Finite Group Databasebeta

← #27388510

Subgroups — #27388510

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283511C211yesmaximal, characteristic
2#20283358C211yesmaximal, characteristic, complemented
3#20283355C211yesmaximal, characteristic, complemented
4#20275605C211yesmaximal, characteristic
5#20275602C211yesmaximal, characteristic
6#20275467C211yesmaximal, characteristic, complemented
7#20275460C211yesmaximal, characteristic, complemented
8#20275428C211yesmaximal, characteristic, complemented
9#20275424C211yesmaximal, characteristic, complemented
10#20275419C211yesmaximal, characteristic, complemented
11#20274620C211yesmaximal, characteristic, complemented
12#20274527C211yesmaximal, characteristic, complemented
13#20274524C211yesmaximal, characteristic, complemented
14#20274401C211yesmaximal, characteristic, complemented
15C53×#1937C211yesmaximal, characteristic, complemented
16C2×#13133237C2211yescharacteristic
17#41×Dic53C2211yescharacteristic
18#16811091C2211yescharacteristic
19#16810732C412yescomplemented
20#16810727C412yescomplemented
21#16810726C412yescomplemented
22#16810720C412yescomplemented
23#16810683—22no—
24#16810681—22no—
25#16810675—22no—
26#16810673—22no—
27C2×#13133152C2211yescharacteristic
28C2×#13133152C2211yescharacteristic
29#16810659C2211yescharacteristic
30#16810651C2211yescharacteristic
31#16810650C2211yescharacteristic
32#16810649C2211yescharacteristic
33C2×#13133145C2211yescharacteristic
34C2×#13133144C2211yescharacteristic
35C2×#13133143C2211yescharacteristic
36#16810620C2211yescharacteristic
37#16810591C2211yescharacteristic
38#16810590C2211yescharacteristic
39#16810589C2211yescharacteristic
40M16×Dic53C2211yescharacteristic
41M16×Dic53C2211yescharacteristic
42C2×#13133128C2211yescharacteristic
43C2×#13133127C2211yescharacteristic
44C2×#13133127C2211yescharacteristic
45C2×C8×Dic53C2211yescharacteristic
46C2×C8×Dic53C2211yescharacteristic
47#16810469C2211yescharacteristic
48#16810388C2211yescharacteristic
49#16810384C2211yescharacteristic
50#16810383C2211yescharacteristic
51#16810382C2211yescharacteristic
52C53×#424C2211yescharacteristic
53C2×#104×C53C2211yescharacteristic
54C53×#416C2211yescharacteristic
55C53×#408C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C53×#341C2211yescharacteristic
58C53×#339C2211yescharacteristic
59C2×#11719945C2311yescharacteristic
60#13133237—22no—
61#13133237—22no—
62C2×#11719925C2×C411yescharacteristic
63C2×#11719923C2×C411yescharacteristic
64C2×#11719924C2×C411yescharacteristic
65C2×#11719922C2×C411yescharacteristic
66C2×#11719917C2311yescharacteristic
67C2×#11719917C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70Q8×Dic53C2×C412yes—
71#13133182C2×C412yes—
72D4×Dic53—22no—
73D4×Dic53C2×C412yes—
74#13133179C2×C412yes—
75#13133177—22no—
76#13133176—21no—
77#13133176C2311yescharacteristic
78#13133174—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53C2311yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82C42×D53—21no—
83#13133152—22no—
84#13133152—22no—
85#13133150C2×C412yes—
86#13133148C2×C412yes—
87C4⋊#88322—22no—
88#13133146—22no—
89#13133146—22no—
90#13133145—22no—
