Group #36457458 — subgroups

Subgroups — #36457458

Order 22784.

416 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 123 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	#26025494	C₂	1	1	yes	maximal, characteristic
2	#26021201	C₂	1	2	yes	maximal
3	#26018135	C₂	1	1	yes	maximal, characteristic
4	#26018092	C₂	1	2	yes	maximal
5	C₈₉×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	#19151537	—	2	2	no	—
8	#19151497	—	2	2	no	—
9	#19151463	—	2	2	no	—
10	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	#19151360	—	2	2	no	—
12	#19151360	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#103×C₈₉	—	2	2	no	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#384×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×#10623494	—	2	2	no	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
21	C₂×C₄×Dic₈₉	—	2	2	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	#15667919	—	2	2	no	—
25	#15667919	—	2	4	no	—
26	#15667919	—	2	2	no	—
27	C₂²×Q₈×C₈₉	—	2	1	no	—
28	C₂²×C₈×C₈₉	—	2	1	no	—
29	C₄⋊Q₈×C₈₉	—	2	2	no	—
30	C₄⋊Q₈×C₈₉	—	2	1	no	—
31	C₄⋊Q₈×C₈₉	Q₈	1	2	yes	—
32	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
36	#103×C₈₉	—	4	4	no	—
37	C₄×C₈×C₈₉	—	2	2	no	—
38	C₂²×Dic₈₉	—	2	2	no	—
39	C₂²×Dic₈₉	—	2	2	no	—
40	C₂×#10623494	—	4	2	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₄⋊Dic₈₉	—	2	2	no	—
44	C₄⋊Dic₈₉	—	2	2	no	—
45	C₄⋊Dic₈₉	—	2	2	no	—
46	C₄⋊Dic₈₉	—	2	2	no	—
47	C₄⋊Dic₈₉	—	8	2	no	—
48	C₄×Dic₈₉	—	4	2	no	—
49	C₄×Dic₈₉	—	4	2	no	—
50	C₂×Q₈×C₈₉	—	2	2	no	—
51	C₂×Q₈×C₈₉	—	4	2	no	—
52	C₂×Q₈×C₈₉	—	4	2	no	—
53	C₂×Q₈×C₈₉	—	4	1	no	—
54	C₂²×C₄×C₈₉	—	2	2	no	—
55	C₂²×C₄×C₈₉	—	4	1	no	—
56	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₈₉	—	2	2	no	—
59	C₂×C₈×C₈₉	—	4	1	no	—
60	C₂×C₈×C₈₉	—	4	1	no	—
61	C₄⋊C₄×C₈₉	—	2	4	no	—
62	C₄⋊C₄×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	#41	1	2	yes	—
68	C₄²×C₈₉	#41	1	1	yes	characteristic
69	C₂³×C₈₉	#120	1	1	yes	characteristic
70	Q₈×C₈₉	—	4	4	no	—
71	Q₈×C₈₉	—	8	2	no	—
72	C₂×C₄×C₈₉	—	2	1	no	—
73	C₂×C₄×C₈₉	—	2	1	no	—
74	C₂×C₄×C₈₉	—	4	2	no	—
75	C₂×C₄×C₈₉	—	4	4	no	—
76	C₂×C₄×C₈₉	—	4	2	no	—
77	C₂×C₄×C₈₉	—	8	1	no	—
78	C₂×C₄×C₈₉	—	8	1	no	—
79	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#124	1	2	yes	—
82	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#135	1	2	yes	—
84	C₂×C₄×C₈₉	#135	1	2	yes	—
85	C₂×C₄×C₈₉	#137	1	2	yes	—
86	C₂×C₄×C₈₉	#137	1	2	yes	—
87	C₂×Dic₈₉	—	4	2	no	—
88	C₂×Dic₈₉	—	4	2	no	—
89	C₂×Dic₈₉	—	4	2	no	—
90	C₂×Dic₈₉	—	4	2	no	—
91	C₂×Dic₈₉	—	4	2	no	—
92	C₂×Dic₈₉	—	4	2	no	—
93	C₈×C₈₉	—	4	2	no	—
94	#10623494	—	8	2	no	—
95	C₂²×C₈₉	#393	1	1	yes	characteristic
96	C₂²×C₈₉	#452	1	2	yes	—
97	C₂²×C₈₉	#452	1	2	yes	—
98	C₂²×C₈₉	#453	1	1	yes	characteristic
99	C₂²×C₈₉	#454	1	1	yes	characteristic
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	8	2	no	—
103	C₄×C₈₉	—	8	2	no	—
104	C₄×C₈₉	#479	1	2	yes	—
105	C₄×C₈₉	#479	1	2	yes	—
106	C₄×C₈₉	#488	1	2	yes	—
107	C₄×C₈₉	#491	1	2	yes	—
108	Dic₈₉	—	8	2	no	—
109	Dic₈₉	—	8	2	no	—
