Finite Group Databasebeta

← #27469665

Subgroups — #27469665

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280540C211yesmaximal, characteristic
2#20277335C211yesmaximal, characteristic
3#20276202C211yesmaximal, characteristic, complemented
4#20276199C211yesmaximal, characteristic, complemented
5#20276194C211yesmaximal, characteristic, complemented
6#20276190C211yesmaximal, characteristic, complemented
7#20275686C211yesmaximal, characteristic, complemented
8#20275674C211yesmaximal, characteristic, complemented
9#20274914C211yesmaximal, characteristic
10#20274907C211yesmaximal, characteristic
11#20274905C211yesmaximal, characteristic
12#20274880C211yesmaximal, characteristic, complemented
13#20274878C211yesmaximal, characteristic, complemented
14#20274483C211yesmaximal, characteristic
15C53×#1854C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×#13133127C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28#16810565—21no—
29#16810559—21no—
30#16810544—21no—
31#16810535—21no—
32#16810522C2211yescharacteristic
33#16810521—21no—
34#16810521—21no—
35#16810519C2211yescharacteristic
36#16810495—21no—
37#16810495—21no—
38#16810492C2211yescharacteristic
39#16810492C2211yescharacteristic
40#16810488C2211yescharacteristic
41#16810488C2211yescharacteristic
42#16810488C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133216—22no—
64#13133216—22no—
65#13133176—22no—
66#13133176—22no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—41no—
70C2×C4⋊Dic53—41no—
71C2×C4⋊Dic53D411yescharacteristic
72C2×C4⋊Dic53D411yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53D411yescharacteristic
85C2×C22⋊Dic53D411yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133151—41no—
89#13133151—41no—
90#13133150—41no—
91#13133148—41no—
92#13133148—41no—
93#13133148—41no—
94C22⋊#88322Q814yescomplemented
95#13133127—22no—
96C8×Dic53—22no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109#117×C53—41no—
110#117×C53—41no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C42×C53—21no—
118#107×C53—41no—
119#106×C53—41no—
120C22⋊C8×C53Q814yescomplemented
121#95×C53—21no—
122C24×C53C2×D411yescharacteristic
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×Q812yes—
133C22×C4×C53C2×Q812yes—
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—42no—
140C22×Dic53—81no—
141C22×Dic53—81no—
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C2×Dic106—42no—
147C2×Dic106—81no—
148C2×Dic106—81no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53C2×Q812yes—
154C2×C8×C53C2×Q812yes—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C22⋊C4×C53—42no—
164C22⋊C4×C53—81no—
165C42×C53—42no—
166C22⋊Dic53—22no—
167C22⋊Dic53—22no—
168C22⋊Dic53—81no—
169C4⋊Dic53—22no—
170C4⋊Dic53—22no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—42no—
180C4⋊Dic53—81no—
181#11719919—42no—
182#11719919—81no—
183#11719919—81no—
184#11719919—81no—
185#11719919—81no—
186#11719919—81no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53#4112yes—
190C4×Dic53#4112yes—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322C2×Q812yes—
196C2×#88322C2×Q812yes—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53C4⋊Q812yes—
204C23×C53C22×D411yescharacteristic
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53C4⋊Q812yes—
224C2×C4×C53C22×Q811yescharacteristic
225C2×C4×C53C22×Q811yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53H(5,𝔽2)11yescharacteristic
229C2×C4×C53H(5,𝔽2)11yescharacteristic
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—84no—
253C2×Dic53#12312yes—
254C2×Dic53#12312yes—
255Dic106—84no—
256C8×C53—22no—
257C8×C53—22no—
258#88322—22no—
259#88322—22no—
260#27959—531nomaximal, Sylow
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53C2×#12311yescharacteristic
269C22×C53C2×C4⋊Q811yescharacteristic
270C22×C53#53511yescharacteristic
271C22×C53#57611yescharacteristic
272C22×C53#57711yescharacteristic
273C22×C53#57711yescharacteristic
274C22×C53#57811yescharacteristic
275C4×C53—22no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53#55411yescharacteristic
281C4×C53#55411yescharacteristic
282C4×C53#55411yescharacteristic
283C4×C53#55411yescharacteristic
284Dic53—22no—
285Dic53—22no—
286Dic53—41no—
287Dic53—41no—
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292#2628—531no—
293#2317—531no—
294#2085—531no—
295#2085—531no—
296#2084—531no—
297#2083—531no—
298#1993—531no—
299#1990—531no—
300#1854—531no—
301#1854D5311yescharacteristic
302#1853—531no—
303#1852—531no—
304#1847—531no—
305#1846—531no—
306#1750—531no—
307C2×C53—42no—
308C2×C53#250211yescharacteristic
309C2×C53#309611yescharacteristic
310C2×C53#309711yescharacteristic
311C2×C53#316311yescharacteristic
312C2×C53#316411yescharacteristic
