Finite Group Databasebeta

← C53×#18973 #27226485

Subgroups — C53×#18973 #27226485

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3555C211yesmaximal, characteristic
2C53×#3230C211yesmaximal, characteristic, complemented
3C53×#3229C211yesmaximal, characteristic, complemented
4C53×#3218C211yesmaximal, characteristic
5C53×#1650C212yesmaximal, complemented
6C53×#1648C212yesmaximal, complemented
7C53×#1645C212yesmaximal, complemented
8C53×#1643C212yesmaximal, complemented
9C53×#1554C211yesmaximal, characteristic, complemented
10C53×#1497C211yesmaximal, characteristic, complemented
11C53×#1482C211yesmaximal, characteristic
12C53×#563—22no—
13Q82×C53C2211yescharacteristic
14C53×#557—21no—
15C53×#557C2211yescharacteristic
16C53×#556—21no—
17C53×#556—21no—
18C53×#556C2211yescharacteristic
19C53×#554C2211yescharacteristic
20C53×#552—21no—
21C53×#544—21no—
22C53×#542—21no—
23C53×#541—21no—
24C53×#541—21no—
25C53×#541—22no—
26C53×#533C2211yescharacteristic
27C53×#533C2211yescharacteristic
28C2×C4⋊Q8×C53C2211yescharacteristic
29C53×#529C2211yescharacteristic
30C53×#500C2212yes—
31C53×#498—21no—
32C53×#497—21no—
33C53×#491—21no—
34C53×#487—21no—
35C53×#487C2212yes—
36C53×#484—22no—
37C53×#483—21no—
38C53×#483—21no—
39C53×#477—22no—
40C53×#475C2212yes—
41C53×#475C2212yes—
42C53×#462C2212yes—
43C53×#461C2212yes—
44C53×#451—21no—
45C53×#448—21no—
46C53×#432C2211yescharacteristic
47C53×#431C2211yescharacteristic
48C53×#431C2211yescharacteristic
49C53×#333C2212yes—
50C53×#332C2212yes—
51C53×Q8⋊C8C2212yes—
52C53×Q8⋊C8C2212yes—
53C53×Q8⋊C8C2212yes—
54C53×D4⋊C8C2212yes—
55C22×Q8×C53—21no—
56C2×Q16×C53—22no—
57C2×SD16×C53—22no—
58C4⋊Q8×C53—22no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—22no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53—21no—
66C4⋊Q8×C53D411yescharacteristic
67C4⋊Q8×C53D411yescharacteristic
68C4⋊Q8×C53C2312yes—
69C4⋊Q8×C53C2311yescharacteristic
70C4⋊Q8×C53C2311yescharacteristic
71#126×C53—22no—
72#126×C53—22no—
73#126×C53—22no—
74#126×C53—22no—
75#126×C53—22no—
76#126×C53—22no—
77#126×C53—21no—
78#125×C53—22no—
79#125×C53—22no—
80#125×C53—22no—
81#125×C53—22no—
82#125×C53—21no—
83#125×C53—22no—
84#125×C53—21no—
85#125×C53—21no—
86#125×C53—21no—
87#125×C53D411yescharacteristic
88#124×C53—22no—
89#124×C53—21no—
90#124×C53D411yescharacteristic
91#123×C53—22no—
92#123×C53—21no—
93C22⋊Q8×C53—22no—
94C22⋊Q8×C53—22no—
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53—21no—
97C22⋊Q8×C53—22no—
98C22⋊Q8×C53—21no—
99C22⋊Q8×C53—21no—
100C22⋊Q8×C53D411yescharacteristic
101C22⋊Q8×C53D411yescharacteristic
102C22⋊Q8×C53D411yescharacteristic
103C22⋊Q8×C53D412yes—
104#121×C53D411yescharacteristic
105C4×Q8×C53—22no—
106C4×Q8×C53—21no—
107C4×Q8×C53—21no—
108C4×Q8×C53—21no—
109C4×Q8×C53—21no—
110C4×Q8×C53—21no—
111C4×Q8×C53—21no—
112C4×Q8×C53—21no—
113C4×Q8×C53D412yes—
114C4×Q8×C53C2311yescharacteristic
115C4×Q8×C53C2311yescharacteristic
116C4×Q8×C53C2311yescharacteristic
117C4×D4×C53—21no—
118C4×D4×C53—21no—
119C4×D4×C53C2311yescharacteristic
120#117×C53—22no—
121#117×C53—21no—
122#117×C53—21no—
123#117×C53—21no—
124#117×C53—21no—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—21no—
128C2×C42×C53C2311yescharacteristic
129#107×C53—24no—
130#106×C53—22no—
131#106×C53—22no—
132C4⋊C8×C53—21no—
133C4⋊C8×C53C2312yes—
134C4⋊C8×C53C2312yes—
135#103×C53—22no—
136#103×C53—22no—
137#103×C53—22no—
138#103×C53—22no—
139#103×C53—24no—
140C22⋊C8×C53—21no—
141C22⋊C8×C53—21no—
142C22⋊C8×C53—21no—
143#97×C53—21no—
144C4×C8×C53—21no—
145C4×C8×C53C2312yes—
146C2×Q8×C53—22no—
147C2×Q8×C53—21no—
148C2×Q8×C53—21no—
149C2×Q8×C53—22no—
150C2×Q8×C53—41no—
151C2×Q8×C53—82no—
152C2×Q8×C53C2×D411yescharacteristic
153C2×Q8×C53C2×D411yescharacteristic
154C2×Q8×C53C2×D411yescharacteristic
155C2×Q8×C53C2×D412yes—
156C2×D4×C53—21no—
157C2×D4×C53C2×D411yescharacteristic
158C22×C4×C53—22no—
159C22×C4×C53—21no—
160C22×C4×C53—41no—