110	#10352	—	89	1	no	maximal, Sylow
111	C₂×C₈₉	#1630	1	1	yes	characteristic
112	C₂×C₈₉	#1634	1	1	yes	characteristic
113	C₂×C₈₉	#1638	1	2	yes	—
114	C₂×C₈₉	#2026	1	2	yes	—
115	C₂×C₈₉	#2027	1	1	yes	characteristic
116	#2571	—	89	1	no	—
117	#1987	—	89	2	no	—
118	#1956	D₈₉	1	1	yes	characteristic
119	#1296	—	89	1	no	—
120	#1279	—	89	2	no	—
121	C₈₉	#10352	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
123	C₂×C₄⋊C₈	—	89	2	no	—
124	C₂×#103	—	2	2	no	—
125	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
126	#400	—	178	2	no	—
127	#389	—	178	2	no	—
128	#384	C₂×D₈₉	1	1	yes	characteristic
129	#382	—	178	2	no	—
130	C₄×C₄⋊C₄	—	89	2	no	—
131	#340	—	178	2	no	—
132	#340	—	178	2	no	—
133	C₂²×Q₈	—	2	1	no	—
134	C₂²×C₈	—	2	1	no	—
135	C₂²×C₈	—	178	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	2	1	no	—
138	C₄⋊Q₈	Dic₁₇₈	1	2	yes	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	89	2	no	—
142	C₂×C₄⋊C₄	—	89	2	no	—
143	C₂×C₄⋊C₄	—	178	2	no	—
144	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
145	C₂×C₄²	—	178	2	no	—
146	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
147	C₄⋊C₈	—	178	2	no	—
148	C₄⋊C₈	—	178	2	no	—
149	#103	—	4	4	no	—
150	C₄×C₈	—	2	2	no	—
151	#95	—	178	2	no	—
152	#95	—	178	4	no	—
153	#95	—	178	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	178	2	no	—
161	C₂²×C₄	—	178	2	no	—
162	C₂²×C₄	D₄×D₈₉	1	2	yes	—
163	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	356	2	no	—
168	C₂×C₈	—	356	2	no	—
169	C₂×C₈	—	356	2	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	178	2	no	—
176	C₄⋊C₄	—	178	2	no	—
177	C₄⋊C₄	—	178	2	no	—
178	C₄⋊C₄	—	178	2	no	—
179	C₄⋊C₄	—	712	2	no	—
180	C₄²	—	356	2	no	—
181	C₄²	—	356	2	no	—
182	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
183	C₄²	#13083573	1	2	yes	—
184	C₄²	#13083574	1	1	yes	characteristic
185	C₂³	#15667980	1	1	yes	characteristic
186	Q₈	—	4	4	no	—
187	Q₈	—	8	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	356	2	no	—
196	C₂×C₄	—	356	2	no	—
197	C₂×C₄	—	356	2	no	—
198	C₂×C₄	—	356	2	no	—
199	C₂×C₄	—	356	2	no	—
200	C₂×C₄	—	356	2	no	—
201	C₂×C₄	#15667989	1	2	yes	—
202	C₂×C₄	#15667991	1	1	yes	characteristic
203	C₂×C₄	#15667998	1	2	yes	—
204	C₂×C₄	#15667999	1	1	yes	characteristic
205	C₂×C₄	#15668033	1	2	yes	—
206	C₂×C₄	#15668035	1	2	yes	—
207	C₂×C₄	#15668041	1	2	yes	—
208	C₂×C₄	#15668044	1	2	yes	—
209	C₈	—	4	2	no	—
210	C₈	—	712	2	no	—
211	C₂²	#19151510	1	1	yes	characteristic
212	C₂²	#19151757	1	2	yes	—
213	C₂²	#19151759	1	2	yes	—
214	C₂²	#19151765	1	1	yes	characteristic
215	C₂²	#19151770	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	2	no	—
220	C₄	—	712	2	no	—
221	C₄	—	712	2	no	—
222	C₄	#19151903	1	2	yes	—
223	C₄	#19151907	1	2	yes	—
224	C₄	#19151954	1	2	yes	—
225	C₄	#19151965	1	2	yes	—
226	C₂	#26019371	1	1	yes	characteristic
227	C₂	#26019395	1	1	yes	characteristic
228	C₂	#26019421	1	2	yes	—
229	C₂	#26021401	1	2	yes	—
230	C₂	#26021408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26025494

