Finite Group Databasebeta

← #28316756

Subgroups — #28316756

Order 13568.

624 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#20291519C211yesmaximal, characteristic
2#20291503C211yesmaximal, characteristic, complemented
3#20290329C211yesmaximal, characteristic, complemented
4#20290327C211yesmaximal, characteristic, complemented
5#20290321C211yesmaximal, characteristic, complemented
6#20290319C211yesmaximal, characteristic, complemented
7#20290310C211yesmaximal, characteristic, complemented
8#20290309C211yesmaximal, characteristic, complemented
9#20290307C211yesmaximal, characteristic, complemented
10#20290305C211yesmaximal, characteristic, complemented
11#20290291C211yesmaximal, characteristic, complemented
12#20290052C211yesmaximal, characteristic
13#20290037C211yesmaximal, characteristic, complemented
14#20289926C211yesmaximal, characteristic, complemented
15#20284781C211yesmaximal, characteristic
16#16811675—21no—
17#16811674C2211yescharacteristic
18#16811673C2211yescharacteristic
19#16811672—21no—
20#16811672C2211yescharacteristic
21#16811672C2211yescharacteristic
22#16811672C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811651C2211yescharacteristic
25#16811651C2211yescharacteristic
26#16811650—21no—
27#16811650—21no—
28#16811641—21no—
29#16811639C2211yescharacteristic
30#16811629—21no—
31#16811626C2211yescharacteristic
32#16811587—21no—
33C4×#11719940C2211yescharacteristic
34C4×D53⋊C8—21no—
35C2×C4×#88324C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42#16811473C2211yescharacteristic
43#16811473C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46#16811471C2211yescharacteristic
47#16811471C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51#16811464C2211yescharacteristic
52#16811175C2211yescharacteristic
53#16811091C2211yescharacteristic
54#16810964C411yescharacteristic
55#16810907C411yescharacteristic
56#16810902C411yescharacteristic
57#16810832C411yescharacteristic
58#16810827C411yescharacteristic
59#16810787C411yescharacteristic
60#16810783C411yescharacteristic
61#16810717C2211yescharacteristic
62#16810715C2211yescharacteristic
63#16810710C2211yescharacteristic
64#16810708C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#485C411yescharacteristic
67C2×#11719940—21no—
68C2×#11719940—21no—
69C2×D53⋊C8—21no—
70C2×D53⋊C8—21no—
71C22×#88324D411yescharacteristic
72C22×#88324D411yescharacteristic
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76#13133284—21no—
77#13133284—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85#13133274—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×#11719945C2×C411yescharacteristic
102C2×Dic212C2×C411yescharacteristic
103C2×#11719925C2×C411yescharacteristic
104C2×#11719913C2×C411yescharacteristic
105C2×#11719924C2×C411yescharacteristic
106#13133216C2×C411yescharacteristic
107#13133215C2×C411yescharacteristic
108#13133207—21no—
109#13133196C2×C411yescharacteristic
110#13133194—21no—
111Q8×Dic53C2311yescharacteristic
112#13133182—21no—
113#13133182C2×C411yescharacteristic
114#13133182C2311yescharacteristic
115#13133182C2311yescharacteristic
116#13133179C2×C411yescharacteristic
117#13133179C2311yescharacteristic
118#13133177—21no—
119#13133177—21no—
120#13133174—21no—
121#13133173—21no—
122C2×C4×Dic53C2311yescharacteristic
123C42×D53—21no—
124#13133150C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133140C2×C411yescharacteristic
130#13133140C2×C411yescharacteristic
131#13133140C2×C411yescharacteristic
132#13133139C2×C411yescharacteristic
133#13133137C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C22⋊Q8×C53C2×C411yescharacteristic
138#117×C53C2×C411yescharacteristic
139#107×C53C2×C411yescharacteristic
140#106×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C22×C4×C53C22×C411yescharacteristic
146C2×C22⋊D53—21no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53#4111yescharacteristic
149#11719945—22no—
150#11719945—22no—
151C2×D212C22×C411yescharacteristic
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×Dic106—21no—
156C2×Dic106C22×C411yescharacteristic
157C2×Dic106C22×C411yescharacteristic
158#11719940—42no—
159#11719940—42no—
160C2×#88324—22no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—21no—
167C2×#88324—41no—
168C2×#88324C2×D411yescharacteristic
169C2×#88324C2×D411yescharacteristic
170C2×#88324C2×D411yescharacteristic
171C2×#88324C2×D411yescharacteristic
172D53⋊C8—42no—
173D53⋊C8—42no—
174C2×C8×C53C22×C411yescharacteristic
175C2×C8×C53C22×C411yescharacteristic
176C4⋊C4×C53—21no—
177C4⋊C4×C53C22×C411yescharacteristic
178C4⋊C4×C53C22×C411yescharacteristic
179C4⋊C4×C53C22×C411yescharacteristic
180C22⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C42×C53—21no—
183C22⋊Dic53—21no—
184#11719925—22no—
185#11719924—22no—
186#11719921—21no—
187#11719921—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53C22×C411yescharacteristic
190#11719919—21no—
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×Dic53—21no—
201C4×Dic53C22×C411yescharacteristic
202C4×Dic53C2×D411yescharacteristic
203C4×Dic53#4111yescharacteristic
