Finite Group Databasebeta

← #27420227

Subgroups — #27420227

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
1C22×#13133208C211yesmaximal, characteristic, complemented
2C2×#16810926C212yesmaximal, complemented
3#20277221C211yesmaximal, characteristic
4#20276051C212yesmaximal, complemented
5#20275860C212yesmaximal, complemented
6#20275669C212yesmaximal, complemented
7#20275032C211yesmaximal, characteristic
8#20275030C211yesmaximal, characteristic
9#20274907C211yesmaximal, characteristic, complemented
10#20274693C211yesmaximal, characteristic, complemented
11C53×#1852C211yesmaximal, characteristic, complemented
12C2×#13133208—28no—
13C2×#13133208—21no—
14C2×#13133208—21no—
15C2×#13133208C2212yes—
16C2×#13133208C2212yes—
17C2×#13133191C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×#13133177—21no—
20C22×C4⋊Dic53—21no—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×#11719919—21no—
23C22×C4×Dic53C2211yescharacteristic
24#16810926—28no—
25C2×#13133148C2212yes—
26C2×#13133146C2212yes—
27C2×#13133139C2212yes—
28C2×#13133137C2212yes—
29C2×C22⋊#88322C2211yescharacteristic
30#16810524C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810512—21no—
33#16810500—21no—
34C4⋊C4×Dic53C2211yescharacteristic
35#16810470—21no—
36#16810386C2212yes—
37#16810385C2212yes—
38#16810385C2212yes—
39#16810377C2212yes—
40C22×C4⋊C4×C53C2211yescharacteristic
41C2×#107×C53C2212yes—
42C2×C22⋊C8×C53C2211yescharacteristic
43C53×#380C2211yescharacteristic
44C53×#340C2212yes—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C23×Dic53—21no—
48C23×Dic53D411yescharacteristic
49C23×Dic53D411yescharacteristic
50C22×C4×D53—21no—
51C22×#88322C2312yes—
52#13133208—28no—
53#13133208—28no—
54#13133208—24no—
55#13133208—24no—
56#13133208—24no—
57#13133208D418yescomplemented
58#13133191—22no—
59#13133191Q814yes—
60#13133179—24no—
61#13133177—44no—
62C2×C4⋊Dic53—28no—
63C2×C4⋊Dic53—24no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53—22no—
67C2×C4⋊Dic53C2312yes—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×#11719919—28no—
70C2×#11719919—22no—
71C2×#11719919—22no—
72C2×#11719919—21no—
73C2×#11719921—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×C4×Dic53—24no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53D411yescharacteristic
80C2×C4×Dic53D411yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83#13133148—24no—
84#13133146—24no—
85#13133139—24no—
86#13133137—24no—
87C22⋊#88322—22no—
88#13133124—21no—
89#13133124—21no—
90C23×C4×C53—21no—
91C23×C4×C53C2311yescharacteristic
92C22×C8×C53C2312yes—
93C2×C4⋊C4×C53—24no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53C2311yescharacteristic
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C4⋊C4×C53C2312yes—
98C2×C22⋊C4×C53—21no—
99#107×C53—24no—
100C22⋊C8×C53—22no—
101#95×C53—21no—
102C24×C53C2×D411yescharacteristic
103C23×D53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—24no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—42no—
111C22×C4×C53—42no—
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×Q812yes—
114C22×C4×C53#4112yes—
115C22×C4×C53C2411yescharacteristic
116C2×C22⋊D53—24no—
117C2×C22⋊D53—42no—
118C2×C22⋊D53—42no—
119C22×Dic53—24no—
120C22×Dic53—22no—
121C22×Dic53—22no—
122C22×Dic53—21no—
123C22×Dic53—22no—
124C22×Dic53—22no—
125C22×Dic53—22no—
126C22×Dic53—41no—
127C22×Dic53—41no—
128C22×Dic53—42no—
129C22×Dic53—42no—
130C22×Dic53C2×D411yescharacteristic
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C2×D212—22no—
135C2×D212—22no—
136C2×D212—21no—
