Finite Group Databasebeta

← #27420218

Subgroups — #27420218

Order 13568.

1162 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810920C212yesmaximal, complemented
2C4×#13133191C211yesmaximal, characteristic
3#20276050C212yesmaximal, complemented
4#20275858C212yesmaximal, complemented
5#20275673C212yesmaximal, complemented
6#20275032C211yesmaximal, characteristic
7#20275032C211yesmaximal, characteristic
8#20274908C211yesmaximal, characteristic, complemented
9#20274903C211yesmaximal, characteristic, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C22×#125×C53C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14C2×C4×C22⋊D53—21no—
15#16810920—28no—
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×C4⋊#88322C2212yes—
19C2×#13133137C2212yes—
20C2×#13133137C2212yes—
21C2×C22⋊#88322C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23#16810525C2211yescharacteristic
24#16810521C2211yescharacteristic
25#16810514—21no—
26#16810502—21no—
27#16810483C2211yescharacteristic
28C4×C4⋊Dic53C2211yescharacteristic
29C4×#11719921—21no—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810378C2212yes—
35C2×#125×C53—28no—
36C2×#125×C53—21no—
37C2×#125×C53—21no—
38C2×#125×C53C2212yes—
39C2×#125×C53C2212yes—
40C22×C4⋊C4×C53—21no—
41C22×C4⋊C4×C53—21no—
42C22×C4⋊C4×C53C2211yescharacteristic
43C22×C4⋊C4×C53C2211yescharacteristic
44C22×C42×C53C2211yescharacteristic
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C22×C4×D53—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133191—22no—
51#13133191Q814yes—
52C4×D212—24no—
53C4×C22⋊D53—44no—
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2311yescharacteristic
56C2×C4⋊Dic53C2311yescharacteristic
57C2×#11719919—21no—
58C2×C22⋊Dic53—21no—
59C2×#11719921—21no—
60C2×#11719921—21no—
61C2×#11719921—21no—
62C2×#11719921—21no—
63C2×C4×Dic53—21no—
64#13133150—24no—
65#13133148—24no—
66C4⋊#88322—24no—
67#13133137—24no—
68#13133137—24no—
69C22⋊#88322—22no—
70C22⋊#88322—22no—
71#13133124—21no—
72#13133124—21no—
73#13133124—21no—
74C23×C4×C53—21no—
75C23×C4×C53—21no—
76C23×C4×C53D411yescharacteristic
77C23×C4×C53D411yescharacteristic
78C23×C4×C53C2311yescharacteristic
79#125×C53—28no—
80#125×C53—28no—
81#125×C53—24no—
82#125×C53—24no—
83#125×C53—24no—
84#125×C53D418yescomplemented
85C2×C4⋊C4×C53—28no—
86C2×C4⋊C4×C53—28no—
87C2×C4⋊C4×C53—24no—
88C2×C4⋊C4×C53—24no—
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53C2312yes—
96C2×C4⋊C4×C53C2312yes—
97C2×C42×C53—24no—
98C2×C42×C53D411yescharacteristic
99C2×C42×C53D411yescharacteristic
100C2×C42×C53C2311yescharacteristic
101C2×C42×C53C2311yescharacteristic
102C24×C53C2×D411yescharacteristic
103C23×D53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—24no—
107C22×C4×C53—24no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—41no—
115C22×C4×C53—41no—
116C22×C4×C53—42no—
117C22×C4×C53—42no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×Q812yes—
126C22×C4×C53#4112yes—
127C22×C4×C53C2411yescharacteristic
128C2×C22⋊D53—24no—
129C2×C22⋊D53—42no—
130C2×C22⋊D53—42no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C2×D212—22no—
135C2×D212—22no—
136C2×D212—21no—
137C2×D212C2×Q812yes—
138C2×C4×D53—42no—
139C2×C4×D53—42no—
140C2×C4×D53—42no—
141C4⋊C4×C53—24no—
142C4⋊C4×C53—24no—
143C4⋊C4×C53—28no—
144C4⋊C4×C53—28no—
145C4⋊C4×C53—28no—
146C4⋊C4×C53—28no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53C2×D414yes—
150C4⋊C4×C53C2×D414yes—
151C42×C53—22no—
152C42×C53—22no—
153C42×C53—22no—
154C42×C53C2×D414yes—
155C22⋊Dic53—42no—
156#11719921—24no—
157#11719921—42no—
158#11719921—42no—
159#11719921—42no—
160C4⋊Dic53—21no—
161C4⋊Dic53—22no—
162C4⋊Dic53—22no—
163C4⋊Dic53C2×Q812yes—
164#11719919—42no—