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×D411yescharacteristic
164Q16×C53—82no—
165SD16×C53—82no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×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—22no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—22no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53—21no—
193C4⋊C4×C53—22no—
194C4⋊C4×C53—22no—
195C4⋊C4×C53—41no—
196C4⋊C4×C53—41no—
197C4⋊C4×C53C2×D411yescharacteristic
198C4⋊C4×C53C2×D411yescharacteristic
199C4⋊C4×C53C2×D411yescharacteristic
200C4⋊C4×C53C2×D411yescharacteristic
201C4⋊C4×C53C2×D412yes—
202C22⋊C4×C53—22no—
203C22⋊C4×C53—22no—
204C22⋊C4×C53—22no—
205C22⋊C4×C53—21no—
206C22⋊C4×C53—22no—
207C22⋊C4×C53—21no—
208C22⋊C4×C53—21no—
209C22⋊C4×C53—21no—
210C22⋊C4×C53—21no—
211C22⋊C4×C53—21no—
212C22⋊C4×C53—21no—
213C42×C53—22no—
214C42×C53—22no—
215C42×C53—22no—
216C42×C53—22no—
217C42×C53—21no—
218C42×C53—21no—
219C42×C53—21no—
220C42×C53C2×D411yescharacteristic
221C42×C53C2×D411yescharacteristic
222C42×C53C2×D411yescharacteristic
223C42×C53C2411yescharacteristic
224C23×C53—21no—
225C23×C53#12011yescharacteristic
226Q8×C53—22no—
227Q8×C53—22no—
228Q8×C53—22no—
229Q8×C53—42no—
230Q8×C53—42no—
231Q8×C53—42no—
232Q8×C53—42no—
233Q8×C53—81no—
234Q8×C53—82no—
235Q8×C53—82no—
236D4×C53—22no—
237D4×C53—81no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—22no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—22no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—41no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—82no—
253C2×C4×C53—82no—
254C2×C4×C53#12011yescharacteristic
255C2×C4×C53#12011yescharacteristic
256C2×C4×C53#12011yescharacteristic
257C2×C4×C53C22×D411yescharacteristic
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C22×D411yescharacteristic
260C8×C53—42no—
261C8×C53—42no—
262C8×C53—42no—
263#18973C5311yesmaximal, characteristic, direct product factor, Sylow
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53C2×#12011yescharacteristic
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—41no—
276C4×C53—42no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53—42no—
282C4×C53—41no—
283C4×C53—41no—
284C4×C53—82no—
285C4×C53#57712yes—
286C4×C53#57812yes—
287#3555C2×C5311yescharacteristic
288#3230C2×C5311yescharacteristic, complemented
289#3229C2×C5311yescharacteristic, complemented
290#3218C2×C5311yescharacteristic
291#1650C2×C5312yescomplemented
292#1648C2×C5312yescomplemented
293#1645C2×C5312yescomplemented
294#1643C2×C5312yescomplemented
295#1554C2×C5311yescharacteristic, complemented
296#1497C2×C5311yescharacteristic, complemented
297#1482C2×C5311yescharacteristic
298C2×C53—41no—
299C2×C53—81no—
300C2×C53#301011yescharacteristic
301C2×C53#301111yescharacteristic
302C2×C53#302211yescharacteristic
303#563—22no—
304Q82C22×C5311yescharacteristic
305#557—21no—
306#557C22×C5311yescharacteristic
307#556—21no—
308#556—21no—
309#556C22×C5311yescharacteristic
310#554C22×C5311yescharacteristic
311#552—21no—
312#544—21no—
313#542—21no—
314#541—21no—
315#541—21no—
316#541—22no—
317#533C22×C5311yescharacteristic
318#533C22×C5311yescharacteristic
319C2×C4⋊Q8C22×C5311yescharacteristic
320#529C22×C5311yescharacteristic
321#500C22×C5312yes—
322#498—21no—
323#497—21no—
324#491—21no—
325#487—21no—
326#487C22×C5312yes—
327#484—22no—
328#483—21no—
329#483—21no—
330#477—22no—
331#475C22×C5312yes—
332#475C22×C5312yes—
333#462C22×C5312yes—
334#461C22×C5312yes—
335#451—21no—
336#448—21no—
337#432C22×C5311yescharacteristic
338#431C22×C5311yescharacteristic
339#431C22×C5311yescharacteristic
340#333C22×C5312yes—
341#332C22×C5312yes—
342Q8⋊C8C22×C5312yes—
343Q8⋊C8C22×C5312yes—
344Q8⋊C8C22×C5312yes—
345D4⋊C8C22×C5312yes—
346C53#1897311yescharacteristic, direct product factor, Sylow
347C22×Q8—21no—
348C2×Q16—22no—
349C2×SD16—22no—
350C4⋊Q8—22no—
351C4⋊Q8—21no—
352C4⋊Q8—22no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8—21no—
358C4⋊Q8D4×C5311yescharacteristic
