Finite Group Databasebeta

← #27442978

Subgroups — #27442978

Order 13568.

674 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#20285245C211yesmaximal, characteristic, complemented
2#20284570C211yesmaximal, characteristic
3#20284566C211yesmaximal, characteristic
4#20283308C211yesmaximal, characteristic
5#20276224C211yesmaximal, characteristic, complemented
6#20276223C211yesmaximal, characteristic, complemented
7#20276065C211yesmaximal, characteristic, complemented
8#20276064C211yesmaximal, characteristic, complemented
9#20275494C211yesmaximal, characteristic, complemented
10#20275462C211yesmaximal, characteristic, complemented
11#20274950C211yesmaximal, characteristic
12#20274877C211yesmaximal, characteristic, complemented
13#20274772C211yesmaximal, characteristic
14#20274553C211yesmaximal, characteristic
15C53×#1860C211yesmaximal, characteristic
16C2×#13133224C2211yescharacteristic
17#16811254C2211yescharacteristic
18#16811251C2211yescharacteristic
19#16811106C2211yescharacteristic
20#16811031—21no—
21#16811030—21no—
22#16811024—21no—
23#16811020—21no—
24#16810930—21no—
25#16810928—21no—
26#16810925—21no—
27#16810923—21no—
28#16810918—21no—
29#16810916—21no—
30#16810903—21no—
31#16810901—21no—
32C2×#13133152C2211yescharacteristic
33#16810666—21no—
34#16810666C2211yescharacteristic
35#16810665C2211yescharacteristic
36#16810665C2211yescharacteristic
37#16810663—21no—
38#16810663C2211yescharacteristic
39#16810663C2211yescharacteristic
40#16810659—21no—
41#16810659C2211yescharacteristic
42C2×#13133150C2211yescharacteristic
43C2×#13133150C2211yescharacteristic
44C2×#13133148—21no—
45C2×#13133148C2211yescharacteristic
46C2×#13133148C2211yescharacteristic
47#16810651C2211yescharacteristic
48#16810650C2211yescharacteristic
49#16810624C2211yescharacteristic
50#16810619C2211yescharacteristic
51#16810581C2211yescharacteristic
52C2×C4×#88322C2211yescharacteristic
53#16810489C2211yescharacteristic
54#16810392C2211yescharacteristic
55#16810391C2211yescharacteristic
56#16810391C2211yescharacteristic
57#16810387C2211yescharacteristic
58#16810383C2211yescharacteristic
59#16810383C2211yescharacteristic
60C53×#520C2211yescharacteristic
61C53×#428C2211yescharacteristic
62C2×#104×C53C2211yescharacteristic
63C53×#417C2211yescharacteristic
64C53×#382C2211yescharacteristic
65C53×#343C2211yescharacteristic
66C53×#339C2211yescharacteristic
67#13133224—22no—
68#13133224Q814yescomplemented
69C2×#11719917—21no—
70C2×#11719917C2311yescharacteristic
71C2×#11719917C2311yescharacteristic
72C22×#88322—21no—
73C22×#88322C2311yescharacteristic
74C22×#88322C2311yescharacteristic
75#13133217—21no—
76#13133216—21no—
77#13133216—21no—
78#13133216—21no—
79#13133209—21no—
80#13133208—21no—
81#13133208—21no—
82#13133208—21no—
83#13133196—41no—
84#13133194—41no—
85Q8×Dic53—21no—
86D4×Dic53—21no—
87D4×Dic53—41no—
88#13133176—21no—
89#13133176C2311yescharacteristic
90#13133176C2311yescharacteristic
91C2×C4⋊Dic53—21no—
92C2×C4⋊Dic53C2311yescharacteristic
93C2×C4⋊Dic53C2311yescharacteristic
94#13133152—22no—
95#13133151—21no—
96#13133151—21no—
97#13133150—21no—
98#13133150—21no—
99#13133150—21no—
100#13133150—21no—
101#13133150—21no—
102#13133150—21no—
103#13133150—21no—
104#13133150D411yescharacteristic
105#13133150D411yescharacteristic
106#13133149—21no—
