Finite Group Databasebeta

← #27369183

Subgroups — #27369183

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285933C211yesmaximal, characteristic
2#20285262C211yesmaximal, characteristic, complemented
3#20283720C211yesmaximal, characteristic
4#20274241C211yesmaximal, characteristic, complemented
5#20274236C211yesmaximal, characteristic, complemented
6#20274219C211yesmaximal, characteristic, complemented
7#20274212C211yesmaximal, characteristic, complemented
8#20274117C211yesmaximal, characteristic, complemented
9#20274112C211yesmaximal, characteristic, complemented
10#20274095C211yesmaximal, characteristic, complemented
11#20274088C211yesmaximal, characteristic, complemented
12#20273480C211yesmaximal, characteristic, complemented
13#20273425C211yesmaximal, characteristic, complemented
14#20273397C211yesmaximal, characteristic, complemented
15C53×#3252C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17#16811033—21no—
18#16811033—21no—
19#16811033—21no—
20#16811026C2211yescharacteristic
21#16811026C2211yescharacteristic
22#16811019C2211yescharacteristic
23#16811019C2211yescharacteristic
24#16810982—21no—
25#16810951—21no—
26#16810934—21no—
27#16810918C2211yescharacteristic
28#16810918C2211yescharacteristic
29#16810913C2211yescharacteristic
30#16810913C2211yescharacteristic
31#16810754C2211yescharacteristic
32Q8×#88322C2211yescharacteristic
33#16810701—21no—
34#16810701C2211yescharacteristic
35#16810695—21no—
36D4×#88322C2211yescharacteristic
37#16810684—21no—
38#16810680C2211yescharacteristic
39#16810676C2211yescharacteristic
40#16810666—21no—
41#16810666—21no—
42#16810579—21no—
43C4×#11719917C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810371C2211yescharacteristic
53#16810371C2211yescharacteristic
54#16810369C2211yescharacteristic
55#16810369C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C53×#529C2211yescharacteristic
58C53×#479—21no—
59C53×#477C2211yescharacteristic
60C53×#476C2211yescharacteristic
61C53×#470—21no—
62C53×#468—21no—
63C53×#464—21no—
64C53×#462C2211yescharacteristic
65C53×#461C2211yescharacteristic
66C53×#423C2211yescharacteristic
67C22×Dic106—21no—
68C2×#11719917—21no—
69C2×#11719917D411yescharacteristic
70C2×#11719917D411yescharacteristic
71C22×#88322—21no—
72#13133217—21no—
73#13133217—21no—
74#13133217—21no—
75#13133217C2311yescharacteristic
76#13133217C2311yescharacteristic
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79#13133150—21no—
80#13133150—21no—
81#13133150—21no—
82#13133150—21no—
83#13133150—21no—
84#13133150—21no—
85#13133148—21no—
86#13133148—21no—
87#13133148—21no—
88#13133148—21no—
89#13133148—21no—
90#13133148—21no—
91C4⋊#88322—21no—
92C4⋊#88322—21no—
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133141—21no—
98#13133141—21no—
99#13133138—21no—
100#13133138—21no—
101C22⋊#88322—21no—
102C22⋊#88322—21no—
103C22⋊#88322—21no—
104C22⋊#88322—21no—
105#13133128—21no—
106#13133128—21no—
107#13133128D411yescharacteristic
108#13133128D411yescharacteristic
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×Q16×C53—21no—
114C2×SD16×C53—21no—
115C2×M16×C53—21no—
116C4⋊Q8×C53C2311yescharacteristic
117C4⋊Q8×C53C2311yescharacteristic
118#126×C53—21no—
119#125×C53—21no—
120#125×C53D411yescharacteristic
121#124×C53D411yescharacteristic
122#123×C53—21no—
123C22⋊Q8×C53—21no—
124C22⋊Q8×C53—21no—
125C22⋊Q8×C53D411yescharacteristic
126#121×C53D411yescharacteristic
127C4×Q8×C53—21no—
128C4×Q8×C53—21no—
129C4×Q8×C53C2311yescharacteristic
130C4×D4×C53—21no—
131C4×D4×C53C2311yescharacteristic
132#117×C53—21no—
133#117×C53—21no—
134C2×C42×C53C2311yescharacteristic
135#107×C53—21no—
136#106×C53—21no—
137C4⋊C8×C53—21no—
138C4⋊C8×C53C2311yescharacteristic
139C4⋊C8×C53C2311yescharacteristic
140#103×C53—21no—
