Finite Group Databasebeta

← #28317035

Subgroups — #28317035

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291531C211yesmaximal, characteristic
2#20291503C211yesmaximal, characteristic, complemented
3#20290362C211yesmaximal, characteristic, complemented
4#20290360C211yesmaximal, characteristic, complemented
5#20290359C211yesmaximal, characteristic, complemented
6#20290357C211yesmaximal, characteristic, complemented
7#20290302C211yesmaximal, characteristic, complemented
8Q16×#88324C211yesmaximal, characteristic, complemented
9#20290300C211yesmaximal, characteristic, complemented
10#20290299C211yesmaximal, characteristic, complemented
11#20290279C211yesmaximal, characteristic, complemented
12#20290077C211yesmaximal, characteristic, complemented
13#20290072C211yesmaximal, characteristic
14M16×#88324C211yesmaximal, characteristic, complemented
15#20285316C211yesmaximal, characteristic
16#16811675—21no—
17#16811674C2211yescharacteristic
18#16811674C2211yescharacteristic
19#16811673C2211yescharacteristic
20#16811672—21no—
21#16811672C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811657C2211yescharacteristic
25#16811652—21no—
26#16811651C2211yescharacteristic
27#16811650—21no—
28#16811648—21no—
29#16811644C2211yescharacteristic
30#16811639C2211yescharacteristic
31#16811629—21no—
32#16811626C2211yescharacteristic
33#16811588—21no—
34#16811587—21no—
35C2×C4×#88324C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46#16811471C2211yescharacteristic
47#16811471C2211yescharacteristic
48#16811464C2211yescharacteristic
49#16811464C2211yescharacteristic
50C8×#88324C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811175C2211yescharacteristic
53#16811159C2211yescharacteristic
54#16811030C411yescharacteristic
55#16811016C411yescharacteristic
56#16810990C411yescharacteristic
57#16810984C411yescharacteristic
58#16810879C411yescharacteristic
59#16810871C411yescharacteristic
60#16810869C411yescharacteristic
61Q16×Dic53C2211yescharacteristic
62#16810715C2211yescharacteristic
63#16810710C2211yescharacteristic
64#16810708C2211yescharacteristic
65M16×Dic53C2211yescharacteristic
66C53×#470C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719940—21no—
69C2×D53⋊C8—21no—
70C22×#88324D411yescharacteristic
71C22×#88324D411yescharacteristic
72#13133284—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85#13133274—21no—
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88#13133272—21no—
89#13133272—21no—
90#13133272—21no—
91#13133272—21no—
92C4×#88324—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324D411yescharacteristic
96C4×#88324D411yescharacteristic
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719925C2×C411yescharacteristic
102C2×#11719913C2×C411yescharacteristic
103C2×#11719924C2×C411yescharacteristic
104C2×#11719917C2×C411yescharacteristic
105#13133216C2×C411yescharacteristic
106#13133215C2×C411yescharacteristic
107#13133212—21no—
108#13133208C2×C411yescharacteristic
109#13133207—21no—
110#13133206C2×C411yescharacteristic
111#13133201—21no—
112#13133196C2×C411yescharacteristic
113#13133194—21no—
114#13133191C2×C411yescharacteristic
115Q8×Dic53C2311yescharacteristic
116Q8×Dic53C2311yescharacteristic
117#13133182—21no—
118#13133182C2311yescharacteristic
119#13133179C2311yescharacteristic
120#13133177—21no—
121#13133175—21no—
122#13133173—21no—
123C2×C4×Dic53C2311yescharacteristic
124#13133151C2×C411yescharacteristic
125#13133150C2×C411yescharacteristic
126#13133148C2×C411yescharacteristic
127#13133142C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133140C2×C411yescharacteristic
131#13133139C2×C411yescharacteristic
132#13133137C2×C411yescharacteristic
133#13133127C2×C411yescharacteristic
134#13133127C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×Q16×C53C2×C411yescharacteristic
138C2×M16×C53C2×C411yescharacteristic
139C22⋊Q8×C53C2×C411yescharacteristic
140C22⋊Q8×C53C2×C411yescharacteristic
