Finite Group Databasebeta

← #28317815

Subgroups — #28317815

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291667C211yesmaximal, characteristic
2#20291626C211yesmaximal, characteristic, complemented
3#20290357C211yesmaximal, characteristic, complemented
4#20290354C211yesmaximal, characteristic, complemented
5#20290339C211yesmaximal, characteristic, complemented
6#20290331C211yesmaximal, characteristic, complemented
7#20290327C211yesmaximal, characteristic, complemented
8#20290323C211yesmaximal, characteristic, complemented
9#20290310C211yesmaximal, characteristic, complemented
10#20290300C211yesmaximal, characteristic, complemented
11#20290267C211yesmaximal, characteristic, complemented
12#20290029C211yesmaximal, characteristic
13#20290025C211yesmaximal, characteristic, complemented
14#20289935C211yesmaximal, characteristic, complemented
15#20285323C211yesmaximal, characteristic
16#16811674—21no—
17#16811673C2211yescharacteristic
18#16811673C2211yescharacteristic
19#16811673C2211yescharacteristic
20#16811672C2211yescharacteristic
21#16811672C2211yescharacteristic
22#16811672C2211yescharacteristic
23Q8×#88324—21no—
24#16811657C2211yescharacteristic
25#16811653—21no—
26#16811651—21no—
27#16811651C2211yescharacteristic
28#16811648—21no—
29#16811648—21no—
30#16811648—21no—
31#16811642C2211yescharacteristic
32#16811642C2211yescharacteristic
33#16811642C2211yescharacteristic
34#16811630—21no—
35D53⋊(C4⋊C8)C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42#16811473C2211yescharacteristic
43#16811473C2211yescharacteristic
44#16811473C2211yescharacteristic
45#16811473C2211yescharacteristic
46#16811471C2211yescharacteristic
47#16811471C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811178C2211yescharacteristic
53#16811158C2211yescharacteristic
54#16811017C411yescharacteristic
55#16810993C411yescharacteristic
56#16810985C411yescharacteristic
57#16810882C411yescharacteristic
58#16810882C411yescharacteristic
59#16810870C411yescharacteristic
60#16810870C411yescharacteristic
61#16810717C2211yescharacteristic
62#16810717C2211yescharacteristic
63#16810710C2211yescharacteristic
64#16810710C2211yescharacteristic
65#16810575C2211yescharacteristic
66C53×#501C411yescharacteristic
67C2×#11719936—21no—
68C2×D53⋊C8—21no—
69C2×D53⋊C8—21no—
70#13133284—21no—
71#13133284—21no—
72#13133284—21no—
73#13133284—21no—
74C4⋊#88324—21no—
75C4⋊#88324—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324D411yescharacteristic
79C4⋊#88324D411yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85#13133274—21no—
86#13133274—21no—
87#13133274—21no—
88#13133274D411yescharacteristic
89#13133274D411yescharacteristic
90#13133272—21no—
91#13133272—21no—
92#13133272—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×Dic212C2×C411yescharacteristic
102C2×Dic212C2×C411yescharacteristic
103C2×#11719913C2×C411yescharacteristic
104C2×#11719913C2×C411yescharacteristic
105C2×#11719912C2×C411yescharacteristic
106C2×#11719912C2×C411yescharacteristic
107#13133217C2×C411yescharacteristic
108#13133212—21no—
109#13133208—21no—
110#13133205C2×C411yescharacteristic
111#13133202—21no—
112#13133197C2×C411yescharacteristic
113#13133197C2×C411yescharacteristic
114#13133197C2×C411yescharacteristic
115#13133193—21no—
116#13133192C2×C411yescharacteristic
117Q8×Dic53—21no—
118#13133182C2311yescharacteristic
119#13133182C2311yescharacteristic
120#13133182C2311yescharacteristic
121#13133179—21no—
122#13133179C2311yescharacteristic
123#13133175—21no—
124#13133175—21no—
125C42×D53C2311yescharacteristic
126#13133140C2×C411yescharacteristic
127#13133140C2×C411yescharacteristic
128#13133140C2×C411yescharacteristic
129#13133140C2×C411yescharacteristic
130#13133140C2×C411yescharacteristic
131#13133140C2×C411yescharacteristic
132#13133137C2×C411yescharacteristic
133#13133137C2×C411yescharacteristic
134#13133128C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C4⋊Q8×C53C2×C411yescharacteristic
138#125×C53C2×C411yescharacteristic
139#107×C53C2×C411yescharacteristic
140#107×C53C2×C411yescharacteristic
141#106×C53C2×C411yescharacteristic
142#106×C53C2×C411yescharacteristic
