Finite Group Databasebeta

← #27366900

Subgroups — #27366900

Order 13568.

758 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285089C211yesmaximal, characteristic, complemented
2#20284723C211yesmaximal, characteristic, complemented
3D4×#11719917C211yesmaximal, characteristic, complemented
4#20274274C212yesmaximal, complemented
5#20274246C212yesmaximal, complemented
6#20274150C212yesmaximal, complemented
7#20274122C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273403C211yesmaximal, characteristic, complemented
10#20273383C211yesmaximal, characteristic, complemented
11C2×C53×#459C211yesmaximal, characteristic
12C22×#11719917—21no—
13#16811180C2211yescharacteristic
14#16810987—22no—
15#16810969C2212yescomplemented
16#16810969C2212yescomplemented
17#16810953—21no—
18#16810913C2211yescharacteristic, complemented
19#16810913C2211yescharacteristic, complemented
20#16810828C2211yescharacteristic, complemented
21#16810828C2211yescharacteristic, complemented
22#16810774—21no—
23C4×#11719923—21no—
24C4×#11719923—21no—
25#16810701C2212yescomplemented
26#16810695—22no—
27D4×#88322C2212yescomplemented
28#16810676C2211yescharacteristic
29#16810647C2211yescharacteristic
30#16810628—21no—
31#16810628—21no—
32#16810587—21no—
33C4×#11719917C2211yescharacteristic
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810371C2212yescomplemented
37#16810370C2212yescomplemented
38#16810367C2212yescomplemented
39#16810367C2212yescomplemented
40C22×D8×C53—21no—
41C2×#127×C53C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#459—21no—
44C53×#459—21no—
45C53×#459C2212yes—
46C53×#459C2212yes—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#102×C53—21no—
49C2×#11719923—21no—
50C2×#11719923—21no—
51C2×#11719917—22no—
52C2×#11719917—21no—
53C2×#11719917—44no—
54C2×#11719917D411yescharacteristic
55C2×#11719917D411yescharacteristic
56C22×#88322—22no—
57#13133217C2311yescharacteristic
58#13133217C2311yescharacteristic
59#13133209—21no—
60#13133196—21no—
61#13133196—21no—
62C4×Dic106—21no—
63C4×Dic106—21no—
64#13133176—21no—
65#13133148—21no—
66#13133148—21no—
67C4⋊#88322—21no—
68C4⋊#88322C2311yescharacteristic
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2312yes—
71#13133140—21no—
72#13133140—21no—
73#13133138—22no—
74#13133138—22no—
75#13133138—22no—
76#13133138—22no—
77#13133138—21no—
78#13133138—21no—
79C22⋊#88322—22no—
80C22⋊#88322—22no—
81C22⋊#88322—21no—
82#13133128D412yes—
83C4×#88322—21no—
84C4×#88322C2312yes—
85C22×D4×C53—21no—
86C22×D4×C53—21no—
87C22×D4×C53—21no—
88C22×D4×C53D411yescharacteristic
89C23×C4×C53—21no—
90C2×D8×C53—22no—
91C2×D8×C53—22no—
92C2×D8×C53—44no—
93C22×C8×C53—21no—
94#127×C53—21no—
95#127×C53—21no—
96#127×C53—21no—
97#127×C53C2312yes—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53D411yescharacteristic
101C4×D4×C53D411yescharacteristic
102C4×D4×C53C2311yescharacteristic
103C4×D4×C53C2311yescharacteristic
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—21no—
108C4⋊C8×C53C2312yes—
109#102×C53—22no—
110#102×C53—22no—
111C24×C53—21no—
112C24×C53—41no—
113C2×D4×C53—22no—
114C2×D4×C53—21no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—41no—
119C2×D4×C53—42no—
120C2×D4×C53—42no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53—42no—
125C2×D4×C53—42no—
126C2×D4×C53—82no—
127C2×D4×C53C2×D411yescharacteristic
128C2×D4×C53C2×D411yescharacteristic
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53#4111yescharacteristic
139C2×Dic106—21no—
140C2×Dic106—21no—
141D8×C53—44no—
142D8×C53—44no—
143C2×C8×C53—22no—
144C2×C8×C53—42no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53C2×D411yescharacteristic
147C4⋊C4×C53C2×D411yescharacteristic
148C22⋊C4×C53—21no—
149C22⋊C4×C53—21no—
150C22⋊C4×C53—21no—
151C22⋊C4×C53—21no—
152C42×C53C2×D411yescharacteristic
