Finite Group Databasebeta

← #27540952

Subgroups — #27540952

Order 13568.

580 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#20287118C211yesmaximal, characteristic, complemented
2#20276946C211yesmaximal, characteristic, complemented
3#20276915C211yesmaximal, characteristic, complemented
4#20276903C211yesmaximal, characteristic, complemented
5#20276872C211yesmaximal, characteristic, complemented
6#20276813C211yesmaximal, characteristic, complemented
7#20276799C211yesmaximal, characteristic, complemented
8#20276767C211yesmaximal, characteristic, complemented
9#20276638C211yesmaximal, characteristic, complemented
10#20276627C211yesmaximal, characteristic, complemented
11C4×#13133158C211yesmaximal, characteristic, complemented
12D4×#11719908C211yesmaximal, characteristic, complemented
13#20276422C211yesmaximal, characteristic, complemented
14C2×#16810424C211yesmaximal, characteristic, complemented
15C53×#3280C211yesmaximal, characteristic, complemented
16#16811319—21no—
17C2×#13133160C2211yescharacteristic, complemented
18C2×#13133158C2211yescharacteristic
19C2×#13133158C2211yescharacteristic
20C22×#11719908C2211yescharacteristic
21C22×#11719908C2211yescharacteristic, complemented
22#16811025C2211yescharacteristic, complemented
23#16810948—21no—
24#16810942—21no—
25#16810924—21no—
26#16810859C2211yescharacteristic
27#16810859C2211yescharacteristic, complemented
28C4×D424C2211yescharacteristic, complemented
29#16810661C2211yescharacteristic, complemented
30C2×#13133151C2211yescharacteristic
31#16810450C2211yescharacteristic, complemented
32#16810448C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810427C2211yescharacteristic, complemented
36#16810427C2211yescharacteristic, complemented
37#16810427C2211yescharacteristic, complemented
38#16810424C2211yescharacteristic
39#16810424C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810423C2211yescharacteristic, complemented
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic, complemented
47C4×#11719908C2211yescharacteristic, complemented
48C2×#135×C53C2211yescharacteristic
49C22×D8×C53C2211yescharacteristic
50C53×#546—21no—
51C53×#495C2211yescharacteristic
52C53×#467C2211yescharacteristic
53C53×#466C2211yescharacteristic
54C53×#463—21no—
55C53×#450—21no—
56C53×#449—21no—
57C4×D8×C53C2211yescharacteristic
58C8×D4×C53C2211yescharacteristic
59C2×D424C2311yescharacteristic
60#13133212—41no—
61#13133209—21no—
62#13133199—41no—
63#13133191—21no—
64#13133191—21no—
65#13133189—41no—
66C4×D212—21no—
67#13133177—41no—
68#13133176—21no—
69C2×C4⋊Dic53—21no—
70#13133160—41no—
71#13133158—22no—
72#13133158—41no—
73C2×#11719908—41no—
74C2×#11719908—41no—
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79#13133151—21no—
80#13133151C2311yescharacteristic
81#13133151C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133149—21no—
84#13133149—21no—
85#13133139—21no—
86#13133139—21no—
87#13133139—21no—
88C2×#41×C53—21no—
89C22×D4×C53—21no—
90#135×C53—22no—
91#135×C53—41no—
92C2×Q16×C53C2311yescharacteristic
93C2×SD16×C53—21no—
94C2×SD16×C53—21no—
95C2×D8×C53—21no—
96C2×D8×C53—22no—
97C2×D8×C53—41no—
98C2×D8×C53C2311yescharacteristic
99C2×D8×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101C22×C8×C53C2311yescharacteristic
102#124×C53—21no—
103#123×C53—41no—
104C22⋊Q8×C53—21no—
105#121×C53—21no—
106#121×C53—41no—
107#120×C53—41no—
108C4×D4×C53—21no—
109C4×D4×C53D411yescharacteristic
110C2×C22⋊C4×C53—41no—
111#107×C53C2311yescharacteristic
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#102×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C24×C53—41no—
120#41×C53—42no—
121#41×C53—81no—
122C2×Q8×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—42no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C2×D4×C53—41no—
129C2×D4×C53—81no—
130C2×D4×C53D811yescharacteristic
131C22×C4×C53—41no—
132C22×C4×C53—41no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C2×C22⋊D53—41no—
136C2×C22⋊D53—41no—
137C22×Dic53—41no—
138C2×D212—21no—
139C2×C4×D53—41no—
140Q16×C53—21no—
141Q16×C53—21no—
142SD16×C53—42no—
143SD16×C53—81no—
144D8×C53—21no—
145D8×C53—21no—
146D8×C53—21no—
147D8×C53—42no—
