Finite Group Databasebeta

← C89×#62817 #36440665

Subgroups — C89×#62817 #36440665

Order 22784.

9494 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3532C2124yesmaximal, complemented
2C89×#3490C2118yesmaximal, complemented
3C89×#3484C2118yesmaximal, complemented
4C22×#120×C89C213yesmaximal, complemented
5C26×C89C2211yescharacteristic, complemented
6C2×H(5,𝔽2)×C89C2216yescomplemented
7C23×D4×C89C2219yescomplemented
8H(3,𝔽4)×C89C2218yescomplemented
9#560×C89C22124yescomplemented
10#551×C89C22172yescomplemented
11#546×C89C221144yescomplemented
12D42×C89C22172yescomplemented
13#543×C89C2216yescomplemented
14#538×C89C22172yescomplemented
15#535×C89C22136yescomplemented
16#534×C89C22118yescomplemented
17#534×C89C22136yescomplemented
18#525×C89C22136yescomplemented
19C2×#120×C89C22136yescomplemented
20C2×#120×C89C22136yescomplemented
21C2×#120×C89C22112yescomplemented
22#518×C89C22118yescomplemented
23C22×C22⋊C4×C89C2219yescomplemented
24C25×C89—412no—
25C25×C89C2319yes—
26C25×C89C2316yes—
27C25×C89C2313yescomplemented
28H(5,𝔽2)×C89—424no—
29C2×#41×C89C23136yescomplemented
30C22×D4×C89—436no—
31C22×D4×C89C23118yes—
32C22×D4×C89C23118yescomplemented
33C22×D4×C89C23172yescomplemented
34C22×D4×C89C23136yescomplemented
35C23×C4×C89C2319yescomplemented
36#127×C89C23124yescomplemented
37#126×C89C23148yescomplemented
38#124×C89C23172yescomplemented
39#123×C89C23172yescomplemented
40#123×C89C23172yescomplemented
41C22⋊Q8×C89C23172yescomplemented
42#121×C89C231144yescomplemented
43#121×C89C23172yescomplemented
44#120×C89C23172yes—
45#120×C89C23124yescomplemented
46#120×C89C23172yescomplemented
47#120×C89C23124yescomplemented
48#120×C89C23172yescomplemented
49#120×C89C23124yescomplemented
50C4×D4×C89C231144yescomplemented
51#117×C89C23136yescomplemented
52C2×C22⋊C4×C89C23118yes—
53C2×C22⋊C4×C89C23118yescomplemented
54C2×C22⋊C4×C89C23172yescomplemented
55C2×C22⋊C4×C89C23136yescomplemented
56C24×C89—236no—
57C24×C89—436no—
58C24×C89—436no—
59C24×C89—418no—
60C24×C89—464no—
61C24×C89C2413yes—
62C24×C89C24118yes—
63C24×C89C2419yes—
64C24×C89C24118yes—
65C24×C89C2412yes—
66C24×C89C2413yes—
67C24×C89C2414yescomplemented
68#41×C89—472no—
69C2×Q8×C89C24112yescomplemented
70C2×D4×C89—272no—
71C2×D4×C89—436no—
72C2×D4×C89—4144no—
73C2×D4×C89—472no—
74C2×D4×C89C24172yes—
75C2×D4×C89C24172yes—
76C2×D4×C89C24136yescomplemented
77C22×C4×C89—418no—
78C22×C4×C89C24118yes—
79C22×C4×C89C24136yes—
80C22×C4×C89C24136yescomplemented
81C4⋊C4×C89C24172yescomplemented
82C22⋊C4×C89C24172yes—
83C22⋊C4×C89C24172yes—
84C22⋊C4×C89C24172yescomplemented
85C42×C89C24124yescomplemented
86C23×C89—218no—
87C23×C89—272no—
88C23×C89—218no—
89C23×C89—236no—
90C23×C89—46no—
91C23×C89—418no—
92C23×C89—418no—
93C23×C89—46no—
94C23×C89—436no—
95C23×C89—412no—
96C23×C89—4144no—
97C23×C89—496no—
98C23×C89—824no—
99C23×C89H(5,𝔽2)112yescomplemented
100C23×C89C2516yes—
101C23×C89C25112yes—
102C23×C89C2519yes—
103Q8×C89—412no—
104D4×C89—2144no—
105D4×C89—472no—
106D4×C89—436no—
107C2×C4×C89—236no—
108C2×C4×C89—436no—
109C2×C4×C89—436no—
110C2×C4×C89C25136yes—
111C22×C89—224no—
112C22×C89—218no—
113C22×C89—236no—
114C22×C89—218no—
115C22×C89—412no—
116C22×C89—472no—
117C22×C89—436no—
118C22×C89—48no—
119C22×C89—836no—
120C22×C89—164no—
121C22×C89C2×H(5,𝔽2)118yes—
122C22×C89C2611yescharacteristic
123C4×C89—236no—
