Finite Group Databasebeta

← #28316736

Subgroups — #28316736

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#88324C211yesmaximal, characteristic
2#20291503C211yesmaximal, characteristic, complemented
3#20290340C211yesmaximal, characteristic, complemented
4#20290338C211yesmaximal, characteristic, complemented
5#20290332C211yesmaximal, characteristic, complemented
6#20290330C211yesmaximal, characteristic, complemented
7#20290302C211yesmaximal, characteristic, complemented
8Q16×#88324C211yesmaximal, characteristic, complemented
9#20290298C211yesmaximal, characteristic, complemented
10SD16×#88324C211yesmaximal, characteristic, complemented
11#20290291C211yesmaximal, characteristic, complemented
12#20290050C211yesmaximal, characteristic
13#20290037C211yesmaximal, characteristic, complemented
14#20289916C211yesmaximal, characteristic, complemented
15#20284779C211yesmaximal, characteristic
16#16811675—21no—
17#16811674C2211yescharacteristic
18#16811673C2211yescharacteristic
19#16811672—21no—
20#16811672C2211yescharacteristic
21Q8×#88324C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24D4×#88324—21no—
25D4×#88324—21no—
26D4×#88324C2211yescharacteristic
27D4×#88324C2211yescharacteristic
28#16811639—21no—
29#16811639C2211yescharacteristic
30#16811629—21no—
31#16811626C2211yescharacteristic
32#16811587—21no—
33C4×#11719940C2211yescharacteristic
34C2×C4×#88324—21no—
35C2×C4×#88324C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50C8×#88324C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811175C2211yescharacteristic
53#41×Dic53C2211yescharacteristic
54#16810965C411yescharacteristic
55#16810961C411yescharacteristic
56#16810906C411yescharacteristic
57#16810901C411yescharacteristic
58#16810831C411yescharacteristic
59#16810826C411yescharacteristic
60#16810785C411yescharacteristic
61Q16×Dic53C2211yescharacteristic
62#16810715C2211yescharacteristic
63#16810714C2211yescharacteristic
64SD16×Dic53C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#452C411yescharacteristic
67C2×#11719940—21no—
68C2×#11719940—21no—
69C22×#88324—21no—
70C22×#88324—21no—
71C22×#88324D411yescharacteristic
72C22×#88324D411yescharacteristic
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
84C22⋊#88324—21no—
85C22⋊#88324—21no—
86C22⋊#88324—21no—
87#13133272—21no—
88#13133272—21no—
89#13133272—21no—
90C4×#88324—21no—
91C4×#88324—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×#11719923C2×C411yescharacteristic
103#13133216C2×C411yescharacteristic
104#13133215C2×C411yescharacteristic
105#13133207—21no—
106#13133196C2×C411yescharacteristic
107#13133194—21no—
108Q8×Dic53C2×C411yescharacteristic
109Q8×Dic53C2311yescharacteristic
110Q8×Dic53C2311yescharacteristic
111#13133182—21no—
112#13133182C2311yescharacteristic
113D4×Dic53—21no—
114D4×Dic53—21no—
115D4×Dic53C2×C411yescharacteristic
116D4×Dic53C2311yescharacteristic
117#13133176—21no—
118#13133176C2×C411yescharacteristic
119#13133173—21no—
120C2×C4×Dic53—21no—
121C2×C4×Dic53C2311yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133150C2×C411yescharacteristic
124#13133149C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133142C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133141C2×C411yescharacteristic
132#13133141C2×C411yescharacteristic
133#13133138C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×#41×C53C2×C411yescharacteristic
138C2×Q16×C53C2×C411yescharacteristic
139C2×SD16×C53C2×C411yescharacteristic
140C22⋊Q8×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#102×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144#41×C53—22no—
145#41×C53—22no—
146C2×Q8×C53C22×C411yescharacteristic
147C2×Q8×C53C22×C411yescharacteristic
148C2×D4×C53—21no—
149C2×D4×C53C22×C411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53C22×C411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53#4111yescharacteristic
156C2×Dic106—21no—
157C2×Dic106C22×C411yescharacteristic
158#11719940—24no—