137C2×D212C2×Q812yes—
138C2×C4×D53—42no—
139C2×C4×D53—42no—
140C2×C4×D53—42no—
141C2×C8×C53—22no—
142C2×C8×C53—22no—
143C2×C8×C53—22no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53C2×D414yes—
149C22⋊C4×C53—42no—
150#11719921—24no—
151#11719921—42no—
152#11719921—42no—
153#11719921—42no—
154C4⋊Dic53—24no—
155C4⋊Dic53—28no—
156C4⋊Dic53—22no—
157C4⋊Dic53—21no—
158C4⋊Dic53—28no—
159C4⋊Dic53C2×D414yes—
160C4⋊Dic53C2×Q812yes—
161#11719919—28no—
162#11719919—28no—
163#11719919—42no—
164C4×Dic53—22no—
165C4×Dic53—22no—
166C4×Dic53—22no—
167C4×Dic53—42no—
168C4×Dic53C2×D414yes—
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—24no—
190C2×C4×C53—24no—
191C2×C4×C53—24no—
192C2×C4×C53—48no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53C22⋊Q812yes—
202C2×C4×C53C22×D412yes—
203C2×C4×C53C22×Q811yescharacteristic
204C2×C4×C53C2×#4111yescharacteristic
205C2×C4×C532−1+412yes—
206C22⋊D53—48no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—21no—
210C2×Dic53—21no—
211C2×Dic53—24no—
212C2×Dic53—21no—
213C2×Dic53—21no—
214C2×Dic53—22no—
215C2×Dic53—21no—
216C2×Dic53—21no—
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53—44no—
220C2×Dic53—44no—
221C2×Dic53—48no—
222C2×Dic53—42no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—42no—
226C2×Dic53#12012yes—
227C2×Dic53#12012yes—
228D212—24no—
229D212—42no—
230C4×D53—84no—
231C8×C53—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—44no—
257C4×C53—81no—
258C4×C53—81no—
259C4×C53#54812yes—
260C4×C53D4×Q812yes—
261Dic53—22no—
262Dic53—22no—
263Dic53—42no—
264Dic53—41no—
265Dic53—44no—
266Dic53—41no—
267C22×#125—531no—
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#1852D5311yescharacteristic, complemented
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—532no—
288C2×#125—532no—
289C2×#125—1061no—
290C2×#125—1061no—
291C2×#125—1068no—
292C2×#121—531no—
293C2×C4×D4—531no—
294C2×C4×D4—1061no—
295C22×C4⋊C4—531no—
296C22×C4⋊C4—1061no—
297C22×C4⋊C4—1061no—
298C22×C4⋊C4C2×D5311yescharacteristic
299C22×C42—531no—
300#478—1068no—
301C2×#107C2×D5312yes—
302C2×#106—532no—
303C2×C4⋊C8—532no—
304C2×#102—532no—
305C2×#102—532no—
306C2×C22⋊C8—531no—
307C2×C22⋊C8C2×D5311yescharacteristic
308#390—531no—
309#389—531no—
310#388—1061no—
311#386—1061no—
312#380C2×D5311yescharacteristic
313C4×C4⋊C4—531no—
314C4×C22⋊C4—1061no—
315#340—532no—
316#340—532no—
317#340—532no—
318#340C2×D5312yes—
319#336—532no—
320C53#2430511yescharacteristic, complemented, Sylow
321C22×D4—531no—
322C22×D4—1061no—
323C23×C4—21no—
324C23×C4—531no—
325C23×C4—531no—
326C23×C4—1061no—
327C23×C4—1061no—
328C23×C4C22×D5311yescharacteristic
329C22×C8—532no—
330C22×C8C22×D5312yes—
331#125—538no—
332#125—1064no—
333#125—1064no—
334#125—1068no—
335#125—1068no—
336#125—1064no—
337#121—534no—
338#121—1062no—
339C4×D4—1064no—
340C4×D4—2124no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—532no—
344C2×C4⋊C4—531no—
345C2×C4⋊C4—1064no—
346C2×C4⋊C4—1062no—
347C2×C4⋊C4—1062no—
348C2×C4⋊C4—1062no—
349C2×C4⋊C4—1062no—
350C2×C4⋊C4—1068no—
351C2×C4⋊C4—1068no—
352C2×C4⋊C4—1061no—
353C2×C4⋊C4—1061no—
354C2×C4⋊C4C22×D5311yescharacteristic
355C2×C4⋊C4C22×D5311yescharacteristic
356C2×C4⋊C4C22×D5312yes—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—1061no—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1061no—
362C2×C42—531no—
363C2×C42—531no—
364C2×C42—531no—
365C2×C42—531no—
366C2×C42—1064no—
367C2×C42—1061no—
368#107—24no—
369#106—1064no—
370C4⋊C8—1064no—
371#102—1064no—
372#102—1064no—
373C22⋊C8—22no—
