Finite Group Databasebeta

← C53×#18381 #27225893

Subgroups — C53×#18381 #27225893

Order 13568.

1266 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3496C211yesmaximal, characteristic
2C53×#3187C212yesmaximal, complemented
3C53×#2564C211yesmaximal, characteristic, complemented
4C53×#1616C212yesmaximal, complemented
5C53×#1613C212yesmaximal, complemented
6C53×#1604C212yesmaximal, complemented
7C53×#1601C212yesmaximal, complemented
8C2×C53×#327C212yesmaximal, complemented
9C53×#1459C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C53—21no—
11C53×#547—22no—
12C53×#546—24no—
13D42×C53—22no—
14C53×#537C2212yes—
15C53×#534—21no—
16C53×#534C2212yes—
17C53×#532—21no—
18C2×C22⋊Q8×C53C2211yescharacteristic, complemented
19C2×#121×C53—21no—
20C2×#121×C53C2212yes—
21C2×C4×D4×C53C2211yescharacteristic
22C53×#483—22no—
23C53×#481C2214yescomplemented
24C53×#465—24no—
25C53×#450—22no—
26C53×#448C2214yescomplemented
27C2×C22⋊C8×C53C2212yescomplemented
28C53×#400C2211yescharacteristic, complemented
29C53×#399C2211yescharacteristic, complemented
30C53×#395C2211yescharacteristic, complemented
31C53×#393C2211yescharacteristic, complemented
32C53×#387—21no—
33C53×#386—21no—
34C53×#385—21no—
35C53×#380—21no—
36C2×#95×C53C2211yescharacteristic, complemented
37C53×#327C2214yescomplemented
38C53×#327C2214yescomplemented
39C53×#324C2212yescomplemented
40C53×#323C2212yescomplemented
41H(5,𝔽2)×C53—44no—
42H(5,𝔽2)×C53—44no—
43C2×#41×C53—24no—
44C2×#41×C53—22no—
45C22×Q8×C53C2311yescharacteristic
46C22×D4×C53—24no—
47C22×D4×C53—22no—
48C22×D4×C53—22no—
49C22×D4×C53—21no—
50C22×D4×C53D411yescharacteristic
51C22×D4×C53D412yes—
52C23×C4×C53—21no—
53C23×C4×C53D411yescharacteristic
54C23×C4×C53C2311yescharacteristic
55C2×SD16×C53—24no—
56C22×C8×C53—22no—
57C4⋊Q8×C53—21no—
58#127×C53—21no—
59#124×C53—22no—
60#123×C53—24no—
61#123×C53—24no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—22no—
64C22⋊Q8×C53—22no—
65#121×C53—22no—
66#121×C53—22no—
67#121×C53—24no—
68#121×C53—22no—
69#121×C53—22no—
70#121×C53—22no—
71#121×C53—22no—
72#121×C53D414yes—
73#121×C53C2314yes—
74#120×C53—24no—
75#120×C53—24no—
76C4×D4×C53—22no—
77C4×D4×C53—22no—
78C4×D4×C53—22no—
79C4×D4×C53—22no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53D412yes—
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53—21no—
89C2×C22⋊C4×C53—21no—
90C2×C22⋊C4×C53—21no—
91C2×C22⋊C4×C53—22no—
92C2×C22⋊C4×C53D411yescharacteristic
93C2×C22⋊C4×C53C2311yescharacteristic
94C2×C42×C53D411yescharacteristic
95#106×C53—24no—
96#103×C53—24no—
97#102×C53—24no—
98C22⋊C8×C53—22no—
99C22⋊C8×C53C2314yes—
100#95×C53—21no—
101#95×C53—21no—
102#95×C53—21no—
103#95×C53—21no—
104#95×C53—21no—
105#95×C53C2311yescharacteristic
106#95×C53C2311yescharacteristic
107C24×C53—22no—
108C24×C53—22no—
109C24×C53C2×D411yescharacteristic
110C24×C53C2×D411yescharacteristic
111#41×C53—44no—
112#41×C53—416no—
113#41×C53—44no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53—81no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—24no—
120C2×D4×C53—22no—
121C2×D4×C53—24no—
122C2×D4×C53—24no—
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—416no—
127C2×D4×C53—44no—
128C2×D4×C53—44no—
129C2×D4×C53—44no—
130C2×D4×C53—44no—
131C2×D4×C53—82no—
132C2×D4×C53—82no—
133C2×D4×C53SD1614yescomplemented
134C2×D4×C53C2×D414yes—
135C22×C4×C53—22no—
136C22×C4×C53—24no—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53—81no—
147C22×C4×C53—81no—
148C22×C4×C53—81no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D412yes—
155C22×C4×C53C2411yescharacteristic
156SD16×C53—84no—
157C2×C8×C53—24no—
158C2×C8×C53—82no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—41no—
166C4⋊C4×C53—41no—
167C4⋊C4×C53—81no—
168C4⋊C4×C53C2×D414yes—
169C22⋊C4×C53—24no—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—24no—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—81no—
178C42×C53—21no—