143#97×C53C2×C411yescharacteristic
144C2×Q8×C53—21no—
145C2×D212—21no—
146C2×D212C22×C411yescharacteristic
147C2×C4×D53—21no—
148C2×C4×D53C22×C411yescharacteristic
149C2×C4×D53C2×D411yescharacteristic
150C2×C4×D53#4111yescharacteristic
151C2×Dic106C22×C411yescharacteristic
152C2×Dic106C22×C411yescharacteristic
153C2×Dic106C22×C411yescharacteristic
154C2×#88324—21no—
155C2×#88324—21no—
156C2×#88324—21no—
157C2×#88324—21no—
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324C2×D411yescharacteristic
161C2×#88324C2×D411yescharacteristic
162C2×#88324C2×D411yescharacteristic
163C2×#88324C2×D411yescharacteristic
164#11719936—81no—
165D53⋊C8—81no—
166D53⋊C8—81no—
167C2×C8×C53C22×C411yescharacteristic
168C2×C8×C53C22×C411yescharacteristic
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53C22×C411yescharacteristic
173C4⋊C4×C53C22×C411yescharacteristic
174C4⋊C4×C53C22×C411yescharacteristic
175C4⋊C4×C53C22×C411yescharacteristic
176C42×C53C22×C411yescharacteristic
177#11719921—21no—
178#11719921—21no—
179#11719921—21no—
180#11719921—21no—
181#11719921—21no—
182#11719921—21no—
183#11719921—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189#11719919—21no—
190#11719919—21no—
191#11719919—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53C2×D411yescharacteristic
196C4×Dic53#4111yescharacteristic
197C4×Dic53C2411yescharacteristic
198C2×#88322C22×C411yescharacteristic
199C2×#88322C22×C411yescharacteristic
200Dic212—22no—
201Dic212—41no—
202#11719913—22no—
203#11719913—41no—
204#11719912—22no—
205#11719912—41no—
206C22×D53—21no—
207C22×D53C4×D411yescharacteristic
208Q8×C53—42no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53C4×D411yescharacteristic
214C2×C4×C53C4×D411yescharacteristic
215C2×C4×C53C23×C411yescharacteristic
216C2×Dic53—21no—
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53C4×D411yescharacteristic
220C2×Dic53C22×D411yescharacteristic
221C2×Dic53C2×#4111yescharacteristic
222D212—21no—
223D212—21no—
224D212—42no—
225C4×D53—21no—
226C4×D53—21no—
227C4×D53—21no—
228C4×D53—21no—
229C4×D53—41no—
230C4×D53—41no—
231C4×D53—41no—
232Dic106—21no—
233Dic106—21no—
234Dic106—21no—
235Dic106—21no—
236Dic106—21no—
237Dic106—21no—
238#88324—21no—
239#88324—21no—
240#88324—21no—
241#88324—21no—
242#88324—41no—
243#88324—41no—
244#88324—41no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248C8×C53—21no—
249C8×C53#13112yes—
250#88322—21no—
251#88322—21no—
252#18081—531nomaximal, Sylow
253C22×C53C2×C4×D411yescharacteristic
254C2×D53—21no—
255C2×D53—21no—
256C2×D53—21no—
257C2×D53—41no—
258C2×D53—41no—
259C2×D53—41no—
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53C2×#13111yescharacteristic
267C4×C53#56811yescharacteristic
268Dic53—21no—
269Dic53—21no—
270Dic53—41no—
271Dic53—41no—
272Dic53—41no—
273Dic53#57711yescharacteristic
274Dic53#57811yescharacteristic
275#3172—531no—
276#2994—531no—
277#2986—531no—
278#1592—531no—
279#1591—531no—
280#1587—531no—
281#1585—531no—
282#1584—531no—
283#1583—531no—
284#1579—531no—
285C8×SD16—531no—
286#1565—531no—
287#1482—531no—
288#1481—531no—
289#1453—531no—
290C2×C53#293511yescharacteristic
291C2×C53#294711yescharacteristic
292C2×C53#295511yescharacteristic
293D53—41no—
294D53—81no—
295#533—531no—
296#529—531no—
297#501#615711yescharacteristic
298#497—531no—
299#491—531no—
300#489—531no—
301#470—531no—
302#470—531no—
303#468—531no—
304#468—531no—
305#446—531no—
306#446—531no—
307#446—531no—
308#446—1061no—
309C8×Q8—531no—
310C8×Q8—531no—
311C8×Q8—531no—
312C8×Q8—1061no—
313C4×Q16—531no—
314C4×Q16—531no—
315C4×SD16—531no—
316C4×SD16—531no—
317#436—531no—
318#435—1061no—
319C8×D4—531no—
320C8×D4—1061no—
321#433—1061no—
322#433—1061no—
323#433—1061no—
324#431—531no—
325#431—531no—
326#431—531no—
327#424—1061no—
328C2×C4⋊C8—531no—
329C4×M16—531no—
330#335—531no—
331#335—531no—
332#334—531no—
333#334—531no—
334Q8⋊C8—531no—
335Q8⋊C8—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340D4⋊C8—531no—
341D4⋊C8—531no—
342#322—531no—
343#322—531no—
344#322—531no—
345C82—531no—
346C53#1808111yescharacteristic, complemented, Sylow
347C2×Q16—531no—