359C4⋊Q8D4×C5311yescharacteristic
360C4⋊Q8C23×C5312yes—
361C4⋊Q8C23×C5311yescharacteristic
362C4⋊Q8C23×C5311yescharacteristic
363#126—22no—
364#126—22no—
365#126—22no—
366#126—22no—
367#126—22no—
368#126—22no—
369#126—21no—
370#125—22no—
371#125—22no—
372#125—22no—
373#125—22no—
374#125—21no—
375#125—22no—
376#125—21no—
377#125—21no—
378#125—21no—
379#125D4×C5311yescharacteristic
380#124—22no—
381#124—21no—
382#124D4×C5311yescharacteristic
383#123—22no—
384#123—21no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—22no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8D4×C5311yescharacteristic
393C22⋊Q8D4×C5311yescharacteristic
394C22⋊Q8D4×C5311yescharacteristic
395C22⋊Q8D4×C5312yes—
396#121D4×C5311yescharacteristic
397C4×Q8—22no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8D4×C5312yes—
406C4×Q8C23×C5311yescharacteristic
407C4×Q8C23×C5311yescharacteristic
408C4×Q8C23×C5311yescharacteristic
409C4×D4—21no—
410C4×D4—21no—
411C4×D4C23×C5311yescharacteristic
412#117—22no—
413#117—21no—
414#117—21no—
415#117—21no—
416#117—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C42C23×C5311yescharacteristic
421#107—24no—
422#106—22no—
423#106—22no—
424C4⋊C8—21no—
425C4⋊C8C23×C5312yes—
426C4⋊C8C23×C5312yes—
427#103—22no—
428#103—22no—
429#103—24no—
430#103—22no—
431#103—22no—
432C22⋊C8—21no—
433C22⋊C8—21no—
434C22⋊C8—21no—
435#97—21no—
436C4×C8—21no—
437C4×C8C23×C5312yes—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—82no—
444C2×Q8C2×D4×C5311yescharacteristic
445C2×Q8C2×D4×C5311yescharacteristic
446C2×Q8C2×D4×C5311yescharacteristic
447C2×Q8C2×D4×C5312yes—
448C2×D4—21no—
449C2×D4C2×D4×C5311yescharacteristic
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4C2×D4×C5311yescharacteristic
454C22×C4C2×D4×C5311yescharacteristic
455C22×C4C2×D4×C5311yescharacteristic
456Q16—82no—
457SD16—82no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
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—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×D4×C5311yescharacteristic
490C4⋊C4C2×D4×C5311yescharacteristic
491C4⋊C4C2×D4×C5311yescharacteristic
492C4⋊C4C2×D4×C5311yescharacteristic
493C4⋊C4C2×D4×C5312yes—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42C2×D4×C5311yescharacteristic
513C42C2×D4×C5311yescharacteristic
514C42C2×D4×C5311yescharacteristic
515C42C24×C5311yescharacteristic
516C23—21no—
517C23#120×C5311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—22no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—81no—
526Q8—82no—
527Q8—82no—
528D4—22no—
529D4—81no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4#120×C5311yescharacteristic
547C2×C4#120×C5311yescharacteristic
548C2×C4#120×C5311yescharacteristic
549C2×C4C22×D4×C5311yescharacteristic
550C2×C4C22×D4×C5311yescharacteristic
551C2×C4C22×D4×C5311yescharacteristic
552C8—42no—
553C8—42no—
554C8—42no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22C2×#120×C5311yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—82no—
576C4C53×#57712yes—
577C4C53×#57812yes—
578C2—41no—
579C2—81no—
580C2C53×#301011yescharacteristic
581C2C53×#301111yescharacteristic
582C2C53×#302211yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC42C24×C53
prime radical#121×C53D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#3230C2semidirect
3. C53×#3229C2semidirect
5. C53×#1650C2semidirect
6. C53×#1648C2semidirect
7. C53×#1645C2semidirect
8. C53×#1643C2semidirect
9. C53×#1554C2semidirect
10. C53×#1497C2semidirect
263. #18973C53direct (N × H)
288. #3230C2×C53semidirect
289. #3229C2×C53semidirect
291. #1650C2×C53semidirect
292. #1648C2×C53semidirect
293. #1645C2×C53semidirect
294. #1643C2×C53semidirect
295. #1554C2×C53semidirect
296. #1497C2×C53semidirect
346. C53#18973direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35552
C53×#32302
C53×#32292
C53×#32182
C53×#16502
C53×#16482
C53×#16452
C53×#16432
C53×#15542
C53×#14972
C53×#14822
#1897353