Finite Group Databasebeta

← C53×#12960 #27220472

Subgroups — C53×#12960 #27220472

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2375C211yesmaximal, characteristic
2C2×C53×AΣL(1,𝔽16)C211yesmaximal, characteristic
3C53×#1889C211yesmaximal, characteristic
4C53×#1867C211yesmaximal, characteristic, complemented
5C53×#1856C211yesmaximal, characteristic, complemented
6C53×#1393C211yesmaximal, characteristic, complemented
7C53×#1390C211yesmaximal, characteristic, complemented
8C2×#120×C53C2211yescharacteristic
9C53×#518C2211yescharacteristic
10C53×#428—21no—
11C2×#100×C53C2211yescharacteristic
12C2×#99×C53C2211yescharacteristic
13C2×#99×C53C2211yescharacteristic
14C53×#390—21no—
15C53×#390C2211yescharacteristic
16C53×#388—21no—
17C53×#388—21no—
18C53×#388—21no—
19C53×#386—21no—
20C53×#386C2211yescharacteristic
21C53×#385—21no—
22C53×#379—21no—
23C53×#379—21no—
24C53×AΣL(1,𝔽16)C414yescomplemented
25C53×#342—21no—
26C53×#342—21no—
27C53×#339—21no—
28C25×C53D411yescharacteristic
29C22×D4×C53—21no—
30C22×D4×C53C2311yescharacteristic
31C23×C4×C53D411yescharacteristic
32C2×M16×C53—21no—
33#120×C53—21no—
34#120×C53—21no—
35#120×C53C2×C412yes—
36#120×C53Q812yes—
37C4×D4×C53—41no—
38C4×D4×C53—41no—
39#117×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—41no—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—41no—
56C2×C22⋊C4×C53—41no—
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C22⋊C4×C53D411yescharacteristic
59C2×C42×C53—21no—
60C2×C42×C53D411yescharacteristic
61#107×C53—41no—
62#106×C53—41no—
63#100×C53—21no—
64#100×C53C2×C412yes—
65#99×C53—21no—
66#99×C53—21no—
67#99×C53—21no—
68#99×C53C2×C412yes—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72#95×C53—41no—
73C24×C53—21no—
74C24×C53—21no—
75C24×C53—41no—
76C24×C53—41no—
77C24×C53—81no—
78C24×C53—81no—
79C24×C53C4⋊C412yescomplemented
80C24×C53C2×D411yescharacteristic
81C24×C53C2×D411yescharacteristic
82C2×D4×C53—22no—
83C2×D4×C53—22no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—81no—
87C2×D4×C53—81no—
88C2×D4×C53C22×C411yescharacteristic
89C2×D4×C53C2×Q811yescharacteristic
90C2×D4×C53#4111yescharacteristic
91C2×D4×C53#4111yescharacteristic
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53—81no—
105C22×C4×C53C2×D411yescharacteristic
106M16×C53—22no—
107M16×C53—41no—
108C2×C8×C53—41no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—81no—
113C4⋊C4×C53—82no—
114C22⋊C4×C53—21no—
115C22⋊C4×C53—22no—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—81no—
126C22⋊C4×C53—81no—
127C22⋊C4×C53—81no—
128C22⋊C4×C53—81no—
129C22⋊C4×C53—81no—
130C22⋊C4×C53—81no—
131C22⋊C4×C53—81no—
132C22⋊C4×C53—82no—
133C22⋊C4×C53—82no—
134C22⋊C4×C53C4⋊C412yescomplemented
135C42×C53—41no—
136C42×C53—41no—
137C42×C53—81no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—22no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—42no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—42no—
153C23×C53—42no—
154C23×C53—41no—
155C23×C53—41no—
156C23×C53—81no—
157C23×C53—82no—
158C23×C53—82no—
159C23×C53—81no—
160C23×C53—81no—
161C23×C53—84no—
162C23×C53C2×C4⋊C411yescharacteristic
163C23×C53C4×D411yescharacteristic
164C23×C53#12011yescharacteristic
165C23×C53C22⋊Q811yescharacteristic
166C23×C53#12311yescharacteristic
167D4×C53—41no—
168D4×C53—41no—
169D4×C53—41no—
170D4×C53—41no—
171D4×C53—82no—
172D4×C53—82no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—22no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—81no—
195C2×C4×C53—81no—
196C2×C4×C53—81no—
197C2×C4×C53—81no—
198C2×C4×C53—81no—
199C2×C4×C53—81no—
200C2×C4×C53—81no—
201C2×C4×C53—81no—
202C2×C4×C53—81no—
203C2×C4×C53—81no—
204C2×C4×C53C4×D411yescharacteristic
205C2×C4×C53C22⋊Q811yescharacteristic
206C8×C53—42no—
207#12960C5311yesmaximal, characteristic, direct product factor, Sylow
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—42no—
219C22×C53—42no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—44no—
225C22×C53—42no—
226C22×C53—81no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53—81no—
230C22×C53—82no—
231C22×C53—82no—
232C22×C53—82no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53#38511yescharacteristic
236C22×C53B(4,𝔽2)11yescharacteristic
237C22×C53#45811yescharacteristic
238C4×C53—21no—
239C4×C53—21no—
240C4×C53—42no—
241C4×C53—81no—
242C4×C53—81no—
243C4×C53—82no—
244C4×C53—81no—
245C4×C53—81no—
246C4×C53—81no—
247C4×C53—81no—
248C4×C53—161no—
249#2375C2×C5311yescharacteristic
250C2×AΣL(1,𝔽16)C2×C5311yescharacteristic
251#1889C2×C5311yescharacteristic