148D8×C53#4112yes—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53C2×D411yescharacteristic
154C2×C8×C53C2×D411yescharacteristic
155C2×C8×C53C2×D411yescharacteristic
156C2×C8×C53C2411yescharacteristic
157C4⋊C4×C53—21no—
158C4⋊C4×C53—41no—
159C4⋊C4×C53D811yescharacteristic
160C22⋊C4×C53—41no—
161C22⋊C4×C53—41no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53—41no—
164C22⋊C4×C53—41no—
165C22⋊C4×C53D811yescharacteristic
166C22⋊C4×C53D811yescharacteristic
167C42×C53C2×D411yescharacteristic
168C22⋊Dic53—41no—
169#11719921—41no—
170#11719921—41no—
171#11719921—41no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—41no—
176#11719919—41no—
177#11719919—41no—
178C4×Dic53—41no—
179D424—21no—
180D424—21no—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—82no—
188C23×C53—82no—
189C23×C53—161no—
190C23×C53C2×D811yescharacteristic
191C23×C53C2×D811yescharacteristic
192C22×D53—41no—
193Q8×C53—41no—
194Q8×C53—41no—
195D4×C53—22no—
196D4×C53—22no—
197D4×C53—41no—
198D4×C53—41no—
199D4×C53—41no—
200D4×C53—82no—
201D4×C53—82no—
202D4×C53—161no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—82no—
210C2×C4×C53—161no—
211C2×C4×C53C2×D811yescharacteristic
212C2×C4×C53C2×D811yescharacteristic
213C2×C4×C53C2×D811yescharacteristic
214C2×C4×C53C2×D811yescharacteristic
215C2×C4×C53C22×D411yescharacteristic
216C22⋊D53—82no—
217C22⋊D53—161no—
218C2×Dic53—41no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—161no—
222D212—41no—
223D212—41no—
224C4×D53—161no—
225C8×C53—21no—
226C8×C53—21no—
227C8×C53—21no—
228C8×C53C2×#4111yescharacteristic
229C8×C53H(5,𝔽2)11yescharacteristic
230#33864—531nomaximal, Sylow
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—42no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53—164no—
243C22×C53#50812yes—
244C22×C53C22×D811yescharacteristic
245C2×D53—81no—
246C2×D53—81no—
247C2×D53—81no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—81no—
253C4×C53—81no—
254C4×C53#54611yescharacteristic
255C4×C53#57711yescharacteristic
256Dic53—81no—
257Dic53—81no—
258Dic53—81no—
259#3339—531no—
260#3280D5311yescharacteristic, complemented
261#2242—531no—
262#2236—531no—
263#2234—531no—
264#2228—531no—
265#2217—531no—
266#2215—531no—
267#2210—531no—
268#2186—531no—
269#2184—531no—
270C4×D16—531no—
271D4×C16—531no—
272#2138—531no—
273C2×#357—531no—
274C2×C53—22no—
275C2×C53—41no—
276C2×C53—82no—
277C2×C53—161no—
278C2×C53#329411yescharacteristic
279C2×C53C2×#50811yescharacteristic
280C2×C53#341511yescharacteristic
281D53—161no—
282C2×#135C2×D5311yescharacteristic
283C22×D8C2×D5311yescharacteristic
284#553—1061no—
285#546—21no—
286C2×SD32—531no—
287C2×D16—531no—
288C2×D16—531no—
289C22×C16—531no—
290C22×C16—531no—
291#499—531no—
292#495C2×D5311yescharacteristic
293#482—1061no—
294#481—1061no—
295#478—1061no—
296#467C2×D5311yescharacteristic
297#466—531no—
298#466—531no—
299#466C2×D5311yescharacteristic
300#463—21no—
301#450—21no—
302#449—21no—
303C4×D8—531no—
304C4×D8C2×D5311yescharacteristic
305C8×D4C2×D5311yescharacteristic
306#427—531no—
307C2×#107—531no—
308#367—531no—
309#366—531no—
310#366—531no—
311C4⋊C16—531no—
312#358—531no—
313#358—531no—
314#358—531no—
315#357—531no—
316#357—531no—
317#357—531no—
318#357—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322C22⋊C16—531no—
323C22⋊C16—531no—
324C4×C16—531no—
325C53#3386411yescharacteristic, complemented, Sylow
326C2×#41—21no—
327C22×D4—21no—
328#135—22no—
329#135—41no—
330C2×Q16C22×D5311yescharacteristic
331C2×SD16—21no—
332C2×SD16—21no—
333C2×D8—22no—
334C2×D8—21no—
335C2×D8—41no—
336C2×D8—531no—
337C2×D8C22×D5311yescharacteristic
338C2×D8C22×D5311yescharacteristic
339C22×C8C22×D5311yescharacteristic
340C22×C8C22×D5311yescharacteristic
341#126—2121no—
342#125—1061no—
343#124—21no—
344#123—41no—
345#123—2121no—
346C22⋊Q8—21no—
347#121—21no—
348#121—41no—
349#121—1061no—
350#121—1061no—
351#121—2121no—
352#120—41no—
353C4×D4—21no—
354C4×D4—1061no—
355C4×D4—2121no—
356C4×D4C22⋊D5311yescharacteristic
357#117—1061no—
358C2×C4⋊C4—1061no—
359C2×C22⋊C4—41no—
360SD32—2121no—
361D16—1062no—
362D16—2121no—
363C2×C16—531no—
364C2×C16—531no—
365C2×C16—531no—
366C2×C16—531no—
367C2×C16—2121no—
368C2×C16—2121no—