313C2×C53#322411yescharacteristic
314C2×C53#322511yescharacteristic
315C2×C4⋊Q8—531no—
316C2×#117—531no—
317C2×#117C2×D5311yescharacteristic
318C22×C4⋊C4—531no—
319C2×#107—531no—
320C2×#107—531no—
321C2×#107—531no—
322C2×#107C2×D5311yescharacteristic
323C2×#106—531no—
324C2×#106—531no—
325C2×#106—531no—
326C2×#106C2×D5311yescharacteristic
327C2×C22⋊C8—531no—
328C2×C22⋊C8C2×D5311yescharacteristic
329C2×#97—531no—
330C2×C4×C8—531no—
331#400—1061no—
332#399—1061no—
333#395—1061no—
334#393—1061no—
335#389—531no—
336#389—531no—
337#389—1061no—
338#389—1061no—
339#385—1061no—
340#385—1061no—
341#384—531no—
342#384—531no—
343#382—531no—
344#382—531no—
345#382—531no—
346#380—531no—
347#380C2×D5311yescharacteristic
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340C2×D5311yescharacteristic
355#340C2×D5311yescharacteristic
356#336—531no—
357#336—531no—
358C53#2795911yescharacteristic, complemented, Sylow
359C22×Q8—1061no—
360C23×C4—1061no—
361C23×C4C22×D5311yescharacteristic
362C22×C8—531no—
363C22×C8—531no—
364C22×C8C22×D5311yescharacteristic
365C22×C8C22×D5311yescharacteristic
366C4⋊Q8—1062no—
367C4⋊Q8—1062no—
368#117—41no—
369#117—41no—
370#117—1062no—
371#117—1062no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—2121no—
387C2×C4⋊C4—2121no—
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—531no—
395C2×C22⋊C4—531no—
396C2×C22⋊C4—1061no—
397C2×C42—21no—
398C2×C42—531no—
399C2×C42—531no—
400#107—41no—
401#107—2121no—
402#107—2121no—
403#107—2121no—
404#106—41no—
405#106—2121no—
406#106—2121no—
407#106—2121no—
408C22⋊C8—534no—
409C22⋊C8Dic10614yescomplemented
410#97—1062no—
411C4×C8—1062no—
412#95—21no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422C24D4×D5311yescharacteristic
423C2×Q8—4241no—
424C2×Q8—4241no—
425C2×Q8—2122no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—4241no—
443C22×C4—4241no—
444C22×C4—2122no—
445C22×C4C2×Dic10612yes—
446C22×C4D4×D5311yescharacteristic
447C22×C4Q8×D5312yes—
448C22×C4C23×D5311yescharacteristic
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—532no—
454C2×C8—532no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8C2×Dic10612yes—
460C2×C8C2×Dic10612yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—1062no—
470C4⋊C4—1062no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—4241no—
480C4⋊C4—4241no—
481C4⋊C4—4241no—
482C4⋊C4—4241no—
483C4⋊C4—4241no—
484C4⋊C4—4241no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C22⋊C4—42no—
488C22⋊C4—81no—
489C22⋊C4—1062no—
490C22⋊C4—1062no—
491C22⋊C4—4241no—
492C42—42no—
493C42—532no—
494C42—532no—
495C42—1061no—
496C42—1061no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23#1313321612yes—
504C23C2×D4×D5311yescharacteristic
505Q8—4244no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—532no—
525C2×C4—532no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—4244no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4#1313321612yes—
550C2×C4C2×Q8×D5311yescharacteristic
551C2×C4C22×Dic10611yescharacteristic
552C2×C4#41×D5311yescharacteristic
553C2×C4#41×D5311yescharacteristic
554C2×C4#1313326211yescharacteristic
555C2×C4#1313326211yescharacteristic
556C8—22no—
557C8—22no—
558C8—1062no—
559C8—1062no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#123×D5311yescharacteristic
568C22C2×#1313321611yescharacteristic
569C22#1681118811yescharacteristic
570C22#1681142511yescharacteristic
571C22#1681143211yescharacteristic
572C22#1681143311yescharacteristic
573C22#1681143711yescharacteristic
574C4—22no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—1062no—
580C4—1062no—
581C4—4241no—
582C4—4241no—
583C4—4241no—
584C4—4241no—
585C4—2121no—
586C4—2121no—
587C4#1681132711yescharacteristic
588C4#1681132711yescharacteristic
589C4#1681133211yescharacteristic
590C4#1681133211yescharacteristic
591C2—42no—
592C2#2027919111yescharacteristic
593C2#2028467011yescharacteristic
594C2#2028467911yescharacteristic
595C2#2028524511yescharacteristic
596C2#2028525511yescharacteristic
597C2#2028581511yescharacteristic
598C2#2028582011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1854C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1854D53
hypocenterC53#27959
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
3. #20276202C2semidirect
4. #20276199C2semidirect
5. #20276194C2semidirect
6. #20276190C2semidirect
7. #20275686C2semidirect
8. #20275674C2semidirect
12. #20274880C2semidirect
13. #20274878C2semidirect
94. C22⋊#88322Q8semidirect
120. C22⋊C8×C53Q8semidirect
358. C53#27959semidirect
409. C22⋊C8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202805402
#202773352
#202762022
#202761992
#202761942
#202761902
#202756862
#202756742
#202749142
#202749072
#202749052
#202748802
#202748782
#202744832
C53×#18542
#2795953