Finite Group Databasebeta

← C53×#8468 #27215980

Subgroups — C53×#8468 #27215980

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3495C211yesmaximal, characteristic
2C53×#1996C211yesmaximal, characteristic, complemented
3C53×#1826C211yesmaximal, characteristic, complemented
4C53×#1825C211yesmaximal, characteristic, complemented
5C53×#1513C211yesmaximal, characteristic
6C53×#1501C211yesmaximal, characteristic
7C53×#1472C211yesmaximal, characteristic, complemented
8C53×#556—21no—
9C53×#554—21no—
10C53×#552—21no—
11C53×#552—21no—
12C53×#552—22no—
13C53×#551—22no—
14C53×#544C411yescharacteristic, complemented
15C53×#542—21no—
16C53×#541—21no—
17C53×#536C411yescharacteristic, complemented
18C53×#533—21no—
19C53×#529—21no—
20C53×#525C2211yescharacteristic
21C2×C22⋊Q8×C53C2211yescharacteristic
22C53×#520C411yescharacteristic, complemented
23C53×#519C411yescharacteristic, complemented
24C2×#117×C53C2211yescharacteristic
25C53×#407C2211yescharacteristic
26C2×C22⋊C8×C53C2211yescharacteristic
27C53×#380C2211yescharacteristic
28C4×C22⋊C4×C53C2211yescharacteristic
29C53×#342—22no—
30C53×#340—21no—
31C53×#340—21no—
32C53×#339—21no—
33C53×#339—21no—
34C53×#328—21no—
35C53×#328—21no—
36C53×#327—21no—
37C53×#327—21no—
38C53×#324—22no—
39C2×#41×C53C2×C411yescharacteristic
40C22×Q8×C53C2×C411yescharacteristic
41C23×C4×C53C2311yescharacteristic
42C2×M16×C53—21no—
43C22×C8×C53—21no—
44C4⋊Q8×C53—21no—
45C4⋊Q8×C53—21no—
46C4⋊Q8×C53—21no—
47#126×C53—22no—
48#126×C53—22no—
49#126×C53—22no—
50#126×C53—22no—
51#125×C53—22no—
52#125×C53—22no—
53#125×C53—22no—
54#125×C53—21no—
55#125×C53—21no—
56#124×C53—22no—
57#124×C53—21no—
58#123×C53—22no—
59#123×C53—22no—
60#123×C53—22no—
61#123×C53—21no—
62C22⋊Q8×C53—22no—
63C22⋊Q8×C53—22no—
64C22⋊Q8×C53—21no—
65C22⋊Q8×C53—21no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53—21no—
70C22⋊Q8×C53—21no—
71C22⋊Q8×C53D411yescharacteristic
72C22⋊Q8×C53D411yescharacteristic
73#121×C53D411yescharacteristic
74#121×C53D411yescharacteristic
75#120×C53—21no—
76C4×Q8×C53—22no—
77C4×Q8×C53—21no—
78C4×Q8×C53—21no—
79C4×Q8×C53—21no—
80C4×Q8×C53—21no—
81C4×Q8×C53—21no—
82C4×D4×C53—22no—
83C4×D4×C53—21no—
84C4×D4×C53—22no—
85C4×D4×C53—21no—
86C4×D4×C53—21no—
87#117×C53—22no—
88#117×C53—22no—
89#117×C53—21no—
90#117×C53—22no—
91#117×C53—21no—
92#117×C53—21no—
93#117×C53C2×C411yescharacteristic
94#117×C53C2×C411yescharacteristic
95#117×C53C2×C411yescharacteristic
96#117×C53D411yescharacteristic
97#117×C53D411yescharacteristic
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53C2×C411yescharacteristic
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53D412yes—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53D412yes—
108C2×C42×C53—21no—
109C2×C42×C53—21no—
110C22⋊C8×C53—21no—
111C22⋊C8×C53—21no—
112C22⋊C8×C53—21no—
113C22⋊C8×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116C24×C53C2×D411yescharacteristic
117#41×C53—42no—
118C2×Q8×C53—22no—
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53—42no—
122C2×Q8×C53C22⋊C411yescharacteristic
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53C22⋊C411yescharacteristic
127C22×C4×C53—21no—
128C22×C4×C53—22no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—42no—
132C22×C4×C53—41no—
133C22×C4×C53—41no—
134C22×C4×C53C22⋊C412yes—
135C22×C4×C53C22⋊C412yes—
136C22×C4×C53C22×C411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142M16×C53—42no—
143C2×C8×C53—42no—
144C2×C8×C53—41no—
145C2×C8×C53—41no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53C22⋊C411yescharacteristic