153C42×C53C2×D411yescharacteristic
154C42×C53#4111yescharacteristic
155C42×C53C2411yescharacteristic
156C22⋊Dic53—41no—
157#11719923—42no—
158#11719923—42no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163#11719919—41no—
164#11719919—41no—
165C4×Dic53—41no—
166#11719917—22no—
167#11719917—22no—
168#11719917—44no—
169#11719917—42no—
170C2×#88322—21no—
171C2×#88322—21no—
172C2×#88322—22no—
173C2×#88322—44no—
174C2×#88322C2×D412yes—
175C2×#88322C2×D412yes—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—81no—
182C23×C53—82no—
183C23×C53—82no—
184C23×C53—82no—
185C23×C53#12111yescharacteristic
186D4×C53—21no—
187D4×C53—21no—
188D4×C53—21no—
189D4×C53—42no—
190D4×C53—42no—
191D4×C53—42no—
192D4×C53—42no—
193D4×C53—42no—
194D4×C53—82no—
195D4×C53—82no—
196D4×C53—82no—
197D4×C53—82no—
198D4×C53AGL(1,ℤ/8)12yes—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—41no—
204C2×C4×C53—42no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53#12111yescharacteristic
212C2×C4×C53#12111yescharacteristic
213C2×C4×C53#12111yescharacteristic
214C2×C4×C53#13512yes—
215C2×C4×C53AGL(1,ℤ/8)12yes—
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53C22×D411yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×Dic53—41no—
220C2×Dic53—41no—
221Dic106—41no—
222Dic106—41no—
223Dic106—81no—
224C8×C53—42no—
225#88322—22no—
226#88322—22no—
227#88322—42no—
228#88322—41no—
229#88322—41no—
230#19166—531nomaximal, Sylow
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—22no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53—41no—
239C22×C53—84no—
240C22×C53—82no—
241C22×C53—82no—
242C22×C53—82no—
243C22×C53—164no—
244C22×C53#46312yes—
245C22×C53C2×#12111yescharacteristic
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—21no—
249C4×C53—41no—
250C4×C53—41no—
251C4×C53#46312yes—
252C4×C53C2×#13511yescharacteristic
253C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C53#57511yescharacteristic
256Dic53—81no—
257Dic53—81no—
258#3138—531no—
259#3101—531no—
260C2×#459D5311yescharacteristic
261D4×M16—531no—
262#1701—532no—
263#1697—532no—
264#1681—532no—
265#1677—532no—
266#1565—531no—
267#1550—531no—
268#1545—531no—
269C2×C53—22no—
270C2×C53—42no—
271C2×C53—81no—
272C2×C53—162no—
273C2×C53C2×#46311yescharacteristic
274C2×C53#305511yescharacteristic
275C2×C53#306811yescharacteristic
276C22×D8—21no—
277C22×M16—1061no—
278#533—531no—
279C2×#127C2×D5311yescharacteristic
280C2×C4×D4C2×D5311yescharacteristic
281#490—1062no—
282#486—532no—
283#486—532no—
284#483—1061no—
285#476—531no—
286#476—531no—
287#461—531no—
288#461—531no—
289#459—21no—
290#459—21no—
291#459C2×D5312yes—
292#459C2×D5312yes—
293#450—1061no—
294C4×SD16—1061no—
295C4×SD16—1061no—
296#436—532no—
297#435—1062no—
298C8×D4—532no—
299#431—531no—
300#423—531no—
301C2×C4⋊C8C2×D5311yescharacteristic
302#418—1061no—
303#418—1061no—
304C2×#102—21no—
305#407—1061no—
306C4×M16—531no—
307#335—532no—
308#334—532no—
309#333—532no—
310#332—532no—
311#331—532no—
312#331—532no—
313C53#1916611yescharacteristic, complemented, Sylow
314C22×D4—21no—
315C22×D4—21no—
316C22×D4—21no—
317C22×D4C22⋊D5311yescharacteristic
318C23×C4—21no—
319C2×SD16—1061no—
320C2×SD16—1061no—
321C2×D8—22no—
322C2×D8—22no—
323C2×D8—44no—
324C2×M16—531no—
325C2×M16—531no—
326C2×M16—1062no—
327C2×M16—1061no—
328C2×M16—2124no—
329C22×C8—21no—
330C22×C8—1062no—
331C4⋊Q8—531no—
332C4⋊Q8—531no—
333#127—21no—
334#127—21no—
335#127—21no—
336#127C22×D5312yes—
337#125—1061no—
338C22⋊Q8—1061no—
339C22⋊Q8—1061no—
340C4×Q8—1061no—
341C4×Q8—1061no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4C22⋊D5311yescharacteristic
345C4×D4C22⋊D5311yescharacteristic
346C4×D4C22×D5311yescharacteristic
347C4×D4C22×D5311yescharacteristic
348#117—1061no—
349C2×C4⋊C4C22⋊D5311yescharacteristic
350C2×C22⋊C4—21no—