165C4×Dic53—42no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—22no—
170C2×#88322—22no—
171C2×#88322—22no—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53C22⋊Q812yes—
177C23×C53#13512yes—
178C23×C53AGL(1,ℤ/8)12yes—
179C23×C53C22×D411yescharacteristic
180C22×D53—22no—
181C22×D53—41no—
182C22×D53—41no—
183C22×D53—41no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—41no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—24no—
194C2×C4×C53—24no—
195C2×C4×C53—24no—
196C2×C4×C53—24no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—44no—
204C2×C4×C53—44no—
205C2×C4×C53—48no—
206C2×C4×C53—48no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53#12012yes—
212C2×C4×C53#12012yes—
213C2×C4×C53C22⋊Q812yes—
214C2×C4×C53C22×D412yes—
215C2×C4×C53C22×Q811yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217C2×C4×C532−1+412yes—
218C22⋊D53—48no—
219C2×Dic53—22no—
220C2×Dic53—41no—
221C2×Dic53—41no—
222C2×Dic53—41no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228D212—24no—
229D212—42no—
230C4×D53—84no—
231#88322—42no—
232#88322—42no—
233#24305—531nomaximal, Sylow
234C22×C53—22no—
235C22×C53—22no—
236C22×C53—22no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53#47814yescomplemented
241C22×C53C2×#12011yescharacteristic
242C22×C53C2×C22⋊Q811yescharacteristic
243C22×C53#53611yescharacteristic
244C22×C53C2×#13511yescharacteristic
245C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C53#57511yescharacteristic
247C22×C53#57511yescharacteristic
248C2×D53—42no—
249C2×D53—42no—
250C2×D53—42no—
251C2×D53—81no—
252C2×D53—81no—
253C2×D53—81no—
254C2×D53—81no—
255C4×C53—22no—
256C4×C53—22no—
257C4×C53—22no—
258C4×C53—41no—
259C4×C53—44no—
260C4×C53—41no—
261C4×C53—44no—
262C4×C53#54812yes—
263C4×C53D4×Q812yes—
264Dic53—42no—
265Dic53—81no—
266Dic53—81no—
267C22×#125D5311yescharacteristic, complemented
268C2×#478—532no—
269C4×#121—531no—
270#2059—532no—
271#2025—532no—
272#1989—532no—
273#1874—531no—
274#1874—531no—
275#1853—531no—
276#1852—531no—
277#1797—531no—
278C2×C53—24no—
279C2×C53#240911yescharacteristic
280C2×C53#300211yescharacteristic
281C2×C53#300611yescharacteristic
282C2×C53C2×#47811yescharacteristic
283C2×C53#308011yescharacteristic
284C2×C53#323211yescharacteristic
285C2×C53#323411yescharacteristic
286D53—82no—
287C2×#125—28no—
288C2×#125—21no—
289C2×#125—21no—
290C2×#125C2×D5312yes—
291C2×#125C2×D5312yes—
292C2×#121—531no—
293C2×C4×D4—531no—
294C2×C4×D4—1061no—
295C22×C4⋊C4—21no—
296C22×C4⋊C4—21no—
297C22×C4⋊C4C2×D5311yescharacteristic
298C22×C4⋊C4C2×D5311yescharacteristic
299C22×C42C2×D5311yescharacteristic
300#478—1068no—
301C2×#107—532no—
302C2×#106—532no—
303C2×C4⋊C8—532no—
304C2×#102—532no—
305C2×#102—532no—
306C2×C22⋊C8—531no—
307C2×C22⋊C8—531no—
308#390—531no—
309#389—531no—
310#388—1061no—
311#386—1061no—
312#380—531no—
313C4×C4⋊C4—531no—
314C4×C22⋊C4—1061no—
315#340—532no—
316#340—532no—
317#340—532no—
318#340—532no—
319#336—532no—
320C53#2430511yescharacteristic, complemented, Sylow
321C22×D4—531no—
322C22×D4—1061no—
323C23×C4—21no—
324C23×C4—21no—
325C23×C4—1061no—
326C23×C4C22⋊D5311yescharacteristic
327C23×C4C22⋊D5311yescharacteristic
328C23×C4C22×D5311yescharacteristic
329C22×C8—532no—
330C22×C8—532no—
331#125—28no—
332#125—28no—
333#125—24no—
334#125—24no—
335#125—24no—
336#125C22⋊D5318yescomplemented
337#121—534no—
338#121—1062no—
339C4×D4—1064no—
340C4×D4—2124no—
341C2×C4⋊C4—28no—
342C2×C4⋊C4—28no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—531no—
352C2×C4⋊C4—531no—
353C2×C4⋊C4—531no—
354C2×C4⋊C4—1061no—
355C2×C4⋊C4C22×D5312yes—
356C2×C4⋊C4C22×D5312yes—
357C2×C22⋊C4—1061no—
358C2×C22⋊C4—1061no—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1061no—
362C2×C42—24no—
363C2×C42—1061no—
364C2×C42C22⋊D5311yescharacteristic
