Finite Group Databasebeta

← #27468300

Subgroups — #27468300

Order 13568.

644 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279115C211yesmaximal, characteristic
2#20276283C211yesmaximal, characteristic, complemented
3#20276277C211yesmaximal, characteristic, complemented
4#20276152C211yesmaximal, characteristic
5#20276146C211yesmaximal, characteristic
6#20275450C211yesmaximal, characteristic, complemented
7#20275445C211yesmaximal, characteristic, complemented
8#20275290C211yesmaximal, characteristic, complemented
9#20275286C211yesmaximal, characteristic, complemented
10#20274881C211yesmaximal, characteristic, complemented
11#20274746C211yesmaximal, characteristic
12#20274746C211yesmaximal, characteristic
13C4×#13133148C211yesmaximal, characteristic, complemented
14#20274499C211yesmaximal, characteristic
15C53×#2496C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133150C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133128C2211yescharacteristic
25C2×C4×#88322C2211yescharacteristic
26C53⋊#401—21no—
27#16810551—21no—
28#16810521C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810517—21no—
31#16810505—21no—
32#16810489C2211yescharacteristic
33#16810483C2211yescharacteristic
34#16810483C2211yescharacteristic
35C4×C4⋊Dic53C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#126×C53—21no—
49C53×#400—21no—
50C53×#397—21no—
51C53×#389C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#387—21no—
54C53×#384C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C2×#11719917—22no—
60C2×#11719917—22no—
61C2×#11719917—22no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C4×Dic53—21no—
75#13133150—22no—
76#13133150—22no—
77#13133150—22no—
78#13133148—22no—
79#13133148—22no—
80#13133148—22no—
81C22⋊#88322—22no—
82#13133128—22no—
83C4×#88322—22no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C23×C4×C53C2311yescharacteristic
92#126×C53—44no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53D411yescharacteristic
104C2×C22⋊C4×C53D411yescharacteristic
105C2×C42×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C2×C42×C53C2311yescharacteristic
108#95×C53—21no—
109#95×C53—21no—
110#95×C53—21no—
111#95×C53—21no—
112#95×C53—21no—
113#95×C53D411yescharacteristic
114#95×C53D411yescharacteristic
115C24×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—42no—
145C4⋊C4×C53—42no—
146C4⋊C4×C53—42no—
147C4⋊C4×C53—42no—
148C22⋊C4×C53—22no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—42no—
151C22⋊C4×C53—42no—
152C42×C53—21no—
153C42×C53—21no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53—42no—
157C22⋊Dic53—42no—
158C22⋊Dic53—81no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167#11719919—42no—
168#11719919—81no—
169C4×Dic53—81no—
170#11719917—44no—
171C2×#88322—21no—
172C2×#88322—21no—
173C2×#88322—21no—
174C2×#88322—21no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—21no—
179C2×#88322#4112yes—
180C2×#88322#4112yes—
181C2×#88322#4112yes—
182C2×#88322#4112yes—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53C22×D411yescharacteristic
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
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—21no—
204C2×C4×C53—21no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
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×C53AGL(1,ℤ/8)12yes—
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×C4×C532−1+411yescharacteristic
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240#88322—22no—
241#88322—41no—
242#88322—41no—
243#88322#12614yescomplemented
244#27842—531nomaximal, Sylow
245C22×C53—41no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53#53211yescharacteristic
253C22×C53#54711yescharacteristic
254C22×C53#54711yescharacteristic
255C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C53#57511yescharacteristic
257C22×C53#57711yescharacteristic
258C22×C53#57811yescharacteristic
259C4×C53—22no—
260C4×C53—41no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—81no—
265C4×C53—81no—
266C4×C53—81no—
267C4×C53—81no—
268C4×C53C2×#12611yescharacteristic
269C4×C53#54111yescharacteristic
270C4×C53#55211yescharacteristic
271C4×C53#55611yescharacteristic
272Dic53—81no—
273Dic53—81no—
274Dic53—81no—
275Dic53—81no—
276#2496D5311yescharacteristic
277#2494—531no—
278#2101—531no—
279#2100—531no—
280#2077—531no—
281#2076—531no—
282#1943—531no—
283#1942—531no—
284#1919—531no—
285#1918—531no—
286#1847—531no—
287#1809—531no—
288#1809—531no—
289C4×#106—531no—
290#1753—531no—
291C2×C53—81no—
292C2×C53#249611yescharacteristic
293C2×C53#317111yescharacteristic
294C2×C53#317811yescharacteristic
295C2×C53#334511yescharacteristic
296C2×C53#334911yescharacteristic
297C2×C53#335111yescharacteristic
298C2×C53#335311yescharacteristic
299C22×M16—531no—
300C2×#126—21no—
301C2×#107—531no—
302C2×#107—531no—
303C2×#107—531no—
304C2×#106—531no—
305C2×#106—531no—