141#106×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#103×C53C2×C411yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C22×C4×C53C22×C411yescharacteristic
147C2×C22⋊D53—21no—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53#4111yescharacteristic
150C2×D212C22×C411yescharacteristic
151C2×C4×D53—21no—
152C2×Dic106—21no—
153C2×Dic106C22×C411yescharacteristic
154#11719940—42no—
155C2×#88324—22no—
156C2×#88324—21no—
157C2×#88324—21no—
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—41no—
163C2×#88324C2×D411yescharacteristic
164C2×#88324C2×D411yescharacteristic
165C2×#88324C2×D411yescharacteristic
166C2×#88324C2×D411yescharacteristic
167#11719936—42no—
168D53⋊C8—42no—
169Q16×C53—22no—
170M16×C53—22no—
171C2×C8×C53C22×C411yescharacteristic
172C2×C8×C53C22×C411yescharacteristic
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53C22×C411yescharacteristic
176C4⋊C4×C53C22×C411yescharacteristic
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊Dic53—21no—
180#11719925—22no—
181#11719924—22no—
182#11719921—21no—
183#11719921—21no—
184#11719921—21no—
185#11719921—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53C22×C411yescharacteristic
190C4⋊Dic53C22×C411yescharacteristic
191#11719919—21no—
192#11719919—21no—
193#11719919—21no—
194#11719919—21no—
195#11719919—21no—
196#11719919—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53C22×C411yescharacteristic
201C4×Dic53C2×D411yescharacteristic
202C4×Dic53#4111yescharacteristic
203C4×Dic53C2411yescharacteristic
204#11719917—22no—
205C2×#88322C22×C411yescharacteristic
206C2×#88322C22×C411yescharacteristic
207#11719913—22no—
208C23×C53C4×D411yescharacteristic
209C22×D53—21no—
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53C4×D411yescharacteristic
221C2×C4×C53C23×C411yescharacteristic
222C22⋊D53—42no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—41no—
229C2×Dic53C4×D411yescharacteristic
230C2×Dic53C4×D411yescharacteristic
231C2×Dic53C22×D411yescharacteristic
232C2×Dic53C2×#4111yescharacteristic
233D212—21no—
234D212—21no—
235C4×D53—41no—
236C4×D53—41no—
237Dic106—21no—
238Dic106—21no—
239Dic106—42no—
240#88324—21no—
241#88324—21no—
242#88324—21no—
243#88324—41no—
244#88324—41no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248#88324—41no—
249#88324#13712yes—
250C8×C53—21no—
251C8×C53#13112yes—
252#88322—21no—
253#88322#13112yes—
254#17965—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53C2×C4×D411yescharacteristic
259C2×D53—41no—
260C2×D53—41no—
261C2×D53—41no—
262C4×C53—21no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53C2×#13111yescharacteristic
268C4×C53C2×#13111yescharacteristic
269Dic53—21no—
270Dic53—21no—
271Dic53—21no—
272Dic53—41no—
273Dic53—41no—
274Dic53—41no—
275Dic53C2×#13711yescharacteristic
276Dic53#57511yescharacteristic
277#3172—531no—
278#2966—531no—
279#2960—531no—
280#1593—531no—
281#1593—531no—
282#1592—531no—
283#1592—531no—
284C8×Q16—531no—
285C8×Q16—531no—
286C8×SD16—531no—
287C8×SD16—531no—
288#1569—531no—
289#1497—531no—
290#1495—531no—
291C8×M16—531no—
292C2×C53—41no—
293C2×C53#293111yescharacteristic
294C2×C53C4×#13711yescharacteristic
295C2×C53#295511yescharacteristic
296D53—81no—
297#533—531no—
298#529—531no—
299#501—531no—
300#497—531no—
301#491—531no—
302#489—531no—
303#470—531no—
304#470#615711yescharacteristic
305#468—531no—
306#468—531no—
307#446—531no—
308#446—531no—
309#446—531no—
310#446—1061no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8—1061no—
315C4×Q16—531no—
316C4×Q16—531no—
317C4×SD16—531no—
318C4×SD16—531no—
319#436—531no—
320#435—1061no—
321C8×D4—531no—
322C8×D4—1061no—
323#433—1061no—
324#432—531no—
325#431—531no—
326#424—1061no—
327#423—531no—
328#405—1061no—
329#405—1061no—
330C4×M16—531no—
331C2×C4×C8—531no—
332#335—531no—
333#335—531no—
334#334—531no—
335#334—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342D4⋊C8—531no—
343D4⋊C8—531no—
344#322—531no—
345#322—531no—
346C82—531no—
347C82—531no—
348C53#1796511yescharacteristic, complemented, Sylow
