Finite Group Databasebeta

← C53×#3532 #20272469

Subgroups — C53×#3532 #20272469

Order 6784.

1336 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×H(3,𝔽4)C211yesmaximal, characteristic, complemented
2C53×#560C213yesmaximal, complemented
3C53×#551C213yesmaximal, complemented
4C53×#546C216yesmaximal, complemented
5D42×C53C213yesmaximal, complemented
6C53×#538C216yesmaximal, complemented
7C53×#535C213yesmaximal, complemented
8C53×#534C213yesmaximal, complemented
9C2×#120×C53C213yesmaximal, complemented
10C25×C53C2211yescharacteristic, complemented
11C2×#41×C53C2213yescomplemented
12C22×D4×C53C2213yescomplemented
13C22×D4×C53C2216yescomplemented
14C22×D4×C53C2213yescomplemented
15#127×C53C2211yescharacteristic, complemented
16#127×C53C2213yescomplemented
17#126×C53C2212yescomplemented
18#126×C53C2216yescomplemented
19#124×C53C2213yescomplemented
20#124×C53C2213yescomplemented
21#124×C53C2216yescomplemented
22#123×C53C2216yescomplemented
23#123×C53C2213yescomplemented
24#123×C53C2213yescomplemented
25#123×C53C2216yescomplemented
26C22⋊Q8×C53C2216yescomplemented
27#121×C53C2216yescomplemented
28#121×C53C2216yescomplemented
29#121×C53C2216yescomplemented
30#121×C53C2216yescomplemented
31#121×C53C2216yescomplemented
32#120×C53C2213yes—
33#120×C53C2216yescomplemented
34#120×C53C2216yescomplemented
35#120×C53C2213yescomplemented
36#120×C53C2213yescomplemented
37#120×C53C2213yescomplemented
38#120×C53C2211yescharacteristic, complemented
39#120×C53C2213yescomplemented
40#120×C53C2216yescomplemented
41C4×D4×C53C2216yescomplemented
42C4×D4×C53C2216yescomplemented
43#117×C53C2213yescomplemented
44C2×C22⋊C4×C53C2213yescomplemented
45C2×C22⋊C4×C53C2216yescomplemented
46C2×C22⋊C4×C53C2213yescomplemented
47C24×C53—46no—
48C24×C53C2313yes—
49C24×C53C2313yes—
50C24×C53C2313yes—
51C24×C53C2311yescharacteristic
52C24×C53C2311yescharacteristic, complemented
53#41×C53—46no—
54C2×Q8×C53C2313yescomplemented
55C2×D4×C53—46no—
56C2×D4×C53—412no—
57C2×D4×C53—46no—
58C2×D4×C53C2316yes—
59C2×D4×C53C2316yes—
60C2×D4×C53C2313yes—
61C2×D4×C53C2313yes—
62C2×D4×C53C2316yes—
63C2×D4×C53C2313yes—
64C2×D4×C53C2316yes—
65C2×D4×C53C2313yes—
66C2×D4×C53C2313yescomplemented
67C2×D4×C53C2316yescomplemented
68C22×C4×C53C2313yes—
69C22×C4×C53C2316yes—
70C22×C4×C53C2316yescomplemented
71C22×C4×C53C2313yescomplemented
72C4⋊C4×C53C2316yescomplemented
73C4⋊C4×C53C2316yescomplemented
74C4⋊C4×C53C2316yescomplemented
75C22⋊C4×C53C2316yes—
76C22⋊C4×C53C2316yes—
77C22⋊C4×C53C2313yes—
78C22⋊C4×C53C2316yes—
79C22⋊C4×C53C2316yes—
80C22⋊C4×C53C2316yes—
81C22⋊C4×C53C2313yes—
82C22⋊C4×C53C2313yescomplemented
83C22⋊C4×C53C2313yescomplemented
84C22⋊C4×C53C2316yescomplemented
85C22⋊C4×C53C2316yescomplemented
86C42×C53C2311yescharacteristic, complemented
87C42×C53C2313yescomplemented
88C42×C53C2312yescomplemented