91#13133144—22no—
92#13133143—22no—
93#13133142C2×C412yes—
94#13133139C2×C412yes—
95C22⋊#88322—22no—
96#13133129—22no—
97#13133129—22no—
98#13133128—22no—
99#13133127—21no—
100#13133127—21no—
101#13133127—22no—
102#13133127C2×C412yes—
103#13133127C2×C412yes—
104C8×Dic53—21no—
105C8×Dic53—21no—
106C8×Dic53—22no—
107C8×Dic53D412yes—
108C8×Dic53D412yes—
109#13133124—22no—
110C2×#41×C53C2311yescharacteristic
111C2×M16×C53C2311yescharacteristic
112C2×M16×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115#117×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C4⋊C8×C53—22no—
118#104×C53—22no—
119#103×C53C2×C412yes—
120#102×C53C2×C412yes—
121C22⋊C8×C53—22no—
122C4×C8×C53—22no—
123#41×C53—21no—
124#41×C53—21no—
125#41×C53—21no—
126#41×C53—21no—
127C2×Q8×C53C22×C411yescharacteristic
128C2×D4×C53—21no—
129C2×D4×C53C22×C411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2411yescharacteristic
133C2×C22⋊D53—21no—
134C22×Dic53—21no—
135C22×Dic53—22no—
136C22×Dic53#4112yes—
137#11719945—21no—
138#11719945—21no—
139#11719945—21no—
140#11719945—21no—
141C2×D212C22×C411yescharacteristic
142C2×C4×D53—21no—
143C2×C4×D53—22no—
144C2×Dic106C22×C411yescharacteristic
145M16×C53—21no—
146M16×C53—21no—
147M16×C53—21no—
148M16×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53C22×C412yes—
156C2×C8×C53C2×D412yes—
157C4⋊C4×C53C22×C412yes—
158C22⋊C4×C53—21no—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53#4112yes—
162C22⋊Dic53—21no—
163C22⋊Dic53—22no—
164#11719925—22no—
165#11719924—22no—
166#11719923—22no—
167#11719922—22no—
168#11719921—22no—
169C4⋊Dic53—22no—
170C4⋊Dic53C22×C412yes—
171#11719919—22no—
172C4×Dic53—21no—
173C4×Dic53—21no—
174C4×Dic53—22no—
175C4×Dic53—22no—
176C4×Dic53C22×C412yes—
177C4×Dic53C2×D411yescharacteristic
178C4×Dic53C2×D411yescharacteristic
179C4×Dic53#4112yes—
180#11719917—21no—
181#11719917—21no—
182#11719917—21no—
183#11719917—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322C22×C411yescharacteristic
189C2×#88322C22×C411yescharacteristic
190C2×#88322C2×D412yes—
191C2×#88322#4112yes—
192C2×#88322#4112yes—
193C23×C53—21no—
194C23×C53C2×#4111yescharacteristic
195C22×D53—21no—
196Q8×C53—21no—
197Q8×C53—21no—
198D4×C53—21no—
199D4×C53—21no—
200D4×C53—41no—
201D4×C53—41no—
202C2×C4×C53—22no—
203C2×C4×C53—21no—
204C2×C4×C53—22no—
205C2×C4×C53—42no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53C23×C411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C2×C4×C53C2×#4111yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C22⋊D53—41no—
215C22⋊D53—41no—
216C2×Dic53—21no—
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—42no—
220C2×Dic53—44no—
221C2×Dic53—42no—
222C2×Dic53C4×D412yes—
223C2×Dic53C4×D412yes—
224D212—21no—
225D212—21no—
226C4×D53—44no—
227C4×D53—41no—
228C4×D53—41no—
229Dic106—21no—
230Dic106—21no—
231C8×C53—22no—
232C8×C53—22no—
233C8×C53—41no—
234C8×C53—41no—
235#88322—22no—
236#88322—41no—
237#88322—41no—
238#88322#11712yes—
239#88322#11712yes—
240#21296—531nomaximal, Sylow
241C22×C53—22no—
242C22×C53—42no—
243C22×C53—41no—
244C22×C53C2×C4×D411yescharacteristic
245C22×C53#52911yescharacteristic
246C22×C53#52911yescharacteristic