306C2×#106—531no—
307C2×C22⋊C8—531no—
308C2×#97—531no—
309C2×C4×C8—531no—
310#401—1061no—
311#400—21no—
312#397—21no—
313#397—1061no—
314#389—531no—
315#389—531no—
316#389C2×D5311yescharacteristic
317#389C2×D5311yescharacteristic
318#388—1061no—
319#387—21no—
320#387—1061no—
321#384C2×D5311yescharacteristic
322#382—531no—
323#380—531no—
324#380—531no—
325#380C2×D5311yescharacteristic
326#380C2×D5311yescharacteristic
327C4×C4⋊C4—531no—
328C4×C4⋊C4C2×D5311yescharacteristic
329C4×C22⋊C4C2×D5311yescharacteristic
330#340—531no—
331#340—531no—
332#340—531no—
333#340—531no—
334#340—531no—
335#340—531no—
336#340—531no—
337#340—531no—
338#340—531no—
339#340—531no—
340#336—531no—
341#336—531no—
342C53#2784211yescharacteristic, complemented, Sylow
343C23×C4C22×D5311yescharacteristic
344C2×M16—1062no—
345C2×M16—1062no—
346C2×M16—1062no—
347C22×C8—531no—
348C22×C8—531no—
349C22×C8—531no—
350C22×C8—531no—
351#126—44no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—531no—
357C2×C4⋊C4—531no—
358C2×C4⋊C4—531no—
359C2×C4⋊C4—531no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4C22×D5311yescharacteristic
363C2×C4⋊C4C22×D5311yescharacteristic
364C2×C4⋊C4C22×D5311yescharacteristic
365C2×C4⋊C4C22×D5311yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—1061no—
369C2×C22⋊C4—1061no—
370C2×C22⋊C4C22⋊D5311yescharacteristic
371C2×C22⋊C4C22⋊D5311yescharacteristic
372C2×C42—21no—
373C2×C42—1061no—
374C2×C42C22×D5311yescharacteristic
375C2×C42C22×D5311yescharacteristic
376#107—1062no—
377#107—1062no—
378#107—1062no—
379#106—1062no—
380#106—1062no—
381#106—1062no—
382C22⋊C8—1062no—
383#97—1062no—
384C4×C8—1062no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—1061no—
391#95—1061no—
392#95—1061no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396#95—1061no—
397#95C22⋊D5311yescharacteristic
398#95C22⋊D5311yescharacteristic
399C24C2×C22⋊D5311yescharacteristic
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—1061no—
413C22×C4—1061no—
414C22×C4C2×C22⋊D5311yescharacteristic
415C22×C4C2×C22⋊D5311yescharacteristic
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C2×C22⋊D5311yescharacteristic
419C22×C4C23×D5311yescharacteristic
420M16—2124no—
421C2×C8—532no—
422C2×C8—532no—
423C2×C8—532no—
424C2×C8—532no—
425C2×C8—1061no—
426C2×C8—1061no—
427C2×C8—1061no—
428C2×C8—1061no—
429C2×C8—1061no—
430C2×C8—1061no—
431C2×C8—1061no—
432C2×C8—1061no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—1061no—
446C4⋊C4—1061no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C4⋊C4—1061no—
450C4⋊C4—1061no—
451C4⋊C4—1061no—
452C4⋊C4—1061no—
453C4⋊C4—4241no—
454C4⋊C4—2122no—
455C22⋊C4—22no—
456C22⋊C4—41no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—4241no—
460C22⋊C4—2122no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—42no—
466C42—4241no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23C22×C22⋊D5311yescharacteristic
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—2121no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4#1313324312yes—
518C2×C4#41×D5311yescharacteristic
519C2×C4#41×D5311yescharacteristic
520C2×C4#41×D5311yescharacteristic
521C2×C4#41×D5311yescharacteristic
522C2×C4#1313326411yescharacteristic
523C2×C4#1313326411yescharacteristic
524C8—534no—
525C8—1062no—
526C8—2121no—
527C8—2121no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22#1681116911yescharacteristic
536C22#1681127211yescharacteristic
537C22#1681127211yescharacteristic
538C22C2×#1313324311yescharacteristic
539C22#1681142111yescharacteristic
540C22#1681143111yescharacteristic
541C22#1681143811yescharacteristic
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—4241no—
552C4—4241no—
553C4—4241no—
554C4—4241no—
555C4#126×D5311yescharacteristic
556C4#1681123011yescharacteristic
557C4#1681130911yescharacteristic
558C4#1681134011yescharacteristic
559C2—81no—
560C2#2027913411yescharacteristic
561C2#2028530411yescharacteristic
562C2#2028538511yescharacteristic
563C2#2028718511yescharacteristic
564C2#2028723911yescharacteristic
565C2#2028727511yescharacteristic
566C2#2028729911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2496C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2496D53
hypocenterC53#27842
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2496

Complements

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

NH ≅ G/Ntype
2. #20276283C2semidirect
3. #20276277C2semidirect
6. #20275450C2semidirect
7. #20275445C2semidirect
8. #20275290C2semidirect
9. #20275286C2semidirect
10. #20274881C2semidirect
13. C4×#13133148C2semidirect
243. #88322#126semidirect
342. C53#27842semidirect

Maximal subgroups

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

subgroupindex
#202791152
#202762832
#202762772
#202761522
#202761462
#202754502
#202754452
#202752902
#202752862
#202748812
#202747462
C4×#131331482
#202744992
C53×#24962
#2784253