89C23×C53—212no—
90C23×C53—26no—
91C23×C53—26no—
92C23×C53—43no—
93C23×C53—46no—
94C23×C53—43no—
95C23×C53—46no—
96C23×C53—43no—
97C23×C53—416no—
98C23×C53C2413yes—
99C23×C53C2413yes—
100C23×C53C2413yes—
101C23×C53C2411yescharacteristic
102C23×C53C2413yes—
103Q8×C53—43no—
104D4×C53—212no—
105D4×C53—212no—
106D4×C53—43no—
107D4×C53—43no—
108D4×C53—43no—
109D4×C53—43no—
110D4×C53—46no—
111D4×C53—46no—
112D4×C53—46no—
113D4×C53—43no—
114C2×C4×C53—26no—
115C2×C4×C53—46no—
116C2×C4×C53—43no—
117C2×C4×C53—46no—
118C2×C4×C53C2413yes—
119C2×C4×C53C2416yes—
120C2×C4×C53C2416yes—
121C2×C4×C53C2413yes—
122C22×C53—26no—
123C22×C53—26no—
124C22×C53—23no—
125C22×C53—26no—
126C22×C53—23no—
127C22×C53—23no—
128C22×C53—26no—
129C22×C53—23no—
130C22×C53—412no—
131C22×C53—412no—
132C22×C53—44no—
133C22×C53—86no—
134C22×C53—82no—
135C22×C53H(5,𝔽2)16yes—
136C22×C53C2511yescharacteristic
137C4×C53—23no—
138C4×C53—23no—
139C4×C53—26no—
140C4×C53—26no—
141#3532C5311yesmaximal, characteristic, direct product factor, Sylow
142C2×C53—26no—
143C2×C53—43no—
144C2×C53—41no—
145C2×C53—43no—
146C2×C53—43no—
147C2×C53C2×H(5,𝔽2)13yes—
148H(3,𝔽4)C2×C5311yescharacteristic, complemented
149#560C2×C5313yescomplemented
150#551C2×C5313yescomplemented
151#546C2×C5316yescomplemented
152D42C2×C5313yescomplemented
153#538C2×C5316yescomplemented
154#535C2×C5313yescomplemented
155#534C2×C5313yescomplemented
156C2×#120C2×C5313yescomplemented
157C53#353211yescharacteristic, direct product factor, Sylow
158C25C22×C5311yescharacteristic, complemented
159C2×#41C22×C5313yescomplemented
160C22×D4C22×C5313yescomplemented
161C22×D4C22×C5316yescomplemented
162C22×D4C22×C5313yescomplemented
163#127C22×C5311yescharacteristic, complemented
164#127C22×C5313yescomplemented
165#126C22×C5312yescomplemented
166#126C22×C5316yescomplemented
167#124C22×C5313yescomplemented
168#124C22×C5313yescomplemented
169#124C22×C5316yescomplemented
170#123C22×C5313yescomplemented
171#123C22×C5313yescomplemented
172#123C22×C5316yescomplemented
173#123C22×C5316yescomplemented
174C22⋊Q8C22×C5316yescomplemented
175#121C22×C5316yescomplemented
176#121C22×C5316yescomplemented
177#121C22×C5316yescomplemented
178#121C22×C5316yescomplemented
179#121C22×C5316yescomplemented
180#120C22×C5313yes—
181#120C22×C5316yescomplemented
182#120C22×C5316yescomplemented
183#120C22×C5313yescomplemented
184#120C22×C5313yescomplemented
185#120C22×C5313yescomplemented
186#120C22×C5316yescomplemented
187#120C22×C5311yescharacteristic, complemented
188#120C22×C5313yescomplemented
189C4×D4C22×C5316yescomplemented
190C4×D4C22×C5316yescomplemented
191#117C22×C5313yescomplemented
192C2×C22⋊C4C22×C5313yescomplemented
193C2×C22⋊C4C22×C5316yescomplemented
194C2×C22⋊C4C22×C5313yescomplemented
195C24—46no—
196C24C23×C5313yes—
197C24C23×C5313yes—