159#11719940—42no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—22no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—41no—
168C2×#88324—42no—
169C2×#88324—42no—
170C2×#88324C2×D411yescharacteristic
171C2×#88324C2×D411yescharacteristic
172C2×#88324C2×D411yescharacteristic
173C2×#88324C2×D411yescharacteristic
174Q16×C53—22no—
175SD16×C53—22no—
176C2×C8×C53C22×C411yescharacteristic
177C2×C8×C53C22×C411yescharacteristic
178C4⋊C4×C53—21no—
179C4⋊C4×C53C22×C411yescharacteristic
180C22⋊C4×C53—21no—
181C22⋊Dic53—21no—
182C22⋊Dic53—21no—
183C22⋊Dic53—21no—
184C22⋊Dic53—21no—
185#11719925—22no—
186#11719923—22no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53C22×C411yescharacteristic
191C4⋊Dic53C22×C411yescharacteristic
192C4⋊Dic53C22×C411yescharacteristic
193#11719919—21no—
194#11719919—21no—
195#11719919—21no—
196#11719919—21no—
197#11719919—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53C22×C411yescharacteristic
204C4×Dic53C2×D411yescharacteristic
205C4×Dic53#4111yescharacteristic
206C4×Dic53C2411yescharacteristic
207C2×#88322C22×C411yescharacteristic
208C2×#88322C22×C411yescharacteristic
209C23×C53—21no—
210C23×C53C4×D411yescharacteristic
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—21no—
214Q8×C53#13112yes—
215D4×C53—24no—
216D4×C53—21no—
217D4×C53#13112yes—
218C2×C4×C53—24no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C23×C411yescharacteristic
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—22no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—42no—
236C2×Dic53C4×D411yescharacteristic
237C2×Dic53C4×D411yescharacteristic
238C2×Dic53C22×D411yescharacteristic
239C2×Dic53C2×#4111yescharacteristic
240Dic106—21no—
241Dic106—21no—
242Dic106—42no—
243#88324—22no—
244#88324—22no—
245#88324—21no—
246#88324—21no—
247#88324—21no—
248#88324—41no—
249#88324—41no—
250#88324—41no—
251#88324—41no—
252#88324#13712yes—
253C8×C53—21no—
254C8×C53—21no—
255#88322—21no—
256#88322—21no—
257#17913—531nomaximal, Sylow
258C22×C53—21no—
259C22×C53—22no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53C2×C4×D411yescharacteristic
265C4×C53—21no—
266C4×C53—22no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53C2×#13111yescharacteristic
270C4×C53C2×#13111yescharacteristic
271Dic53—21no—
272Dic53—21no—
273Dic53—41no—
274Dic53—41no—
275Dic53—41no—
276Dic53—41no—
277Dic53#44312yes—
278Dic53C2×#13711yescharacteristic
279Dic53#57511yescharacteristic
280#3107—531no—
281C8×#41—531no—
282#2960—531no—
283#1587—531no—
284#1587—531no—
285#1585—531no—
286#1585—531no—
287C8×Q16—531no—
288C8×Q16—531no—
289C8×SD16—531no—
290C8×SD16—531no—
291#1574—531no—
292#1489—531no—
293#1485—531no—
294C8×M16—531no—
295C2×C53—41no—
296C2×C53—42no—
297C2×C53#292911yescharacteristic
298C2×C53C4×#13711yescharacteristic
299C2×C53C2×#44311yescharacteristic
300#533—531no—
301C4×#41—531no—
302#485—531no—
303#485—531no—
304#475—531no—
305#474—531no—
306#462—531no—
307#461—531no—
308#452—531no—
309#452#615711yescharacteristic
310#446—531no—
311#446—531no—
312#446—1061no—
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8—531no—
316C8×Q8—531no—
317C8×Q8—1061no—
318C4×Q16—531no—
319C4×Q16—531no—
320C4×SD16—531no—
321C4×SD16—531no—
322C8×D4—531no—
323C8×D4—531no—
324C8×D4—1061no—
325C8×D4—1061no—
326#431—531no—
327#431—531no—
328#431—1061no—
329#424—1061no—
330#423—531no—
331#405—1061no—
332C4×M16—531no—
333C2×C4×C8—531no—
334C2×C4×C8—1061no—
335#335—531no—
336#335—531no—
337#334—531no—
338#334—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342Q8⋊C8—531no—
343Q8⋊C8—531no—
344Q8⋊C8—531no—
345D4⋊C8—531no—
346D4⋊C8—531no—
347#322—531no—
348#322—531no—
349C82—531no—
350C82—531no—
351C53#1791311yescharacteristic, complemented, Sylow
352C2×#41C2×#615711yescharacteristic
353C2×Q16—531no—
354C2×Q16C2×#615711yescharacteristic
355C2×SD16—531no—
356C2×SD16C2×#615711yescharacteristic
357C2×M16—1061no—
358C2×M16—1061no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8—1061no—
362C22×C8—1061no—
