Finite Group Databasebeta

← #28316320

Subgroups — #28316320

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291643C211yesmaximal, characteristic
2#20291640C211yesmaximal, characteristic
3#20290290C211yesmaximal, characteristic, complemented
4#20290288C211yesmaximal, characteristic, complemented
5#20290287C211yesmaximal, characteristic, complemented
6#20290286C211yesmaximal, characteristic, complemented
7#20290283C211yesmaximal, characteristic, complemented
8#20290282C211yesmaximal, characteristic, complemented
9#20290280C211yesmaximal, characteristic, complemented
10#20290279C211yesmaximal, characteristic, complemented
11#20290277C211yesmaximal, characteristic, complemented
12#20290267C211yesmaximal, characteristic, complemented
13#20290266C211yesmaximal, characteristic, complemented
14C2×#16811484C211yesmaximal, characteristic, complemented
15#20283675C211yesmaximal, characteristic
16#16811675—21no—
17#16811673—21no—
18#16811655—21no—
19#16811654—21no—
20#16811652—21no—
21#16811652—21no—
22#16811650—21no—
23D4×#88324—21no—
24#16811645C2211yescharacteristic
25#16811644C2211yescharacteristic
26#16811644C2211yescharacteristic
27#16811644C2211yescharacteristic
28#16811642C2211yescharacteristic
29#16811639C2211yescharacteristic
30#16811627C2211yescharacteristic
31#16811626C2211yescharacteristic
32D53⋊(C4⋊C8)C2211yescharacteristic
33C2×C4⋊#88324C2211yescharacteristic
34C2×C4⋊#88324C2211yescharacteristic
35C2×C4⋊#88324C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811487C2211yescharacteristic
38#16811486C2211yescharacteristic
39#16811486C2211yescharacteristic
40#16811486C2211yescharacteristic
41#16811486C2211yescharacteristic
42#16811485C2211yescharacteristic
43#16811485C2211yescharacteristic
44#16811485C2211yescharacteristic
45#16811485C2211yescharacteristic
46#16811484C2211yescharacteristic
47#16811484C2211yescharacteristic
48#16811484C2211yescharacteristic
49#16811484C2211yescharacteristic
50#16811484C2211yescharacteristic
51#16811484C2211yescharacteristic
52C4×#11719947C2211yescharacteristic
53#16810752C411yescharacteristic
54#16810751C411yescharacteristic
55#16810699C411yescharacteristic
56#16810698C411yescharacteristic
57#16810695C411yescharacteristic
58#16810694C411yescharacteristic
59#16810691C411yescharacteristic
60#16810675C2211yescharacteristic
61#16810673C2211yescharacteristic
62#16810673C2211yescharacteristic
63#16810575C2211yescharacteristic
64M16×Dic53C2211yescharacteristic
65C2×C8×Dic53C2211yescharacteristic
66C53×#435C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719940—21no—
69C2×D53⋊C8—21no—
70C22×#88324—21no—
71C22×#88324—21no—
72C22×#88324—21no—
73#13133284—21no—
74#13133284—21no—
75C4⋊#88324—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324—21no—
79C4⋊#88324C2311yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84C4⋊#88324C2311yescharacteristic
85C4⋊#88324C2311yescharacteristic
86C4⋊#88324C2311yescharacteristic
87#13133274—21no—
88#13133274—21no—
89C22⋊#88324—21no—
90C22⋊#88324—21no—
91C22⋊#88324—21no—
92C22⋊#88324—21no—
93#13133272—21no—
94#13133272—21no—
95#13133272—21no—
96#13133272—21no—
97C4×#88324—21no—
98C4×#88324—21no—
99C2×#11719947Q811yescharacteristic
100C2×#11719917C2×C411yescharacteristic
101C2×#11719912C2×C411yescharacteristic
102C2×#11719912C2×C411yescharacteristic
103C22×#88322C2×C411yescharacteristic
104C4×Dic106C2×C411yescharacteristic
105#13133182D411yescharacteristic
106D4×Dic53D411yescharacteristic
