Finite Group Databasebeta

← C53×#36607 #27244119

Subgroups — C53×#36607 #27244119

Order 13568.

2524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2847C213yesmaximal, complemented
2C53×#2569C213yesmaximal, complemented
3C53×#2543C213yesmaximal, complemented
4C53×#2510C216yesmaximal, complemented
5C53×#2444C216yesmaximal, complemented
6C53×#2364C213yesmaximal, complemented
7C53×#2353C213yesmaximal, complemented
8C4×#120×C53C213yesmaximal, complemented
9C22×#95×C53C211yesmaximal, characteristic, complemented
10C24×C4×C53C2213yes—
11C2×#121×C53C2213yes—
12C2×#121×C53C2216yes—
13C2×#120×C53C2213yes—
14C2×C4×D4×C53C2213yes—
15C2×C4×D4×C53C2216yes—
16C53×#401C2212yescomplemented
17C53×#400C2216yescomplemented
18C53×#398C2213yescomplemented
19C53×#397C2216yescomplemented
20C53×#397C2213yescomplemented
21C53×#394C2213yescomplemented
22C53×#392C2211yescharacteristic, complemented
23C53×#388C2216yescomplemented
24C53×#388C2216yescomplemented
25C53×#387C2216yescomplemented
26C53×#387C2216yescomplemented
27C53×#386C2213yes—
28C53×#386C2216yes—
29C53×#386C2216yes—
30C53×#386C2216yes—
31C53×#386C2216yes—
32C53×#386C2216yescomplemented
33C53×#385C2216yes—
34C53×#385C2213yes—
35C53×#383C2213yescomplemented
36C53×#381C2213yescomplemented
37C53×#380C2216yes—
38C53×#380C2213yes—
39C53×#379C2211yescharacteristic
40C4×C22⋊C4×C53C2216yes—
41C4×C22⋊C4×C53C2213yes—
42C4×C22⋊C4×C53C2213yescomplemented
43C53×#376C2211yescharacteristic, complemented
44C2×#95×C53C2216yes—
45C2×#95×C53C2213yes—
46C2×#95×C53C2213yes—
47C25×C53C2311yescharacteristic
48C22×D4×C53C2313yes—
49C22×D4×C53C2316yes—
50C22×D4×C53C2313yes—
51C23×C4×C53—212no—
52C23×C4×C53—412no—
53C23×C4×C53C2316yes—
54C23×C4×C53C2316yes—
55C23×C4×C53C2313yes—
56C23×C4×C53C2313yes—
57C23×C4×C53C2313yes—
58#121×C53—212no—
59#121×C53—412no—
60#120×C53—212no—
61C4×D4×C53—212no—
62C4×D4×C53—412no—
63C2×C4⋊C4×C53C2313yes—
64C2×C4⋊C4×C53C2313yes—
65C2×C4⋊C4×C53C2316yes—
66C2×C4⋊C4×C53C2316yes—
67C2×C22⋊C4×C53C2313yes—
68C2×C22⋊C4×C53C2313yes—
69C2×C22⋊C4×C53C2313yes—
70C2×C22⋊C4×C53C2316yes—
71C2×C22⋊C4×C53C2313yes—
72C2×C22⋊C4×C53C2316yes—
73C2×C22⋊C4×C53C2316yes—
74C2×C22⋊C4×C53C2316yes—
75C2×C22⋊C4×C53C2316yes—
76C2×C42×C53C2313yes—
77C2×C42×C53C2316yes—
78#95×C53C2311yescharacteristic
79#95×C53C2313yes—
80#95×C53C2316yes—
81#95×C53C2311yescharacteristic
82#95×C53C2313yes—
83#95×C53C2312yes—
84#95×C53C2313yes—
85#95×C53C2316yes—
86#95×C53C2316yes—
87#95×C53C2316yes—
88#95×C53C2313yes—
89#95×C53C2316yes—
90#95×C53C2316yes—
91C24×C53—26no—
92C24×C53—26no—
93C24×C53#4114yes—
94C24×C53C2413yes—
95C24×C53C2413yes—
96C2×D4×C53—26no—
97C2×D4×C53—26no—
98C2×D4×C53—212no—
99C2×D4×C53—212no—
100C2×D4×C53—212no—