107#13133149—21no—
108#13133148—21no—
109#13133148—21no—
110#13133148—21no—
111#13133148—21no—
112#13133148—21no—
113#13133148—21no—
114#13133148—21no—
115#13133148D411yescharacteristic
116#13133148D411yescharacteristic
117C4⋊#88322—21no—
118C4⋊#88322—21no—
119#13133146—21no—
120#13133146—21no—
121#13133141—21no—
122#13133141D411yescharacteristic
123#13133141D411yescharacteristic
124#13133138—21no—
125#13133138D411yescharacteristic
126#13133138D411yescharacteristic
127#13133127—21no—
128C4×#88322—22no—
129C8×Dic53—21no—
130#13133124—41no—
131C2×#41×C53C2311yescharacteristic
132C2×M16×C53C2311yescharacteristic
133C2×M16×C53C2311yescharacteristic
134C4×Q8×C53—21no—
135C4×D4×C53—21no—
136C4×D4×C53—41no—
137#117×C53—21no—
138#117×C53C2311yescharacteristic
139C2×C4⋊C4×C53—21no—
140C2×C4⋊C4×C53C2311yescharacteristic
141C2×C4⋊C4×C53C2311yescharacteristic
142C2×C42×C53C2311yescharacteristic
143#107×C53—21no—
144#106×C53—21no—
145#104×C53Q814yescomplemented
146#103×C53—21no—
147#102×C53—21no—
148#95×C53—41no—
149#41×C53—21no—
150#41×C53—21no—
151#41×C53C2×Q812yes—
152#41×C53C2×Q812yes—
153C2×Q8×C53C2×D411yescharacteristic
154C2×D4×C53—21no—
155C2×D4×C53C2×D411yescharacteristic
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—41no—
161C22×C4×C53C2411yescharacteristic
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—41no—
165C2×Dic106—41no—
166C2×Dic106—41no—
167M16×C53—21no—
168M16×C53—21no—
169M16×C53—21no—
170M16×C53C2×Q812yes—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—41no—
178C4⋊C4×C53C2×D411yescharacteristic
179C4⋊C4×C53C2×D411yescharacteristic
180C22⋊C4×C53—21no—
181C22⋊C4×C53—41no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—41no—
185C42×C53C2×Q812yes—
186C22⋊Dic53—21no—
187C22⋊Dic53—21no—
188C22⋊Dic53—41no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—41no—
194C4⋊Dic53—41no—
195C4⋊Dic53—41no—
196C4⋊Dic53C2×D411yescharacteristic
197C4⋊Dic53C2×D411yescharacteristic
198C4⋊Dic53C2×D411yescharacteristic
199C4⋊Dic53C2×D411yescharacteristic
200#11719919—41no—
201#11719919—41no—
202#11719919—41no—
203#11719919—41no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—41no—
207#11719917—21no—
208#11719917—24no—
209#11719917—21no—
210#11719917—21no—
211#11719917C2×Q812yes—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—24no—
216C2×#88322—21no—
217C2×#88322C2×D411yescharacteristic
218C2×#88322C2×D411yescharacteristic
219C2×#88322C2×D411yescharacteristic
220C2×#88322C2×D411yescharacteristic
221C2×#88322C2×Q812yes—
222C2×#88322#4112yes—
223C23×C53—21no—
224C23×C53H(5,𝔽2)11yescharacteristic
225Q8×C53—21no—
226Q8×C53C4⋊Q812yes—
227D4×C53—22no—
228D4×C53—22no—
229D4×C53—21no—
230D4×C53C4⋊Q812yes—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—22no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—81no—
242C2×C4×C53—81no—
243C2×C4×C53—161no—
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C22×Q811yescharacteristic
247C2×C4×C53C22×Q811yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—81no—
257C2×Dic53—81no—
258C2×Dic53—161no—
259Dic106—161no—
260Dic106—161no—
261C8×C53—22no—
262C8×C53—41no—
263#88322—22no—
264#88322—21no—
265#88322—21no—
266#88322—41no—
267#88322C22⋊Q812yes—
268#88322C22⋊Q812yes—
269#26080—531nomaximal, Sylow
270C22×C53—21no—
271C22×C53—21no—