204C4×Dic53C2411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206C2×#88322C22×C411yescharacteristic
207Dic212—22no—
208#11719913—22no—
209C23×C53C4×D411yescharacteristic
210C22×D53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53C4×D411yescharacteristic
220C2×C4×C53C23×C411yescharacteristic
221C22⋊D53—24no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—22no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53C4×D411yescharacteristic
229C2×Dic53C4×D411yescharacteristic
230C2×Dic53C22×D411yescharacteristic
231C2×Dic53C2×#4111yescharacteristic
232D212—21no—
233D212#13112yes—
234C4×D53—24no—
235C4×D53—41no—
236C4×D53—41no—
237C4×D53—42no—
238Dic106—21no—
239Dic106—21no—
240Dic106—21no—
241Dic106—42no—
242Dic106#13112yes—
243#88324—21no—
244#88324—21no—
245#88324—21no—
246#88324—41no—
247#88324—41no—
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#17916—531nomaximal, Sylow
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53C2×C4×D411yescharacteristic
262C2×D53—22no—
263C2×D53—41no—
264C2×D53—41no—
265C4×C53—21no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53C2×#13111yescharacteristic
271C4×C53C2×#13111yescharacteristic
272Dic53—22no—
273Dic53—21no—
274Dic53—21no—
275Dic53—41no—
276Dic53—41no—
277Dic53#44412yes—
278Dic53C2×#13711yescharacteristic
279Dic53#57511yescharacteristic
280#3107—531no—
281C8×#41—531no—
282#2960—531no—
283#1584—531no—
284#1584—531no—
285#1582—531no—
286#1582—531no—
287#1579—531no—
288#1579—531no—
289SD16⋊C8—531no—
290SD16⋊C8—531no—
291#1574—531no—
292#1489—531no—
293#1485—531no—
294#1451—531no—
295C2×C53—41no—
296C2×C53#292911yescharacteristic
297C2×C53C4×#13711yescharacteristic
298C2×C53C2×#44411yescharacteristic
299D53—42no—
300#533—531no—
301C4×#41—531no—
302#485—531no—
303#485#615711yescharacteristic
304#475—531no—
305#474—531no—
306#462—531no—
307#461—531no—
308#452—531no—
309#452—531no—
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—
349#322—531no—
350#322—531no—
351C53#1791611yescharacteristic, complemented, Sylow
352C2×#41—531no—
353C2×Q16—531no—
354C2×Q16—531no—
355C2×SD16—531no—
356C2×SD16—531no—
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—1061no—
379#117—1061no—
380#117C2×#615711yescharacteristic
381C2×C42—531no—
382C2×C42—1061no—
383#107—531no—
384#107C2×#615711yescharacteristic
385#106—531no—
386#106C2×#615711yescharacteristic
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#103C2×#615711yescharacteristic
405#103C2×#615711yescharacteristic
406#102—531no—
407#102—531no—
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—1062no—
430#41—1062no—
431C2×Q8—531no—
432C2×Q8—531no—
433C2×Q8—1061no—
434C2×Q8C22×#615711yescharacteristic
435C2×D4—531no—
436C2×D4—1061no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4C22×#615711yescharacteristic
443Q16—1062no—
444Q16—1062no—
445SD16—1062no—
446SD16—1062no—
447M16—2122no—
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—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4C22×#615711yescharacteristic
478C4⋊C4C22×#615711yescharacteristic
479C4⋊C4C22×#615711yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—1061no—
483C22⋊C4—1061no—
484C22⋊C4—1061no—
485C42—21no—
486C42—531no—
487C42—531no—
488C42—531no—
489C42—531no—
490C42—1061no—
491C42—1061no—
492C42—1061no—
493C42—1061no—
494C23—1061no—
495C23D4×#615711yescharacteristic
496Q8—21no—
497Q8—21no—
498Q8—532no—
499Q8—1061no—
500Q8—1061no—
501Q8—1061no—
502Q8—2122no—
503D4—532no—
504D4—1064no—
505D4—1061no—
506C2×C4—21no—
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—1064no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1062no—
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—1061no—
535C8—1061no—
536C8—2121no—
537C8—2121no—
538C8—2121no—
539C8—2121no—
540C8—2121no—
541C8—2121no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—1062no—
546C22—2121no—
547C22—2121no—
548C22C2×D4×#615711yescharacteristic
549C4—21no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—531no—
555C4—531no—
556C4—532no—
557C4—1062no—
558C4—1061no—
559C4—1061no—
560C4—2121no—
561C4—2121no—
562C4D53⋊#13111yescharacteristic
563C4D53⋊#13111yescharacteristic
564C2—41no—
565C2—2122no—
566C2#2029143211yescharacteristic
567C2#137×#615711yescharacteristic
568C2D53⋊#44411yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#485C4
socleC22×C53C2×C4×D4
prime radicalC2×#11719945C2×C4
cosocleC4×Dic53C24
hypercenter#485#6157
hypocenterC53#17916
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
2. #20291503C2semidirect
3. #20290329C2semidirect
4. #20290327C2semidirect
5. #20290321C2semidirect
6. #20290319C2semidirect
7. #20290310C2semidirect
8. #20290309C2semidirect
9. #20290307C2semidirect
10. #20290305C2semidirect
11. #20290291C2semidirect
13. #20290037C2semidirect
14. #20289926C2semidirect
351. C53#17916semidirect

Maximal subgroups

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

subgroupindex
#202915192
#202915032
#202903292
#202903272
#202903212
#202903192
#202903102
#202903092
#202903072
#202903052
#202902912
#202900522
#202900372
#202899262
#202847812
#1791653