363C4⋊Q8—531no—
364C4⋊Q8—531no—
365#125—1061no—
366C22⋊Q8—531no—
367C22⋊Q8—1061no—
368C22⋊Q8C2×#615711yescharacteristic
369C4×Q8—531no—
370C4×Q8—531no—
371C4×Q8—531no—
372C4×Q8—531no—
373C4×Q8—1061no—
374C4×D4—531no—
375C4×D4—531no—
376C4×D4—1061no—
377C4×D4—1061no—
378#117—531no—
379#117—1061no—
380#117—1061no—
381C2×C42—531no—
382C2×C42—1061no—
383#107—531no—
384#107—531no—
385#106—531no—
386#106—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—531no—
391C4⋊C8—531no—
392C4⋊C8—531no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400#103—531no—
401#103—531no—
402#103—531no—
403#103—531no—
404#103—531no—
405#103C2×#615711yescharacteristic
406#102—531no—
407#102C2×#615711yescharacteristic
408C22⋊C8—531no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8C2×#615711yescharacteristic
413#97—1061no—
414#97—1061no—
415#97—1061no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8—1061no—
427C4×C8—1061no—
428C4×C8—1061no—
429#41—22no—
430#41—22no—
431C2×Q8—531no—
432C2×Q8—1061no—
433C2×Q8C22×#615711yescharacteristic
434C2×Q8C22×#615711yescharacteristic
435C2×D4—21no—
436C2×D4C22×#615711yescharacteristic
437C22×C4—21no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4C22×#615711yescharacteristic
443Q16—22no—
444Q16—1062no—
445SD16—22no—
446SD16—1062no—
447M16—1064no—
448M16—2122no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—1062no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—2122no—
463C2×C8—2121no—
464C2×C8—2122no—
465C2×C8C22×#615711yescharacteristic
466C2×C8C22×#615711yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—531no—
469C4⋊C4—531no—
470C4⋊C4—531no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4C22×#615711yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—1061no—
482C22⋊C4—1061no—
483C22⋊C4—1061no—
484C22⋊C4—1061no—
485C42—531no—
486C42—531no—
487C42—531no—
488C42—531no—
489C42—1061no—
490C42—1061no—
491C42—1061no—
492C42—1061no—
493C42—1061no—
494C23—21no—
495C23D4×#615711yescharacteristic
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—1061no—
500Q8—1061no—
501Q8—2122no—
502Q8#1313330412yes—
503D4—24no—
504D4—21no—
505D4#1313330412yes—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—531no—
513C2×C4—531no—
514C2×C4—531no—
515C2×C4—531no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1062no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2122no—
526C2×C4D4×#615711yescharacteristic
527C2×C4C23×#615711yescharacteristic
528C8—21no—
529C8—21no—
530C8—532no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—1062no—
535C8—1062no—
536C8—1061no—
537C8—1061no—
538C8—2121no—
539C8—2121no—
540C8—2121no—
541C8—2121no—
542C22—21no—
543C22—22no—
544C22—21no—
545C22—21no—
546C22—41no—
547C22—41no—
548C22C2×D4×#615711yescharacteristic
549C4—21no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—531no—
554C4—531no—
555C4—532no—
556C4—1061no—
557C4—1061no—
558C4—2121no—
559C4—2121no—
560C4—2121no—
561C4—2121no—
562C4C2×#1313330411yescharacteristic
563C4C2×#1313330411yescharacteristic
564C2—41no—
565C2—42no—
566C2#2029143011yescharacteristic
567C2#137×#615711yescharacteristic
568C2D53⋊#44311yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#452C4
socleC22×C53C2×C4×D4
prime radicalC2×#41×C53C2×C4
cosocleC4×Dic53C24
hypercenter#452#6157
hypocenterC53#17913
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #20291503C2semidirect
3. #20290340C2semidirect
4. #20290338C2semidirect
5. #20290332C2semidirect
6. #20290330C2semidirect
7. #20290302C2semidirect
8. Q16×#88324C2semidirect
9. #20290298C2semidirect
10. SD16×#88324C2semidirect
11. #20290291C2semidirect
13. #20290037C2semidirect
14. #20289916C2semidirect
351. C53#17913semidirect

Maximal subgroups

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

subgroupindex
#41×#883242
#202915032
#202903402
#202903382
#202903322
#202903302
#202903022
Q16×#883242
#202902982
SD16×#883242
#202902912
#202900502
#202900372
#202899162
#202847792
#1791353