272C22×C53—22no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53#54411yescharacteristic
276C22×C53D4211yescharacteristic
277C22×C53#55411yescharacteristic
278C4×C53—22no—
279C4×C53—42no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53C2×C22⋊Q811yescharacteristic
285C4×C53C2×C4⋊Q811yescharacteristic
286C4×C53#54711yescharacteristic
287C4×C53D4×Q811yescharacteristic
288Dic53—41no—
289Dic53—41no—
290Dic53—41no—
291Dic53—41no—
292Dic53—81no—
293Dic53—81no—
294#3163—531no—
295#3083—531no—
296#3083—531no—
297#2912—531no—
298#2089—531no—
299#2089—531no—
300#2061—531no—
301#2061—531no—
302#1951—531no—
303#1945—531no—
304#1860—531no—
305#1860D5311yescharacteristic
306#1846—531no—
307#1815—531no—
308#1764—531no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53—42no—
312C2×C53#244011yescharacteristic
313C2×C53#329011yescharacteristic
314C2×C53#329011yescharacteristic
315C2×#131—531no—
316#544—531no—
317#544—531no—
318#520—531no—
319#520C2×D5311yescharacteristic
320#501—1061no—
321#501—1061no—
322#499—1061no—
323#498—1061no—
324#479—1061no—
325#479—1061no—
326#478—1061no—
327#478—1061no—
328#477—1061no—
329#477—1061no—
330#474—1061no—
331#474—1061no—
332C2×#108—531no—
333#428—531no—
334#428—531no—
335#428—531no—
336#428—531no—
337#428—531no—
338#428—1061no—
339#428—1061no—
340#428C2×D5311yescharacteristic
341#427—531no—
342#427—1061no—
343C2×#107—531no—
344C2×#107—531no—
345C2×#106—531no—
346C2×#106—531no—
347C2×#106—1061no—
348#424—531no—
349#424—531no—
350C2×#104C2×D5311yescharacteristic
351#417—531no—
352#417C2×D5311yescharacteristic
353#416—531no—
354#405—531no—
355C2×C4×C8—531no—
356#382—531no—
357#382C2×D5311yescharacteristic
358#343—531no—
359#343—531no—
360#343—531no—
361#343C2×D5311yescharacteristic
362#341—531no—
363#339—531no—
364#339—531no—
365#339C2×D5311yescharacteristic
366C53#2608011yescharacteristic, complemented, Sylow
367C2×#41C22×D5311yescharacteristic
368#131—534no—
369#131—1062no—
370C2×M16—531no—
371C2×M16—531no—
372C2×M16—1061no—
373C2×M16C22×D5311yescharacteristic
374C2×M16C22×D5311yescharacteristic
375C22×C8—531no—
376C22×C8—531no—
377C22×C8—1061no—
378C4⋊Q8—1061no—
379C4⋊Q8—1061no—
380C4⋊Q8—1061no—
381C4⋊Q8—1061no—
382#125—1061no—
383#125—1061no—
384#125—1061no—
385#125—1061no—
386C22⋊Q8—2121no—
387C22⋊Q8—2121no—
388C4×Q8—21no—
389C4×Q8—1061no—
390C4×D4—21no—
391C4×D4—41no—
392C4×D4—1061no—
393C4×D4—2121no—
394#117—21no—
395#117—531no—
396#117—531no—
397#117—1061no—
398#117C22×D5311yescharacteristic
399C2×C4⋊C4—21no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C42C22×D5311yescharacteristic
406#108—1062no—
407#107—21no—
408#107—531no—
409#107—531no—
410#107—1061no—
411#107—1061no—
412#107—1061no—
413#107—1061no—
414#107—1061no—
415#107—1061no—
416#107—1061no—
417#107—1061no—
418#107—1061no—
419#106—21no—
420#106—531no—
421#106—531no—
422#106—1061no—
423#106—1061no—
424#106—1061no—
425#106—1061no—
426#106—1061no—
427#106—1061no—
428#106—1061no—
429#106—1061no—
430#106—1061no—
431C4⋊C8—1061no—
432C4⋊C8—1061no—
433C4⋊C8—1061no—
434C4⋊C8—1061no—
435#104Dic10614yescomplemented
436#103—21no—
437#103—531no—
438#103—531no—
439#103—1061no—
440#102—21no—
441#102—531no—
442#102—531no—
443#102—1061no—
444#97—1061no—
445C4×C8—1062no—
446C4×C8—1061no—
447#95—41no—
448#95—2121no—
449#41—21no—
450#41—21no—