252#1867C2×C5311yescharacteristic, complemented
253#1856C2×C5311yescharacteristic, complemented
254#1393C2×C5311yescharacteristic, complemented
255#1390C2×C5311yescharacteristic, complemented
256C2×C53—21no—
257C2×C53—21no—
258C2×C53—41no—
259C2×C53—41no—
260C2×C53—41no—
261C2×C53—41no—
262C2×C53—82no—
263C2×C53#186711yescharacteristic
264C2×C53#219911yescharacteristic
265C2×C53#220011yescharacteristic
266C2×#120C22×C5311yescharacteristic
267#518C22×C5311yescharacteristic
268#428—21no—
269C2×#100C22×C5311yescharacteristic
270C2×#99C22×C5311yescharacteristic
271C2×#99C22×C5311yescharacteristic
272#390—21no—
273#390C22×C5311yescharacteristic
274#388—21no—
275#388—21no—
276#388—21no—
277#386—21no—
278#386C22×C5311yescharacteristic
279#385—21no—
280#379—21no—
281#379—21no—
282AΣL(1,𝔽16)C4×C5314yescomplemented
283#342—21no—
284#342—21no—
285#339—21no—
286C53#1296011yescharacteristic, direct product factor, Sylow
287C25D4×C5311yescharacteristic
288C22×D4—21no—
289C22×D4C23×C5311yescharacteristic
290C23×C4D4×C5311yescharacteristic
291C2×M16—21no—
292#120—21no—
293#120—21no—
294#120C2×C4×C5312yes—
295#120Q8×C5312yes—
296C4×D4—41no—
297C4×D4—41no—
298#117—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—41no—
301C2×C4⋊C4D4×C5311yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—41no—
315C2×C22⋊C4—41no—
316C2×C22⋊C4D4×C5311yescharacteristic
317C2×C22⋊C4D4×C5311yescharacteristic
318C2×C42—21no—
319C2×C42D4×C5311yescharacteristic
320#107—41no—
321#106—41no—
322#100—21no—
323#100C2×C4×C5312yes—
324#99—21no—
325#99—21no—
326#99—21no—
327#99C2×C4×C5312yes—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—41no—
332C24—21no—
333C24—21no—
334C24—41no—
335C24—41no—
336C24—81no—
337C24—81no—
338C24C4⋊C4×C5312yescomplemented
339C24C2×D4×C5311yescharacteristic
340C24C2×D4×C5311yescharacteristic
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—81no—
346C2×D4—81no—
347C2×D4C22×C4×C5311yescharacteristic
348C2×D4C2×Q8×C5311yescharacteristic
349C2×D4#41×C5311yescharacteristic
350C2×D4#41×C5311yescharacteristic
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—81no—
364C22×C4C2×D4×C5311yescharacteristic
365M16—22no—
366M16—41no—
367C2×C8—41no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—81no—
372C4⋊C4—82no—
373C22⋊C4—21no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—81no—
385C22⋊C4—81no—
386C22⋊C4—81no—
387C22⋊C4—81no—
388C22⋊C4—81no—
389C22⋊C4—81no—
390C22⋊C4—81no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4C4⋊C4×C5312yescomplemented
394C42—41no—
395C42—41no—
396C42—81no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—42no—
412C23—42no—
413C23—41no—
414C23—41no—
415C23—81no—
416C23—82no—
417C23—82no—
418C23—81no—
419C23—81no—
420C23—84no—
421C23C2×C4⋊C4×C5311yescharacteristic
422C23C4×D4×C5311yescharacteristic
423C23#120×C5311yescharacteristic
424C23C22⋊Q8×C5311yescharacteristic
425C23#123×C5311yescharacteristic
426D4—41no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—82no—
431D4—82no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4C4×D4×C5311yescharacteristic
464C2×C4C22⋊Q8×C5311yescharacteristic
465C8—42no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—42no—
477C22—42no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—44no—
483C22—42no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—81no—
492C22—81no—
493C22C53×#38511yescharacteristic
494C22C53×B(4,𝔽2)11yescharacteristic
495C22C53×#45811yescharacteristic
496C4—21no—
497C4—21no—
498C4—42no—
499C4—81no—
500C4—81no—
501C4—82no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—161no—
507C2—21no—
508C2—21no—
509C2—41no—
510C2—41no—
511C2—41no—
512C2—41no—
513C2—82no—
514C2C53×#186711yescharacteristic
515C2C53×#219911yescharacteristic
516C2C53×#220011yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutatorC2×D4C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC2×#120×C53C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C22×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1867C2semidirect
5. C53×#1856C2semidirect
6. C53×#1393C2semidirect
7. C53×#1390C2semidirect
24. C53×AΣL(1,𝔽16)C4semidirect
79. C24×C53C4⋊C4semidirect
134. C22⋊C4×C53C4⋊C4semidirect
207. #12960C53direct (N × H)
252. #1867C2×C53semidirect
253. #1856C2×C53semidirect
254. #1393C2×C53semidirect
255. #1390C2×C53semidirect
282. AΣL(1,𝔽16)C4×C53semidirect
286. C53#12960direct (N × H)
338. C24C4⋊C4×C53semidirect
393. C22⋊C4C4⋊C4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#23752
C2×C53×AΣL(1,𝔽16)2
C53×#18892
C53×#18672
C53×#18562
C53×#13932
C53×#13902
#1296053