Finite Group Databasebeta

← #27369164

Subgroups — #27369164

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286088C211yesmaximal, characteristic
2#20285739C211yesmaximal, characteristic
3#20285261C211yesmaximal, characteristic, complemented
4#20274233C211yesmaximal, characteristic, complemented
5#20274228C211yesmaximal, characteristic, complemented
6#20274221C211yesmaximal, characteristic, complemented
7#20274214C211yesmaximal, characteristic, complemented
8#20274109C211yesmaximal, characteristic, complemented
9#20274104C211yesmaximal, characteristic, complemented
10#20274097C211yesmaximal, characteristic, complemented
11#20274090C211yesmaximal, characteristic, complemented
12#20273482C211yesmaximal, characteristic, complemented
13#20273423C211yesmaximal, characteristic, complemented
14#20273386C211yesmaximal, characteristic, complemented
15C53×#2969C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18#16811032—21no—
19#16811032—21no—
20#16811028C2211yescharacteristic
21#16811028C2211yescharacteristic
22#16811021C2211yescharacteristic
23#16811021C2211yescharacteristic
24#16811018—21no—
25#16810993—21no—
26#16810927—21no—
27#16810918C2211yescharacteristic
28#16810913C2211yescharacteristic
29#16810879—21no—
30#16810867—21no—
31#16810842—21no—
32#16810833C2211yescharacteristic
33#16810833C2211yescharacteristic
34#16810833C2211yescharacteristic
35#16810828C2211yescharacteristic
36#16810828C2211yescharacteristic
37#16810828C2211yescharacteristic
38#16810778—21no—
39#16810761—21no—
40#16810680C2211yescharacteristic
41#16810665—21no—
42#16810665—21no—
43#16810647C2211yescharacteristic
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810368C2211yescharacteristic
51#16810368C2211yescharacteristic
52#16810368C2211yescharacteristic
53#16810368C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C53×#529C2211yescharacteristic
57C53×#446C2211yescharacteristic
58C8×Q8×C53C2211yescharacteristic
59C53×#436—21no—
60C53×#436C2211yescharacteristic
61C53×#435—21no—
62C8×D4×C53C2211yescharacteristic
63C53×#433—21no—
64C53×#431C2211yescharacteristic
65C53×#405—21no—
66C4×M16×C53C2211yescharacteristic
67C22×Dic106—21no—
68C2×#11719925—21no—
69C2×#11719925—21no—
70C2×#11719925—21no—
71C2×#11719923—21no—
72C2×#11719923—21no—
73C2×#11719923—21no—
74C2×#11719917—21no—
75#13133217—21no—
76#13133217—21no—
77#13133217—21no—
78#13133217C2311yescharacteristic
79#13133217C2311yescharacteristic
80#13133217C2311yescharacteristic
81#13133217C2311yescharacteristic
82#13133209—21no—
83#13133196—21no—
84#13133196—21no—
85C4×Dic106—21no—
86C4×Dic106—21no—
87#13133176—21no—
88C2×C4⋊Dic53—21no—
89C2×C4⋊Dic53—21no—
90#13133151—21no—
91#13133151—21no—
92#13133151—21no—
93#13133151—21no—
94#13133150—21no—
95#13133149—21no—
96#13133149—21no—
97#13133149—21no—
98#13133149—21no—
99#13133148—21no—
100C4⋊#88322—21no—
101C4⋊#88322C2311yescharacteristic
102C4⋊#88322C2311yescharacteristic
103C4⋊#88322C2311yescharacteristic
104C4⋊#88322C2311yescharacteristic
105#13133141—21no—
106#13133140—21no—
107#13133140—21no—
108#13133140—21no—
109#13133140—21no—
110#13133140—21no—
111#13133140—21no—
112#13133138—21no—
113C22⋊#88322—21no—
114#13133128—21no—
115C2×M16×C53—21no—
116C2×M16×C53D411yescharacteristic
117C2×M16×C53D411yescharacteristic
118C22×C8×C53—21no—
119#126×C53—21no—
120#125×C53D411yescharacteristic
121#124×C53D411yescharacteristic
122#123×C53—21no—
123C22⋊Q8×C53D411yescharacteristic
124#121×C53D411yescharacteristic
125C4×Q8×C53C2311yescharacteristic
126C4×D4×C53—21no—
127C4×D4×C53C2311yescharacteristic
128#117×C53—21no—
129C2×C42×C53C2311yescharacteristic
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53C2311yescharacteristic
133C4⋊C8×C53C2311yescharacteristic
134C22⋊C8×C53—21no—
135C22⋊C8×C53—21no—
136C22⋊C8×C53—21no—
137#97×C53—21no—
138#97×C53D411yescharacteristic
139#97×C53D411yescharacteristic
140C4×C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C2×D411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53C2×D411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53#4111yescharacteristic
