Finite Group Databasebeta

← #27369615

Subgroups — #27369615

Order 13568.

616 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285847C211yesmaximal, characteristic, complemented
2#20284877C211yesmaximal, characteristic, complemented
3#20283826C211yesmaximal, characteristic, complemented
4#20274157C211yesmaximal, characteristic, complemented
5#20274150C211yesmaximal, characteristic, complemented
6#20274129C211yesmaximal, characteristic, complemented
7#20274122C211yesmaximal, characteristic, complemented
8#20274117C211yesmaximal, characteristic, complemented
9#20274112C211yesmaximal, characteristic, complemented
10#20274099C211yesmaximal, characteristic, complemented
11#20274092C211yesmaximal, characteristic, complemented
12#20273493C211yesmaximal, characteristic, complemented
13#20273410C211yesmaximal, characteristic, complemented
14#20273370C211yesmaximal, characteristic, complemented
15C53×#3106C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811026C2211yescharacteristic
18#16811019C2211yescharacteristic
19#16810969C2211yescharacteristic
20#16810969C2211yescharacteristic
21#16810964—21no—
22#16810944—21no—
23#16810944—21no—
24#16810923—21no—
25#16810923—21no—
26#16810918C2211yescharacteristic
27#16810913C2211yescharacteristic
28#16810833C2211yescharacteristic
29#16810828C2211yescharacteristic
30#16810785—21no—
31Q8×#88322C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33#16810728—21no—
34#16810723—21no—
35D4×#88322—21no—
36D4×#88322—21no—
37D4×#88322C2211yescharacteristic
38D4×#88322C2211yescharacteristic
39#16810680C2211yescharacteristic
40#16810676—21no—
41#16810676C2211yescharacteristic
42C2×C4⋊#88322—21no—
43C2×C4⋊#88322C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810370C2211yescharacteristic
53#16810370C2211yescharacteristic
54#16810367C2211yescharacteristic
55#16810367C2211yescharacteristic
56C53×#532C2211yescharacteristic
57C4×#41×C53C2211yescharacteristic
58C53×#482—21no—
59C53×#477C2211yescharacteristic
60C53×#476C2211yescharacteristic
61C53×#460C2211yescharacteristic
62C53×#459C2211yescharacteristic
63C53×#449—21no—
64C4×SD16×C53—21no—
65C4×D8×C53—21no—
66C53×#431C2211yescharacteristic
67C2×#11719925—21no—
68C2×#11719923—21no—
69C22×#88322—21no—
70C22×#88322—21no—
71C22×#88322—21no—
72#13133217C2311yescharacteristic
73#13133217C2311yescharacteristic
74#13133209—21no—
75#13133196—21no—
76#13133196—21no—
77C4×Dic106—21no—
78C4×Dic106—21no—
79#13133176—21no—
80#13133150—21no—
81#13133150—21no—
82#13133150—21no—
83#13133148—21no—
84#13133148—21no—
85#13133148—21no—
86C4⋊#88322—21no—
87C4⋊#88322—21no—
88C4⋊#88322—21no—
89C4⋊#88322D411yescharacteristic
90C4⋊#88322D411yescharacteristic
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133141—21no—
96#13133140—21no—
97#13133140—21no—
98#13133138—21no—
99#13133138—21no—
100#13133138—21no—
101#13133138—21no—
102#13133138—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106C22⋊#88322D411yescharacteristic
107C22⋊#88322D411yescharacteristic
108#13133128—21no—
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×#41×C53—21no—
114C2×#41×C53D411yescharacteristic
115C2×SD16×C53—21no—
116C2×D8×C53—21no—
117C4⋊Q8×C53C2311yescharacteristic
118#127×C53C2311yescharacteristic
119#124×C53—21no—
120#121×C53—21no—
121#121×C53—21no—
122C4×Q8×C53D411yescharacteristic
123C4×Q8×C53C2311yescharacteristic
124C4×D4×C53—21no—
125C4×D4×C53—21no—
126C4×D4×C53—21no—
127C4×D4×C53—21no—
128C4×D4×C53D411yescharacteristic
129C4×D4×C53C2311yescharacteristic
130#117×C53—21no—
131#117×C53D411yescharacteristic
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#107×C53—21no—
135#106×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53—21no—
139#102×C53—21no—
140#102×C53—21no—
141#102×C53—21no—
142C22⋊C8×C53—21no—
143C4×C8×C53—21no—
144#41×C53—22no—
145#41×C53—22no—
146#41×C53—82no—
147C2×Q8×C53—21no—
148C2×Q8×C53C2×D411yescharacteristic
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—21no—
