Finite Group Databasebeta

← #27440737

Subgroups — #27440737

Order 13568.

1116 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133208C211yesmaximal, characteristic, complemented
2C2×#16810979C212yesmaximal, complemented
3#20283018C211yesmaximal, characteristic
4#20275864C212yesmaximal, complemented
5#20275860C212yesmaximal, complemented
6#20275678C212yesmaximal, complemented
7#20275032C211yesmaximal, characteristic
8#20275030C211yesmaximal, characteristic
9#20275023C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C53×#1872C211yesmaximal, characteristic, complemented
12C2×#13133208—28no—
13C2×#13133208—21no—
14C2×#13133208—21no—
15C2×#13133208C2212yes—
16C2×#13133208C2212yes—
17C2×#13133206C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19C22×C4⋊Dic53—21no—
20C22×C4⋊Dic53C2211yescharacteristic
21C22×#11719919—21no—
22C22×C4×Dic53C2211yescharacteristic
23#16810979—28no—
24C2×#13133141C2212yes—
25C2×#13133139C2212yes—
26C2×#13133137C2212yes—
27C2×C22⋊#88322C2211yescharacteristic
28C2×C8×Dic53C2212yes—
29#16810530C2211yescharacteristic
30#16810524C2211yescharacteristic
31#16810513—21no—
32#16810513—21no—
33#16810500—21no—
34#16810500—21no—
35#16810488C2211yescharacteristic
36#16810386C2212yes—
37#16810385C2212yes—
38#16810385C2212yes—
39#16810377C2212yes—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C22×C4⋊C4×C53C2211yescharacteristic
42C2×#103×C53C2212yes—
43C2×C22⋊C8×C53C2211yescharacteristic
44C53×#340C2212yes—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C23×Dic53—21no—
48C23×Dic53D411yescharacteristic
49C23×Dic53D411yescharacteristic
50C22×#88322C2312yes—
51#13133208—28no—
52#13133208—28no—
53#13133208—24no—
54#13133208—24no—
55#13133208—24no—
56#13133208D418yescomplemented
57#13133206—24no—
58#13133191—22no—
59#13133191—22no—
60C2×C4⋊Dic53—28no—
61C2×C4⋊Dic53—24no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—22no—
64C2×C4⋊Dic53—22no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2312yes—
67C2×#11719919—28no—
68C2×#11719919—22no—
69C2×#11719919—22no—
70C2×#11719921—21no—
71C2×#11719921—21no—
72C2×#11719921—21no—
73C2×#11719921—21no—
74C2×#11719921—21no—
75C2×C4×Dic53—24no—
76C2×C4×Dic53D411yescharacteristic
77C2×C4×Dic53D411yescharacteristic
78C2×C4×Dic53C2311yescharacteristic
79C2×C4×Dic53C2311yescharacteristic
80#13133141—24no—
81#13133139—24no—
82#13133137—24no—
83C22⋊#88322—22no—
84C8×Dic53—24no—
85#13133124—21no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89C22×Q8×C53C2311yescharacteristic
90C23×C4×C53—21no—
91C23×C4×C53C2311yescharacteristic
92C22×C8×C53C2312yes—
93C22⋊Q8×C53—22no—
94C22⋊Q8×C53—22no—
95C2×C4⋊C4×C53—24no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2312yes—
100C2×C22⋊C4×C53—21no—
101#103×C53—24no—
102C22⋊C8×C53—22no—
103C24×C53C2×D411yescharacteristic
104C23×D53—21no—
105C2×Q8×C53—22no—
106C2×Q8×C53—21no—
107C2×Q8×C53—21no—
108C2×Q8×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—24no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—42no—
116C22×C4×C53—42no—
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53#4112yes—
119C22×C4×C53#4112yes—
120C22×C4×C53C2411yescharacteristic
121C2×C22⋊D53—42no—
122C2×C22⋊D53—42no—
123C2×C22⋊D53—42no—
124C22×Dic53—24no—
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—21no—
131C22×Dic53—41no—
132C22×Dic53—41no—
133C22×Dic53—42no—
134C22×Dic53—42no—
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C2×D212—22no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—22no—
143C2×C8×C53—22no—
144C2×C8×C53—22no—
145C2×C8×C53—22no—
146C4⋊C4×C53—24no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—42no—
151C4⋊C4×C53C2×D414yes—
152C22⋊C4×C53—42no—
153#11719921—42no—
154#11719921—42no—
155#11719921—42no—
156#11719921—42no—
157#11719921—42no—
158C4⋊Dic53—24no—
159C4⋊Dic53—28no—
160C4⋊Dic53—22no—
161C4⋊Dic53—28no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53C2×D414yes—