451#41C2×Dic10612yes—
452#41Q8×D5312yes—
453C2×Q8—2121no—
454C2×Q8—2121no—
455C2×Q8D4×D5311yescharacteristic
456C2×D4—21no—
457C2×D4D4×D5311yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—2121no—
466C22×C4C23×D5311yescharacteristic
467M16—21no—
468M16—21no—
469M16—21no—
470M16—532no—
471M16—1064no—
472M16—1061no—
473M16—1061no—
474M16—1061no—
475M16C2×Dic10612yes—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—531no—
479C2×C8—531no—
480C2×C8—532no—
481C2×C8—532no—
482C2×C8—531no—
483C2×C8—531no—
484C2×C8—1064no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—531no—
495C4⋊C4—531no—
496C4⋊C4—531no—
497C4⋊C4—531no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—2121no—
503C4⋊C4—2121no—
504C4⋊C4—2121no—
505C4⋊C4—2121no—
506C4⋊C4—2121no—
507C4⋊C4—2121no—
508C4⋊C4—2121no—
509C4⋊C4D4×D5311yescharacteristic
510C4⋊C4D4×D5311yescharacteristic
511C22⋊C4—21no—
512C22⋊C4—41no—
513C22⋊C4—1061no—
514C22⋊C4—1061no—
515C22⋊C4—2121no—
516C42—21no—
517C42—21no—
518C42—41no—
519C42—1061no—
520C42—1061no—
521C42—2121no—
522C42C2×Dic10612yes—
523C23—21no—
524C23#1313326211yescharacteristic
525Q8—21no—
526Q8—8481no—
527Q8—8481no—
528Q8#1313321612yes—
529D4—22no—
530D4—22no—
531D4—21no—
532D4#1313321612yes—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—161no—
546C2×C4—8481no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—4241no—
552C2×C4—4241no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4C2×D4×D5311yescharacteristic
556C2×C4C2×D4×D5311yescharacteristic
557C2×C4C2×Q8×D5311yescharacteristic
558C2×C4C22×Dic10611yescharacteristic
559C2×C4#41×D5311yescharacteristic
560C2×C4#1313326411yescharacteristic
561C8—22no—
562C8—41no—
563C8—532no—
564C8—532no—
565C8—1061no—
566C8—1061no—
567C8—1062no—
568C8—2121no—
569C22—21no—
570C22—21no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22#1681125211yescharacteristic
575C22#1681125511yescharacteristic
576C22#1681133211yescharacteristic
577C4—22no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—4241no—
584C4—4241no—
585C4—2121no—
586C4—2121no—
587C4—2121no—
588C4—2121no—
589C4C22⋊Q8×D5311yescharacteristic
590C4C2×#1313321611yescharacteristic
591C4#1681127111yescharacteristic
592C4D4×Dic10611yescharacteristic
593C2—21no—
594C2—21no—
595C2—42no—
596C2#2027841611yescharacteristic
597C2#2028649411yescharacteristic
598C2#2028649911yescharacteristic
subgroupG/H
centerC22#16811255
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1860C2
socleC22×C53D42
prime radicalC2×#41×C53C23
hypercenter#1860D53
hypocenterC53#26080
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1860

Complements

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

NH ≅ G/Ntype
1. #20285245C2semidirect
5. #20276224C2semidirect
6. #20276223C2semidirect
7. #20276065C2semidirect
8. #20276064C2semidirect
9. #20275494C2semidirect
10. #20275462C2semidirect
12. #20274877C2semidirect
68. #13133224Q8semidirect
145. #104×C53Q8semidirect
366. C53#26080semidirect
435. #104Dic106semidirect

Maximal subgroups

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

subgroupindex
#202852452
#202845702
#202845662
#202833082
#202762242
#202762232
#202760652
#202760642
#202754942
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C53×#18602
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