Finite Group Databasebeta

← #27541117

Subgroups — #27541117

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D424C211yesmaximal, characteristic
2#20276958C211yesmaximal, characteristic, complemented
3#20276906C211yesmaximal, characteristic, complemented
4#20276895C211yesmaximal, characteristic, complemented
5#20276869C211yesmaximal, characteristic, complemented
6#20276830C211yesmaximal, characteristic, complemented
7#20276813C211yesmaximal, characteristic, complemented
8#20276755C211yesmaximal, characteristic, complemented
9#20276643C211yesmaximal, characteristic, complemented
10#20276643C211yesmaximal, characteristic, complemented
11#20276577C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276430C211yesmaximal, characteristic, complemented
14C2×#16810423C211yesmaximal, characteristic, complemented
15C53×#3336C211yesmaximal, characteristic, complemented
16C22×D424C2211yescharacteristic
17C22×D424C2211yescharacteristic
18D4×D212—21no—
19C2×#13133160C2211yescharacteristic
20C2×#13133160C2211yescharacteristic, complemented
21C2×#13133160C2211yescharacteristic, complemented
22C2×#13133157C2211yescharacteristic, complemented
23C22×#11719908C2211yescharacteristic, complemented
24#16811000C2211yescharacteristic
25#16810859C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810819—21no—
28#16810759—21no—
29#16810759—21no—
30C4×D424C2211yescharacteristic
31#16810450C2211yescharacteristic, complemented
32#16810448C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810427C2211yescharacteristic, complemented
36#16810423C2211yescharacteristic
37#16810423C2211yescharacteristic
38#16810423C2211yescharacteristic
39#16810423C2211yescharacteristic, complemented
40#16810423C2211yescharacteristic, complemented
41#16810423C2211yescharacteristic, complemented
42#16810423C2211yescharacteristic, complemented
43#16810423C2211yescharacteristic, complemented
44#16810423C2211yescharacteristic, complemented
45C22⋊#11719908C2211yescharacteristic, complemented
46C22⋊#11719908C2211yescharacteristic, complemented
47#16810399C2211yescharacteristic, complemented
48C53×#552—21no—
49C53×#499C2211yescharacteristic, complemented
50C53×#485—21no—
51C53×#483—21no—
52C53×#479—21no—
53C53×#467C2211yescharacteristic, complemented
54C53×#467C2211yescharacteristic, complemented
55C4×Q16×C53C2211yescharacteristic, complemented
56C8×D4×C53C2211yescharacteristic
57C53×#427C2211yescharacteristic, complemented
58C2×#107×C53C2211yescharacteristic, complemented
59C22×D212—21no—
60C22×D212—21no—
61C2×D4×D53—41no—
62C2×D424—21no—
63C2×D424—21no—
64C2×D424—21no—
65C2×D424—22no—
66C2×D424—22no—
67C2×D424—41no—
68C2×D424—41no—
69C2×D424C2311yescharacteristic
70C2×D424C2311yescharacteristic
71C2×D424C2311yescharacteristic
72#13133214—21no—
73#13133192—41no—
74#13133191—21no—
75#13133191—21no—
76#13133187—41no—
77#13133187—41no—
78C4×D212—21no—
79#13133160—22no—
80#13133160—41no—
81#13133160—41no—
82#13133157—41no—
83C2×#11719908—41no—
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133139—21no—
90#13133139—21no—
91#13133139—21no—
92C2×Q16×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95#126×C53—41no—
96#125×C53—21no—
97#125×C53—41no—
98#123×C53—41no—
99C22⋊Q8×C53—21no—
100C22⋊Q8×C53—21no—
101C4×Q8×C53—21no—