151C22×Dic53—41no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—41no—
157C2×Dic106—81no—
158C2×Dic106—81no—
159M16×C53—22no—
160M16×C53—22no—
161M16×C53—42no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—42no—
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C2×C8×C53C2×D411yescharacteristic
168C2×C8×C53C2×D411yescharacteristic
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53C2×D411yescharacteristic
174C4⋊C4×C53C2×D411yescharacteristic
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C42×C53—21no—
181C42×C53C2×D411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53#4111yescharacteristic
184C42×C53C2411yescharacteristic
185C22⋊Dic53—41no—
186#11719925—81no—
187#11719925—81no—
188#11719925—81no—
189#11719923—81no—
190#11719923—81no—
191#11719923—81no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—41no—
201C4⋊Dic53—41no—
202#11719919—41no—
203#11719919—41no—
204C4×Dic53—41no—
205#11719917—81no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C23×C53—21no—
211C23×C53#12111yescharacteristic
212Q8×C53—21no—
213Q8×C53—21no—
214D4×C53—21no—
215D4×C53—21no—
216D4×C53—81no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—82no—
238Dic106—41no—
239Dic106—41no—
240Dic106—41no—
241Dic106—41no—
242Dic106—41no—
243Dic106—41no—
244Dic106—82no—
245Dic106—81no—
246C8×C53—21no—
247C8×C53—21no—
248C8×C53—41no—
249C8×C53—41no—
250C8×C53#13712yes—
251C8×C53#13712yes—
252#88322—41no—
253#88322—41no—
254#88322—41no—
255#88322—41no—
256#19378—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53C2×#12111yescharacteristic
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53C2×#13711yescharacteristic
272C4×C53C2×#13711yescharacteristic
273C4×C53#57611yescharacteristic
274C4×C53#57711yescharacteristic
275Dic53—81no—
276Dic53—81no—
277Dic53—81no—
278Dic53—81no—
279#3252—531no—
280#3218—531no—
281#3165—531no—
282#2969D5311yescharacteristic
283#1694—531no—
284#1693—531no—
285#1692—531no—
286#1691—531no—
287#1674—531no—
288#1673—531no—
289#1672—531no—
290#1671—531no—
291#1569—531no—
292#1554—531no—
293#1546—531no—
294C2×C53—41no—
295C2×C53—81no—
296C2×C53#303911yescharacteristic
297C2×C53#306211yescharacteristic
298C2×C53#306911yescharacteristic
299#533—531no—
300C2×C4⋊Q8—531no—
301#529C2×D5311yescharacteristic
302#501—1061no—
303#501—1061no—
304#500—531no—
305#500—531no—
306#498—531no—
307#498—531no—
308#497—1061no—
309#491—1061no—
310#479—1061no—
311#477—531no—
312#476—531no—
313#470—1061no—
314#468—1061no—
315#464—1061no—
316#462—531no—
317#462—531no—
318#462—531no—
319#461—531no—
320#461—531no—
321#461—531no—
322#451—1061no—
323#448—1061no—
324#446C2×D5311yescharacteristic
325C8×Q8C2×D5311yescharacteristic
326#436—21no—
327#436C2×D5311yescharacteristic
328#435—21no—
329C8×D4C2×D5311yescharacteristic
330#433—21no—
331#432—531no—
332#431C2×D5311yescharacteristic
333#428—1061no—
334#428—1061no—
335#423—531no—
336#405—21no—
337C4×M16C2×D5311yescharacteristic
338#335—531no—
339#335—531no—
340#334—531no—
341#334—531no—
342#333—531no—
343#333—531no—
344#332—531no—
345#332—531no—
346#332—531no—
347#332—531no—
348#332—531no—
349#332—531no—
350C53#1937811yescharacteristic, complemented, Sylow
351C22×Q8—1061no—
352C2×Q16—1061no—
353C2×Q16—1061no—
354C2×Q16—1061no—
355C2×SD16—1061no—
356C2×SD16—1061no—
357C2×SD16—1061no—
358C2×M16—21no—
359C2×M16—1061no—
360C2×M16C22⋊D5311yescharacteristic
361C2×M16C22⋊D5311yescharacteristic
362C22×C8—21no—
363C4⋊Q8—531no—
364C4⋊Q8—531no—
365C4⋊Q8—531no—
366C4⋊Q8—531no—
367C4⋊Q8—1061no—
368C4⋊Q8—1061no—
369C4⋊Q8—1061no—
370#126—21no—
371#125—1061no—
372#125D21211yescharacteristic
373#124D21211yescharacteristic
374#123—21no—
375C22⋊Q8—1061no—
376C22⋊Q8—1061no—
377C22⋊Q8D21211yescharacteristic
378#121D21211yescharacteristic
379C4×Q8—1061no—
380C4×Q8—1061no—
381C4×Q8C22×D5311yescharacteristic
382C4×D4—21no—
383C4×D4C22×D5311yescharacteristic