247C2×D53—42no—
248C2×D53—41no—
249C4×C53—42no—
250C4×C53—42no—
251C4×C53—41no—
252C4×C53C2×#11711yescharacteristic
253C4×C53C4×#4111yescharacteristic
254C4×C53#53211yescharacteristic
255C4×C53#53211yescharacteristic
256Dic53—22no—
257Dic53—22no—
258Dic53—42no—
259Dic53—42no—
260Dic53—41no—
261#2943—531no—
262#2922—531no—
263#2922—531no—
264#1975—531no—
265#1974—531no—
266#1946—531no—
267#1945—531no—
268#1938—531no—
269#1938—531no—
270#1937—531no—
271#1937D5311yescharacteristic, complemented
272#1778—531no—
273#1759—531no—
274#1759—531no—
275#1726—531no—
276C2×C53—21no—
277C2×C53—21no—
278C2×C53—81no—
279C2×C53#229311yescharacteristic
280C2×C53#295512yes—
281D53—81no—
282C2×#135—531no—
283C4×#41—531no—
284C4×#41—531no—
285#442—532no—
286#441—532no—
287#440—532no—
288#440—532no—
289#433—1062no—
290#433—1062no—
291#431—1062no—
292#431—1062no—
293C2×#108—531no—
294C2×#108—531no—
295#427—531no—
296#424—531no—
297#424—531no—
298#424—531no—
299#424C2×D5311yescharacteristic
300C2×#104—531no—
301C2×#104—531no—
302C2×#104—531no—
303C2×#104C2×D5311yescharacteristic
304#416—531no—
305#416C2×D5311yescharacteristic
306#408—531no—
307#408—531no—
308#408—531no—
309#408C2×D5311yescharacteristic
310C4×M16—531no—
311C4×M16—531no—
312C2×#97—531no—
313C2×#97—531no—
314C2×#97—531no—
315C2×C4×C8—531no—
316C2×C4×C8—531no—
317C2×C4×C8C2×D5311yescharacteristic
318#376—531no—
319#341—531no—
320#341C2×D5311yescharacteristic
321#339—531no—
322#339—531no—
323#339—531no—
324#339C2×D5311yescharacteristic
325C53#2129611yescharacteristic, complemented, Sylow
326C2×#41—531no—
327C2×#41C22×D5311yescharacteristic
328#135—1062no—
329#135—1062no—
330C2×Q16—531no—
331C2×SD16—531no—
332C2×SD16—531no—
333C2×D8—531no—
334C2×M16—531no—
335C2×M16—531no—
336C2×M16C22×D5311yescharacteristic
337C2×M16C22×D5311yescharacteristic
338C22×C8—531no—
339C22×C8—531no—
340C22×C8C22×D5311yescharacteristic
341C22×C8C22×D5311yescharacteristic
342C4×Q8—532no—
343C4×Q8—532no—
344C4×D4—532no—
345C4×D4—532no—
346C4×D4—1062no—
347C4×D4—1062no—
348#117—531no—
349#117—1061no—
350#117—1061no—
351#117C22×D5311yescharacteristic
352C2×C42—531no—
353C2×C42—531no—
354C2×C42—1061no—
355C2×C42—1061no—
356C2×C42C22×D5311yescharacteristic
357#108—1062no—
358#108—1062no—
359#107—532no—
360#106—532no—
361C4⋊C8—22no—
362C4⋊C8—1062no—
363C4⋊C8—1062no—
364C4⋊C8—1062no—
365#104—22no—
366#104—1062no—
367#104—1062no—
368#104—1062no—
369#103—532no—
370#103C4×D5312yes—
371#102—532no—
372#102C4×D5312yes—
373C22⋊C8—22no—
374C22⋊C8—1062no—
375C22⋊C8—1062no—
376C22⋊C8—1062no—
377#97—532no—
378#97—532no—
379#97—1062no—
380#97—1062no—
381#97—1061no—
382#97—1061no—
383C4×C8—22no—
384C4×C8—532no—
385C4×C8—532no—
386C4×C8—1061no—
387C4×C8—1061no—
388C4×C8—1062no—
389#95—1062no—
390#41—21no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—1061no—
395#41—1061no—
396#41—1061no—
397#41—1061no—
398C2×Q8—531no—
399C2×Q8C2×C4×D5311yescharacteristic
400C2×D4—21no—
401C2×D4—531no—
402C2×D4—1061no—
403C2×D4C2×C4×D5311yescharacteristic
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—532no—
407C22×C4—1062no—