102C4×D4×C53D411yescharacteristic
103#117×C53—21no—
104#117×C53—41no—
105C2×C4⋊C4×C53—21no—
106#107×C53—21no—
107#107×C53C2311yescharacteristic
108#107×C53C2311yescharacteristic
109#107×C53C2311yescharacteristic
110#106×C53—21no—
111#106×C53—21no—
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#103×C53—21no—
115#103×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C23×D53—41no—
120C23×D53—41no—
121C2×Q8×C53—21no—
122C2×D4×C53D811yescharacteristic
123C22×C4×C53—41no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C2×C22⋊D53—41no—
127C2×C22⋊D53—41no—
128D4×D53—84no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—42no—
133C2×D212—42no—
134C2×D212—41no—
135C2×D212—41no—
136C2×D212—41no—
137C2×D212—41no—
138C2×D212—81no—
139C2×D212—81no—
140C2×C4×D53—41no—
141Q16×C53—21no—
142Q16×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53C2×D411yescharacteristic
148C2×C8×C53C2×D411yescharacteristic
149C2×C8×C53C2×D411yescharacteristic
150C2×C8×C53C2411yescharacteristic
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—41no—
155C4⋊C4×C53—41no—
156C4⋊C4×C53—41no—
157C4⋊C4×C53—41no—
158C4⋊C4×C53—41no—
159C4⋊C4×C53—41no—
160C4⋊C4×C53D811yescharacteristic
161C22⋊C4×C53—41no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53—41no—
164C22⋊C4×C53D811yescharacteristic
165C22⋊C4×C53D811yescharacteristic
166C42×C53—41no—
167C42×C53—41no—
168C42×C53C2×D411yescharacteristic
169#11719921—41no—
170#11719921—41no—
171#11719921—41no—
172C4⋊Dic53—21no—
173D424—21no—
174D424—21no—
175D424—21no—
176D424—21no—
177D424—21no—
178D424—42no—
179D424—42no—
180D424—81no—
181D424#4112yes—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53C2×D811yescharacteristic
187C23×C53C2×D811yescharacteristic
188C22×D53—41no—
189C22×D53—41no—
190C22×D53—41no—
191C22×D53—82no—
192C22×D53—82no—
193C22×D53—82no—
194C22×D53—82no—
195C22×D53—161no—
196C22×D53—161no—
197Q8×C53—41no—
198Q8×C53—41no—
199D4×C53—22no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—82no—
209C2×C4×C53C2×D811yescharacteristic
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C2×D811yescharacteristic
212C2×C4×C53C2×D811yescharacteristic
213C2×C4×C53C22×D411yescharacteristic
214C22⋊D53—82no—
215C22⋊D53—82no—
216C2×Dic53—41no—
217D212—22no—
218D212—41no—
219D212—41no—
220D212—41no—
221D212—41no—
222D212—41no—
223D212—82no—
224D212—82no—
225D212—82no—
226D212—161no—
227C4×D53—82no—
228C8×C53—21no—
229C8×C53—21no—
230C8×C53—21no—
231C8×C53C2×#4111yescharacteristic
232C8×C53H(5,𝔽2)11yescharacteristic
233#33877—531nomaximal, Sylow
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53#50912yes—
240C22×C53C22×D811yescharacteristic
241C2×D53—42no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—81no—
247C2×D53—81no—
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—164no—
251C2×D53—164no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—81no—
257C4×C53—81no—
258C4×C53—81no—
259C4×C53—81no—
260C4×C53#54611yescharacteristic
261C4×C53#57711yescharacteristic
262Dic53—81no—
263#3336D5311yescharacteristic, complemented
264D4×D8—531no—
265#2244—531no—
266#2234—531no—
267#2232—531no—
268#2228—531no—
269#2220—531no—
270#2217—531no—
271#2207—531no—
272#2187—531no—
273#2187—531no—
274#2175—531no—
275#2168—531no—