164C4⋊C4×C53C22⋊C411yescharacteristic
165C4⋊C4×C53C22⋊C411yescharacteristic
166C4⋊C4×C53C22⋊C411yescharacteristic
167C4⋊C4×C53C22⋊C411yescharacteristic
168C4⋊C4×C53C22⋊C411yescharacteristic
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C22⋊C4×C53—21no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—41no—
185C22⋊C4×C53—42no—
186C42×C53—22no—
187C42×C53—22no—
188C42×C53—22no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—21no—
192C42×C53—21no—
193C42×C53—21no—
194C42×C53—42no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53C2×C22⋊C411yescharacteristic
201C23×C53#12011yescharacteristic
202C23×C53#12011yescharacteristic
203Q8×C53—42no—
204Q8×C53—41no—
205Q8×C53—42no—
206Q8×C53—82no—
207D4×C53—41no—
208D4×C53—42no—
209D4×C53—41no—
210D4×C53—41no—
211D4×C53—81no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—81no—
234C2×C4×C53—82no—
235C2×C4×C53—81no—
236C2×C4×C53—81no—
237C2×C4×C53—81no—
238C2×C4×C53—81no—
239C2×C4×C53—81no—
240C2×C4×C53C2×C22⋊C411yescharacteristic
241C2×C4×C53C2×C22⋊C411yescharacteristic
242C2×C4×C53#12011yescharacteristic
243C2×C4×C53#12011yescharacteristic
244C8×C53—81no—
245C8×C53—81no—
246#8468C5311yesmaximal, characteristic, direct product factor, Sylow
247C22×C53—21no—
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—81no—
259C22×C53—81no—
260C22×C53#37911yescharacteristic
261C22×C53#45411yescharacteristic
262C22×C53#45611yescharacteristic
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—42no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53—42no—
275C4×C53—82no—
276C4×C53—161no—
277C4×C53#41012yes—
278#3495C2×C5311yescharacteristic
279#1996C2×C5311yescharacteristic, complemented
280#1826C2×C5311yescharacteristic, complemented
281#1825C2×C5311yescharacteristic, complemented
282#1513C2×C5311yescharacteristic
283#1501C2×C5311yescharacteristic
284#1472C2×C5311yescharacteristic, complemented
285C2×C53—21no—
286C2×C53—21no—
287C2×C53—41no—
288C2×C53—41no—
289C2×C53—81no—
290C2×C53#178211yescharacteristic
291C2×C53#179311yescharacteristic
292C2×C53#179411yescharacteristic
293#556—21no—
294#554—21no—
295#552—21no—
296#552—22no—
297#552—21no—
298#551—22no—
299#544C4×C5311yescharacteristic, complemented
300#542—21no—
301#541—21no—
302#536C4×C5311yescharacteristic, complemented
303#533—21no—
304#529—21no—
305#525C22×C5311yescharacteristic
306C2×C22⋊Q8C22×C5311yescharacteristic
307#520C4×C5311yescharacteristic, complemented
308#519C4×C5311yescharacteristic, complemented
309C2×#117C22×C5311yescharacteristic
310#407C22×C5311yescharacteristic
311C2×C22⋊C8C22×C5311yescharacteristic
312#380C22×C5311yescharacteristic
313C4×C22⋊C4C22×C5311yescharacteristic
314#342—22no—
315#340—21no—
316#340—21no—
317#339—21no—
318#339—21no—
319#328—21no—
320#328—21no—
321#327—21no—
322#327—21no—
323#324—22no—
324C53#846811yescharacteristic, direct product factor, Sylow
325C2×#41C2×C4×C5311yescharacteristic
326C22×Q8C2×C4×C5311yescharacteristic
327C23×C4C23×C5311yescharacteristic
328C2×M16—21no—
329C22×C8—21no—
330C4⋊Q8—21no—
331C4⋊Q8—21no—
332C4⋊Q8—21no—
333#126—22no—
334#126—22no—
335#126—22no—
336#126—22no—
337#125—22no—
338#125—22no—
339#125—22no—
340#125—21no—
341#125—21no—
342#124—22no—
343#124—21no—
344#123—22no—
345#123—22no—
346#123—21no—
347#123—22no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353C22⋊Q8—21no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8D4×C5311yescharacteristic
358C22⋊Q8D4×C5311yescharacteristic
