Finite Group Databasebeta

← C53×#18268 #27225780

Subgroups — C53×#18268 #27225780

Order 13568.

940 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3538C211yesmaximal, characteristic
2C53×#3092C211yesmaximal, characteristic
3C2×C53×#483C211yesmaximal, characteristic, complemented
4C53×#2502C211yesmaximal, characteristic, complemented
5C53×#1611C214yesmaximal, complemented
6C53×#1609C214yesmaximal, complemented
7C53×#1508C211yesmaximal, characteristic, complemented
8C53×#1501C211yesmaximal, characteristic
9C53×#1472C211yesmaximal, characteristic, complemented
10C53×#563—22no—
11C53×#552—21no—
12C53×#552—21no—
13C53×#551—22no—
14D4×Q8×C53—21no—
15D4×Q8×C53—21no—
16C53×#544—21no—
17C53×#544C2212yes—
18C53×#541—21no—
19C53×#539—21no—
20C53×#539C2212yes—
21C53×#536—21no—
22C53×#525C2211yescharacteristic
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#484—21no—
27C53×#483—22no—
28C53×#483C2214yes—
29C53×#483C2212yes—
30C53×#481—21no—
31C53×#480C2212yes—
32C53×#428—21no—
33C2×#106×C53—21no—
34C53×#417—21no—
35C2×#103×C53—21no—
36C53×#407C2211yescharacteristic
37C2×C22⋊C8×C53C2211yescharacteristic
38C53×#398C2214yes—
39C53×#391—21no—
40C53×#385—22no—
41C53×#384—21no—
42C4×C22⋊C4×C53C2211yescharacteristic
43C53×#328C2212yes—
44C53×#328C2214yes—
45C53×#327C2212yes—
46C53×#324C2214yes—
47C2×#41×C53D411yescharacteristic
48C22×Q8×C53—21no—
49C22×Q8×C53D411yescharacteristic
50C22×Q8×C53D411yescharacteristic
51C23×C4×C53—21no—
52C23×C4×C53C2311yescharacteristic
53C2×M16×C53—21no—
54C22×C8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—22no—
57C4⋊Q8×C53—22no—
58C4⋊Q8×C53—22no—
59#126×C53—22no—
60#126×C53—22no—
61#126×C53—22no—
62#126×C53—22no—
63#125×C53—21no—
64#125×C53—22no—
65#125×C53—22no—
66#125×C53—22no—
67#125×C53—22no—
68#124×C53—22no—
69#123×C53—22no—
70#123×C53—22no—
71#123×C53—22no—
72#123×C53—21no—
73#123×C53—22no—
74C22⋊Q8×C53—21no—
75C22⋊Q8×C53—21no—
76C22⋊Q8×C53—22no—
77C22⋊Q8×C53—22no—
78C22⋊Q8×C53—22no—
79C22⋊Q8×C53—21no—
80C22⋊Q8×C53—21no—
81C22⋊Q8×C53—21no—
82C22⋊Q8×C53—21no—
83C22⋊Q8×C53—21no—
84C22⋊Q8×C53—21no—
85C22⋊Q8×C53—22no—
86C22⋊Q8×C53D412yes—
87C22⋊Q8×C53D411yescharacteristic
88C22⋊Q8×C53C2312yes—
89C22⋊Q8×C53C2311yescharacteristic
90#121×C53D411yescharacteristic
91#121×C53C2311yescharacteristic
92#120×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×D4×C53—22no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—22no—
99#117×C53—21no—
100#117×C53—22no—
101#117×C53—22no—
102#117×C53D411yescharacteristic
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—24no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C4⋊C4×C53C2312yes—
113C2×C4⋊C4×C53C2312yes—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53D412yes—
117C2×C42×C53—21no—
118#107×C53—22no—
119#106×C53—24no—
120#106×C53—22no—
121#103×C53—24no—
122#103×C53—22no—
123#102×C53—22no—
124C22⋊C8×C53—21no—
125C22⋊C8×C53—21no—
126C22⋊C8×C53C2312yes—
127C22⋊C8×C53C2312yes—
128#95×C53—24no—
129#95×C53C2312yes—
130C24×C53C2×D411yescharacteristic
131#41×C53—42no—
132C2×Q8×C53—22no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—42no—
137C2×Q8×C53—42no—
138C2×Q8×C53—82no—
139C2×Q8×C53C2×D412yes—
140C2×Q8×C53C2×D411yescharacteristic
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53C2×D411yescharacteristic
145C22×C4×C53—21no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—22no—
151C22×C4×C53—42no—
152C22×C4×C53—41no—
153C22×C4×C53—41no—
154C22×C4×C53—82no—
155C22×C4×C53C2×D412yes—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D412yes—
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53C2411yescharacteristic