107C4×C22⋊D53C2×C411yescharacteristic
108C4×C22⋊D53C2×C411yescharacteristic
109#13133177D411yescharacteristic
110#13133177D411yescharacteristic
111#13133175Q811yescharacteristic
112#13133173Q811yescharacteristic
113#13133173Q811yescharacteristic
114C2×C4×Dic53C2311yescharacteristic
115C2×C4×Dic53C2311yescharacteristic
116C42×D53C2311yescharacteristic
117C4⋊#88322C2×C411yescharacteristic
118#13133146C2×C411yescharacteristic
119#13133146C2×C411yescharacteristic
120#13133146C2×C411yescharacteristic
121#13133146C2×C411yescharacteristic
122C22⋊#88322C2×C411yescharacteristic
123C22⋊#88322C2×C411yescharacteristic
124#13133129C2×C411yescharacteristic
125#13133129C2×C411yescharacteristic
126#13133128C2×C411yescharacteristic
127#13133127C2×C411yescharacteristic
128#13133127C2×C411yescharacteristic
129C8×Dic53C2×C411yescharacteristic
130C8×Dic53C2×C411yescharacteristic
131C8×Dic53C2311yescharacteristic
132C8×Dic53C2311yescharacteristic
133C8×Dic53C2311yescharacteristic
134C8×Dic53C2311yescharacteristic
135C2×M16×C53C2×C411yescharacteristic
136C22×C8×C53C2×C411yescharacteristic
137C4×D4×C53C2×C411yescharacteristic
138C4⋊C8×C53C2×C411yescharacteristic
139C22⋊C8×C53C2×C411yescharacteristic
140C22⋊C8×C53C2×C411yescharacteristic
141#97×C53C2×C411yescharacteristic
142C2×D4×C53C4⋊C411yescharacteristic
143C22×C4×C53C22×C411yescharacteristic
144C22×C4×C53C22×C411yescharacteristic
145C2×C22⋊D53C4⋊C411yescharacteristic
146C2×C22⋊D53C4⋊C411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×Q811yescharacteristic
150C22×Dic53C2×Q811yescharacteristic
151#11719947—42no—
152C2×C4×D53C22×C411yescharacteristic
153C2×C4×D53C2×D411yescharacteristic
154C2×C4×D53C2×Q811yescharacteristic
155C2×Dic106C4⋊C411yescharacteristic
156#11719940—81no—
157C2×#88324—21no—
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—81no—
166C2×#88324—81no—
167C2×#88324—81no—
168#11719936—81no—
169D53⋊C8—81no—
170M16×C53—22no—
171C2×C8×C53—22no—
172C2×C8×C53C22×C411yescharacteristic
173C2×C8×C53C22×C411yescharacteristic
174C2×C8×C53C22×C411yescharacteristic
175C2×C8×C53C22×C411yescharacteristic
176C4⋊C4×C53C4⋊C411yescharacteristic
177C22⋊C4×C53C4⋊C411yescharacteristic
178C22⋊C4×C53C4⋊C411yescharacteristic
179C42×C53C22×C411yescharacteristic
180C22⋊Dic53C4⋊C411yescharacteristic
181C22⋊Dic53C4⋊C411yescharacteristic
182#11719921C4⋊C411yescharacteristic
183#11719921C4⋊C411yescharacteristic
184C4⋊Dic53C4⋊C411yescharacteristic
185#11719919C4⋊C411yescharacteristic
186#11719919C4⋊C411yescharacteristic
187#11719919C4⋊C411yescharacteristic
188#11719919C4⋊C411yescharacteristic
189C4×Dic53C22×C411yescharacteristic
190C4×Dic53C22×C411yescharacteristic
191C4×Dic53C2×D411yescharacteristic
192C4×Dic53C2×D411yescharacteristic
193C4×Dic53C2×D411yescharacteristic
194C4×Dic53C2×Q811yescharacteristic
195C4×Dic53C2×Q811yescharacteristic
196C4×Dic53C2×Q811yescharacteristic
197C4×Dic53C2411yescharacteristic
198#11719917—41no—
199C2×#88322—41no—
200C2×#88322C22×C411yescharacteristic
201C2×#88322C22×C411yescharacteristic
202C2×#88322C22×C411yescharacteristic
203C2×#88322C22×C411yescharacteristic
204#11719912—22no—
205#11719912—41no—
206C23×C53C2×C4⋊C411yescharacteristic
207C23×C53C2×C4⋊C411yescharacteristic
208C22×D53C2×C4⋊C411yescharacteristic
209D4×C53—22no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53C2×C4⋊C411yescharacteristic
215C2×C4×C53C2×C4⋊C411yescharacteristic
216C2×C4×C53C2×C4⋊C411yescharacteristic
217C2×C4×C53C2×C4⋊C411yescharacteristic