153C2×D4×C53—41no—
154C2×D4×C53C2×D411yescharacteristic
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—41no—
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53#4111yescharacteristic
162C2×Dic106—21no—
163C2×Dic106—21no—
164SD16×C53—42no—
165D8×C53—42no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53C2×D411yescharacteristic
173C4⋊C4×C53C2×D411yescharacteristic
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—41no—
178C22⋊C4×C53—41no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53C2×D411yescharacteristic
183C42×C53C2×D411yescharacteristic
184C42×C53#4111yescharacteristic
185C42×C53C2411yescharacteristic
186C22⋊Dic53—41no—
187#11719925—42no—
188#11719923—42no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193#11719919—41no—
194#11719919—41no—
195C4×Dic53—41no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—81no—
201C2×#88322—81no—
202C2×#88322—81no—
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C23×C53—21no—
208C23×C53—41no—
209C23×C53#12111yescharacteristic
210Q8×C53—21no—
211Q8×C53—81no—
212Q8×C53#13512yes—
213D4×C53—24no—
214D4×C53—21no—
215D4×C53—41no—
216D4×C53—41no—
217D4×C53—41no—
218D4×C53—41no—
219D4×C53—81no—
220D4×C53—82no—
221D4×C53#13512yes—
222C2×C4×C53—24no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—81no—
235C2×C4×C53—82no—
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53#12111yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×Dic53—41no—
243C2×Dic53—41no—
244Dic106—41no—
245Dic106—41no—
246Dic106—81no—
247C8×C53—41no—
248C8×C53—41no—
249#88322—21no—
250#88322—21no—
251#88322—21no—
252#88322—21no—
253#88322—41no—
254#88322—41no—
255#88322—41no—
256#88322—41no—
257#19419—531nomaximal, Sylow
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—22no—
261C22×C53—21no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53C2×#12111yescharacteristic
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—22no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—81no—
276C4×C53C2×#13511yescharacteristic
277C4×C53C2×#13511yescharacteristic
278C4×C53#57711yescharacteristic
279C4×C53#57811yescharacteristic
280Dic53—81no—
281Dic53—81no—
282#3228—531no—
283#3116—531no—
284#3106D5311yescharacteristic
285#2988—531no—
286#1682—531no—
287#1681—531no—
288#1678—531no—
289#1677—531no—
290#1676—531no—
291#1675—531no—
292#1672—531no—
293#1671—531no—
294#1572—531no—
295#1551—531no—
296#1542—531no—
297C2×C53—41no—
298C2×C53—42no—
299C2×C53—161no—
300C2×C53#304611yescharacteristic
301C2×C53#305011yescharacteristic
302C2×C53#306811yescharacteristic
303#533—531no—
304#532C2×D5311yescharacteristic
305C4×#41C2×D5311yescharacteristic
306#500—531no—
307#498—531no—
308#486—531no—
309#486—531no—
310#485—1061no—
311#482—21no—
312#482—1061no—
313#482—1061no—
314#478—1061no—
315#478—1061no—
316#477—531no—
317#477C2×D5311yescharacteristic
318#476—531no—
319#476C2×D5311yescharacteristic
320#462—531no—
321#461—531no—
322#460C2×D5311yescharacteristic
323#459C2×D5311yescharacteristic
324#452—1061no—
325#449—21no—
326C8×Q8—531no—
327C8×Q8—531no—
328#441—1061no—
329#440—1061no—
330C4×SD16—21no—
331C4×D8—21no—
332C8×D4—531no—
333C8×D4—531no—
334C8×D4—1061no—
335C8×D4—1061no—
336#432—531no—
337#431—531no—
338#431—1061no—
339#431C2×D5311yescharacteristic
340C2×C4⋊C8—531no—
341C2×C4⋊C8—1061no—
342#335—531no—
343#335—531no—
344#334—531no—
345#334—531no—
346#333—531no—
347#333—531no—
348#333—531no—
349#333—531no—
350#332—531no—
351#332—531no—
352#331—531no—
353#331—531no—
354C53#1941911yescharacteristic, complemented, Sylow
355C2×#41—21no—
356C2×#41C22⋊D5311yescharacteristic
357C2×Q16—1061no—
358C2×SD16—21no—
359C2×SD16—1061no—
360C2×D8—21no—
361C22×C8—1061no—
362C22×C8—1061no—
363C22×C8—1061no—
364C4⋊Q8—531no—
365C4⋊Q8—531no—
366C4⋊Q8C22×D5311yescharacteristic
367#127C22×D5311yescharacteristic
368#125—1061no—
369#124—21no—
370C22⋊Q8—1061no—
371C22⋊Q8—1061no—
372#121—21no—
373#121—21no—
374C4×Q8—1061no—