359#121D4×C5311yescharacteristic
360#121D4×C5311yescharacteristic
361#120—21no—
362C4×Q8—22no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×D4—22no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—22no—
373#117—22no—
374#117—22no—
375#117—21no—
376#117—21no—
377#117—22no—
378#117—21no—
379#117C2×C4×C5311yescharacteristic
380#117C2×C4×C5311yescharacteristic
381#117C2×C4×C5311yescharacteristic
382#117D4×C5311yescharacteristic
383#117D4×C5311yescharacteristic
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4C2×C4×C5311yescharacteristic
387C2×C4⋊C4D4×C5312yes—
388C2×C4⋊C4D4×C5311yescharacteristic
389C2×C4⋊C4D4×C5311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4D4×C5312yes—
394C2×C42—21no—
395C2×C42—21no—
396C22⋊C8—21no—
397C22⋊C8—21no—
398C22⋊C8—21no—
399C22⋊C8—21no—
400#95—21no—
401#95—21no—
402C24C2×D4×C5311yescharacteristic
403#41—42no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—42no—
408C2×Q8C22⋊C4×C5311yescharacteristic
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4C22⋊C4×C5311yescharacteristic
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4C22⋊C4×C5312yes—
421C22×C4C22⋊C4×C5312yes—
422C22×C4C22×C4×C5311yescharacteristic
423C22×C4C2×D4×C5311yescharacteristic
424C22×C4C2×D4×C5311yescharacteristic
425C22×C4C2×D4×C5311yescharacteristic
426C22×C4C2×D4×C5311yescharacteristic
427C22×C4C2×D4×C5311yescharacteristic
428M16—42no—
429C2×C8—42no—
430C2×C8—41no—
431C2×C8—41no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—42no—
449C4⋊C4C22⋊C4×C5311yescharacteristic
450C4⋊C4C22⋊C4×C5311yescharacteristic
451C4⋊C4C22⋊C4×C5311yescharacteristic
452C4⋊C4C22⋊C4×C5311yescharacteristic
453C4⋊C4C22⋊C4×C5311yescharacteristic
454C4⋊C4C22⋊C4×C5311yescharacteristic
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—22no—
470C22⋊C4—41no—
471C22⋊C4—42no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—42no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—41no—
485C23—41no—
486C23C2×C22⋊C4×C5311yescharacteristic
487C23#120×C5311yescharacteristic
488C23#120×C5311yescharacteristic
489Q8—41no—
490Q8—42no—
491Q8—42no—
492Q8—82no—
493D4—41no—
494D4—42no—
495D4—41no—
496D4—41no—
497D4—81no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4C2×C22⋊C4×C5311yescharacteristic
527C2×C4C2×C22⋊C4×C5311yescharacteristic
528C2×C4#120×C5311yescharacteristic
529C2×C4#120×C5311yescharacteristic
530C8—81no—
531C8—81no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—81no—
544C22—81no—
545C22C53×#37911yescharacteristic
546C22C53×#45411yescharacteristic
547C22C53×#45611yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—82no—
561C4—161no—
562C4C53×#41012yes—
563C2—21no—
564C2—21no—
565C2—41no—
566C2—41no—
567C2—81no—
568C2C53×#178211yescharacteristic
569C2C53×#179311yescharacteristic
570C2C53×#179411yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×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
2. C53×#1996C2semidirect
3. C53×#1826C2semidirect
4. C53×#1825C2semidirect
7. C53×#1472C2semidirect
14. C53×#544C4semidirect
17. C53×#536C4semidirect
22. C53×#520C4semidirect
23. C53×#519C4semidirect
246. #8468C53direct (N × H)
279. #1996C2×C53semidirect
280. #1826C2×C53semidirect
281. #1825C2×C53semidirect
284. #1472C2×C53semidirect
299. #544C4×C53semidirect
302. #536C4×C53semidirect
307. #520C4×C53semidirect
308. #519C4×C53semidirect
324. C53#8468direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#34952
C53×#19962
C53×#18262
C53×#18252
C53×#15132
C53×#15012
C53×#14722
#846853