179C42×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—24no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—44no—
188C23×C53—44no—
189C23×C53—82no—
190C23×C53—82no—
191C23×C53—82no—
192C23×C53—82no—
193C23×C53#12012yes—
194C23×C53C2×SD1612yes—
195C23×C53C2×SD1612yes—
196C23×C53AGL(1,ℤ/8)12yes—
197C23×C53C22×D411yescharacteristic
198Q8×C53—44no—
199Q8×C53—44no—
200D4×C53—28no—
201D4×C53—44no—
202D4×C53—44no—
203D4×C53—44no—
204D4×C53—48no—
205D4×C53—44no—
206D4×C53—44no—
207D4×C53—48no—
208D4×C53—82no—
209D4×C53—82no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—24no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—41no—
220C2×C4×C53—42no—
221C2×C4×C53—48no—
222C2×C4×C53—41no—
223C2×C4×C53—44no—
224C2×C4×C53—41no—
225C2×C4×C53—44no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—81no—
238C2×C4×C53—84no—
239C2×C4×C53#12011yescharacteristic
240C2×C4×C53#12011yescharacteristic
241C2×C4×C53C2×SD1612yes—
242C2×C4×C53C22×D412yes—
243C8×C53—44no—
244#18381C5311yesmaximal, characteristic, direct product factor, Sylow
245C22×C53—21no—
246C22×C53—22no—
247C22×C53—21no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—42no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—44no—
256C22×C53—44no—
257C22×C53—44no—
258C22×C53—81no—
259C22×C53—164no—
260C22×C53—164no—
261C22×C53#45014yes—
262C22×C53C2×#12011yescharacteristic
263C22×C53C22×SD1611yescharacteristic
264C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
265C4×C53—22no—
266C4×C53—22no—
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—42no—
270C4×C53—44no—
271C4×C53—41no—
272C4×C53—41no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53—81no—
276#3496C2×C5311yescharacteristic
277#3187C2×C5312yescomplemented
278#2564C2×C5311yescharacteristic, complemented
279#1616C2×C5312yescomplemented
280#1613C2×C5312yescomplemented
281#1604C2×C5312yescomplemented
282#1601C2×C5312yescomplemented
283C2×#327C2×C5312yescomplemented
284#1459C2×C5311yescharacteristic, complemented
285C2×C53—21no—
286C2×C53—21no—
287C2×C53—24no—
288C2×C53—42no—
289C2×C53—84no—
290C2×C53C2×#45011yescharacteristic
291C2×C53#301811yescharacteristic
292C2×C53#302611yescharacteristic
293C2×H(5,𝔽2)—21no—
294#547—22no—
295#546—24no—
296D42—22no—
297#537C22×C5312yes—
298#534—21no—
299#534C22×C5312yes—
300#532—21no—
301C2×C22⋊Q8C22×C5311yescharacteristic, complemented
302C2×#121—21no—
303C2×#121C22×C5312yes—
304C2×C4×D4C22×C5311yescharacteristic
305#483—22no—
306#481C22×C5314yescomplemented
307#465—24no—
308#450—22no—
309#448C22×C5314yescomplemented
310C2×C22⋊C8C22×C5312yescomplemented
311#400C22×C5311yescharacteristic, complemented
312#399C22×C5311yescharacteristic, complemented
313#395C22×C5311yescharacteristic, complemented
314#393C22×C5311yescharacteristic, complemented
315#387—21no—
316#386—21no—
317#385—21no—
318#380—21no—
319C2×#95C22×C5311yescharacteristic, complemented
320#327C22×C5314yescomplemented
321#327C22×C5314yescomplemented
322#324C22×C5312yescomplemented
323#323C22×C5312yescomplemented
324C53#1838111yescharacteristic, direct product factor, Sylow
325H(5,𝔽2)—44no—
326H(5,𝔽2)—44no—
327C2×#41—24no—
328C2×#41—22no—
329C22×Q8C23×C5311yescharacteristic
330C22×D4—24no—
331C22×D4—22no—
332C22×D4—22no—
333C22×D4—21no—
334C22×D4D4×C5311yescharacteristic
335C22×D4D4×C5312yes—
336C23×C4—21no—
337C23×C4D4×C5311yescharacteristic
338C23×C4C23×C5311yescharacteristic
339C2×SD16—24no—
340C22×C8—22no—
341C4⋊Q8—21no—
342#127—21no—
343#124—22no—
344#123—24no—
345#123—24no—
346C22⋊Q8—24no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349#121—22no—
350#121—22no—
351#121—24no—
352#121—22no—
353#121—22no—
354#121—22no—
355#121—22no—
356#121D4×C5314yes—
357#121C23×C5314yes—
358#120—24no—
359#120—24no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4D4×C5312yes—
369C2×C4⋊C4C23×C5311yescharacteristic
370C2×C4⋊C4C23×C5311yescharacteristic
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4D4×C5311yescharacteristic