101C2×D4×C53—26no—
102C2×D4×C53—26no—
103C2×D4×C53—46no—
104C22×C4×C53—26no—
105C22×C4×C53—212no—
106C22×C4×C53—212no—
107C22×C4×C53—212no—
108C22×C4×C53—212no—
109C22×C4×C53—26no—
110C22×C4×C53—46no—
111C22×C4×C53—46no—
112C22×C4×C53—46no—
113C22×C4×C53—46no—
114C22×C4×C53—46no—
115C22×C4×C53—448no—
116C22×C4×C53#41112yes—
117C22×C4×C53C2416yes—
118C22×C4×C53C2411yescharacteristic
119C22×C4×C53C2413yes—
120C22×C4×C53C2416yes—
121C22×C4×C53C2413yes—
122C22×C4×C53C2413yes—
123C22×C4×C53C2413yes—
124C4⋊C4×C53—26no—
125C4⋊C4×C53—26no—
126C4⋊C4×C53—46no—
127C4⋊C4×C53—46no—
128C22⋊C4×C53—26no—
129C22⋊C4×C53—212no—
130C22⋊C4×C53—26no—
131C22⋊C4×C53—26no—
132C22⋊C4×C53—26no—
133C22⋊C4×C53—212no—
134C22⋊C4×C53—46no—
135C22⋊C4×C53—46no—
136C22⋊C4×C53—46no—
137C42×C53—26no—
138C42×C53—46no—
139C23×C53—224no—
140C23×C53—28no—
141C23×C53—224no—
142C23×C53—26no—
143C23×C53—26no—
144C23×C53—23no—
145C23×C53—23no—
146C23×C53—26no—
147C23×C53—26no—
148C23×C53—23no—
149C23×C53C2×#4116yes—
150C23×C53C2×#4116yes—
151C23×C53H(5,𝔽2)16yes—
152C23×C53C2511yescharacteristic
153D4×C53—412no—
154D4×C53—424no—
155D4×C53—86no—
156C2×C4×C53—26no—
157C2×C4×C53—26no—
158C2×C4×C53—26no—
159C2×C4×C53—23no—
160C2×C4×C53—23no—
161C2×C4×C53—23no—
162C2×C4×C53—26no—
163C2×C4×C53—26no—
164C2×C4×C53—23no—
165C2×C4×C53—26no—
166C2×C4×C53—26no—
167C2×C4×C53—26no—
168C2×C4×C53—26no—
169C2×C4×C53—23no—
170C2×C4×C53—412no—
171C2×C4×C53—424no—
172C2×C4×C53—424no—
173C2×C4×C53—412no—
174C2×C4×C53—86no—
175C2×C4×C53C2×#4116yes—
176C2×C4×C53C2×#4116yes—
177#36607C5311yesmaximal, characteristic, direct product factor, Sylow
178C22×C53—232no—
179C22×C53—26no—
180C22×C53—212no—
181C22×C53—26no—
182C22×C53—212no—
183C22×C53—43no—
184C22×C53—43no—
185C22×C53—46no—
186C22×C53—43no—
187C22×C53—46no—
188C22×C53#553112yes—
189C22×C53C22×#4113yes—
190C22×C53C22×#4111yescharacteristic
191C22×C53C2×H(5,𝔽2)13yes—
192C4×C53—26no—
193C4×C53—26no—
194C4×C53—41no—
195C4×C53—43no—
196C4×C53—46no—
197C4×C53—46no—
198C4×C53—43no—
199#2847C2×C5313yescomplemented
200#2569C2×C5313yescomplemented
201#2543C2×C5313yescomplemented
202#2510C2×C5316yescomplemented
203#2444C2×C5316yescomplemented
204#2364C2×C5313yescomplemented
205#2353C2×C5313yescomplemented
206C4×#120C2×C5313yescomplemented
207C22×#95C2×C5311yescharacteristic, complemented
208C2×C53—212no—
209C2×C53—83no—
210C2×C53#344411yescharacteristic
211C2×C53C2×#55313yes—
212C2×C53#349013yes—
213C24×C4C22×C5313yes—
214C2×#121C22×C5313yes—
215C2×#121C22×C5316yes—
216C2×#120C22×C5313yes—
217C2×C4×D4C22×C5313yes—
218C2×C4×D4C22×C5316yes—
219#401C22×C5312yescomplemented
220#400C22×C5316yescomplemented
221#398C22×C5313yescomplemented
222#397C22×C5316yescomplemented
