Finite Group Databasebeta

← #27468298

Subgroups — #27468298

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#20279135C211yesmaximal, characteristic
2#20279118C211yesmaximal, characteristic
3#20276279C211yesmaximal, characteristic, complemented
4#20276153C211yesmaximal, characteristic
5#20276147C211yesmaximal, characteristic
6#20275450C211yesmaximal, characteristic, complemented
7#20275447C211yesmaximal, characteristic, complemented
8#20275291C211yesmaximal, characteristic, complemented
9#20275289C211yesmaximal, characteristic, complemented
10#20274884C211yesmaximal, characteristic, complemented
11#20274747C211yesmaximal, characteristic
12#20274747C211yesmaximal, characteristic
13#20274603C211yesmaximal, characteristic, complemented
14#20274497C211yesmaximal, characteristic
15C53×#2101C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C2×#13133212—21no—
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133129C2211yescharacteristic
25C2×#13133128C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27C53⋊#401—21no—
28#16810562—21no—
29#16810552—21no—
30#16810550—21no—
31#16810522C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810515—21no—
35#16810506—21no—
36#16810506—21no—
37#16810493C2211yescharacteristic
38#16810480C2211yescharacteristic
39#16810480C2211yescharacteristic
40#16810480C2211yescharacteristic
41#16810480C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4×#11719921C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#382C2211yescharacteristic
54C53×#340C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C2×#11719912—22no—
61C2×#11719912—22no—
62C2×#11719912—22no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133212—44no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921D411yescharacteristic
80C2×#11719921D411yescharacteristic
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53C2311yescharacteristic
84#13133150—22no—
85#13133150—22no—
86#13133150—22no—
87#13133148—22no—
88#13133148—22no—
89#13133148—22no—
90#13133129—22no—
91#13133128—22no—
92C8×Dic53—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124D411yescharacteristic
103#13133124D411yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112#95×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115C23×D53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2411yescharacteristic
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—21no—
134C2×C4×D53—21no—
135C2×C4×D53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—42no—
151C42×C53—21no—
152C42×C53—21no—
153#11719921—22no—
154#11719921—41no—
155#11719921—42no—
156#11719921—42no—
157#11719921—42no—
158#11719921—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—
167C4⋊Dic53—42no—
168C4⋊Dic53—42no—
169C4⋊Dic53—42no—
170#11719919—42no—
171#11719919—81no—
172C4×Dic53—21no—
173C4×Dic53—21no—
174C4×Dic53—42no—
175C4×Dic53—81no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322#4112yes—
179C2×#88322#4112yes—
180C2×#88322#4112yes—
181C2×#88322#4112yes—
182#11719912—44no—
183C23×C53C22×D411yescharacteristic
184C22×D53—21no—
185C22×D53—21no—
186C22×D53—21no—
187C22×D53—21no—
188C22×D53—21no—
189C22×D53—21no—
190C22×D53—21no—
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—41no—
198C2×C4×C53—41no—
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—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53C2×#4111yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C2×C4×C53C2×#4111yescharacteristic
213C2×C4×C532−1+411yescharacteristic
214C2×C4×C532−1+411yescharacteristic
215C2×Dic53—21no—
216C2×Dic53—21no—
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53AGL(1,ℤ/8)12yes—
236C4×D53—41no—
237C4×D53—41no—
238C4×D53—41no—
239C4×D53—41no—
240C8×C53—41no—
241C8×C53—41no—
242#88322—22no—
243#88322#12614yescomplemented
244#27842—531nomaximal, Sylow
245C22×C53#53211yescharacteristic
246C22×C53#54711yescharacteristic
247C22×C53#54711yescharacteristic
248C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
249C22×C53#57511yescharacteristic
250C22×C53#57711yescharacteristic
251C22×C53#57811yescharacteristic
252C2×D53—41no—
253C2×D53—41no—
254C2×D53—41no—
255C2×D53—41no—
256C2×D53—41no—
257C2×D53—41no—
258C2×D53—41no—
259C4×C53—41no—
260C4×C53—41no—
261C4×C53—81no—
262C4×C53—81no—
263C4×C53—81no—
264C4×C53—81no—
265C4×C53C2×#12611yescharacteristic
266C4×C53#54111yescharacteristic
267C4×C53#55211yescharacteristic
268C4×C53#55611yescharacteristic
269Dic53—22no—
270Dic53—41no—
271Dic53—41no—
272Dic53—81no—
273Dic53—81no—
274Dic53—81no—
275Dic53—81no—
276#2496—531no—
277#2494—531no—
278#2101D5311yescharacteristic, complemented
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#249611yescharacteristic
292C2×C53#317111yescharacteristic
293C2×C53#317811yescharacteristic
294C2×C53#334511yescharacteristic
295C2×C53#334911yescharacteristic
296C2×C53#335111yescharacteristic
297C2×C53#335311yescharacteristic
298D53—81no—