163M16×C53—42no—
164C2×C8×C53—22no—
165C2×C8×C53—22no—
166C2×C8×C53—42no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—22no—
186C4⋊C4×C53—41no—
187C4⋊C4×C53—41no—
188C4⋊C4×C53—84no—
189C4⋊C4×C53C2×D412yes—
190C4⋊C4×C53C2×D411yescharacteristic
191C4⋊C4×C53C2×D411yescharacteristic
192C22⋊C4×C53—24no—
193C22⋊C4×C53—22no—
194C22⋊C4×C53—22no—
195C22⋊C4×C53—22no—
196C22⋊C4×C53—22no—
197C22⋊C4×C53—22no—
198C22⋊C4×C53—22no—
199C22⋊C4×C53—21no—
200C22⋊C4×C53—21no—
201C22⋊C4×C53—22no—
202C22⋊C4×C53—21no—
203C22⋊C4×C53—21no—
204C22⋊C4×C53—21no—
205C42×C53—22no—
206C42×C53—22no—
207C42×C53—22no—
208C42×C53—21no—
209C42×C53—42no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—21no—
213C23×C53—41no—
214C23×C53#12011yescharacteristic
215C23×C53#12011yescharacteristic
216C23×C53C22×D411yescharacteristic
217Q8×C53—42no—
218Q8×C53—41no—
219Q8×C53—42no—
220Q8×C53—42no—
221Q8×C53—81no—
222Q8×C53—82no—
223D4×C53—41no—
224D4×C53—42no—
225D4×C53—42no—
226D4×C53—81no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—24no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—44no—
249C2×C4×C53—81no—
250C2×C4×C53—82no—
251C2×C4×C53—82no—
252C2×C4×C53—82no—
253C2×C4×C53#12011yescharacteristic
254C2×C4×C53#12011yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C22×D411yescharacteristic
257C8×C53—42no—
258C8×C53—42no—
259#18268C5311yesmaximal, characteristic, direct product factor, Sylow
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—44no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53#45612yes—
272C22×C53C2×#12011yescharacteristic
273C22×C53#57712yes—
274C4×C53—21no—
275C4×C53—21no—
276C4×C53—21no—
277C4×C53—44no—
278C4×C53—42no—
279C4×C53—41no—
280C4×C53—42no—
281C4×C53—41no—
282C4×C53—41no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53—82no—
286C4×C53#45812yes—
287#3538C2×C5311yescharacteristic
288#3092C2×C5311yescharacteristic
289C2×#483C2×C5311yescharacteristic, complemented
290#2502C2×C5311yescharacteristic, complemented
291#1611C2×C5314yescomplemented
292#1609C2×C5314yescomplemented
293#1508C2×C5311yescharacteristic, complemented
294#1501C2×C5311yescharacteristic
295#1472C2×C5311yescharacteristic, complemented
296C2×C53—22no—
297C2×C53—22no—
298C2×C53—41no—
299C2×C53—81no—
300C2×C53#301011yescharacteristic
301C2×C53C2×#45611yescharacteristic
302C2×C53C2×#45811yescharacteristic
303#563—22no—
304#552—21no—
305#552—21no—
306#551—22no—
307D4×Q8—21no—
308D4×Q8—21no—
309#544—21no—
310#544C22×C5312yes—
311#541—21no—
312#539—21no—
313#539C22×C5312yes—
314#536—21no—
315#525C22×C5311yescharacteristic
316C2×C22⋊Q8C22×C5311yescharacteristic
317C2×C22⋊Q8C22×C5311yescharacteristic
318C22×C4⋊C4C22×C5311yescharacteristic
319#484—21no—
320#483—22no—
321#483C22×C5314yes—
322#483C22×C5312yes—
323#481—21no—
324#480C22×C5312yes—
325#428—21no—
326C2×#106—21no—
327#417—21no—
328C2×#103—21no—
329#407C22×C5311yescharacteristic
330C2×C22⋊C8C22×C5311yescharacteristic
331#398C22×C5314yes—
332#391—21no—
333#385—22no—
334#384—21no—
335C4×C22⋊C4C22×C5311yescharacteristic
336#328C22×C5314yes—
337#328C22×C5312yes—
338#327C22×C5312yes—
339#324C22×C5314yes—
340C53#1826811yescharacteristic, direct product factor, Sylow
341C2×#41D4×C5311yescharacteristic
342C22×Q8—21no—
343C22×Q8D4×C5311yescharacteristic
344C22×Q8D4×C5311yescharacteristic
345C23×C4—21no—
346C23×C4C23×C5311yescharacteristic
347C2×M16—21no—
348C22×C8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—22no—
351C4⋊Q8—22no—
352C4⋊Q8—22no—
353#126—22no—
354#126—22no—
355#126—22no—
356#126—22no—
357#125—21no—
358#125—22no—
359#125—22no—
360#125—22no—
361#125—22no—
