Finite Group Databasebeta

← C53×#8419 #27215931

Subgroups — C53×#8419 #27215931

Order 13568.

778 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3522C211yesmaximal, characteristic
2C53×#1980C211yesmaximal, characteristic, complemented
3C53×#1763C211yesmaximal, characteristic, complemented
4C53×#1530C211yesmaximal, characteristic, complemented
5C53×#1524C211yesmaximal, characteristic, complemented
6C53×#1509C211yesmaximal, characteristic, complemented
7C53×#1491C211yesmaximal, characteristic, complemented
8C53×#559—21no—
9C53×#559C411yescharacteristic, complemented
10Q82×C53C411yescharacteristic, complemented
11C53×#554—21no—
12C53×#554C411yescharacteristic, complemented
13C53×#550C411yescharacteristic, complemented
14C53×#548—21no—
15C53×#547—22no—
16C53×#547—21no—
17C53×#542—22no—
18C53×#542—21no—
19C53×#532—21no—
20C53×#532C2211yescharacteristic
21C53×#529—21no—
22C53×#529C2211yescharacteristic
23C53×#520—21no—
24C2×C4×Q8×C53C2211yescharacteristic
25C53×#432C2211yescharacteristic
26C2×#104×C53—22no—
27C2×#103×C53—21no—
28C4×M16×C53C2211yescharacteristic
29C2×#97×C53C2211yescharacteristic
30C4×C4⋊C4×C53C2211yescharacteristic
31C53×#340—21no—
32C53×#330—21no—
33C53×#329—21no—
34C53×#329—22no—
35C53×#328—21no—
36C53×#327—21no—
37C53×Q8⋊C8—21no—
38C53×Q8⋊C8—21no—
39C2×#41×C53—22no—
40C2×#41×C53—21no—
41C2×#41×C53D412yes—
42C22×Q8×C53D411yescharacteristic
43C2×M16×C53—21no—
44C22×C8×C53—21no—
45C4⋊Q8×C53—21no—
46C4⋊Q8×C53—21no—
47C4⋊Q8×C53—22no—
48C4⋊Q8×C53C2×C411yescharacteristic
49#127×C53—21no—
50#127×C53C2×C411yescharacteristic
51#126×C53—22no—
52#126×C53—21no—
53#125×C53—21no—
54#125×C53D411yescharacteristic
55#124×C53—22no—
56#124×C53—22no—
57#124×C53—21no—
58#124×C53—21no—
59#124×C53D411yescharacteristic
60#124×C53D412yes—
61#123×C53—22no—
62#123×C53—22no—
63#123×C53—22no—
64#123×C53—22no—
65#123×C53—21no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—22no—
68C22⋊Q8×C53—22no—
69C22⋊Q8×C53—22no—
70C22⋊Q8×C53—21no—
71C22⋊Q8×C53D411yescharacteristic
72#121×C53—22no—
73#121×C53—21no—
74#121×C53—22no—
75#121×C53—21no—
76#121×C53—21no—
77#121×C53D411yescharacteristic
78C4×Q8×C53—21no—
79C4×Q8×C53—21no—
80C4×Q8×C53—21no—
81C4×Q8×C53—21no—
82C4×Q8×C53C2×C411yescharacteristic
83C4×Q8×C53C2×C411yescharacteristic
84C4×Q8×C53C2×C411yescharacteristic
85C4×Q8×C53D411yescharacteristic
86C4×Q8×C53D411yescharacteristic
87C4×D4×C53—22no—
88C4×D4×C53—22no—
89C4×D4×C53—22no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53C2×C411yescharacteristic
98#117×C53—22no—
99#117×C53—21no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C42×C53—21no—
106C2×C42×C53—21no—
107C2×C42×C53C2311yescharacteristic
108C4⋊C8×C53—21no—
109#104×C53—44no—
110#103×C53—24no—
111C22⋊C8×C53—21no—
112#97×C53—21no—
113#97×C53—21no—
114#97×C53—21no—
115#97×C53—21no—
116C4×C8×C53—21no—
117#95×C53—21no—
118#41×C53—42no—
119#41×C53—42no—
120#41×C53—42no—
121#41×C53—42no—
122#41×C53—82no—
123#41×C53—82no—
124C2×Q8×C53—22no—
125C2×Q8×C53—21no—
126C2×Q8×C53—22no—
127C2×Q8×C53—41no—
128C2×Q8×C53C22⋊C411yescharacteristic
129C2×Q8×C53C22⋊C411yescharacteristic
130C2×Q8×C53C22⋊C411yescharacteristic
131C2×Q8×C53C22⋊C412yes—
132C2×D4×C53—22no—
133C2×D4×C53—22no—
134C2×D4×C53—21no—
135C2×D4×C53—22no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53C22⋊C411yescharacteristic
139C2×D4×C53C22⋊C412yes—
140C22×C4×C53—22no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—21no—
146C22×C4×C53—22no—
147C22×C4×C53—41no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151M16×C53—22no—
152M16×C53—41no—
153C2×C8×C53—22no—
154C2×C8×C53—41no—
155C2×C8×C53—41no—
156C2×C8×C53—41no—
157C2×C8×C53—41no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—41no—
175C4⋊C4×C53C22⋊C411yescharacteristic
176C4⋊C4×C53C22⋊C411yescharacteristic
177C4⋊C4×C53C22⋊C411yescharacteristic
