maximal, characteristic

#26018091

C₂

yes

maximal

C₈₉×#1296

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×C₄⋊Dic₈₉

C₂²

yes

—

#19151361

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

#384×C₈₉

C₂²

yes

characteristic

C₂²×Dic₁₇₈

—

C₂²×#10623494

—

#15668011

—

#15668011

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

D₄

yes

—

C₂×C₄⋊Dic₈₉

D₄

yes

—

C₂×#13083553

—

C₂×C₄×Dic₈₉

—

—

—

—

Q₈

yes

complemented

#15667919

—

#15667919

—

C₂²×C₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

Q₈

yes

complemented

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

SD₁₆

yes

—

C₄⋊Dic₈₉

SD₁₆

yes

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

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

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

yes

C₂³×C₈₉

yes

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

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

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#395

yes

characteristic

C₂²×C₈₉

#448

yes

—

C₂²×C₈₉

#450

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

100

C₄×C₈₉

—

101

C₄×C₈₉

#475

yes

—

102

C₄×C₈₉

#476

yes

—

103

C₄×C₈₉

#498

yes

—

104

C₄×C₈₉

#501

yes

—

105

Dic₈₉

—

106

Dic₈₉

—

107

Dic₈₉

—

108

#10574

—

maximal, Sylow

109

C₂×C₈₉

#1623

yes

characteristic

110

C₂×C₈₉

#1629

yes

characteristic

111

C₂×C₈₉

#1651

yes

—

112

C₂×C₈₉

#2058

yes

—

113

C₂×C₈₉

#2060

yes

characteristic

114

#2551

—

115

#1948

—

116

#1844

—

117

#1296

D₈₉

yes

characteristic

118

#1279

—

119

C₈₉

#10574

yes

characteristic, complemented, Sylow

120

C₂×C₄⋊Q₈

—

178

—

121

C₂×#106

—

178

—

122

C₂×#106

—

178

—

123

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

124

C₂×C₄×C₈

—

125

#395

—

178

—

126

#389

—

178

—

127

#384

—

128

#384

C₂×D₈₉

yes

characteristic

129

C₄×C₄⋊C₄

—

130

#340

—

178

—

131

C₂²×Q₈

—

178

—

132

C₂²×C₈

—

133

C₂²×C₈

—

178

—

134

C₄⋊Q₈

—

356

—

135

C₄⋊Q₈

—

356

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

178

—

140

C₂×C₄⋊C₄

—

178

—

141

C₂×C₄⋊C₄

—

178

—

142

C₂×C₄⋊C₄

D₃₅₆

yes

—

143

C₂×C₄²

—

178

—

144

C₂×C₄²

C₂²×D₈₉

yes

characteristic

145

#106

—

356

—

146

#106

—

356

—

147

C₄⋊C₈

—

148

C₄⋊C₈

Dic₁₇₈

yes

complemented

149

C₄×C₈

—

150

C₄×C₈

—

178

—

151

#95

—

178

—

152

#95

—

178

—

153

C₂×Q₈

—

356

—

154

C₂×Q₈

—

712

—

155

C₂²×C₄

—

156

C₂²×C₄

—

356

—

157

C₂²×C₄

—

178

—

158

C₂²×C₄

—

178

—

159

C₂²×C₄

D₄×D₈₉

yes

—

160

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

178

—

165

C₂×C₈

—

178

—

166

C₂×C₈

—

178

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

356

—

173

C₄⋊C₄

—

356

—

174

C₄⋊C₄

—

356

—

175

C₄⋊C₄

—

356

—

176

C₄⋊C₄

—

178

—

177

C₄⋊C₄

—

178

—

178

C₄²

—

356

—

179

C₄²

—

356

—

180

C₄²

Q₈×D₈₉

yes

characteristic

181

C₄²

C₂×Dic₁₇₈

yes

—

182

C₄²

#13083573

yes

characteristic

183

C₂³

#15667982

yes

characteristic

184

Q₈

—

712

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

356

—

190

C₂×C₄

—

356

—

191

C₂×C₄

—

356

—

192

C₂×C₄

—

356

—

193

C₂×C₄

—

178

—

194

C₂×C₄

—

178

—

195

C₂×C₄

—

712

—

196

C₂×C₄

—

712

—

197

C₂×C₄

#15667989

yes

—

198

C₂×C₄

#15667992

yes

characteristic

199

C₂×C₄

#15668011

yes

—

200

C₂×C₄

#15668012

yes

characteristic

201

C₂×C₄

SD₁₆×D₈₉

yes

—

202

C₂×C₄

SD₁₆×D₈₉

yes

—

203

C₂×C₄

#15668036

yes

—

204

C₂×C₄

#15668044

yes

—

205

C₈

—

206

C₈

—

178

—

207

C₂²

#19151519

yes

characteristic

208

C₂²

#19151738

yes

—

209

C₂²

#19151750

yes

—

210

C₂²

#19151768

yes

characteristic

211

C₂²

#19151783

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

356

—

216

C₄

—

356

—

217

C₄

—

1424

—

218

C₄

#19151881

yes

—

219

C₄

#19151886

yes

—

220

C₄

#19151995

yes

—

221

C₄

#19152006

yes

—

222

C₂

#26019331

yes

characteristic

223

C₂

#26019369

yes

characteristic

224

C₂

#26019516

yes

—

225

C₂

#26021584

yes

—

226

C₂

#26021598

yes

characteristic

	subgroup	G/H
center	C₂³	#15667982
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1296	C₂
socle, prime radical	C₂³×C₈₉	#120
hypercenter	#1296	D₈₉
hypocenter	C₈₉	#10574
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

subgroup

G/H

center

C₂³

#15667982

commutator, cosocle

C₂×C₄²×C₈₉

C₂³

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1296

C₂

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

N	H ≅ G/N	type
30. C₄×#10623494	Q₈	semidirect
38. C₄⋊C₈×C₈₉	Q₈	semidirect
119. C₈₉	#10574	semidirect
148. C₄⋊C₈	Dic₁₇₈	semidirect

H ≅ G/N

type

30. C₄×#10623494

Q₈

semidirect

38. C₄⋊C₈×C₈₉

Q₈

semidirect

119. C₈₉

#10574

semidirect

148. C₄⋊C₈

Dic₁₇₈

semidirect

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

subgroup	index
#26025281	2
#26021016	2
#26020408	2
#26018091	2
C₈₉×#1296	2
#10574	89

subgroup

index

C₈₉×#1296

Subgroups — #36458869

Conjugacy classes of subgroups

Complements

Maximal subgroups