276#2140—531no—
277C2×#357—531no—
278C2×C53—22no—
279C2×C53—41no—
280C2×C53#329411yescharacteristic
281C2×C53C2×#50911yescharacteristic
282C2×C53#341611yescharacteristic
283D53—82no—
284D53—161no—
285D53—161no—
286C22×D8—531no—
287C22×D8—531no—
288#552—21no—
289D42—1061no—
290C2×SD32—531no—
291C2×SD32—531no—
292C2×SD32—531no—
293C2×M32—531no—
294C22×C16—531no—
295#499C2×D5311yescharacteristic, complemented
296#493—531no—
297#485—21no—
298#483—21no—
299#479—21no—
300#467C2×D5311yescharacteristic, complemented
301#467C2×D5311yescharacteristic, complemented
302#466—531no—
303#466—531no—
304#459—1061no—
305#447—1061no—
306#447—1061no—
307C4×Q16C2×D5311yescharacteristic, complemented
308C4×D8—531no—
309C8×D4C2×D5311yescharacteristic
310#427C2×D5311yescharacteristic, complemented
311C2×#107C2×D5311yescharacteristic, complemented
312#367—531no—
313#366—531no—
314#366—531no—
315C4⋊C16—531no—
316#358—531no—
317#357—531no—
318#357—531no—
319#357—531no—
320#357—531no—
321#357—531no—
322#357—531no—
323#357—531no—
324#357—531no—
325#357—531no—
326C22⋊C16—531no—
327C22⋊C16—531no—
328#346—531no—
329C53#3387711yescharacteristic, complemented, Sylow
330C22×D4—1061no—
331C22×D4—1061no—
332C22×D4—2121no—
333C2×Q16C22×D5311yescharacteristic
334C2×D8—531no—
335C2×D8—531no—
336C2×D8—531no—
337C2×D8—1062no—
338C2×D8—1062no—
339C2×D8—1061no—
340C2×D8—1061no—
341C2×D8—1061no—
342C2×D8—2121no—
343C2×D8—2121no—
344C22×C8C22×D5311yescharacteristic
345C22×C8C22×D5311yescharacteristic
346#127—1061no—
347#126—41no—
348#125—21no—
349#125—41no—
350#123—41no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353#121—1061no—
354#121—1061no—
355#121—2121no—
356#120—2121no—
357#120—2121no—
358C4×Q8—21no—
359C4×D4—1061no—
360C4×D4C22⋊D5311yescharacteristic
361#117—21no—
362#117—41no—
363C2×C4⋊C4—21no—
364SD32—1062no—
365SD32—2121no—
366SD32—2121no—
367M32—2121no—
368C2×C16—531no—
369C2×C16—531no—
370C2×C16—531no—
371C2×C16—531no—
372C2×C16—2121no—
373#107—21no—
374#107—531no—
375#107C22×D5311yescharacteristic
376#107C22×D5311yescharacteristic
377#107C22×D5311yescharacteristic
378#106—21no—
379#106—21no—
380C4⋊C8C22⋊D5311yescharacteristic
381#103—21no—
382#103—21no—
383#103—21no—
384#102—1061no—
385#102—1061no—
386#102—1061no—
387C22⋊C8C22⋊D5311yescharacteristic
388C22⋊C8C22⋊D5311yescharacteristic
389C4×C8C22×D5311yescharacteristic
390C24—2121no—
391C24—2121no—
392C2×Q8—21no—
393C2×D4—1061no—
394C2×D4—1061no—
395C2×D4—1061no—
396C2×D4—4241no—
397C2×D4—4241no—
398C2×D4—4244no—
399C2×D4—2122no—
400C2×D4—2122no—
401C2×D4—2121no—
402C2×D4—2121no—
403C2×D4—2121no—
404C2×D4—2121no—
405C2×D4—2121no—
406C2×D4—2121no—
407C2×D4#1171992211yescharacteristic
408C22×C4—41no—
409C22×C4—2121no—
410C22×C4C2×C22⋊D5311yescharacteristic
411C22×C4C2×C22⋊D5311yescharacteristic
412Q16—21no—
413Q16—21no—
414D8—532no—
415D8—1061no—
416D8—1061no—
417D8—1061no—
418D8—1061no—
419D8—1061no—
420D8—4241no—
421D8—2122no—
422D8—2122no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×C22⋊D5311yescharacteristic
428C2×C8C2×C22⋊D5311yescharacteristic
429C2×C8C2×C22⋊D5311yescharacteristic
430C2×C8C23×D5311yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—1061no—
441C4⋊C4#1171992211yescharacteristic
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—2121no—
