Finite Group Databasebeta

← #27370032

Subgroups — #27370032

Order 13568.

744 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#20285633C211yesmaximal, characteristic, complemented
2#20284806C211yesmaximal, characteristic, complemented
3#20283750C211yesmaximal, characteristic, complemented
4#20274246C212yesmaximal, complemented
5#20274192C212yesmaximal, complemented
6#20274122C212yesmaximal, complemented
7#20274068C212yesmaximal, complemented
8#20273498C211yesmaximal, characteristic, complemented
9#20273429C211yesmaximal, characteristic, complemented
10#20273408C211yesmaximal, characteristic, complemented
11C2×C53×#459C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16810999C2212yescomplemented
14#16810969C2212yescomplemented
15#16810941—21no—
16#16810913C2211yescharacteristic, complemented
17#16810913C2211yescharacteristic, complemented
18#16810818C2211yescharacteristic, complemented
19#16810818C2211yescharacteristic, complemented
20#16810815—22no—
21#16810757—21no—
22#16810757—21no—
23#16810757—21no—
24#16810733—21no—
25#16810733—21no—
26#16810701C2212yescomplemented
27#16810695—22no—
28D4×#88322C2212yescomplemented
29#16810680C2211yescharacteristic
30#16810680C2211yescharacteristic
31#16810676C2211yescharacteristic
32#16810587—21no—
33C2×C22⋊#88322—21no—
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810371C2212yescomplemented
37#16810367C2212yescomplemented
38#16810367C2212yescomplemented
39#16810366C2212yescomplemented
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×#11719945—21no—
50C2×#11719922—22no—
51C2×#11719922—22no—
52C2×#11719922—21no—
53C2×#11719922—21no—
54C2×#11719917—22no—
55C22×#88322—22no—
56#13133217C2311yescharacteristic
57#13133214C2311yescharacteristic
58#13133205—21no—
59#13133191—21no—
60#13133191—21no—
61C4×D212—21no—
62C4×D212—21no—
63#13133148—21no—
64#13133148—21no—
65C4⋊#88322D412yes—
66C4⋊#88322C2311yescharacteristic
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2312yes—
69#13133138—22no—
70#13133138—22no—
71#13133138—21no—
72#13133138—21no—
73#13133137—21no—
74#13133137—21no—
75C22⋊#88322—22no—
76C22⋊#88322—22no—
77C22⋊#88322—21no—
78C22⋊#88322—21no—
79C22⋊#88322D411yescharacteristic
80C22⋊#88322D411yescharacteristic
81#13133128—21no—
82#13133128—21no—
83C4×#88322C2312yes—
84C22×D4×C53—21no—
85C22×D4×C53—21no—
86C22×D4×C53—21no—
87C22×D4×C53D411yescharacteristic
88C23×C4×C53—21no—
89C2×D8×C53—22no—
90C2×D8×C53—22no—
91C2×D8×C53—44no—
92C22×C8×C53—21no—
93#127×C53—21no—
94#127×C53—21no—
95#127×C53—21no—
96#127×C53C2312yes—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53D411yescharacteristic
100C4×D4×C53D411yescharacteristic
101C4×D4×C53C2311yescharacteristic
102C4×D4×C53C2311yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C42×C53C2311yescharacteristic
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108#102×C53—22no—
109#102×C53—22no—
110C24×C53—21no—
111C24×C53—41no—
112C2×D4×C53—22no—
113C2×D4×C53—21no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—41no—
118C2×D4×C53—42no—
119C2×D4×C53—42no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—82no—
124C2×D4×C53—82no—
125C2×D4×C53C2×D411yescharacteristic
126C2×D4×C53C2×D411yescharacteristic
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—82no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53#4111yescharacteristic
136C2×C22⋊D53—41no—
137#11719945—82no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×C4×D53—41no—
142C2×Dic106—21no—
143D8×C53—44no—
144D8×C53—44no—
145C2×C8×C53—22no—
146C2×C8×C53—42no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53C2×D411yescharacteristic
149C4⋊C4×C53C2×D411yescharacteristic
150C22⋊C4×C53—21no—
151C22⋊C4×C53—21no—
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C42×C53C2×D411yescharacteristic
155C42×C53C2×D411yescharacteristic