374C22⋊C8—1062no—
375#95—21no—
376#95—1061no—
377#95—1061no—
378C24—1061no—
379C24D4×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—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—531no—
396C22×C4—531no—
397C22×C4—531no—
398C22×C4—531no—
399C22×C4—1062no—
400C22×C4—1062no—
401C22×C4—1064no—
402C22×C4—1062no—
403C22×C4—1062no—
404C22×C4—1061no—
405C22×C4—1062no—
406C22×C4—2122no—
407C22×C4—2122no—
408C22×C4—2122no—
409C22×C4—2121no—
410C22×C4—2121no—
411C22×C4—2122no—
412C22×C4—2122no—
413C22×C4#1171994512yes—
414C22×C4D4×D5311yescharacteristic
415C22×C4Q8×D5312yes—
416C22×C4C23×D5311yescharacteristic
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—1062no—
421C2×C8—1062no—
422C2×C8—1062no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—534no—
428C4⋊C4—532no—
429C4⋊C4—1068no—
430C4⋊C4—1068no—
431C4⋊C4—1068no—
432C4⋊C4—1064no—
433C4⋊C4—1068no—
434C4⋊C4—1062no—
435C4⋊C4—1061no—
436C4⋊C4—2122no—
437C4⋊C4D4×D5314yes—
438C22⋊C4—42no—
439C22⋊C4—1064no—
440C22⋊C4—2122no—
441C22⋊C4—2122no—
442C22⋊C4—2122no—
443C42—534no—
444C42—1062no—
445C42—1062no—
446C42—1062no—
447C42—2122no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—1062no—
453C23—2121no—
454C23—2121no—
455C23—2121no—
456C23—2121no—
457C23—2121no—
458C23—2121no—
459C23#1313319512yes—
460C23#1313324012yes—
461C23#1313324112yes—
462C23C2×D4×D5311yescharacteristic
463D4—1064no—
464D4—2128no—
465D4—2122no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—48no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—532no—
481C2×C4—532no—
482C2×C4—1061no—
483C2×C4—1061no—
484C2×C4—1061no—
485C2×C4—1061no—
486C2×C4—1061no—
487C2×C4—1061no—
488C2×C4—1064no—
489C2×C4—1062no—
490C2×C4—1061no—
491C2×C4—1061no—
492C2×C4—1061no—
493C2×C4—1061no—
494C2×C4—4244no—
495C2×C4—2124no—
496C2×C4—2124no—
497C2×C4—2128no—
498C2×C4—2122no—
499C2×C4—2121no—
500C2×C4—2121no—
501C2×C4—2122no—
502C2×C4#1313319512yes—
503C2×C4C2×D4×D5312yes—
504C2×C4C2×Q8×D5311yescharacteristic
505C2×C4C2×#1171994511yescharacteristic
506C2×C4#1313326412yes—
507C8—42no—
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#1681092614yescomplemented
523C22#120×D5311yescharacteristic
524C22C2×#1313319511yescharacteristic
525C22#1681119511yescharacteristic
526C22C2×#1313324011yescharacteristic
527C22C2×#1313324111yescharacteristic
528C22#1681141911yescharacteristic
529C22#1681142211yescharacteristic
530C4—22no—
531C4—44no—
532C4—81no—
533C4—81no—
534C4—1062no—
535C4—1062no—
536C4—2121no—
537C4—2124no—
538C4—2121no—
539C4—2122no—
540C4#1681128212yes—
541C4#1681128612yes—
542C2—24no—
543C2—4242no—
544C2#2027807911yescharacteristic
545C2#2028393811yescharacteristic
546C2#2028396211yescharacteristic
547C2C2×#1681092611yescharacteristic
548C2#2028453611yescharacteristic
549C2#2028588811yescharacteristic
550C2#2028590211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1852D53
hypocenterC53#24305
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. C22×#13133208C2semidirect
2. C2×#16810926C2semidirect
4. #20276051C2semidirect
5. #20275860C2semidirect
6. #20275669C2semidirect
9. #20274907C2semidirect
10. #20274693C2semidirect
11. C53×#1852C2semidirect
57. #13133208D4semidirect
240. C22×C53#478semidirect
276. #1852D53semidirect
320. C53#24305semidirect
522. C22#16810926semidirect

Maximal subgroups

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

subgroupindex
C22×#131332082
C2×#168109262
#202772212
#202760512
#202758602
#202756692
#202750322
#202750302
#202749072
#202746932
C53×#18522
#2430553