375C4×Q8—1061no—
376C4×Q8C22⋊D5311yescharacteristic
377C4×Q8C22×D5311yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4C22⋊D5311yescharacteristic
383C4×D4C22×D5311yescharacteristic
384#117—21no—
385#117—1061no—
386#117C22⋊D5311yescharacteristic
387C2×C42—21no—
388C2×C42C22×D5311yescharacteristic
389#107—21no—
390#107—1061no—
391#107—1061no—
392#107—1061no—
393#106—21no—
394#106—1061no—
395#106—1061no—
396#106—1061no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8C22×D5311yescharacteristic
407C4⋊C8C22×D5311yescharacteristic
408#103—21no—
409#103—1061no—
410#103—1061no—
411#103—1061no—
412#102—21no—
413#102—21no—
414#102—21no—
415#102—1061no—
416#102—1061no—
417#102—1061no—
418#102—1061no—
419#102—1061no—
420C22⋊C8—21no—
421C22⋊C8—531no—
422C22⋊C8—531no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8—1061no—
426#97—1061no—
427C4×C8—21no—
428C4×C8—531no—
429C4×C8—531no—
430C4×C8—1061no—
431C4×C8—1061no—
432#41—22no—
433#41—22no—
434#41—82no—
435C2×Q8—21no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8C2×C22⋊D5311yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—41no—
444C2×D4C2×C22⋊D5311yescharacteristic
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—41no—
449C22×C4D4×D5311yescharacteristic
450C22×C4#1171994711yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452Q16—2122no—
453SD16—42no—
454SD16—2122no—
455D8—42no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—531no—
459C2×C8—531no—
460C2×C8—531no—
461C2×C8—531no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—4241no—
467C2×C8—4241no—
468C2×C8—4241no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—2121no—
478C4⋊C4—2121no—
479C4⋊C4C2×C22⋊D5311yescharacteristic
480C4⋊C4C2×C22⋊D5311yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—2121no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—2121no—
491C42D4×D5311yescharacteristic
492C42#1171994711yescharacteristic
493C42C2×C22⋊D5311yescharacteristic
494C42C23×D5311yescharacteristic
495C23—21no—
496C23—41no—
497C23#1313318911yescharacteristic
498Q8—21no—
499Q8—81no—
500Q8—4241no—
501Q8—2121no—
502Q8—2121no—
503Q8#1313323712yes—
504D4—24no—
505D4—21no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—81no—
511D4—82no—
512D4#1313323712yes—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4#1313318911yescharacteristic
530C2×C4#1313318911yescharacteristic
531C2×C4#1313318911yescharacteristic
532C2×C4C2×D4×D5311yescharacteristic
533C2×C4C22×C22⋊D5311yescharacteristic
534C2×C4C2×#1171994711yescharacteristic
535C8—41no—
536C8—41no—
537C8—1061no—
538C8—1061no—
539C8—1061no—
540C8—1061no—
541C8—2121no—
542C8—2121no—
543C8—2121no—
544C8—2121no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22C2×#1313318911yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—22no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—4241no—
564C4—4241no—
565C4C2×#1313323711yescharacteristic
566C4C2×#1313323711yescharacteristic
567C4#1681143011yescharacteristic
568C4#1681143711yescharacteristic
569C2—41no—
570C2—42no—
571C2—161no—
572C2#2028426311yescharacteristic
573C2#2028429311yescharacteristic
574C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3106C2
socleC22×C53C2×#121
hypercenter#3106D53
hypocenterC53#19419
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #20285847C2semidirect
2. #20284877C2semidirect
3. #20283826C2semidirect
4. #20274157C2semidirect
5. #20274150C2semidirect
6. #20274129C2semidirect
7. #20274122C2semidirect
8. #20274117C2semidirect
9. #20274112C2semidirect
10. #20274099C2semidirect
11. #20274092C2semidirect
12. #20273493C2semidirect
13. #20273410C2semidirect
14. #20273370C2semidirect
354. C53#19419semidirect

Maximal subgroups

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

subgroupindex
#202858472
#202848772
#202838262
#202741572
#202741502
#202741292
#202741222
#202741172
#202741122
#202740992
#202740922
#202734932
#202734102
#202733702
C53×#31062
#1941953