446C22⋊C4—2121no—
447C22⋊C4—2121no—
448C22⋊C4#1171992211yescharacteristic
449C22⋊C4#1171992211yescharacteristic
450C42—41no—
451C42—41no—
452C42C2×C22⋊D5311yescharacteristic
453C16—1061no—
454C16—1061no—
455C16—1061no—
456C16—1061no—
457C23—8481no—
458C23—8481no—
459C23—4242no—
460C23—4242no—
461C23—4242no—
462C23—4242no—
463C23—2121no—
464C23—2121no—
465C23—2121no—
466C23C2×#1171992211yescharacteristic
467C23C2×#1171992211yescharacteristic
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—8481no—
472D4—1062no—
473D4—4242no—
474D4—4242no—
475D4—4242no—
476D4—4242no—
477D4—4242no—
478D4—2121no—
479D4—2121no—
480D4—2121no—
481D4—2121no—
482D4—2121no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4—4242no—
493C2×C4—2121no—
494C2×C4C2×#1171992211yescharacteristic
495C2×C4C2×#1171992211yescharacteristic
496C2×C4C2×#1171992211yescharacteristic
497C2×C4C2×#1171992211yescharacteristic
498C2×C4C22×C22⋊D5311yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8#41×D5311yescharacteristic
503C8#1313326311yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—8484no—
510C22—8484no—
511C22—4241no—
512C22—4241no—
513C22—4241no—
514C22—4241no—
515C22—4241no—
516C22—4241no—
517C22—4241no—
518C22—4241no—
519C22—2122no—
520C22#1681105812yes—
521C22C22×#1171992211yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—4241no—
531C4#1681126111yescharacteristic
532C4#1681143111yescharacteristic
533C2—22no—
534C2—41no—
535C2—8481no—
536C2—8481no—
537C2—4242no—
538C2#2028652211yescharacteristic
539C2C2×#1681105811yescharacteristic
540C2#2028785911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3336C2
socleC22×C53C22×D8
prime radicalD4×D424C2
hypercenter#3336D53
hypocenterC53#33877
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #20276958C2semidirect
3. #20276906C2semidirect
4. #20276895C2semidirect
5. #20276869C2semidirect
6. #20276830C2semidirect
7. #20276813C2semidirect
8. #20276755C2semidirect
9. #20276643C2semidirect
10. #20276643C2semidirect
11. #20276577C2semidirect
12. #20276541C2semidirect
13. #20276430C2semidirect
14. C2×#16810423C2semidirect
15. C53×#3336C2semidirect
20. C2×#13133160C22semidirect
21. C2×#13133160C22semidirect
22. C2×#13133157C22semidirect
23. C22×#11719908C22semidirect
31. #16810450C22semidirect
32. #16810448C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810427C22semidirect
39. #16810423C22semidirect
40. #16810423C22semidirect
41. #16810423C22semidirect
42. #16810423C22semidirect
43. #16810423C22semidirect
44. #16810423C22semidirect
45. C22⋊#11719908C22semidirect
46. C22⋊#11719908C22semidirect
47. #16810399C22semidirect
49. C53×#499C22semidirect
53. C53×#467C22semidirect
54. C53×#467C22semidirect
55. C4×Q16×C53C22semidirect
57. C53×#427C22semidirect
58. C2×#107×C53C22semidirect
263. #3336D53semidirect
295. #499C2×D53semidirect
300. #467C2×D53semidirect
301. #467C2×D53semidirect
307. C4×Q16C2×D53semidirect
310. #427C2×D53semidirect
311. C2×#107C2×D53semidirect
329. C53#33877semidirect

Maximal subgroups

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

subgroupindex
D4×D4242
#202769582
#202769062
#202768952
#202768692
#202768302
#202768132
#202767552
#202766432
#202765772
#202765412
#202764302
C2×#168104232
C53×#33362
#3387753