178C4⋊C4×C53C22⋊C411yescharacteristic
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—21no—
185C22⋊C4×C53—21no—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—21no—
188C22⋊C4×C53—21no—
189C22⋊C4×C53—22no—
190C22⋊C4×C53—21no—
191C42×C53—22no—
192C42×C53—21no—
193C42×C53—21no—
194C42×C53—21no—
195C42×C53—21no—
196C42×C53—21no—
197C42×C53—42no—
198C42×C53C22×C411yescharacteristic
199C42×C53C2×D411yescharacteristic
200C42×C53C2×D411yescharacteristic
201C42×C53C2×D411yescharacteristic
202C23×C53—22no—
203C23×C53—21no—
204C23×C53#12011yescharacteristic
205Q8×C53—21no—
206Q8×C53—21no—
207Q8×C53—21no—
208Q8×C53—42no—
209Q8×C53—41no—
210Q8×C53—42no—
211Q8×C53—82no—
212Q8×C53#10412yescomplemented
213D4×C53—41no—
214D4×C53—42no—
215D4×C53—42no—
216D4×C53—44no—
217D4×C53—42no—
218D4×C53—41no—
219D4×C53—42no—
220D4×C53—41no—
221D4×C53—41no—
222D4×C53—41no—
223D4×C53—82no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—44no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—42no—
244C2×C4×C53—82no—
245C2×C4×C53—82no—
246C2×C4×C53—82no—
247C2×C4×C53—81no—
248C2×C4×C53—81no—
249C2×C4×C53C2×C22⋊C411yescharacteristic
250C2×C4×C53C2×C22⋊C411yescharacteristic
251C2×C4×C53C2×C22⋊C411yescharacteristic
252C2×C4×C53#12011yescharacteristic
253C2×C4×C53#12011yescharacteristic
254C2×C4×C53#12011yescharacteristic
255C8×C53—42no—
256C8×C53—42no—
257C8×C53—81no—
258#8419C5311yesmaximal, characteristic, direct product factor, Sylow
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—42no—
262C22×C53—41no—
263C22×C53—42no—
264C22×C53—41no—
265C22×C53—42no—
266C22×C53—41no—
267C22×C53#37911yescharacteristic
268C22×C53#45311yescharacteristic
269C22×C53#45511yescharacteristic
270C4×C53—21no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—22no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—42no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—41no—
283C4×C53—82no—
284C4×C53C2×#10411yescharacteristic
285C4×C53#42111yescharacteristic
286#3522C2×C5311yescharacteristic
287#1980C2×C5311yescharacteristic, complemented
288#1763C2×C5311yescharacteristic, complemented
289#1530C2×C5311yescharacteristic, complemented
290#1524C2×C5311yescharacteristic, complemented
291#1509C2×C5311yescharacteristic, complemented
292#1491C2×C5311yescharacteristic, complemented
293C2×C53—21no—
294C2×C53—21no—
295C2×C53—82no—
296C2×C53—81no—
297C2×C53#178911yescharacteristic
298C2×C53#179211yescharacteristic
299C2×C53#179611yescharacteristic
300#559—21no—
301#559C4×C5311yescharacteristic, complemented
302Q82C4×C5311yescharacteristic, complemented
303#554—21no—
304#554C4×C5311yescharacteristic, complemented
305#550C4×C5311yescharacteristic, complemented
306#548—21no—
307#547—22no—
308#547—21no—
309#542—22no—
310#542—21no—
311#532—21no—
312#532C22×C5311yescharacteristic
313#529—21no—
314#529C22×C5311yescharacteristic
315#520—21no—
316C2×C4×Q8C22×C5311yescharacteristic
317#432C22×C5311yescharacteristic
318C2×#104—22no—
319C2×#103—21no—
320C4×M16C22×C5311yescharacteristic
321C2×#97C22×C5311yescharacteristic
322C4×C4⋊C4C22×C5311yescharacteristic
323#340—21no—
324#330—21no—
325#329—21no—
326#329—22no—
327#328—21no—
328#327—21no—
329Q8⋊C8—21no—
330Q8⋊C8—21no—
331C53#841911yescharacteristic, direct product factor, Sylow
332C2×#41—22no—
333C2×#41—21no—
334C2×#41D4×C5312yes—
335C22×Q8D4×C5311yescharacteristic
336C2×M16—21no—
337C22×C8—21no—
338C4⋊Q8—22no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8C2×C4×C5311yescharacteristic
342#127—21no—
343#127C2×C4×C5311yescharacteristic
344#126—22no—
345#126—21no—
346#125—21no—
347#125D4×C5311yescharacteristic
348#124—22no—
349#124—22no—
350#124—21no—
351#124—21no—
352#124D4×C5311yescharacteristic
353#124D4×C5312yes—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—22no—
358#123—21no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—21no—