156C42×C53#4111yescharacteristic
157C42×C53C2411yescharacteristic
158#11719922—42no—
159#11719922—42no—
160#11719922—82no—
161#11719922—82no—
162#11719921—41no—
163#11719921—41no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166#11719917—82no—
167C2×#88322—22no—
168C2×#88322—21no—
169C2×#88322—21no—
170C2×#88322—82no—
171C2×#88322C2×D412yes—
172C2×#88322C2×D412yes—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53—81no—
179C23×C53—82no—
180C23×C53—82no—
181C23×C53—82no—
182C23×C53#12111yescharacteristic
183C22×D53—41no—
184D4×C53—21no—
185D4×C53—21no—
186D4×C53—21no—
187D4×C53—42no—
188D4×C53—42no—
189D4×C53—42no—
190D4×C53—42no—
191D4×C53—42no—
192D4×C53—82no—
193D4×C53—82no—
194D4×C53—82no—
195D4×C53—82no—
196D4×C53AGL(1,ℤ/8)12yes—
197C2×C4×C53—22no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—22no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—81no—
206C2×C4×C53—81no—
207C2×C4×C53—81no—
208C2×C4×C53#12111yescharacteristic
209C2×C4×C53#12111yescharacteristic
210C2×C4×C53#12111yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C22⋊D53—82no—
215C2×Dic53—41no—
216D212—41no—
217D212—41no—
218D212—42no—
219D212—81no—
220C4×D53—82no—
221Dic106—81no—
222C8×C53—42no—
223#88322—22no—
224#88322—22no—
225#88322—41no—
226#88322—41no—
227#88322—42no—
228#19458—531nomaximal, Sylow
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—22no—
232C22×C53—41no—
233C22×C53—41no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—84no—
238C22×C53—82no—
239C22×C53—82no—
240C22×C53—82no—
241C22×C53—164no—
242C22×C53#47212yes—
243C22×C53C2×#12111yescharacteristic
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—81no—
247C4×C53—21no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—41no—
252C4×C53—41no—
253C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C53#57511yescharacteristic
255C4×C53#57612yes—
256Dic53—81no—
257#3205—531no—
258#3109—531no—
259C2×#459D5311yescharacteristic, complemented
260#2974—531no—
261#1697—532no—
262#1687—532no—
263#1677—532no—
264#1667—532no—
265#1573—531no—
266#1555—531no—
267#1551—531no—
268C2×C53—22no—
269C2×C53—42no—
270C2×C53—81no—
271C2×C53—162no—
272C2×C53#304111yescharacteristic
273C2×C53#305511yescharacteristic
274C2×C53C2×#47211yescharacteristic
275D53—161no—
276C22×D8—21no—
277#532—531no—
278C2×#127C2×D5311yescharacteristic
279C2×C4×D4C2×D5311yescharacteristic
280#493—532no—
281#486—532no—
282#481—1061no—
283#476—531no—
284#476—531no—
285#459—21no—
286#459—21no—
287#459—531no—
288#459—531no—
289#459—1062no—
290#459C2×D5312yes—
291#459C2×D5312yes—
292#447—1061no—
293#447—1061no—
294#447—1061no—
295#442—1061no—
296#442—1061no—
297#436—532no—
298#435—1062no—
299C8×D4—532no—
300#432—531no—
301#432—531no—
302#431—531no—
303C2×C4⋊C8C2×D5311yescharacteristic
304C2×#102—21no—
305#407—1061no—
306C2×C22⋊C8—1061no—
307#335—532no—
308#334—532no—
309#333—532no—
310#331—532no—
311#331—532no—
312#331—532no—
313C53#1945811yescharacteristic, complemented, Sylow
314C2×#41—1061no—
315C22×D4—21no—
316C22×D4—21no—
317C22×D4—21no—
318C22×D4C22⋊D5311yescharacteristic
319C23×C4—21no—
320C2×D8—22no—
321C2×D8—22no—
322C2×D8—44no—
323C2×D8—1062no—
324C2×D8—1062no—
325C2×D8—1061no—
326C2×D8—1061no—
327C2×M16—1062no—
328C22×C8—21no—
329C22×C8—1062no—
330C4⋊Q8—531no—
331#127—21no—
332#127—21no—
333#127—21no—
334#127—531no—
335#127C22×D5312yes—
336#124—1061no—
337#121—1061no—
338#121—1061no—
339C4×D4—21no—
340C4×D4—21no—
341C4×D4—1061no—
342C4×D4—1061no—
343C4×D4C22⋊D5311yescharacteristic
344C4×D4C22⋊D5311yescharacteristic
345C4×D4C22×D5311yescharacteristic
346C4×D4C22×D5311yescharacteristic
347C2×C4⋊C4C22⋊D5311yescharacteristic
348C2×C22⋊C4—21no—
349C2×C42C22×D5311yescharacteristic
350#106—1061no—
351#106—1061no—
352C4⋊C8—21no—
353C4⋊C8—532no—
354C4⋊C8—532no—
355C4⋊C8—531no—
356C4⋊C8—531no—