299C22×M16—531no—
300C2×#126—1061no—
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—1061no—
312#397—1061no—
313#397—1061no—
314#389—531no—
315#389—531no—
316#389—531no—
317#389C2×D5311yescharacteristic
318#388—1061no—
319#387—1061no—
320#387—1061no—
321#384—531no—
322#382C2×D5311yescharacteristic
323#380—531no—
324#380—531no—
325#380—531no—
326#380—531no—
327C4×C4⋊C4—531no—
328C4×C4⋊C4—531no—
329C4×C22⋊C4—531no—
330#340—531no—
331#340—531no—
332#340—531no—
333#340—531no—
334#340—531no—
335#340—531no—
336#340C2×D5311yescharacteristic
337#340C2×D5311yescharacteristic
338#340C2×D5311yescharacteristic
339#340C2×D5311yescharacteristic
340#336—531no—
341#336C2×D5311yescharacteristic
342C53#2784211yescharacteristic, complemented, Sylow
343C23×C4—531no—
344C2×M16—1062no—
345C2×M16—1062no—
346C2×M16—1062no—
347C22×C8—531no—
348C22×C8—531no—
349C22×C8C22×D5311yescharacteristic
350C22×C8C22×D5311yescharacteristic
351#126—2124no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—531no—
354C2×C4⋊C4—531no—
355C2×C4⋊C4—531no—
356C2×C4⋊C4—531no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4—1061no—
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—531no—
367C2×C22⋊C4—531no—
368C2×C22⋊C4—1061no—
369C2×C22⋊C4—1061no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C42—531no—
373C2×C42—1061no—
374C2×C42—1061no—
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—531no—
389#95—531no—
390#95—1061no—
391#95—1061no—
392#95—1061no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396#95—1061no—
397#95—1061no—
398#95—1061no—
399C24—531no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—531no—
405C22×C4—531no—
406C22×C4—531no—
407C22×C4—1061no—
408C22×C4—1061no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—1061no—
413C22×C4—1061no—
414C22×C4—1061no—
415C22×C4—1061no—
416C22×C4—1061no—
417C22×C4D4×D5311yescharacteristic
418C22×C4D4×D5311yescharacteristic
419C22×C4C23×D5311yescharacteristic
420M16—2124no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—532no—
428C2×C8—532no—
429C2×C8—532no—
430C2×C8—532no—
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—1061no—
443C4⋊C4—1061no—
444C4⋊C4—1061no—
445C4⋊C4—1061no—
446C4⋊C4—1061no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C4⋊C4—1061no—
450C4⋊C4—4241no—
451C4⋊C4—2122no—
452C4⋊C4—2122no—
453C4⋊C4—2122no—
454C4⋊C4—2122no—
455C22⋊C4—1062no—
456C22⋊C4—4241no—
457C22⋊C4—2121no—
458C22⋊C4—2122no—
459C22⋊C4—2122no—
460C22⋊C4—2122no—
461C42—21no—
462C42—21no—
463C42—1061no—
464C42—1061no—
465C42—4241no—
466C42—2122no—
467C23—1061no—
468C23—1061no—
469C23—1061no—
470C23—1061no—
471C23—1061no—
472C23—1061no—
473C23—1061no—
474C23C2×D4×D5311yescharacteristic
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—532no—
494C2×C4—1061no—
495C2×C4—1061no—
496C2×C4—1061no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1061no—
500C2×C4—1061no—
501C2×C4—1061no—
502C2×C4—2121no—
503C2×C4—2121no—
504C2×C4—2121no—
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—2121no—
518C2×C4#41×D5311yescharacteristic
519C2×C4#41×D5311yescharacteristic
520C2×C4#41×D5311yescharacteristic
521C2×C4#41×D5311yescharacteristic
522C2×C4#1313326411yescharacteristic
523C2×C4#1313326511yescharacteristic
524C8—41no—
525C8—41no—
526C8—534no—
527C8—1062no—
528C22—2121no—
529C22—2121no—
530C22—2121no—
531C22—2121no—
532C22—2121no—
533C22—2121no—
534C22—2121no—
535C22#1681117211yescharacteristic
536C22#1681127111yescharacteristic
537C22#1681127511yescharacteristic
538C22AGL(1,ℤ/8)×D5311yescharacteristic
539C22#1681142211yescharacteristic
540C22#1681143011yescharacteristic
541C22#1681143711yescharacteristic
542C4—41no—
543C4—41no—
544C4—81no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—1062no—
549C4—4241no—
550C4—4241no—
551C4—4241no—
552C4—4241no—
553C4—2121no—
554C4—2121no—
555C4#126×D5311yescharacteristic
556C4#1681123211yescharacteristic
557C4#1681130511yescharacteristic
558C4#1681134011yescharacteristic
559C2—4241no—
560C2#2027913211yescharacteristic
561C2#2028531311yescharacteristic
562C2#2028538211yescharacteristic
563C2#2028718211yescharacteristic
564C2#2028725211yescharacteristic
565C2#2028727311yescharacteristic
566C2#2028729711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2101C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2101D53
hypocenterC53#27842
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
3. #20276279C2semidirect
6. #20275450C2semidirect
7. #20275447C2semidirect
8. #20275291C2semidirect
9. #20275289C2semidirect
10. #20274884C2semidirect
13. #20274603C2semidirect
15. C53×#2101C2semidirect
243. #88322#126semidirect
278. #2101D53semidirect
342. C53#27842semidirect

Maximal subgroups

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

subgroupindex
#202791352
#202791182
#202762792
#202761532
#202761472
#202754502
#202754472
#202752912
#202752892
#202748842
#202747472
#202746032
#202744972
C53×#21012
#2784253