384#117—21no—
385#117—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C42C22×D5311yescharacteristic
389#107—1061no—
390#107—1061no—
391#107—1061no—
392#107—1061no—
393#107—1061no—
394#106—1061no—
395#106—1061no—
396#106—1061no—
397#106—1061no—
398#106—1061no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—531no—
404C4⋊C8—531no—
405C4⋊C8—1061no—
406C4⋊C8C22×D5311yescharacteristic
407C4⋊C8C22×D5311yescharacteristic
408#103—1061no—
409#103—1061no—
410#103—1061no—
411#103—1061no—
412#103—1061no—
413#103—1061no—
414#103—1061no—
415#102—1061no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8—21no—
419C22⋊C8—1061no—
420#97—21no—
421#97—1061no—
422#97C22⋊D5311yescharacteristic
423#97C22⋊D5311yescharacteristic
424C4×C8—21no—
425C4×C8—21no—
426C4×C8C22×D5311yescharacteristic
427C4×C8C22×D5311yescharacteristic
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8—1061no—
431C2×Q8—1061no—
432C2×Q8—4241no—
433C2×Q8—4241no—
434C2×Q8—2121no—
435C2×Q8C2×D21211yescharacteristic
436C2×D4—21no—
437C2×D4C2×D21211yescharacteristic
438C22×C4—21no—
439C22×C4—2121no—
440C22×C4C2×D21211yescharacteristic
441C22×C4#1171994511yescharacteristic
442C22×C4C2×C22⋊D5311yescharacteristic
443Q16—4241no—
444Q16—4241no—
445Q16—4241no—
446SD16—4241no—
447SD16—4241no—
448SD16—4241no—
449M16—22no—
450M16—22no—
451M16—42no—
452M16—4241no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—42no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8C2×C22⋊D5311yescharacteristic
461C2×C8C2×C22⋊D5311yescharacteristic
462C2×C8C2×C22⋊D5311yescharacteristic
463C2×C8C2×C22⋊D5311yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—2121no—
477C4⋊C4—2121no—
478C4⋊C4—2121no—
479C4⋊C4—2121no—
480C4⋊C4C2×D21211yescharacteristic
481C4⋊C4C2×D21211yescharacteristic
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—2121no—
488C42—21no—
489C42—2121no—
490C42C2×D21211yescharacteristic
491C42#1171994511yescharacteristic
492C42C2×C22⋊D5311yescharacteristic
493C42C23×D5311yescharacteristic
494C23—21no—
495C23#1313319111yescharacteristic
496Q8—21no—
497Q8—21no—
498Q8—4241no—
499Q8—4242no—
500Q8—2121no—
501Q8—2121no—
502Q8—2121no—
503Q8—2121no—
504Q8—2121no—
505Q8—2121no—
506D4—21no—
507D4—21no—
508D4—81no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—4242no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4#1313319111yescharacteristic
525C2×C4#1313319111yescharacteristic
526C2×C4#1313319111yescharacteristic
527C2×C4C22×C22⋊D5311yescharacteristic
528C2×C4C22×D21211yescharacteristic
529C2×C4C2×#1171994511yescharacteristic
530C8—21no—
531C8—21no—
532C8—41no—
533C8—41no—
534C8—2121no—
535C8—2121no—
536C8—2121no—
537C8—2121no—
538C8#1313324712yes—
539C8#1313324712yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×#1313319111yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—4241no—
555C4—4241no—
556C4—4241no—
557C4—4241no—
558C4C2×#1313324711yescharacteristic
559C4C2×#1313324711yescharacteristic
560C4#1681142811yescharacteristic
561C4#1681143511yescharacteristic
562C2—41no—
563C2—81no—
564C2#2028420211yescharacteristic
565C2#2028439511yescharacteristic
566C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2969C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#2969D53
hypocenterC53#19378
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2969

Complements

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

NH ≅ G/Ntype
3. #20285261C2semidirect
4. #20274233C2semidirect
5. #20274228C2semidirect
6. #20274221C2semidirect
7. #20274214C2semidirect
8. #20274109C2semidirect
9. #20274104C2semidirect
10. #20274097C2semidirect
11. #20274090C2semidirect
12. #20273482C2semidirect
13. #20273423C2semidirect
14. #20273386C2semidirect
350. C53#19378semidirect

Maximal subgroups

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

subgroupindex
#202860882
#202857392
#202852612
#202742332
#202742282
#202742212
#202742142
#202741092
#202741042
#202740972
#202740902
#202734822
#202734232
#202733862
C53×#29692
#1937853