351C2×C42C22×D5311yescharacteristic
352#106—1061no—
353#106—1061no—
354C4⋊C8—21no—
355C4⋊C8—532no—
356C4⋊C8—531no—
357C4⋊C8—531no—
358C4⋊C8—1061no—
359C4⋊C8C22×D5312yes—
360#103—1061no—
361#103—1061no—
362#102—22no—
363#102—22no—
364#102—1062no—
365#102—1062no—
366#102—1062no—
367#102—1062no—
368#102—1061no—
369#102—1061no—
370C22⋊C8—1062no—
371C22⋊C8—1062no—
372C22⋊C8—1061no—
373#97—532no—
374C4×C8—532no—
375C4×C8—1061no—
376C24—21no—
377C24—41no—
378C2×Q8—1061no—
379C2×Q8—1061no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—41no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—82no—
394C2×D4C2×C22⋊D5311yescharacteristic
395C2×D4C2×C22⋊D5311yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4D4×D5311yescharacteristic
404C22×C4#1171994711yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406SD16—2122no—
407SD16—2122no—
408D8—44no—
409D8—44no—
410M16—1062no—
411M16—1062no—
412M16—2124no—
413M16—2122no—
414C2×C8—22no—
415C2×C8—42no—
416C2×C8—532no—
417C2×C8—532no—
418C2×C8—1062no—
419C2×C8—1061no—
420C2×C8—1061no—
421C2×C8—2124no—
422C4⋊C4—21no—
423C4⋊C4—1061no—
424C4⋊C4—1061no—
425C4⋊C4—1061no—
426C4⋊C4—1061no—
427C4⋊C4—2121no—
428C4⋊C4—2121no—
429C4⋊C4C2×C22⋊D5311yescharacteristic
430C4⋊C4C2×C22⋊D5311yescharacteristic
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—2121no—
436C42—2121no—
437C42D4×D5311yescharacteristic
438C42#1171994711yescharacteristic
439C42C2×C22⋊D5311yescharacteristic
440C42C23×D5311yescharacteristic
441C23—21no—
442C23—21no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—81no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23#1313318911yescharacteristic
451Q8—4241no—
452Q8—2121no—
453Q8—2121no—
454D4—21no—
455D4—21no—
456D4—21no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—82no—
463D4—82no—
464D4—82no—
465D4—82no—
466D4#1313324312yes—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4#1313318911yescharacteristic
482C2×C4#1313318911yescharacteristic
483C2×C4#1313318911yescharacteristic
484C2×C4#1313323612yes—
485C2×C4#1313324112yes—
486C2×C4C2×D4×D5311yescharacteristic
487C2×C4C22×C22⋊D5311yescharacteristic
488C2×C4C2×#1171994711yescharacteristic
489C8—42no—
490C8—1062no—
491C8—1062no—
492C8—2122no—
493C8—2121no—
494C8—2121no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—84no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—164no—
508C22#1681083612yes—
509C22C2×#1313318911yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—4241no—
516C4—4241no—
517C4#1681083612yes—
518C4C2×#1313323611yescharacteristic
519C4C2×#1313324111yescharacteristic
520C4C2×#1313324311yescharacteristic
521C4#1681142111yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—162no—
526C2C2×#1681083611yescharacteristic
527C2#2028433011yescharacteristic
528C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC2×C53×#459C2
socleC22×C53C2×#121
hypercenterC2×#459D53
hypocenterC53#19166
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. #20285089C2semidirect
2. #20284723C2semidirect
3. D4×#11719917C2semidirect
4. #20274274C2semidirect
5. #20274246C2semidirect
6. #20274150C2semidirect
7. #20274122C2semidirect
8. #20273465C2semidirect
9. #20273403C2semidirect
10. #20273383C2semidirect
15. #16810969C22semidirect
16. #16810969C22semidirect
18. #16810913C22semidirect
19. #16810913C22semidirect
20. #16810828C22semidirect
21. #16810828C22semidirect
25. #16810701C22semidirect
27. D4×#88322C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810371C22semidirect
37. #16810370C22semidirect
38. #16810367C22semidirect
39. #16810367C22semidirect
313. C53#19166semidirect

Maximal subgroups

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

subgroupindex
#202850892
#202847232
D4×#117199172
#202742742
#202742462
#202741502
#202741222
#202734652
#202734032
#202733832
C2×C53×#4592
#1916653