408C22×C4—1062no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4C23×D5311yescharacteristic
412Q16—1062no—
413SD16—1062no—
414SD16—1062no—
415D8—1062no—
416M16—21no—
417M16—21no—
418M16—21no—
419M16—21no—
420M16—1061no—
421M16—1061no—
422M16—1061no—
423M16—1061no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—531no—
431C2×C8—531no—
432C2×C8—532no—
433C2×C8—532no—
434C2×C8—532no—
435C2×C8—1061no—
436C2×C8—1061no—
437C2×C8—1061no—
438C2×C8—1061no—
439C2×C8C2×C4×D5312yes—
440C2×C8D4×D5312yes—
441C4⋊C4—532no—
442C4⋊C4—1062no—
443C4⋊C4—1062no—
444C4⋊C4C2×C4×D5312yes—
445C22⋊C4—21no—
446C22⋊C4—1062no—
447C22⋊C4—1062no—
448C22⋊C4—1061no—
449C42—21no—
450C42—21no—
451C42—532no—
452C42—531no—
453C42—531no—
454C42—532no—
455C42—1062no—
456C42—1062no—
457C42—1061no—
458C42—1061no—
459C42#1171994712yes—
460C23—21no—
461C23—1061no—
462C23#41×D5311yescharacteristic
463Q8—21no—
464Q8—21no—
465Q8—1061no—
466Q8—1061no—
467D4—21no—
468D4—21no—
469D4—41no—
470D4—41no—
471D4—1061no—
472D4—1061no—
473D4—2121no—
474D4—2121no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—532no—
482C2×C4—532no—
483C2×C4—1062no—
484C2×C4—1061no—
485C2×C4—1062no—
486C2×C4—2124no—
487C2×C4—2122no—
488C2×C4—2124no—
489C2×C4—2122no—
490C2×C4—2121no—
491C2×C4—2121no—
492C2×C4C22×C4×D5311yescharacteristic
493C2×C4C2×D4×D5311yescharacteristic
494C2×C4#41×D5311yescharacteristic
495C2×C4#41×D5311yescharacteristic
496C2×C4#41×D5311yescharacteristic
497C2×C4C2×#1171994711yescharacteristic
498C8—22no—
499C8—22no—
500C8—41no—
501C8—41no—
502C8—532no—
503C8—532no—
504C8—1062no—
505C8—2121no—
506C8—2121no—
507C22—22no—
508C22—41no—
509C22—42no—
510C22—2121no—
511C22—2122no—
512C22C4×D4×D5311yescharacteristic
513C22#1681115011yescharacteristic
514C22#1681115611yescharacteristic
515C4—42no—
516C4—41no—
517C4—42no—
518C4—1062no—
519C4—1062no—
520C4—2122no—
521C4—2121no—
522C4—2122no—
523C4#117×D5311yescharacteristic
524C4C4×#1171994711yescharacteristic
525C4#1681116811yescharacteristic
526C4#1681117211yescharacteristic
527C2—21no—
528C2—21no—
529C2—81no—
530C2—4241no—
531C2#2027717511yescharacteristic
532C2#2028361312yes—
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1937C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133237C22
cosocleC22×C4×C53C24
hypercenter#1937D53
hypocenterC53#21296
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1937

Complements

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

NH ≅ G/Ntype
2. #20283358C2semidirect
3. #20283355C2semidirect
6. #20275467C2semidirect
7. #20275460C2semidirect
8. #20275428C2semidirect
9. #20275424C2semidirect
10. #20275419C2semidirect
11. #20274620C2semidirect
12. #20274527C2semidirect
13. #20274524C2semidirect
14. #20274401C2semidirect
15. C53×#1937C2semidirect
19. #16810732C4semidirect
20. #16810727C4semidirect
21. #16810726C4semidirect
22. #16810720C4semidirect
271. #1937D53semidirect
325. C53#21296semidirect

Maximal subgroups

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

subgroupindex
#202835112
#202833582
#202833552
#202756052
#202756022
#202754672
#202754602
#202754282
#202754242
#202754192
#202746202
#202745272
#202745242
#202744012
C53×#19372
#2129653