C₂

yes

maximal, characteristic

#26021201

C₂

yes

maximal

#26018135

C₂

yes

maximal, characteristic

#26018092

C₂

yes

maximal

C₈₉×#1956

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

C₂²

yes

—

#19151537

—

#19151497

—

#19151463

—

C₄×C₄⋊Dic₈₉

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

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

D₄

yes

—

C₂×C₄⋊Dic₈₉

D₄

yes

—

C₂×C₄×Dic₈₉

—

—

—

—

—

—

C₂²×Q₈×C₈₉

—

C₂²×C₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

Q₈

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#103×C₈₉

—

C₄×C₈×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

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

—

C₄⋊C₄×C₈₉

—

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

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#393

yes

characteristic

C₂²×C₈₉

#452

yes

—

C₂²×C₈₉

#452

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

100

C₄×C₈₉

—

101

C₄×C₈₉

—

102

C₄×C₈₉

—

103

C₄×C₈₉

—

104

C₄×C₈₉

#479

yes

—

105

C₄×C₈₉

#479

yes

—

106

C₄×C₈₉

#488

yes

—

107

C₄×C₈₉

#491

yes

—

108

Dic₈₉

—

109

Dic₈₉

—

110

#10352

—

maximal, Sylow

111

C₂×C₈₉

#1630

yes

characteristic

112

C₂×C₈₉

#1634

yes

characteristic

113

C₂×C₈₉

#1638

yes

—

114

C₂×C₈₉

#2026

yes

—

115

C₂×C₈₉

#2027

yes

characteristic

116

#2571

—

117

#1987

—

118

#1956

D₈₉

yes

characteristic

119

#1296

—

120

#1279

—

121

C₈₉

#10352

yes

characteristic, complemented, Sylow

122

C₂×C₄⋊Q₈

C₂×D₈₉

yes

characteristic

123

C₂×C₄⋊C₈

—

124

C₂×#103

—

125

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

126

#400

—

178

—

127

#389

—

178

—

128

#384

C₂×D₈₉

yes

characteristic

129

#382

—

178

—

130

C₄×C₄⋊C₄

—

131

#340

—

178

—

132

#340

—

178

—

133

C₂²×Q₈

—

134

C₂²×C₈

—

135

C₂²×C₈

—

178

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

Dic₁₇₈

yes

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄⋊C₄

—

143

C₂×C₄⋊C₄

—

178

—

144

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

145

C₂×C₄²

—

178

—

146

C₂×C₄²

C₂²×D₈₉

yes

characteristic

147

C₄⋊C₈

—

178

—

148

C₄⋊C₈

—

178

—

149

#103

—

150

C₄×C₈

—

151

#95

—

178

—

152

#95

—

178

—

153

#95

—

178

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

178

—

161

C₂²×C₄

—

178

—

162

C₂²×C₄

D₄×D₈₉

yes

—

163

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

356

—

168

C₂×C₈

—

356

—

169

C₂×C₈

—

356

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

178

—

176

C₄⋊C₄

—

178

—

177

C₄⋊C₄

—

178

—

178

C₄⋊C₄

—

178

—

179

C₄⋊C₄

—

712

—

180

C₄²

—

356

—

181

C₄²

—

356

—

182

C₄²

C₂×Dic₁₇₈

yes

characteristic

183

C₄²

#13083573

yes

—

184

C₄²

#13083574

yes

characteristic

185

C₂³

#15667980

yes

characteristic

186

Q₈

—

187

Q₈

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

356

—

196

C₂×C₄

—

356

—

197

C₂×C₄

—

356

—

198

C₂×C₄

—

356

—

199

C₂×C₄

—

356

—

200

C₂×C₄

—

356

—

201

C₂×C₄

#15667989

yes

—

202

C₂×C₄

#15667991

yes

characteristic

203

C₂×C₄

#15667998

yes

—

204

C₂×C₄

#15667999

yes

characteristic

205

C₂×C₄

#15668033

yes

—

206

C₂×C₄

#15668035

yes

—

207

C₂×C₄

#15668041

yes

—

208

C₂×C₄

#15668044

yes

—

209

C₈

—

210

C₈

—

712

—

211

C₂²

#19151510

yes

characteristic

212

C₂²

#19151757

yes

—

213

C₂²

#19151759

yes

—

214

C₂²

#19151765

yes

characteristic

215

C₂²

#19151770

yes

characteristic

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

712

—

221

C₄

—

712

—

222

C₄

#19151903

yes

—

223

C₄

#19151907

yes

—

224

C₄

#19151954

yes

—

225

C₄

#19151965

yes

—

226

C₂

#26019371

yes

characteristic

227

C₂

#26019395

yes

characteristic

228

C₂

#26019421

yes

—

229

C₂

#26021401

yes

—

230

C₂

#26021408

yes

characteristic

	subgroup	G/H
center	C₂³	#15667980
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1956	C₂
socle, prime radical	C₂³×C₈₉	#120
hypercenter	#1956	D₈₉
hypocenter	C₈₉	#10352
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#15667980

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
121. C₈₉	#10352	semidirect

H ≅ G/N

type

121. C₈₉

#10352

semidirect

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

subgroup	index
#26025494	2
#26021201	2
#26018135	2
#26018092	2
C₈₉×#1956	2
#10352	89

subgroup

index

C₈₉×#1956

Subgroups — #36457458

Conjugacy classes of subgroups

Complements

Maximal subgroups