362#124—22no—
363#123—22no—
364#123—22no—
365#123—22no—
366#123—21no—
367#123—22no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8D4×C5312yes—
381C22⋊Q8D4×C5311yescharacteristic
382C22⋊Q8C23×C5312yes—
383C22⋊Q8C23×C5311yescharacteristic
384#121D4×C5311yescharacteristic
385#121C23×C5311yescharacteristic
386#120—21no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×D4—22no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—22no—
393#117—21no—
394#117—22no—
395#117—22no—
396#117D4×C5311yescharacteristic
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4D4×C5311yescharacteristic
405C2×C4⋊C4D4×C5311yescharacteristic
406C2×C4⋊C4C23×C5312yes—
407C2×C4⋊C4C23×C5312yes—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4D4×C5312yes—
411C2×C42—21no—
412#107—22no—
413#106—24no—
414#106—22no—
415#103—24no—
416#103—22no—
417#102—22no—
418C22⋊C8—21no—
419C22⋊C8—21no—
420C22⋊C8C23×C5312yes—
421C22⋊C8C23×C5312yes—
422#95—24no—
423#95C23×C5312yes—
424C24C2×D4×C5311yescharacteristic
425#41—42no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—82no—
433C2×Q8C2×D4×C5312yes—
434C2×Q8C2×D4×C5311yescharacteristic
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4C2×D4×C5311yescharacteristic
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—82no—
449C22×C4C2×D4×C5312yes—
450C22×C4C2×D4×C5311yescharacteristic
451C22×C4C2×D4×C5311yescharacteristic
452C22×C4C2×D4×C5311yescharacteristic
453C22×C4C2×D4×C5312yes—
454C22×C4C2×D4×C5311yescharacteristic
455C22×C4C2×D4×C5311yescharacteristic
456C22×C4C24×C5311yescharacteristic
457M16—42no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—42no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—84no—
483C4⋊C4C2×D4×C5312yes—
484C4⋊C4C2×D4×C5311yescharacteristic
485C4⋊C4C2×D4×C5311yescharacteristic
486C22⋊C4—24no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—22no—
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—21no—
503C42—42no—
504C23—22no—
505C23—22no—
506C23—21no—
507C23—41no—
508C23#120×C5311yescharacteristic
509C23#120×C5311yescharacteristic
510C23C22×D4×C5311yescharacteristic
511Q8—42no—
512Q8—41no—
513Q8—42no—
514Q8—42no—
515Q8—81no—
516Q8—82no—
517D4—41no—
518D4—42no—
519D4—42no—
520D4—81no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#120×C5311yescharacteristic
548C2×C4#120×C5311yescharacteristic
549C2×C4C22×D4×C5311yescharacteristic
550C2×C4C22×D4×C5311yescharacteristic
551C8—42no—
552C8—42no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—44no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22C53×#45612yes—
565C22C2×#120×C5311yescharacteristic
566C22C53×#57712yes—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—44no—
571C4—42no—
572C4—41no—
573C4—42no—
574C4—41no—
575C4—41no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4C53×#45812yes—
580C2—22no—
581C2—22no—
582C2—41no—
583C2—81no—
584C2C53×#301011yescharacteristic
585C2C2×C53×#45611yescharacteristic
586C2C2×C53×#45811yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
prime radicalC53×#525C22
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
3. C2×C53×#483C2semidirect
4. C53×#2502C2semidirect
5. C53×#1611C2semidirect
6. C53×#1609C2semidirect
7. C53×#1508C2semidirect
9. C53×#1472C2semidirect
259. #18268C53direct (N × H)
289. C2×#483C2×C53semidirect
290. #2502C2×C53semidirect
291. #1611C2×C53semidirect
292. #1609C2×C53semidirect
293. #1508C2×C53semidirect
295. #1472C2×C53semidirect
340. C53#18268direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35382
C53×#30922
C2×C53×#4832
C53×#25022
C53×#16112
C53×#16092
C53×#15082
C53×#15012
C53×#14722
#1826853