369#107—531no—
370#107—531no—
371#107—531no—
372#107—1061no—
373#107C22×D5311yescharacteristic
374#106—1061no—
375#106—1061no—
376C4⋊C8C22⋊D5311yescharacteristic
377#103—21no—
378#102—21no—
379#102—21no—
380#102—1061no—
381#102—1061no—
382#102—1061no—
383C22⋊C8C22⋊D5311yescharacteristic
384C22⋊C8C22⋊D5311yescharacteristic
385C4×C8C22×D5311yescharacteristic
386C24—41no—
387#41—42no—
388#41—81no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—42no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—81no—
397C2×D4—1061no—
398C2×D4—2121no—
399C2×D4—2121no—
400C2×D4#1171992211yescharacteristic
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—2121no—
404C22×C4—2121no—
405C22×C4C2×C22⋊D5311yescharacteristic
406C22×C4C2×C22⋊D5311yescharacteristic
407Q16—21no—
408Q16—21no—
409SD16—42no—
410SD16—81no—
411D8—21no—
412D8—21no—
413D8—21no—
414D8—42no—
415D8—1061no—
416D8—1061no—
417D8#1171994712yes—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8C2×C22⋊D5311yescharacteristic
423C2×C8C2×C22⋊D5311yescharacteristic
424C2×C8C2×C22⋊D5311yescharacteristic
425C2×C8C23×D5311yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—1061no—
429C4⋊C4—1061no—
430C4⋊C4—1061no—
431C4⋊C4—2121no—
432C4⋊C4—2121no—
433C4⋊C4—2121no—
434C4⋊C4#1171992211yescharacteristic
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—2121no—
441C22⋊C4—2121no—
442C22⋊C4—2121no—
443C22⋊C4—2121no—
444C22⋊C4#1171992211yescharacteristic
445C22⋊C4#1171992211yescharacteristic
446C42—2121no—
447C42C2×C22⋊D5311yescharacteristic
448C16—1061no—
449C16—1061no—
450C16—1061no—
451C16—1061no—
452C23—41no—
453C23—41no—
454C23—82no—
455C23—82no—
456C23—161no—
457C23—2121no—
458C23C2×#1171992211yescharacteristic
459C23C2×#1171992211yescharacteristic
460Q8—41no—
461Q8—41no—
462D4—22no—
463D4—22no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—82no—
468D4—82no—
469D4—161no—
470D4—8481no—
471D4—4242no—
472D4—2121no—
473D4—2121no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—161no—
482C2×C4—8481no—
483C2×C4—8481no—
484C2×C4—2121no—
485C2×C4—2121no—
486C2×C4—2121no—
487C2×C4C2×#1171992211yescharacteristic
488C2×C4C2×#1171992211yescharacteristic
489C2×C4C2×#1171992211yescharacteristic
490C2×C4C2×#1171992211yescharacteristic
491C2×C4C22×C22⋊D5311yescharacteristic
492C8—21no—
493C8—21no—
494C8—21no—
495C8C2×#1171994711yescharacteristic
496C8#1313326211yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—164no—
509C22—4241no—
510C22—4241no—
511C22—4241no—
512C22#1681105312yes—
513C22C22×#1171992211yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—81no—
519C4—81no—
520C4—4241no—
521C4—4241no—
522C4—4241no—
523C4#1681126711yescharacteristic
524C4#1681143411yescharacteristic
525C2—22no—
526C2—41no—
527C2—82no—
528C2—161no—
529C2—8481no—
530C2#2028652811yescharacteristic
531C2C2×#1681105311yescharacteristic
532C2#2028785411yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3280C2
socleC22×C53C22×D8
hypercenter#3280D53
hypocenterC53#33864
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #20287118C2semidirect
2. #20276946C2semidirect
3. #20276915C2semidirect
4. #20276903C2semidirect
5. #20276872C2semidirect
6. #20276813C2semidirect
7. #20276799C2semidirect
8. #20276767C2semidirect
9. #20276638C2semidirect
10. #20276627C2semidirect
11. C4×#13133158C2semidirect
12. D4×#11719908C2semidirect
13. #20276422C2semidirect
14. C2×#16810424C2semidirect
15. C53×#3280C2semidirect
17. C2×#13133160C22semidirect
21. C22×#11719908C22semidirect
22. #16811025C22semidirect
27. #16810859C22semidirect
28. C4×D424C22semidirect
29. #16810661C22semidirect
31. #16810450C22semidirect
32. #16810448C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810427C22semidirect
36. #16810427C22semidirect
37. #16810427C22semidirect
44. #16810423C22semidirect
46. C22⋊#11719908C22semidirect
47. C4×#11719908C22semidirect
260. #3280D53semidirect
325. C53#33864semidirect

Maximal subgroups

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

subgroupindex
#202871182
#202769462
#202769152
#202769032
#202768722
#202768132
#202767992
#202767672
#202766382
#202766272
C4×#131331582
D4×#117199082
#202764222
C2×#168104242
C53×#32802
#3386453