357C4⋊C8C22×D5312yes—
358#102—22no—
359#102—22no—
360#102—1062no—
361#102—1062no—
362#102—1061no—
363#102—1061no—
364#102—1061no—
365#102—1061no—
366C22⋊C8—531no—
367C22⋊C8—531no—
368C22⋊C8—1062no—
369C22⋊C8—1062no—
370C22⋊C8—1061no—
371C22⋊C8—1061no—
372#97—1061no—
373#97—1061no—
374C4×C8—532no—
375C24—21no—
376C24—41no—
377#41—4242no—
378C2×Q8—1061no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—41no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4—1061no—
393C2×D4—1061no—
394C2×D4—1061no—
395C2×D4—2121no—
396C2×D4C2×C22⋊D5311yescharacteristic
397C2×D4C2×C22⋊D5311yescharacteristic
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—82no—
404C22×C4—2121no—
405C22×C4D4×D5311yescharacteristic
406C22×C4#1171994711yescharacteristic
407C22×C4C2×C22⋊D5311yescharacteristic
408D8—44no—
409D8—44no—
410D8—4242no—
411D8—4242no—
412D8—2122no—
413D8—2122no—
414M16—4242no—
415C2×C8—22no—
416C2×C8—42no—
417C2×C8—532no—
418C2×C8—532no—
419C2×C8—1062no—
420C2×C8—1061no—
421C2×C8—1061no—
422C2×C8—4242no—
423C4⋊C4—21no—
424C4⋊C4—1061no—
425C4⋊C4—1061no—
426C4⋊C4C2×C22⋊D5311yescharacteristic
427C4⋊C4C2×C22⋊D5311yescharacteristic
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—21no—
432C22⋊C4—2121no—
433C22⋊C4—2121no—
434C42D4×D5311yescharacteristic
435C42#1171994711yescharacteristic
436C42C2×C22⋊D5311yescharacteristic
437C42C23×D5311yescharacteristic
438C23—21no—
439C23—21no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—81no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—2121no—
448C23#1313318911yescharacteristic
449Q8—4241no—
450D4—21no—
451D4—21no—
452D4—21no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—82no—
462D4—4241no—
463D4—4242no—
464D4—2121no—
465D4—2121no—
466D4—2122no—
467D4#1313324312yes—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—4242no—
480C2×C4—2121no—
481C2×C4#1313318911yescharacteristic
482C2×C4#1313318911yescharacteristic
483C2×C4#1313318911yescharacteristic
484C2×C4C2×D4×D5311yescharacteristic
485C2×C4C22×C22⋊D5311yescharacteristic
486C2×C4C2×#1171994711yescharacteristic
487C8—42no—
488C8—1062no—
489C8—1062no—
490C8—2121no—
491C8—2121no—
492C8—2122no—
493C22—21no—
494C22—21no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—84no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—164no—
506C22—4241no—
507C22—4241no—
508C22—4241no—
509C22#1681089112yes—
510C22C2×#1313318911yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—41no—
516C4—41no—
517C4—4241no—
518C4C2×#1313324311yescharacteristic
519C4#1681142111yescharacteristic
520C4#1681142412yes—
521C2—22no—
522C2—42no—
523C2—81no—
524C2—162no—
525C2—8481no—
526C2#2028421511yescharacteristic
527C2#2028433011yescharacteristic
528C2C2×#1681089111yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#459C2
socleC22×C53C2×#121
hypercenterC2×#459D53
hypocenterC53#19458
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. #20285633C2semidirect
2. #20284806C2semidirect
3. #20283750C2semidirect
4. #20274246C2semidirect
5. #20274192C2semidirect
6. #20274122C2semidirect
7. #20274068C2semidirect
8. #20273498C2semidirect
9. #20273429C2semidirect
10. #20273408C2semidirect
11. C2×C53×#459C2semidirect
13. #16810999C22semidirect
14. #16810969C22semidirect
16. #16810913C22semidirect
17. #16810913C22semidirect
18. #16810818C22semidirect
19. #16810818C22semidirect
26. #16810701C22semidirect
28. D4×#88322C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810371C22semidirect
37. #16810367C22semidirect
38. #16810367C22semidirect
39. #16810366C22semidirect
259. C2×#459D53semidirect
313. C53#19458semidirect

Maximal subgroups

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

subgroupindex
#202856332
#202848062
#202837502
#202742462
#202741922
#202741222
#202740682
#202734982
#202734292
#202734082
C2×C53×#4592
#1945853