218C2×C4×C53C23×C411yescharacteristic
219C22⋊D53—22no—
220C22⋊D53—41no—
221C2×Dic53—22no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—41no—
227C2×Dic53C2×C4⋊C411yescharacteristic
228C2×Dic53C2×C4⋊C411yescharacteristic
229C2×Dic53C2×C4⋊C411yescharacteristic
230C2×Dic53C2×C4⋊C411yescharacteristic
231C2×Dic53C2×C4⋊C411yescharacteristic
232C2×Dic53C22×D411yescharacteristic
233C2×Dic53C22×Q811yescharacteristic
234C4×D53—21no—
235C4×D53—21no—
236C4×D53—21no—
237C4×D53—21no—
238C4×D53—41no—
239Dic106—41no—
240#88324—41no—
241#88324—41no—
242#88324—41no—
243#88324—41no—
244#88324—41no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248C8×C53—21no—
249C8×C53—21no—
250C8×C53—21no—
251C8×C53#13112yes—
252#88322—21no—
253#88322—21no—
254#88322—21no—
255#88322—21no—
256#17838—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53#43012yes—
263C22×C53C22×C4⋊C411yescharacteristic
264C2×D53—21no—
265C2×D53—21no—
266C2×D53—21no—
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—21no—
271C4×C53C2×#13111yescharacteristic
272C4×C53#56811yescharacteristic
273Dic53—21no—
274Dic53—21no—
275Dic53—21no—
276Dic53—21no—
277Dic53—21no—
278Dic53—21no—
279Dic53#57711yescharacteristic
280Dic53#57711yescharacteristic
281#2990—531no—
282#2990—531no—
283#2965—531no—
284#1573—531no—
285#1573—531no—
286#1572—531no—
287#1572—531no—
288#1570—531no—
289#1570—531no—
290#1569—531no—
291#1569—531no—
292#1568—531no—
293#1565—531no—
294#1565—531no—
295C2×#334—531no—
296C2×C53—22no—
297C2×C53—41no—
298C2×C53#290611yescharacteristic
299C2×C53#291211yescharacteristic
300C2×C53C2×#43011yescharacteristic
301D53—41no—
302C4×#41—531no—
303#446—531no—
304#446—531no—
305#446—1061no—
306#446—1061no—
307#436—531no—
308#436—531no—
309#435—531no—
310#435—531no—
311#435—531no—
312#435—1061no—
313#435—1061no—
314#435—1061no—
315#435—1061no—
316#435#615711yescharacteristic
317C8×D4—1061no—
318C8×D4—1061no—
319#432—531no—
320#432—531no—
321#432—531no—
322#432—531no—
323#431—531no—
324#431—531no—
325#431—531no—
326#431—531no—
327#431—531no—
328#423—531no—
329#423—531no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C4×M16—531no—
335C4×M16—531no—
336C2×C4×C8—531no—
337#335—531no—
338#335—531no—
339#335—531no—
340#335—531no—
341#335—531no—
342#335—531no—
343#334—531no—
344#334—531no—
345#334—531no—
346#334—531no—
347#334—531no—
348#334—531no—
349#334—531no—
350#334—531no—
351#334—531no—
352#334—531no—
353C53#1783811yescharacteristic, complemented, Sylow
354C2×#41—531no—
355C2×M16—531no—
356C2×M16—531no—
357C2×M16—531no—
358C2×M16—1061no—
359C2×M16—1061no—
360C2×M16C2×#615711yescharacteristic
361C22×C8—531no—
362C22×C8—1061no—
363C22×C8—1061no—
364C22×C8—1061no—
365C22×C8—1061no—
366C22×C8C2×#615711yescharacteristic
367C4×Q8—531no—
368C4×Q8—531no—
369C4×D4—531no—
370C4×D4—531no—
371C4×D4—531no—
372C4×D4—531no—
373C4×D4—531no—
374C4×D4C2×#615711yescharacteristic
375#117—531no—
376#117—531no—
377#117—531no—
378C2×C42—531no—
379C2×C42—531no—
380C2×C42—531no—
381C4⋊C8—531no—
382C4⋊C8—531no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—531no—
391C4⋊C8—531no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8C2×#615711yescharacteristic
401C22⋊C8—531no—
402C22⋊C8—531no—
403C22⋊C8—531no—
404C22⋊C8—531no—
405C22⋊C8—1061no—
406C22⋊C8—1061no—
407C22⋊C8—1061no—
408C22⋊C8—1061no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8C2×#615711yescharacteristic
412C22⋊C8C2×#615711yescharacteristic
413#97—531no—
414#97—531no—