124#62817C8911yesmaximal, characteristic, direct product factor, Sylow
125C2×C89—218no—
126C2×C89—46no—
127C2×C89—412no—
128C2×C89C22×H(5,𝔽2)13yes—
129#3532C2×C89124yescomplemented
130#3490C2×C89118yescomplemented
131#3484C2×C89118yescomplemented
132C22×#120C2×C8913yescomplemented
133C89#6281711yescharacteristic, direct product factor, Sylow
134C26C22×C8911yescharacteristic, complemented
135C2×H(5,𝔽2)C22×C8916yescomplemented
136C23×D4C22×C8919yescomplemented
137H(3,𝔽4)C22×C8918yescomplemented
138#560C22×C89124yescomplemented
139#551C22×C89172yescomplemented
140#546C22×C891144yescomplemented
141D42C22×C89172yescomplemented
142#543C22×C8916yescomplemented
143#538C22×C89172yescomplemented
144#535C22×C89136yescomplemented
145#534C22×C89118yescomplemented
146#534C22×C89136yescomplemented
147#525C22×C89136yescomplemented
148C2×#120C22×C89136yescomplemented
149C2×#120C22×C89136yescomplemented
150C2×#120C22×C89112yescomplemented
151#518C22×C89118yescomplemented
152C22×C22⋊C4C22×C8919yescomplemented
153C25—412no—
154C25C23×C8919yes—
155C25C23×C8916yes—
156C25C23×C8913yescomplemented
157H(5,𝔽2)—424no—
158C2×#41C23×C89136yescomplemented
159C22×D4—436no—
160C22×D4C23×C89118yes—
161C22×D4C23×C89118yescomplemented
162C22×D4C23×C89172yescomplemented
163C22×D4C23×C89136yescomplemented
164C23×C4C23×C8919yescomplemented
165#127C23×C89124yescomplemented
166#126C23×C89148yescomplemented
167#124C23×C89172yescomplemented
168#123C23×C89172yescomplemented
169#123C23×C89172yescomplemented
170C22⋊Q8C23×C89172yescomplemented
171#121C23×C891144yescomplemented
172#121C23×C89172yescomplemented
173#120C23×C89172yes—
174#120C23×C89124yescomplemented
175#120C23×C89172yescomplemented
176#120C23×C89124yescomplemented
177#120C23×C89172yescomplemented
178#120C23×C89124yescomplemented
179C4×D4C23×C891144yescomplemented
180#117C23×C89136yescomplemented
181C2×C22⋊C4C23×C89118yes—
182C2×C22⋊C4C23×C89118yescomplemented
183C2×C22⋊C4C23×C89172yescomplemented
184C2×C22⋊C4C23×C89136yescomplemented
185C24—236no—
186C24—436no—
187C24—436no—
188C24—418no—
189C24—464no—
190C24C24×C8913yes—
191C24C24×C89118yes—
192C24C24×C8919yes—
193C24C24×C89118yes—
194C24C24×C8912yes—
195C24C24×C8913yes—
196C24C24×C8914yescomplemented
197#41—472no—
198C2×Q8C24×C89112yescomplemented
199C2×D4—272no—
200C2×D4—436no—
201C2×D4—4144no—
202C2×D4—472no—
203C2×D4C24×C89172yes—
204C2×D4C24×C89172yes—
205C2×D4C24×C89136yescomplemented
206C22×C4—418no—
207C22×C4C24×C89118yes—
208C22×C4C24×C89136yes—
209C22×C4C24×C89136yescomplemented
210C4⋊C4C24×C89172yescomplemented
211C22⋊C4C24×C89172yes—
212C22⋊C4C24×C89172yes—
213C22⋊C4C24×C89172yescomplemented
214C42C24×C89124yescomplemented
215C23—218no—
216C23—272no—
217C23—218no—
218C23—236no—
219C23—46no—
220C23—418no—
221C23—418no—
222C23—46no—
223C23—436no—
224C23—412no—
225C23—4144no—
226C23—496no—
227C23—824no—
228C23H(5,𝔽2)×C89112yescomplemented
229C23C25×C8916yes—
230C23C25×C89112yes—
231C23C25×C8919yes—
232Q8—412no—
233D4—2144no—
234D4—472no—
235D4—436no—
236C2×C4—236no—
237C2×C4—436no—
238C2×C4—436no—
239C2×C4C25×C89136yes—
240C22—224no—
241C22—218no—
242C22—236no—
243C22—218no—
244C22—412no—
245C22—472no—
246C22—436no—
247C22—48no—
248C22—836no—
249C22—164no—
250C22C2×H(5,𝔽2)×C89118yes—
251C22C26×C8911yescharacteristic
252C4—236no—
253C2—218no—
254C2—46no—
255C2—412no—
256C2C22×H(5,𝔽2)×C8913yes—
subgroupG/H
center, socleC22×C89C26