349C2×Q16—531no—
350C2×Q16C2×#615711yescharacteristic
351C2×SD16—531no—
352C2×SD16—531no—
353C2×M16—531no—
354C2×M16—1061no—
355C2×M16—1061no—
356C2×M16C2×#615711yescharacteristic
357C22×C8—531no—
358C22×C8—531no—
359C22×C8—1061no—
360C4⋊Q8—531no—
361C4⋊Q8—531no—
362#126—1061no—
363#125—531no—
364#125—1061no—
365#124—531no—
366#123—1061no—
367C22⋊Q8—531no—
368C22⋊Q8—1061no—
369C22⋊Q8C2×#615711yescharacteristic
370C22⋊Q8C2×#615711yescharacteristic
371#121—531no—
372C4×Q8—531no—
373C4×Q8—531no—
374C4×Q8—531no—
375C4×Q8—1061no—
376C4×D4—531no—
377C4×D4—1061no—
378#117—1061no—
379#117—1061no—
380C2×C42—531no—
381#107—531no—
382#107—531no—
383#106—531no—
384#106C2×#615711yescharacteristic
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103C2×#615711yescharacteristic
402#103C2×#615711yescharacteristic
403#102—531no—
404#102—531no—
405C22⋊C8—1061no—
406C22⋊C8—1061no—
407C22⋊C8—1061no—
408C22⋊C8—1061no—
409#97—531no—
410#97—531no—
411#97—1061no—
412#97—1061no—
413#97—1061no—
414#97—1061no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8—1061no—
426C2×Q8—531no—
427C2×Q8—1061no—
428C2×Q8C22×#615711yescharacteristic
429C2×Q8C22×#615711yescharacteristic
430C2×D4—531no—
431C2×D4—1061no—
432C22×C4—531no—
433C22×C4—531no—
434C22×C4—1061no—
435C22×C4C22×#615711yescharacteristic
436Q16—22no—
437Q16—1062no—
438SD16—1062no—
439SD16—1062no—
440M16—22no—
441M16—1062no—
442M16—2122no—
443M16—2122no—
444C2×C8—531no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—1062no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—2122no—
458C2×C8—2121no—
459C2×C8C22×#615711yescharacteristic
460C2×C8C22×#615711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—531no—
464C4⋊C4—531no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4C22×#615711yescharacteristic
475C4⋊C4C22×#615711yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—1061no—
479C22⋊C4—1061no—
480C22⋊C4—1061no—
481C22⋊C4—1061no—
482C22⋊C4—1061no—
483C42—531no—
484C42—531no—
485C42—531no—
486C42—531no—
487C42—1061no—
488C42—1061no—
489C42—1061no—
490C23—1061no—
491C23D4×#615711yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—1061no—
497Q8—1061no—
498Q8—2122no—
499D4—1061no—
500D4—1061no—
501D4—2122no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—531no—
509C2×C4—531no—
510C2×C4—531no—
511C2×C4—531no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4D4×#615711yescharacteristic
521C2×C4C23×#615711yescharacteristic
522C8—21no—
523C8—532no—
524C8—532no—
525C8—1061no—
526C8—1061no—
527C8—1061no—
528C8—1061no—
529C8—2121no—
530C8—2121no—
531C8—2121no—
532C8—2121no—
533C8—2121no—
534C8—2121no—
535C8#1313330412yes—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—2121no—
540C22—2121no—
541C22—2121no—
542C22C2×D4×#615711yescharacteristic
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—531no—
549C4—531no—
550C4—1061no—
551C4—1061no—
552C4—1061no—
553C4—2121no—
554C4—2121no—
555C4—2121no—
556C4C2×#1313330411yescharacteristic
557C4D53⋊#13111yescharacteristic
558C2—41no—
559C2—4241no—
560C2#2029143811yescharacteristic
561C2#137×#615711yescharacteristic
562C2#2029148811yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#470C4
socleC22×C53C2×C4×D4
prime radical#13133191C2×C4
cosocleC4×Dic53C24
hypercenter#470#6157
hypocenterC53#17965
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #20291503C2semidirect
3. #20290362C2semidirect
4. #20290360C2semidirect
5. #20290359C2semidirect
6. #20290357C2semidirect
7. #20290302C2semidirect
8. Q16×#88324C2semidirect
9. #20290300C2semidirect
10. #20290299C2semidirect
11. #20290279C2semidirect
12. #20290077C2semidirect
14. M16×#88324C2semidirect
348. C53#17965semidirect

Maximal subgroups

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

subgroupindex
#202915312
#202915032
#202903622
#202903602
#202903592
#202903572
#202903022
Q16×#883242
#202903002
#202902992
#202902792
#202900772
#202900722
M16×#883242
#202853162
#1796553