415#97—531no—
416#97—1061no—
417#97—1061no—
418#97—1061no—
419#97—1061no—
420#97C2×#615711yescharacteristic
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—531no—
425C4×C8—531no—
426C4×C8—531no—
427C4×C8—1061no—
428C4×C8—1061no—
429#41—2122no—
430C2×Q8—531no—
431C2×D4—531no—
432C2×D4—531no—
433C2×D4C4⋊#615711yescharacteristic
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4C22×#615711yescharacteristic
442C22×C4C22×#615711yescharacteristic
443M16—22no—
444M16—1062no—
445M16—4241no—
446M16—4241no—
447M16—2121no—
448M16—2121no—
449C2×C8—22no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—4241no—
463C2×C8—4241no—
464C2×C8—4241no—
465C2×C8—4241no—
466C2×C8—2121no—
467C2×C8C22×#615711yescharacteristic
468C2×C8C22×#615711yescharacteristic
469C2×C8C22×#615711yescharacteristic
470C2×C8C22×#615711yescharacteristic
471C4⋊C4—531no—
472C4⋊C4—531no—
473C4⋊C4—531no—
474C4⋊C4—531no—
475C4⋊C4—531no—
476C4⋊C4C4⋊#615711yescharacteristic
477C22⋊C4—531no—
478C22⋊C4—531no—
479C22⋊C4—531no—
480C22⋊C4—531no—
481C22⋊C4C4⋊#615711yescharacteristic
482C22⋊C4C4⋊#615711yescharacteristic
483C42—531no—
484C42—531no—
485C42—531no—
486C42—531no—
487C42—531no—
488C42—531no—
489C42—531no—
490C42—531no—
491C42—531no—
492C42C22×#615711yescharacteristic
493C23—531no—
494C23C2×C4⋊#615711yescharacteristic
495C23C2×C4⋊#615711yescharacteristic
496Q8—2121no—
497D4—22no—
498D4—1062no—
499D4—2121no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—531no—
505C2×C4—531no—
506C2×C4—531no—
507C2×C4—531no—
508C2×C4—531no—
509C2×C4—531no—
510C2×C4—531no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1062no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4C2×C4⋊#615711yescharacteristic
523C2×C4C2×C4⋊#615711yescharacteristic
524C2×C4C2×C4⋊#615711yescharacteristic
525C2×C4C2×C4⋊#615711yescharacteristic
526C2×C4C23×#615711yescharacteristic
527C8—21no—
528C8—21no—
529C8—21no—
530C8—1061no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—2121no—
535C8—2121no—
536C8—2121no—
537C8—2121no—
538C8—2121no—
539C8—2121no—
540C8—2121no—
541C8—2121no—
542C8#1313330412yes—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—1061no—
549C22—1061no—
550C22—1061no—
551C22#1681163812yes—
552C22C22×C4⋊#615711yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—531no—
558C4—531no—
559C4—1061no—
560C4—1061no—
561C4—1061no—
562C4—1061no—
563C4—1061no—
564C4—1061no—
565C4C2×#1313330411yescharacteristic
566C4#1681170211yescharacteristic
567C2—22no—
568C2—41no—
569C2—2121no—
570C2#2029136011yescharacteristic
571C2#2029136911yescharacteristic
572C2C2×#1681163811yescharacteristic
subgroupG/H
centerC22C22×C4⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#435C4
socleC22×C53C22×C4⋊C4
prime radicalC2×#11719947Q8
cosocleC4×Dic53C24
hypercenter#435#6157
hypocenterC53#17838
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
3. #20290290C2semidirect
4. #20290288C2semidirect
5. #20290287C2semidirect
6. #20290286C2semidirect
7. #20290283C2semidirect
8. #20290282C2semidirect
9. #20290280C2semidirect
10. #20290279C2semidirect
11. #20290277C2semidirect
12. #20290267C2semidirect
13. #20290266C2semidirect
14. C2×#16811484C2semidirect
353. C53#17838semidirect

Maximal subgroups

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

subgroupindex
#202916432
#202916402
#202902902
#202902882
#202902872
#202902862
#202902832
#202902822
#202902802
#202902792
#202902772
#202902672
#202902662
C2×#168114842
#202836752
#1783853