377C2×C22⋊C4C23×C5311yescharacteristic
378C2×C42D4×C5311yescharacteristic
379#106—24no—
380#103—24no—
381#102—24no—
382C22⋊C8—22no—
383C22⋊C8C23×C5314yes—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95C23×C5311yescharacteristic
390#95C23×C5311yescharacteristic
391C24—22no—
392C24—22no—
393C24C2×D4×C5311yescharacteristic
394C24C2×D4×C5311yescharacteristic
395#41—44no—
396#41—416no—
397#41—44no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—81no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—416no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4SD16×C5314yescomplemented
418C2×D4C2×D4×C5314yes—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4C2×D4×C5311yescharacteristic
434C22×C4C2×D4×C5311yescharacteristic
435C22×C4C2×D4×C5311yescharacteristic
436C22×C4C2×D4×C5311yescharacteristic
437C22×C4C2×D4×C5312yes—
438C22×C4C2×D4×C5312yes—
439C22×C4C24×C5311yescharacteristic
440SD16—84no—
441C2×C8—24no—
442C2×C8—82no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—81no—
452C4⋊C4C2×D4×C5314yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—81no—
462C42—21no—
463C42—21no—
464C23—21no—
465C23—21no—
466C23—24no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—44no—
472C23—44no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23#120×C5312yes—
478C23C2×SD16×C5312yes—
479C23C2×SD16×C5312yes—
480C23AGL(1,ℤ/8)×C5312yes—
481C23C22×D4×C5311yescharacteristic
482Q8—44no—
483Q8—44no—
484D4—28no—
485D4—44no—
486D4—44no—
487D4—44no—
488D4—48no—
489D4—44no—
490D4—44no—
491D4—48no—
492D4—82no—
493D4—82no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—48no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—81no—
522C2×C4—84no—
523C2×C4#120×C5311yescharacteristic
524C2×C4#120×C5311yescharacteristic
525C2×C4C2×SD16×C5312yes—
526C2×C4C22×D4×C5312yes—
527C8—44no—
528C22—21no—
529C22—22no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—41no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—81no—
542C22—164no—
543C22—164no—
544C22C53×#45014yes—
545C22C2×#120×C5311yescharacteristic
546C22C22×SD16×C5311yescharacteristic
547C22C2×AGL(1,ℤ/8)×C5311yescharacteristic
548C4—22no—
549C4—22no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—44no—
554C4—41no—
555C4—41no—
556C4—81no—
557C4—81no—
558C4—81no—
559C2—21no—
560C2—21no—
561C2—24no—
562C2—42no—
563C2—84no—
564C2C2×C53×#45011yescharacteristic
565C2C53×#301811yescharacteristic
566C2C53×#302611yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
prime radicalC53×#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#3187C2semidirect
3. C53×#2564C2semidirect
4. C53×#1616C2semidirect
5. C53×#1613C2semidirect
6. C53×#1604C2semidirect
7. C53×#1601C2semidirect
8. C2×C53×#327C2semidirect
9. C53×#1459C2semidirect
18. C2×C22⋊Q8×C53C22semidirect
23. C53×#481C22semidirect
26. C53×#448C22semidirect
27. C2×C22⋊C8×C53C22semidirect
28. C53×#400C22semidirect
29. C53×#399C22semidirect
30. C53×#395C22semidirect
31. C53×#393C22semidirect
36. C2×#95×C53C22semidirect
37. C53×#327C22semidirect
38. C53×#327C22semidirect
39. C53×#324C22semidirect
40. C53×#323C22semidirect
133. C2×D4×C53SD16semidirect
244. #18381C53direct (N × H)
277. #3187C2×C53semidirect
278. #2564C2×C53semidirect
279. #1616C2×C53semidirect
280. #1613C2×C53semidirect
281. #1604C2×C53semidirect
282. #1601C2×C53semidirect
283. C2×#327C2×C53semidirect
284. #1459C2×C53semidirect
301. C2×C22⋊Q8C22×C53semidirect
306. #481C22×C53semidirect
309. #448C22×C53semidirect
310. C2×C22⋊C8C22×C53semidirect
311. #400C22×C53semidirect
312. #399C22×C53semidirect
313. #395C22×C53semidirect
314. #393C22×C53semidirect
319. C2×#95C22×C53semidirect
320. #327C22×C53semidirect
321. #327C22×C53semidirect
322. #324C22×C53semidirect
323. #323C22×C53semidirect
324. C53#18381direct (N × H)
417. C2×D4SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#34962
C53×#31872
C53×#25642
C53×#16162
C53×#16132
C53×#16042
C53×#16012
C2×C53×#3272
C53×#14592
#1838153