348C2×Q16—531no—
349C2×SD16—531no—
350C2×SD16—531no—
351C2×M16—531no—
352C2×M16—531no—
353C2×M16—1061no—
354C22×C8—1061no—
355C22×C8—1061no—
356C4⋊Q8—531no—
357C4⋊Q8C2×#615711yescharacteristic
358#126—1061no—
359#125—1061no—
360#125C2×#615711yescharacteristic
361#124—531no—
362#123—1061no—
363C22⋊Q8—531no—
364C22⋊Q8—531no—
365C22⋊Q8—531no—
366C22⋊Q8—1061no—
367#121—531no—
368C4×Q8—531no—
369C4×Q8—531no—
370C4×Q8—531no—
371C4×Q8—1061no—
372C4×D4—531no—
373C4×D4—1061no—
374#117—1061no—
375#117—1061no—
376C2×C42—531no—
377#107C2×#615711yescharacteristic
378#107C2×#615711yescharacteristic
379#106C2×#615711yescharacteristic
380#106C2×#615711yescharacteristic
381C4⋊C8—531no—
382C4⋊C8—531no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—1061no—
388C4⋊C8—1061no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395#103—531no—
396#103—531no—
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#102—531no—
402#102—531no—
403C22⋊C8—531no—
404C22⋊C8—531no—
405C22⋊C8—1061no—
406C22⋊C8—1061no—
407C22⋊C8—1061no—
408C22⋊C8—1061no—
409#97—531no—
410#97—1061no—
411#97—1061no—
412#97—1061no—
413#97C2×#615711yescharacteristic
414C4×C8—531no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—1061no—
421C4×C8—1061no—
422C4×C8—1061no—
423C4×C8—1061no—
424C2×Q8—21no—
425C2×Q8—531no—
426C2×Q8—531no—
427C2×Q8—531no—
428C2×D4—531no—
429C2×D4—1061no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—1061no—
434Q16—1062no—
435Q16—2121no—
436SD16—1062no—
437SD16—2121no—
438M16—1062no—
439M16—4241no—
440M16—2121no—
441C2×C8—531no—
442C2×C8—531no—
443C2×C8—531no—
444C2×C8—531no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—4241no—
454C2×C8—4241no—
455C2×C8C22×#615711yescharacteristic
456C2×C8C22×#615711yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4C22×#615711yescharacteristic
469C4⋊C4C22×#615711yescharacteristic
470C4⋊C4C22×#615711yescharacteristic
471C4⋊C4C22×#615711yescharacteristic
472C22⋊C4—1061no—
473C22⋊C4—1061no—
474C22⋊C4—1061no—
475C22⋊C4—1061no—
476C22⋊C4—1061no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C42—531no—
480C42—531no—
481C42—531no—
482C42—1061no—
483C42—1061no—
484C42—1061no—
485C42C22×#615711yescharacteristic
486C23—531no—
487C23—1061no—
488Q8—42no—
489Q8—1061no—
490Q8—1061no—
491Q8—1061no—
492Q8—1061no—
493Q8—1061no—
494Q8—1061no—
495D4—1061no—
496D4—1061no—
497D4—2122no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—531no—
503C2×C4—531no—
504C2×C4—531no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4D4×#615711yescharacteristic
516C2×C4D4×#615711yescharacteristic
517C2×C4C23×#615711yescharacteristic
518C8—21no—
519C8—1061no—
520C8—1061no—
521C8—1061no—
522C8—1061no—
523C8—1061no—
524C8—1061no—
525C8—2121no—
526C8—2121no—
527C8—2121no—
528C8—2121no—
529C8—2121no—
530C8—2121no—
531C8#1313330512yes—
532C22—1061no—
533C22—1061no—
534C22—1061no—
535C22—2121no—
536C22—2121no—
537C22—2121no—
538C22C2×D4×#615711yescharacteristic
539C4—21no—
540C4—21no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—531no—
546C4—531no—
547C4—1061no—
548C4—1061no—
549C4—2121no—
550C4—2121no—
551C4—2121no—
552C4C2×#1313330511yescharacteristic
553C4#1681170311yescharacteristic
554C2—4241no—
555C2—2121no—
556C2#2029146111yescharacteristic
557C2#2029147311yescharacteristic
558C2#2029148711yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#501C4
socleC22×C53C2×C4×D4
prime radical#13133192C2×C4
cosocleC4×Dic53C24
hypercenter#501#6157
hypocenterC53#18081
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #20291626C2semidirect
3. #20290357C2semidirect
4. #20290354C2semidirect
5. #20290339C2semidirect
6. #20290331C2semidirect
7. #20290327C2semidirect
8. #20290323C2semidirect
9. #20290310C2semidirect
10. #20290300C2semidirect
11. #20290267C2semidirect
13. #20290025C2semidirect
14. #20289935C2semidirect
346. C53#18081semidirect

Maximal subgroups

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

subgroupindex
#202916672
#202916262
#202903572
#202903542
#202903392
#202903312
#202903272
#202903232
#202903102
#202903002
#202902672
#202900292
#202900252
#202899352
#202853232
#1808153