commutator, Frattini, cosocleC22C26×C89
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C89 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C89×#3532C2semidirect
2. C89×#3490C2semidirect
3. C89×#3484C2semidirect
4. C22×#120×C89C2semidirect
5. C26×C89C22semidirect
6. C2×H(5,𝔽2)×C89C22semidirect
7. C23×D4×C89C22semidirect
8. H(3,𝔽4)×C89C22semidirect
9. #560×C89C22semidirect
10. #551×C89C22semidirect
11. #546×C89C22semidirect
12. D42×C89C22semidirect
13. #543×C89C22semidirect
14. #538×C89C22semidirect
15. #535×C89C22semidirect
16. #534×C89C22semidirect
17. #534×C89C22semidirect
18. #525×C89C22semidirect
19. C2×#120×C89C22semidirect
20. C2×#120×C89C22semidirect
21. C2×#120×C89C22semidirect
22. #518×C89C22semidirect
23. C22×C22⋊C4×C89C22semidirect
27. C25×C89C23semidirect
29. C2×#41×C89C23semidirect
32. C22×D4×C89C23semidirect
33. C22×D4×C89C23semidirect
34. C22×D4×C89C23semidirect
35. C23×C4×C89C23semidirect
36. #127×C89C23semidirect
37. #126×C89C23semidirect
38. #124×C89C23semidirect
39. #123×C89C23semidirect
40. #123×C89C23semidirect
41. C22⋊Q8×C89C23semidirect
42. #121×C89C23semidirect
43. #121×C89C23semidirect
45. #120×C89C23semidirect
46. #120×C89C23semidirect
47. #120×C89C23semidirect
48. #120×C89C23semidirect
49. #120×C89C23semidirect
50. C4×D4×C89C23semidirect
51. #117×C89C23semidirect
53. C2×C22⋊C4×C89C23semidirect
54. C2×C22⋊C4×C89C23semidirect
55. C2×C22⋊C4×C89C23semidirect
67. C24×C89C24semidirect
69. C2×Q8×C89C24semidirect
76. C2×D4×C89C24semidirect
80. C22×C4×C89C24semidirect
81. C4⋊C4×C89C24semidirect
84. C22⋊C4×C89C24semidirect
85. C42×C89C24semidirect
99. C23×C89H(5,𝔽2)semidirect
124. #62817C89direct (N × H)
129. #3532C2×C89semidirect
130. #3490C2×C89semidirect
131. #3484C2×C89semidirect
132. C22×#120C2×C89semidirect
133. C89#62817direct (N × H)
134. C26C22×C89semidirect
135. C2×H(5,𝔽2)C22×C89semidirect
136. C23×D4C22×C89semidirect
137. H(3,𝔽4)C22×C89semidirect
138. #560C22×C89semidirect
139. #551C22×C89semidirect
140. #546C22×C89semidirect
141. D42C22×C89semidirect
142. #543C22×C89semidirect
143. #538C22×C89semidirect
144. #535C22×C89semidirect
145. #534C22×C89semidirect
146. #534C22×C89semidirect
147. #525C22×C89semidirect
148. C2×#120C22×C89semidirect
149. C2×#120C22×C89semidirect
150. C2×#120C22×C89semidirect
151. #518C22×C89semidirect
152. C22×C22⋊C4C22×C89semidirect
156. C25C23×C89semidirect
158. C2×#41C23×C89semidirect
161. C22×D4C23×C89semidirect
162. C22×D4C23×C89semidirect
163. C22×D4C23×C89semidirect
164. C23×C4C23×C89semidirect
165. #127C23×C89semidirect
166. #126C23×C89semidirect
167. #124C23×C89semidirect
168. #123C23×C89semidirect
169. #123C23×C89semidirect
170. C22⋊Q8C23×C89semidirect
171. #121C23×C89semidirect
172. #121C23×C89semidirect
174. #120C23×C89semidirect
175. #120C23×C89semidirect
176. #120C23×C89semidirect
177. #120C23×C89semidirect
178. #120C23×C89semidirect
179. C4×D4C23×C89semidirect
180. #117C23×C89semidirect
182. C2×C22⋊C4C23×C89semidirect
183. C2×C22⋊C4C23×C89semidirect
184. C2×C22⋊C4C23×C89semidirect
196. C24C24×C89semidirect
198. C2×Q8C24×C89semidirect
205. C2×D4C24×C89semidirect
209. C22×C4C24×C89semidirect
210. C4⋊C4C24×C89semidirect
213. C22⋊C4C24×C89semidirect
214. C42C24×C89semidirect
228. C23H(5,𝔽2)×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#35322
C89×#34902
C89×#34842
C22×#120×C892
#6281789