223#397C22×C5313yescomplemented
224#394C22×C5313yescomplemented
225#392C22×C5311yescharacteristic, complemented
226#388C22×C5316yescomplemented
227#388C22×C5316yescomplemented
228#387C22×C5316yescomplemented
229#387C22×C5316yescomplemented
230#386C22×C5316yes—
231#386C22×C5316yes—
232#386C22×C5316yes—
233#386C22×C5313yes—
234#386C22×C5316yes—
235#386C22×C5316yescomplemented
236#385C22×C5316yes—
237#385C22×C5313yes—
238#383C22×C5313yescomplemented
239#381C22×C5313yescomplemented
240#380C22×C5313yes—
241#380C22×C5316yes—
242#379C22×C5311yescharacteristic
243C4×C22⋊C4C22×C5313yes—
244C4×C22⋊C4C22×C5316yes—
245C4×C22⋊C4C22×C5313yescomplemented
246#376C22×C5311yescharacteristic, complemented
247C2×#95C22×C5313yes—
248C2×#95C22×C5313yes—
249C2×#95C22×C5316yes—
250C53#3660711yescharacteristic, direct product factor, Sylow
251C25C23×C5311yescharacteristic
252C22×D4C23×C5313yes—
253C22×D4C23×C5316yes—
254C22×D4C23×C5313yes—
255C23×C4—212no—
256C23×C4—412no—
257C23×C4C23×C5316yes—
258C23×C4C23×C5316yes—
259C23×C4C23×C5313yes—
260C23×C4C23×C5313yes—
261C23×C4C23×C5313yes—
262#121—212no—
263#121—412no—
264#120—212no—
265C4×D4—212no—
266C4×D4—412no—
267C2×C4⋊C4C23×C5313yes—
268C2×C4⋊C4C23×C5313yes—
269C2×C4⋊C4C23×C5316yes—
270C2×C4⋊C4C23×C5316yes—
271C2×C22⋊C4C23×C5313yes—
272C2×C22⋊C4C23×C5313yes—
273C2×C22⋊C4C23×C5313yes—
274C2×C22⋊C4C23×C5313yes—
275C2×C22⋊C4C23×C5316yes—
276C2×C22⋊C4C23×C5316yes—
277C2×C22⋊C4C23×C5316yes—
278C2×C22⋊C4C23×C5316yes—
279C2×C22⋊C4C23×C5316yes—
280C2×C42C23×C5313yes—
281C2×C42C23×C5316yes—
282#95C23×C5311yescharacteristic
283#95C23×C5313yes—
284#95C23×C5316yes—
285#95C23×C5311yescharacteristic
286#95C23×C5313yes—
287#95C23×C5312yes—
288#95C23×C5313yes—
289#95C23×C5316yes—
290#95C23×C5316yes—
291#95C23×C5316yes—
292#95C23×C5313yes—
293#95C23×C5316yes—
294#95C23×C5316yes—
295C24—26no—
296C24—26no—
297C24#41×C5314yes—
298C24C24×C5313yes—
299C24C24×C5313yes—
300C2×D4—26no—
301C2×D4—26no—
302C2×D4—212no—
303C2×D4—212no—
304C2×D4—212no—
305C2×D4—26no—
306C2×D4—26no—
307C2×D4—46no—
308C22×C4—26no—
309C22×C4—212no—
310C22×C4—212no—
311C22×C4—212no—
312C22×C4—212no—
313C22×C4—26no—
314C22×C4—46no—
315C22×C4—46no—
316C22×C4—46no—
317C22×C4—46no—
318C22×C4—46no—
319C22×C4—448no—
320C22×C4#41×C53112yes—
321C22×C4C24×C5316yes—
322C22×C4C24×C5311yescharacteristic
323C22×C4C24×C5313yes—
324C22×C4C24×C5316yes—
325C22×C4C24×C5313yes—
326C22×C4C24×C5313yes—
327C22×C4C24×C5313yes—
328C4⋊C4—26no—
329C4⋊C4—26no—
330C4⋊C4—46no—
331C4⋊C4—46no—
332C22⋊C4—26no—
333C22⋊C4—26no—
334C22⋊C4—212no—
335C22⋊C4—26no—
336C22⋊C4—26no—
337C22⋊C4—212no—
338C22⋊C4—46no—
339C22⋊C4—46no—
340C22⋊C4—46no—
341C42—26no—
342C42—46no—
343C23—224no—
344C23—28no—
345C23—224no—