198C24C23×C5313yes—
199C24C23×C5311yescharacteristic
200C24C23×C5311yescharacteristic, complemented
201#41—46no—
202C2×Q8C23×C5313yescomplemented
203C2×D4—46no—
204C2×D4—412no—
205C2×D4—46no—
206C2×D4C23×C5316yes—
207C2×D4C23×C5316yes—
208C2×D4C23×C5313yes—
209C2×D4C23×C5313yes—
210C2×D4C23×C5316yes—
211C2×D4C23×C5313yes—
212C2×D4C23×C5316yes—
213C2×D4C23×C5313yes—
214C2×D4C23×C5313yescomplemented
215C2×D4C23×C5316yescomplemented
216C22×C4C23×C5313yes—
217C22×C4C23×C5316yes—
218C22×C4C23×C5316yescomplemented
219C22×C4C23×C5313yescomplemented
220C4⋊C4C23×C5316yescomplemented
221C4⋊C4C23×C5316yescomplemented
222C4⋊C4C23×C5316yescomplemented
223C22⋊C4C23×C5316yes—
224C22⋊C4C23×C5316yes—
225C22⋊C4C23×C5313yes—
226C22⋊C4C23×C5316yes—
227C22⋊C4C23×C5316yes—
228C22⋊C4C23×C5316yes—
229C22⋊C4C23×C5313yes—
230C22⋊C4C23×C5313yescomplemented
231C22⋊C4C23×C5313yescomplemented
232C22⋊C4C23×C5316yescomplemented
233C22⋊C4C23×C5316yescomplemented
234C42C23×C5311yescharacteristic, complemented
235C42C23×C5313yescomplemented
236C42C23×C5312yescomplemented
237C23—212no—
238C23—26no—
239C23—26no—
240C23—43no—
241C23—46no—
242C23—43no—
243C23—46no—
244C23—43no—
245C23—416no—
246C23C24×C5313yes—
247C23C24×C5311yescharacteristic
248C23C24×C5313yes—
249C23C24×C5313yes—
250C23C24×C5313yes—
251Q8—43no—
252D4—212no—
253D4—212no—
254D4—43no—
255D4—43no—
256D4—43no—
257D4—43no—
258D4—46no—
259D4—46no—
260D4—46no—
261D4—43no—
262C2×C4—26no—
263C2×C4—46no—
264C2×C4—43no—
265C2×C4—46no—
266C2×C4C24×C5313yes—
267C2×C4C24×C5316yes—
268C2×C4C24×C5316yes—
269C2×C4C24×C5313yes—
270C22—26no—
271C22—26no—
272C22—23no—
273C22—26no—
274C22—23no—
275C22—23no—
276C22—26no—
277C22—23no—
278C22—412no—
279C22—412no—
280C22—44no—
281C22—86no—
282C22—82no—
283C22H(5,𝔽2)×C5316yes—
284C22C25×C5311yescharacteristic
285C4—23no—
286C4—23no—
287C4—26no—
288C4—26no—
289C2—26no—
290C2—43no—
291C2—41no—
292C2—43no—
293C2—43no—
294C2C2×H(5,𝔽2)×C5313yes—
subgroupG/H
center, socleC22×C53C25
commutator, Frattini, cosocleC22C25×C53
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×H(3,𝔽4)C2semidirect
2. C53×#560C2semidirect
3. C53×#551C2semidirect
4. C53×#546C2semidirect
5. D42×C53C2semidirect
6. C53×#538C2semidirect
7. C53×#535C2semidirect
8. C53×#534C2semidirect
9. C2×#120×C53C2semidirect
10. C25×C53C22semidirect
11. C2×#41×C53C22semidirect
12. C22×D4×C53C22semidirect
13. C22×D4×C53C22semidirect
14. C22×D4×C53C22semidirect
15. #127×C53C22semidirect
16. #127×C53C22semidirect
17. #126×C53C22semidirect
18. #126×C53C22semidirect
19. #124×C53C22semidirect
20. #124×C53C22semidirect
21. #124×C53C22semidirect
22. #123×C53C22semidirect
23. #123×C53C22semidirect
24. #123×C53C22semidirect
25. #123×C53C22semidirect
26. C22⋊Q8×C53C22semidirect
27. #121×C53C22semidirect
28. #121×C53C22semidirect
29. #121×C53C22semidirect