165#11719919—28no—
166#11719919—28no—
167C4×Dic53—22no—
168C4×Dic53—22no—
169C4×Dic53—22no—
170C4×Dic53C2×D414yes—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53#12412yes—
179C23×C53#13512yes—
180C23×C53#13512yes—
181C23×C53C22×D411yescharacteristic
182C22×D53—41no—
183C22×D53—41no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—41no—
189Q8×C53—42no—
190Q8×C53—42no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—24no—
194C2×C4×C53—24no—
195C2×C4×C53—24no—
196C2×C4×C53—48no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53#12412yes—
206C2×C4×C53C22×D412yes—
207C2×C4×C53C2×#4111yescharacteristic
208C2×C4×C53C2×#4111yescharacteristic
209C2×C4×C532−1+412yes—
210C22⋊D53—84no—
211C2×Dic53—21no—
212C2×Dic53—21no—
213C2×Dic53—21no—
214C2×Dic53—21no—
215C2×Dic53—24no—
216C2×Dic53—21no—
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—44no—
223C2×Dic53—44no—
224C2×Dic53—48no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53#12012yes—
231C2×Dic53#12012yes—
232D212—42no—
233D212—42no—
234C8×C53—42no—
235#88322—42no—
236#25907—531nomaximal, Sylow
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53#48814yescomplemented
244C22×C53C2×#12011yescharacteristic
245C22×C53C2×#12411yescharacteristic
246C22×C53#53611yescharacteristic
247C22×C53C2×#13511yescharacteristic
248C22×C53C2×#13511yescharacteristic
249C22×C53#57511yescharacteristic
250C22×C53#57511yescharacteristic
251C2×D53—81no—
252C2×D53—81no—
253C2×D53—81no—
254C2×D53—81no—
255C2×D53—81no—
256C2×D53—81no—
257C2×D53—81no—
258C4×C53—22no—
259C4×C53—44no—
260C4×C53—81no—
261C4×C53—81no—
262C4×C53#54812yes—
263C4×C53#54812yes—
264Dic53—22no—
265Dic53—22no—
266Dic53—41no—
267Dic53—41no—
268Dic53—44no—
269Dic53—81no—
270C22×#125—531no—
271C2×#488—532no—
272#2850—531no—
273#2026—532no—
274#2025—532no—
275#1991—532no—
276#1874—531no—
277#1874—531no—
278#1872—531no—
279#1872D5311yescharacteristic, complemented
280C4×C22⋊C8—531no—
281C2×C53—24no—
282C2×C53#243611yescharacteristic
283C2×C53#300211yescharacteristic
284C2×C53#300211yescharacteristic
285C2×C53C2×#48811yescharacteristic
286C2×C53#314211yescharacteristic
287C2×C53#323211yescharacteristic
288C2×C53#323211yescharacteristic
289D53—161no—
290C2×#125—532no—
291C2×#125—532no—
292C2×#125—1061no—
293C2×#125—1061no—
294C2×#125—1068no—
295C2×#124—531no—
296C2×C22⋊Q8C2×D5311yescharacteristic
297C2×#121—531no—
298C22×C4⋊C4—531no—
299C22×C4⋊C4—1061no—
300C22×C4⋊C4—1061no—
301C22×C4⋊C4C2×D5311yescharacteristic
302C22×C42—531no—
303#488—1068no—
304C2×#103—532no—
305C2×#103C2×D5312yes—
306C2×#102—532no—
307C2×#102—532no—
308C2×C22⋊C8—531no—
309C2×C22⋊C8C2×D5311yescharacteristic
310C2×C4×C8—532no—
311#391—531no—
312#390—531no—
313#388—1061no—
314#388—1061no—
315#386—1061no—
316#386—1061no—
317#382—531no—
318#340—532no—
319#340—532no—
320#340—532no—
321#340C2×D5312yes—
322#336—532no—
323C53#2590711yescharacteristic, complemented, Sylow
324C22×Q8C22×D5311yescharacteristic
325C22×D4—531no—
326C22×D4—1061no—
327C23×C4—21no—
328C23×C4—531no—
329C23×C4—531no—
330C23×C4—1061no—
331C23×C4C22×D5311yescharacteristic
332C22×C8—532no—
333C22×C8C22×D5312yes—
334#125—538no—
335#125—1064no—
336#125—1064no—
337#125—1068no—
338#125—1068no—
339#125—1064no—
340#124—1064no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343#121—1062no—
344#121—1062no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—532no—
349C2×C4⋊C4—531no—
350C2×C4⋊C4—1064no—
351C2×C4⋊C4—1062no—
352C2×C4⋊C4—1062no—
353C2×C4⋊C4—1062no—
354C2×C4⋊C4—1062no—
355C2×C4⋊C4—1068no—
356C2×C4⋊C4—1068no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4C22×D5311yescharacteristic
359C2×C4⋊C4C22×D5312yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—1061no—
362C2×C22⋊C4—1061no—
363C2×C22⋊C4—1061no—
364C2×C22⋊C4—1061no—
365C2×C22⋊C4—1061no—
366C2×C42—531no—
367C2×C42—531no—
368C2×C42—531no—