364C22⋊Q8D4×C5311yescharacteristic
365#121—22no—
366#121—22no—
367#121—21no—
368#121—21no—
369#121—21no—
370#121D4×C5311yescharacteristic
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×Q8C2×C4×C5311yescharacteristic
376C4×Q8C2×C4×C5311yescharacteristic
377C4×Q8C2×C4×C5311yescharacteristic
378C4×Q8D4×C5311yescharacteristic
379C4×Q8D4×C5311yescharacteristic
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4—21no—
390C4×D4C2×C4×C5311yescharacteristic
391#117—22no—
392#117—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4D4×C5311yescharacteristic
398C2×C42—21no—
399C2×C42—21no—
400C2×C42C23×C5311yescharacteristic
401C4⋊C8—21no—
402#104—44no—
403#103—24no—
404C22⋊C8—21no—
405#97—21no—
406#97—21no—
407#97—21no—
408#97—21no—
409C4×C8—21no—
410#95—21no—
411#41—42no—
412#41—42no—
413#41—42no—
414#41—42no—
415#41—82no—
416#41—82no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—41no—
421C2×Q8C22⋊C4×C5311yescharacteristic
422C2×Q8C22⋊C4×C5311yescharacteristic
423C2×Q8C22⋊C4×C5311yescharacteristic
424C2×Q8C22⋊C4×C5312yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4C22⋊C4×C5311yescharacteristic
432C2×D4C22⋊C4×C5312yes—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4C2×D4×C5311yescharacteristic
442C22×C4C2×D4×C5311yescharacteristic
443C22×C4C2×D4×C5311yescharacteristic
444M16—22no—
445M16—41no—
446C2×C8—22no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4C22⋊C4×C5311yescharacteristic
469C4⋊C4C22⋊C4×C5311yescharacteristic
470C4⋊C4C22⋊C4×C5311yescharacteristic
471C4⋊C4C22⋊C4×C5311yescharacteristic
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—22no—
483C22⋊C4—21no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—42no—
491C42C22×C4×C5311yescharacteristic
492C42C2×D4×C5311yescharacteristic
493C42C2×D4×C5311yescharacteristic
494C42C2×D4×C5311yescharacteristic
495C23—22no—
496C23—21no—
497C23#120×C5311yescharacteristic
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—42no—
502Q8—41no—
503Q8—42no—
504Q8—82no—
505Q8#104×C5312yescomplemented
506D4—41no—
507D4—42no—
508D4—42no—
509D4—44no—
510D4—42no—
511D4—41no—
512D4—42no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—82no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4C2×C22⋊C4×C5311yescharacteristic
543C2×C4C2×C22⋊C4×C5311yescharacteristic
544C2×C4C2×C22⋊C4×C5311yescharacteristic
545C2×C4#120×C5311yescharacteristic
546C2×C4#120×C5311yescharacteristic
547C2×C4#120×C5311yescharacteristic
548C8—42no—
549C8—42no—
550C8—81no—
551C22—21no—
552C22—21no—
553C22—42no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22—42no—
558C22—41no—
559C22C53×#37911yescharacteristic
560C22C53×#45311yescharacteristic
561C22C53×#45511yescharacteristic
562C4—21no—
563C4—22no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—82no—
576C4C2×#104×C5311yescharacteristic
577C4C53×#42111yescharacteristic
578C2—21no—
579C2—21no—
580C2—82no—
581C2—81no—
582C2C53×#178911yescharacteristic
583C2C53×#179211yescharacteristic
584C2C53×#179611yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3522C2
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×#1980C2semidirect
3. C53×#1763C2semidirect
4. C53×#1530C2semidirect
5. C53×#1524C2semidirect
6. C53×#1509C2semidirect
7. C53×#1491C2semidirect
9. C53×#559C4semidirect
10. Q82×C53C4semidirect
12. C53×#554C4semidirect
13. C53×#550C4semidirect
212. Q8×C53#104semidirect
258. #8419C53direct (N × H)
287. #1980C2×C53semidirect
288. #1763C2×C53semidirect
289. #1530C2×C53semidirect
290. #1524C2×C53semidirect
291. #1509C2×C53semidirect
292. #1491C2×C53semidirect
301. #559C4×C53semidirect
302. Q82C4×C53semidirect
304. #554C4×C53semidirect
305. #550C4×C53semidirect
331. C53#8419direct (N × H)
505. Q8#104×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#35222
C53×#19802
C53×#17632
C53×#15302
C53×#15242
C53×#15092
C53×#14912
#841953