365C2×C42C22⋊D5311yescharacteristic
366C2×C42C22×D5311yescharacteristic
367C2×C42C22×D5311yescharacteristic
368#107—1064no—
369#106—1064no—
370C4⋊C8—1064no—
371#102—1064no—
372#102—1064no—
373C22⋊C8—1062no—
374C22⋊C8—1062no—
375#95—1061no—
376#95—1061no—
377#95—1061no—
378C24—1061no—
379C24C2×C22⋊D5311yescharacteristic
380C2×D4—532no—
381C2×D4—1064no—
382C2×D4—1062no—
383C2×D4—1061no—
384C2×D4—1062no—
385C2×D4—2122no—
386C2×D4—2122no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—1061no—
404C22×C4—1061no—
405C22×C4—1061no—
406C22×C4—2122no—
407C22×C4—2122no—
408C22×C4—2122no—
409C22×C4Q8×D5312yes—
410C22×C4#1171994912yes—
411C22×C4C2×C22⋊D5311yescharacteristic
412C22×C4C2×C22⋊D5311yescharacteristic
413C22×C4C2×C22⋊D5311yescharacteristic
414C22×C4C2×C22⋊D5311yescharacteristic
415C22×C4C2×C22⋊D5311yescharacteristic
416C22×C4C23×D5311yescharacteristic
417C2×C8—1062no—
418C2×C8—1062no—
419C2×C8—1062no—
420C2×C8—1062no—
421C2×C8—1062no—
422C2×C8—1062no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—532no—
432C4⋊C4—1061no—
433C4⋊C4—1062no—
434C4⋊C4—1062no—
435C4⋊C4—2122no—
436C4⋊C4C2×C22⋊D5314yes—
437C4⋊C4C2×C22⋊D5314yes—
438C22⋊C4—1064no—
439C22⋊C4—2122no—
440C22⋊C4—2122no—
441C22⋊C4—2122no—
442C22⋊C4—2122no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—2122no—
447C42C2×C22⋊D5314yes—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—1062no—
453C23—2121no—
454C23—2121no—
455C23—2121no—
456C23—2121no—
457C23—2121no—
458C23—2121no—
459C23#1313319312yes—
460C23#1313323712yes—
461C23#1313324212yes—
462C23C22×C22⋊D5311yescharacteristic
463D4—1064no—
464D4—2128no—
465D4—2122no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—1062no—
491C2×C4—4244no—
492C2×C4—2121no—
493C2×C4—2121no—
494C2×C4—2121no—
495C2×C4—2121no—
496C2×C4—2121no—
497C2×C4—2121no—
498C2×C4—2121no—
499C2×C4—2121no—
500C2×C4#1313318612yes—
501C2×C4#1313318612yes—
502C2×C4#1313319312yes—
503C2×C4C22×C22⋊D5312yes—
504C2×C4C2×Q8×D5311yescharacteristic
505C2×C4C2×#1171994911yescharacteristic
506C2×C4#1313326512yes—
507C8—2122no—
508C8—2122no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—4241no—
516C22—4241no—
517C22—4241no—
518C22—4241no—
519C22—2122no—
520C22—2122no—
521C22—2122no—
522C22#1681092014yescomplemented
523C22C2×#1313318611yescharacteristic
524C22C2×#1313319311yescharacteristic
525C22#1681119711yescharacteristic
526C22C2×#1313323711yescharacteristic
527C22C2×#1313324211yescharacteristic
528C22#1681142111yescharacteristic
529C22#1681142111yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—44no—
535C4—41no—
536C4—44no—
537C4—4241no—
538C4—4241no—
539C4—2122no—
540C4#1681128412yes—
541C4Q8×C22⋊D5312yes—
542C2—24no—
543C2—4242no—
544C2#2027807411yescharacteristic
545C2#2028393311yescharacteristic
546C2#2028396711yescharacteristic
547C2C2×#1681092011yescharacteristic
548C2#2028453311yescharacteristic
549C2#2028588311yescharacteristic
550C2#2028590511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×#125×C53C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC22×#125D53
hypocenterC53#24305
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
1. C2×#16810920C2semidirect
3. #20276050C2semidirect
4. #20275858C2semidirect
5. #20275673C2semidirect
8. #20274908C2semidirect
9. #20274903C2semidirect
10. #20274695C2semidirect
11. C22×#125×C53C2semidirect
84. #125×C53D4semidirect
240. C22×C53#478semidirect
267. C22×#125D53semidirect
320. C53#24305semidirect
336. #125C22⋊D53semidirect
522. C22#16810920semidirect

Maximal subgroups

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

subgroupindex
C2×#168109202
C4×#131331912
#202760502
#202758582
#202756732
#202750322
#202749082
#202749032
#202746952
C22×#125×C532
#2430553