369C2×C42—531no—
370C2×C42—1064no—
371#103—24no—
372#103—1064no—
373#102—1064no—
374#102—1064no—
375C22⋊C8—22no—
376C22⋊C8—1062no—
377C4×C8—1064no—
378#95—1061no—
379#95—1061no—
380#95—1061no—
381#95—1061no—
382C24—1061no—
383C24D4×D5311yescharacteristic
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×D4—1062no—
389C2×D4—1061no—
390C2×D4—1061no—
391C2×D4—1062no—
392C2×D4—2122no—
393C2×D4—2122no—
394C2×D4—2122no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—531no—
404C22×C4—531no—
405C22×C4—531no—
406C22×C4—531no—
407C22×C4—1062no—
408C22×C4—1062no—
409C22×C4—1064no—
410C22×C4—1062no—
411C22×C4—1062no—
412C22×C4—1062no—
413C22×C4—1061no—
414C22×C4—2121no—
415C22×C4—2121no—
416C22×C4—2122no—
417C22×C4—2122no—
418C22×C4#1171994512yes—
419C22×C4D4×D5311yescharacteristic
420C22×C4#1171994712yes—
421C22×C4C23×D5311yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—1062no—
426C2×C8—1062no—
427C2×C8—1062no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—42no—
433C4⋊C4—534no—
434C4⋊C4—1068no—
435C4⋊C4—1068no—
436C4⋊C4—1068no—
437C4⋊C4—1064no—
438C4⋊C4—1068no—
439C4⋊C4—1062no—
440C4⋊C4—1061no—
441C4⋊C4—1061no—
442C4⋊C4D4×D5314yes—
443C22⋊C4—42no—
444C22⋊C4—2122no—
445C22⋊C4—2122no—
446C22⋊C4—2122no—
447C22⋊C4—2122no—
448C22⋊C4—2122no—
449C42—534no—
450C42—1062no—
451C42—1062no—
452C42—1062no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—2121no—
458C23—2121no—
459C23—2121no—
460C23—2121no—
461C23—2121no—
462C23—2121no—
463C23—2121no—
464C23#1313320312yes—
465C23#1313323812yes—
466C23#1313324012yes—
467C23C2×D4×D5311yescharacteristic
468Q8—42no—
469Q8—42no—
470D4—4244no—
471D4—2122no—
472D4—2122no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—48no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—532no—
488C2×C4—532no—
489C2×C4—1061no—
490C2×C4—1061no—
491C2×C4—1061no—
492C2×C4—1061no—
493C2×C4—1061no—
494C2×C4—1061no—
495C2×C4—1064no—
496C2×C4—1061no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1061no—
500C2×C4—2124no—
501C2×C4—2124no—
502C2×C4—2121no—
503C2×C4—2121no—
504C2×C4—2121no—
505C2×C4—2128no—
506C2×C4—2122no—
507C2×C4—2122no—
508C2×C4#1313320312yes—
509C2×C4C2×D4×D5312yes—
510C2×C4C2×#1171994711yescharacteristic
511C2×C4C2×#1171994511yescharacteristic
512C2×C4#1313326412yes—
513C8—42no—
514C8—2122no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—4241no—
522C22—4241no—
523C22—4241no—
524C22—4241no—
525C22—4241no—
526C22—4241no—
527C22—4241no—
528C22#1681097914yescomplemented
529C22#120×D5311yescharacteristic
530C22C2×#1313320311yescharacteristic
531C22#1681119511yescharacteristic
532C22C2×#1313323811yescharacteristic
533C22C2×#1313324011yescharacteristic
534C22#1681141911yescharacteristic
535C22#1681142211yescharacteristic
536C4—22no—
537C4—44no—
538C4—81no—
539C4—81no—
540C4—1062no—
541C4—1062no—
542C4—4241no—
543C4—2121no—
544C4—2121no—
545C4—2124no—
546C4#1681127912yes—
547C4#1681128212yes—
548C2—24no—
549C2—8481no—
550C2#2027837111yescharacteristic
551C2#2028393411yescharacteristic
552C2#2028393811yescharacteristic
553C2C2×#1681097911yescharacteristic
554C2#2028511311yescharacteristic
555C2#2028588611yescharacteristic
556C2#2028588811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1872D53
hypocenterC53#25907
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
1. C22×#13133208C2semidirect
2. C2×#16810979C2semidirect
4. #20275864C2semidirect
5. #20275860C2semidirect
6. #20275678C2semidirect
9. #20275023C2semidirect
10. C22⋊C8×Dic53C2semidirect
11. C53×#1872C2semidirect
56. #13133208D4semidirect
243. C22×C53#488semidirect
279. #1872D53semidirect
323. C53#25907semidirect
528. C22#16810979semidirect

Maximal subgroups

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

subgroupindex
C22×#131332082
C2×#168109792
#202830182
#202758642
#202758602
#202756782
#202750322
#202750302
#202750232
C22⋊C8×Dic532
C53×#18722
#2590753