346C23—26no—
347C23—26no—
348C23—23no—
349C23—26no—
350C23—23no—
351C23—26no—
352C23—23no—
353C23C2×#41×C5316yes—
354C23C2×#41×C5316yes—
355C23H(5,𝔽2)×C5316yes—
356C23C25×C5311yescharacteristic
357D4—412no—
358D4—424no—
359D4—86no—
360C2×C4—26no—
361C2×C4—26no—
362C2×C4—23no—
363C2×C4—23no—
364C2×C4—23no—
365C2×C4—26no—
366C2×C4—23no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—26no—
370C2×C4—26no—
371C2×C4—23no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—412no—
375C2×C4—424no—
376C2×C4—424no—
377C2×C4—412no—
378C2×C4—86no—
379C2×C4C2×#41×C5316yes—
380C2×C4C2×#41×C5316yes—
381C22—232no—
382C22—212no—
383C22—26no—
384C22—212no—
385C22—26no—
386C22—43no—
387C22—46no—
388C22—43no—
389C22—46no—
390C22—43no—
391C22C53×#553112yes—
392C22C22×#41×C5311yescharacteristic
393C22C22×#41×C5313yes—
394C22C2×H(5,𝔽2)×C5313yes—
395C4—26no—
396C4—26no—
397C4—43no—
398C4—46no—
399C4—46no—
400C4—43no—
401C4—41no—
402C2—212no—
403C2—83no—
404C2C53×#344411yescharacteristic
405C2C2×C53×#55313yes—
406C2C53×#349013yes—
subgroupG/H
center, socleC23×C53C25
commutator, Frattini, cosocleC23C25×C53
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#2847C2semidirect
2. C53×#2569C2semidirect
3. C53×#2543C2semidirect
4. C53×#2510C2semidirect
5. C53×#2444C2semidirect
6. C53×#2364C2semidirect
7. C53×#2353C2semidirect
8. C4×#120×C53C2semidirect
9. C22×#95×C53C2semidirect
16. C53×#401C22semidirect
17. C53×#400C22semidirect
18. C53×#398C22semidirect
19. C53×#397C22semidirect
20. C53×#397C22semidirect
21. C53×#394C22semidirect
22. C53×#392C22semidirect
23. C53×#388C22semidirect
24. C53×#388C22semidirect
25. C53×#387C22semidirect
26. C53×#387C22semidirect
32. C53×#386C22semidirect
35. C53×#383C22semidirect
36. C53×#381C22semidirect
42. C4×C22⋊C4×C53C22semidirect
43. C53×#376C22semidirect
177. #36607C53direct (N × H)
199. #2847C2×C53semidirect
200. #2569C2×C53semidirect
201. #2543C2×C53semidirect
202. #2510C2×C53semidirect
203. #2444C2×C53semidirect
204. #2364C2×C53semidirect
205. #2353C2×C53semidirect
206. C4×#120C2×C53semidirect
207. C22×#95C2×C53semidirect
219. #401C22×C53semidirect
220. #400C22×C53semidirect
221. #398C22×C53semidirect
222. #397C22×C53semidirect
223. #397C22×C53semidirect
224. #394C22×C53semidirect
225. #392C22×C53semidirect
226. #388C22×C53semidirect
227. #388C22×C53semidirect
228. #387C22×C53semidirect
229. #387C22×C53semidirect
235. #386C22×C53semidirect
238. #383C22×C53semidirect
239. #381C22×C53semidirect
245. C4×C22⋊C4C22×C53semidirect
246. #376C22×C53semidirect
250. C53#36607direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#28472
C53×#25692
C53×#25432
C53×#25102
C53×#24442
C53×#23642
C53×#23532
C4×#120×C532
C22×#95×C532
#3660753