141#103×C53—21no—
142#103×C53—21no—
143#102×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53C2×D411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53C2×D411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53#4111yescharacteristic
153C22×Dic53—41no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—41no—
157C2×Dic106—82no—
158Q16×C53—81no—
159SD16×C53—81no—
160M16×C53—81no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—41no—
172C4⋊C4×C53—41no—
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—
180C22⋊C4×C53—41no—
181C42×C53—21no—
182C42×C53—41no—
183C42×C53C2×D411yescharacteristic
184C42×C53C2×D411yescharacteristic
185C42×C53#4111yescharacteristic
186C42×C53C2411yescharacteristic
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—41no—
192C4⋊Dic53—41no—
193#11719917—22no—
194#11719917—22no—
195#11719917—42no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—42no—
201C2×#88322C2×D411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C23×C53—21no—
206C23×C53#12111yescharacteristic
207Q8×C53—21no—
208Q8×C53—21no—
209Q8×C53—41no—
210Q8×C53—41no—
211Q8×C53—81no—
212D4×C53—21no—
213D4×C53—21no—
214D4×C53—81no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
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—82no—
236Dic106—82no—
237Dic106—81no—
238Dic106—81no—
239C8×C53—41no—
240C8×C53—41no—
241#88322—21no—
242#88322—21no—
243#88322—41no—
244#88322—41no—
245#88322—41no—
246#88322—41no—
247#88322#13712yes—
248#88322#13712yes—
249#19379—531nomaximal, Sylow
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53C2×#12111yescharacteristic
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—81no—
265C4×C53—81no—
266C4×C53C2×#13711yescharacteristic
267C4×C53C2×#13711yescharacteristic
268C4×C53#57711yescharacteristic
269C4×C53#57811yescharacteristic
270Dic53—81no—
271Dic53—81no—
272#3252D5311yescharacteristic
273#3238—531no—
274#3165—531no—
275#2969—531no—
276#1696—531no—
277#1695—531no—
278#1692—531no—
279#1691—531no—
280#1676—531no—
281#1675—531no—
282#1672—531no—
283#1671—531no—
284#1569—531no—
285#1554—531no—
286#1548—531no—
287C2×C53—41no—
288C2×C53—81no—
289C2×C53#303911yescharacteristic
290C2×C53#306311yescharacteristic
291C2×C53#306811yescharacteristic
292#533C2×D5311yescharacteristic
293C2×C4⋊Q8—531no—
294#529C2×D5311yescharacteristic
295#501—1061no—
296#501—1061no—
297#501—1061no—
298#500—531no—
299#500—531no—
300#498—531no—
301#498—531no—
302#489—1061no—
303#483—1061no—
304#480—1061no—
305#479—21no—
306#477—531no—
307#477—531no—
308#477C2×D5311yescharacteristic
309#476—531no—
310#476—531no—
311#476C2×D5311yescharacteristic
312#470—21no—
313#468—21no—
314#464—21no—
315#462C2×D5311yescharacteristic
316#461C2×D5311yescharacteristic
317#446—531no—
318C8×Q8—531no—
319#436—531no—
320#436—1061no—
321#435—1061no—
322C8×D4—531no—
323#433—1061no—
324#432—531no—
325#431—531no—
326#428—1061no—
327#428—1061no—
328#423C2×D5311yescharacteristic
329#405—1061no—
330C4×M16—531no—
331#335—531no—
332#335—531no—
333#334—531no—
334#334—531no—
335#333—531no—
336#333—531no—
337#333—531no—
338#333—531no—
339#333—531no—
340#333—531no—
341#332—531no—
342#332—531no—
343C53#1937911yescharacteristic, complemented, Sylow
344C22×Q8—1061no—
345C2×Q16—21no—
346C2×SD16—21no—
347C2×M16—21no—
348C2×M16—531no—
349C2×M16—531no—
350C2×M16—1061no—
351C22×C8—1061no—
352C4⋊Q8—531no—
353C4⋊Q8—531no—
354C4⋊Q8—1061no—
355C4⋊Q8—1061no—
356C4⋊Q8—1061no—
357C4⋊Q8C22×D5311yescharacteristic
358C4⋊Q8C22×D5311yescharacteristic
359#126—21no—
360#125—21no—
361#125C22⋊D5311yescharacteristic
362#124C22⋊D5311yescharacteristic
363#123—21no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8C22⋊D5311yescharacteristic
367#121C22⋊D5311yescharacteristic