30. #121×C53C22semidirect
31. #121×C53C22semidirect
33. #120×C53C22semidirect
34. #120×C53C22semidirect
35. #120×C53C22semidirect
36. #120×C53C22semidirect
37. #120×C53C22semidirect
38. #120×C53C22semidirect
39. #120×C53C22semidirect
40. #120×C53C22semidirect
41. C4×D4×C53C22semidirect
42. C4×D4×C53C22semidirect
43. #117×C53C22semidirect
44. C2×C22⋊C4×C53C22semidirect
45. C2×C22⋊C4×C53C22semidirect
46. C2×C22⋊C4×C53C22semidirect
52. C24×C53C23semidirect
54. C2×Q8×C53C23semidirect
66. C2×D4×C53C23semidirect
67. C2×D4×C53C23semidirect
70. C22×C4×C53C23semidirect
71. C22×C4×C53C23semidirect
72. C4⋊C4×C53C23semidirect
73. C4⋊C4×C53C23semidirect
74. C4⋊C4×C53C23semidirect
82. C22⋊C4×C53C23semidirect
83. C22⋊C4×C53C23semidirect
84. C22⋊C4×C53C23semidirect
85. C22⋊C4×C53C23semidirect
86. C42×C53C23semidirect
87. C42×C53C23semidirect
88. C42×C53C23semidirect
141. #3532C53direct (N × H)
148. H(3,𝔽4)C2×C53semidirect
149. #560C2×C53semidirect
150. #551C2×C53semidirect
151. #546C2×C53semidirect
152. D42C2×C53semidirect
153. #538C2×C53semidirect
154. #535C2×C53semidirect
155. #534C2×C53semidirect
156. C2×#120C2×C53semidirect
157. C53#3532direct (N × H)
158. C25C22×C53semidirect
159. C2×#41C22×C53semidirect
160. C22×D4C22×C53semidirect
161. C22×D4C22×C53semidirect
162. C22×D4C22×C53semidirect
163. #127C22×C53semidirect
164. #127C22×C53semidirect
165. #126C22×C53semidirect
166. #126C22×C53semidirect
167. #124C22×C53semidirect
168. #124C22×C53semidirect
169. #124C22×C53semidirect
170. #123C22×C53semidirect
171. #123C22×C53semidirect
172. #123C22×C53semidirect
173. #123C22×C53semidirect
174. C22⋊Q8C22×C53semidirect
175. #121C22×C53semidirect
176. #121C22×C53semidirect
177. #121C22×C53semidirect
178. #121C22×C53semidirect
179. #121C22×C53semidirect
181. #120C22×C53semidirect
182. #120C22×C53semidirect
183. #120C22×C53semidirect
184. #120C22×C53semidirect
185. #120C22×C53semidirect
186. #120C22×C53semidirect
187. #120C22×C53semidirect
188. #120C22×C53semidirect
189. C4×D4C22×C53semidirect
190. C4×D4C22×C53semidirect
191. #117C22×C53semidirect
192. C2×C22⋊C4C22×C53semidirect
193. C2×C22⋊C4C22×C53semidirect
194. C2×C22⋊C4C22×C53semidirect
200. C24C23×C53semidirect
202. C2×Q8C23×C53semidirect
214. C2×D4C23×C53semidirect
215. C2×D4C23×C53semidirect
218. C22×C4C23×C53semidirect
219. C22×C4C23×C53semidirect
220. C4⋊C4C23×C53semidirect
221. C4⋊C4C23×C53semidirect
222. C4⋊C4C23×C53semidirect
230. C22⋊C4C23×C53semidirect
231. C22⋊C4C23×C53semidirect
232. C22⋊C4C23×C53semidirect
233. C22⋊C4C23×C53semidirect
234. C42C23×C53semidirect
235. C42C23×C53semidirect
236. C42C23×C53semidirect

Maximal subgroups

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

subgroupindex
C53×H(3,𝔽4)2
C53×#5602
C53×#5512
C53×#5462
D42×C532
C53×#5382
C53×#5352
C53×#5342
C2×#120×C532
#353253