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8C22×D5311yescharacteristic
371C4×D4—21no—
372C4×D4C22×D5311yescharacteristic
373#117—21no—
374#117—21no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C42C22×D5311yescharacteristic
378#107—21no—
379#107—1061no—
380#107—1061no—
381#107—1061no—
382#107—1061no—
383#107—1061no—
384#107—1061no—
385#106—21no—
386#106—1061no—
387#106—1061no—
388#106—1061no—
389#106—1061no—
390#106—1061no—
391#106—1061no—
392C4⋊C8—21no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—1061no—
398C4⋊C8—1061no—
399C4⋊C8C22×D5311yescharacteristic
400C4⋊C8C22×D5311yescharacteristic
401#103—21no—
402#103—21no—
403#103—21no—
404#103—1061no—
405#103—1061no—
406#102—21no—
407#102—1061no—
408#102—1061no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—531no—
414#97—531no—
415#97—1061no—
416#97—1061no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—1061no—
420C4×C8—1061no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—4242no—
426C2×Q8—2121no—
427C2×Q8C2×C22⋊D5311yescharacteristic
428C2×D4—21no—
429C2×D4C2×C22⋊D5311yescharacteristic
430C22×C4—21no—
431C22×C4—2121no—
432C22×C4D4×D5311yescharacteristic
433C22×C4#1171994711yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435Q16—81no—
436SD16—81no—
437M16—81no—
438M16—1062no—
439M16—1062no—
440M16—2122no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—531no—
444C2×C8—531no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—2122no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—2121no—
467C4⋊C4—2121no—
468C4⋊C4C2×C22⋊D5311yescharacteristic
469C4⋊C4C2×C22⋊D5311yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—41no—
476C42—21no—
477C42—41no—
478C42D4×D5311yescharacteristic
479C42#1171994711yescharacteristic
480C42C2×C22⋊D5311yescharacteristic
481C42C23×D5311yescharacteristic
482C23—21no—
483C23#1313318911yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—41no—
487Q8—41no—
488Q8—81no—
489Q8—4241no—
490Q8—4241no—
491Q8—4242no—
492D4—21no—
493D4—21no—
494D4—81no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—4242no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4#1313318911yescharacteristic
511C2×C4#1313318911yescharacteristic
512C2×C4#1313318911yescharacteristic
513C2×C4C2×D4×D5311yescharacteristic
514C2×C4C22×C22⋊D5311yescharacteristic
515C2×C4C2×#1171994711yescharacteristic
516C8—41no—
517C8—41no—
518C8—532no—
519C8—532no—
520C8—1061no—
521C8—1061no—
522C8—2121no—
523C8—2121no—
524C8—2121no—
525C8—2121no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22C2×#1313318911yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—81no—
541C4—81no—
542C4—4241no—
543C4—4241no—
544C4#137×D5311yescharacteristic
545C4#137×D5311yescharacteristic
546C4#1681143411yescharacteristic
547C4#1681143811yescharacteristic
548C2—41no—
549C2—81no—
550C2#2028420011yescharacteristic
551C2#2028440811yescharacteristic
552C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3252C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#3252D53
hypocenterC53#19379
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3252

Complements

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

NH ≅ G/Ntype
2. #20285262C2semidirect
4. #20274241C2semidirect
5. #20274236C2semidirect
6. #20274219C2semidirect
7. #20274212C2semidirect
8. #20274117C2semidirect
9. #20274112C2semidirect
10. #20274095C2semidirect
11. #20274088C2semidirect
12. #20273480C2semidirect
13. #20273425C2semidirect
14. #20273397C2semidirect
343. C53#19379semidirect

Maximal subgroups

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

subgroupindex
#202859332
#202852622
#202837202
#202742412
#202742362
#202742192
#202742122
#202741172
#202741122
#202740952
#202740882
#202734802
#202734252
#202733972
C53×#32522
#1937953