Finite Group Databasebeta

← #27541177

Subgroups — #27541177

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286402C211yesmaximal, characteristic
2#20276954C211yesmaximal, characteristic, complemented
3#20276917C211yesmaximal, characteristic, complemented
4#20276895C211yesmaximal, characteristic, complemented
5#20276849C211yesmaximal, characteristic, complemented
6#20276830C211yesmaximal, characteristic, complemented
7#20276804C211yesmaximal, characteristic, complemented
8#20276777C211yesmaximal, characteristic, complemented
9#20276653C211yesmaximal, characteristic, complemented
10#20276643C211yesmaximal, characteristic, complemented
11#20276577C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276425C211yesmaximal, characteristic, complemented
14C2×#16810423C211yesmaximal, characteristic, complemented
15C53×#3338C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C22×D424C2211yescharacteristic
18#16811269—21no—
19C2×#13133163C2211yescharacteristic, complemented
20C2×#13133160C2211yescharacteristic
21C2×#13133160C2211yescharacteristic
22C2×#13133157C2211yescharacteristic, complemented
23C22×#11719908C2211yescharacteristic
24#16811007C2211yescharacteristic
25#16810864C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810839—21no—
28#16810776—21no—
29#16810770—21no—
30C4×D424C2211yescharacteristic
31#16810448C2211yescharacteristic, complemented
32#16810448C2211yescharacteristic, complemented
33#16810448C2211yescharacteristic, complemented
34C4⋊#11719908C2211yescharacteristic, complemented
35#16810427C2211yescharacteristic, complemented
36#16810426C2211yescharacteristic, complemented
37#16810426C2211yescharacteristic, complemented
38#16810426C2211yescharacteristic, complemented
39#16810423C2211yescharacteristic
40#16810423C2211yescharacteristic
41#16810423C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44#16810423C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic, complemented
47#16810399C2211yescharacteristic, complemented
48C53×#554—21no—
49C53×#500C2211yescharacteristic, complemented
50C53×#484—21no—
51C53×#484—21no—
52C53×#477—21no—
53C53×#467C2211yescharacteristic
54C53×#467C2211yescharacteristic, complemented
55C4×Q16×C53C2211yescharacteristic, complemented
56C8×D4×C53C2211yescharacteristic
57C2×#107×C53C2211yescharacteristic
58C2×#107×C53C2211yescharacteristic, complemented
59C2×#11719945—21no—
60C22×D212—21no—
61#13133239—22no—
62#13133239—41no—
63C2×Dic212C2311yescharacteristic
64C2×#11719913—21no—
65C2×#11719913—21no—
66C2×D424—21no—
67C2×D424—22no—
68C2×D424—41no—
69C2×D424C2311yescharacteristic
70C2×D424C2311yescharacteristic
71#13133205—21no—
72#13133202—41no—
73#13133196—21no—
74#13133191—21no—
75#13133190—41no—
76#13133185—41no—
77C4×D212—21no—
78C22⋊C4×D53—41no—
79#13133163—41no—
80#13133160—22no—
81#13133160—41no—
82#13133157—41no—
83C2×#11719908—41no—
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133142—21no—
90#13133139—21no—
91#13133139—21no—
92C2×Q16×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95C4⋊Q8×C53—21no—
96#125×C53—41no—
97C22⋊Q8×C53—21no—
98C22⋊Q8×C53—21no—
99C22⋊Q8×C53—41no—
100C22⋊Q8×C53—41no—
101C4×Q8×C53—21no—
102C4×D4×C53—41no—
103C4×D4×C53D411yescharacteristic
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106#107×C53—21no—
107#107×C53—21no—
108#107×C53—21no—
109#107×C53C2311yescharacteristic
110#107×C53C2311yescharacteristic
111#107×C53C2311yescharacteristic
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—
120C2×Q8×C53—21no—
121C2×Q8×C53—41no—
122C2×D4×C53D811yescharacteristic
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C2×C22⋊D53—41no—
128C2×C22⋊D53—41no—
129#11719945—42no—
130#11719945—81no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—42no—
134C2×D212—41no—
135C2×D212—81no—
136C2×C4×D53—41no—
137C2×C4×D53—41no—
138C2×Dic106—21no—
139Q16×C53—21no—
140Q16×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53C2×D411yescharacteristic
146C2×C8×C53C2×D411yescharacteristic
147C2×C8×C53C2×D411yescharacteristic
148C2×C8×C53C2411yescharacteristic
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—41no—
153C4⋊C4×C53—41no—
154C4⋊C4×C53—41no—
155C4⋊C4×C53—41no—
156C4⋊C4×C53—41no—
157C4⋊C4×C53—41no—
158C4⋊C4×C53D811yescharacteristic
159C22⋊C4×C53—41no—
160C22⋊C4×C53—41no—
161C22⋊C4×C53D811yescharacteristic
162C22⋊C4×C53D811yescharacteristic
163C42×C53—41no—
164C42×C53C2×D411yescharacteristic
165C22⋊Dic53—41no—
166#11719921—41no—
167#11719921—41no—
168#11719921—41no—
169#11719921—41no—
170C4⋊Dic53—21no—
171#11719919—41no—
172Dic212—21no—
173Dic212—21no—
174D424—21no—
175D424—21no—
176D424—21no—
177D424—42no—
178D424#4112yes—
179#11719913—42no—
180#11719913—81no—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53C2×D811yescharacteristic
186C23×C53C2×D811yescharacteristic
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—82no—
190C22×D53—82no—
191C22×D53—161no—
192Q8×C53—41no—
193Q8×C53—41no—
194Q8×C53—161no—
195D4×C53—22no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—161no—
205C2×C4×C53—161no—
206C2×C4×C53C2×D811yescharacteristic
207C2×C4×C53C2×D811yescharacteristic
208C2×C4×C53C2×D811yescharacteristic
209C2×C4×C53C2×D811yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C22⋊D53—82no—
212C22⋊D53—161no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215D212—22no—
216D212—41no—
217D212—41no—
218D212—41no—
219D212—82no—
220C4×D53—82no—
221C4×D53—161no—
222Dic106—41no—
223Dic106—41no—
224C8×C53—21no—
225C8×C53—21no—
226C8×C53—21no—
227C8×C53C2×#4111yescharacteristic
228C8×C53H(5,𝔽2)11yescharacteristic
229#33881—531nomaximal, Sylow
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53#51012yes—
236C22×C53C22×D811yescharacteristic
237C2×D53—42no—
238C2×D53—81no—
239C2×D53—81no—
240C2×D53—81no—
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—164no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—81no—
249C4×C53—81no—
250C4×C53—81no—
251C4×C53—81no—
252C4×C53#54611yescharacteristic
253C4×C53#57711yescharacteristic
254Dic53—81no—
255Dic53—81no—
256#3338D5311yescharacteristic, complemented
257#3280—531no—
258#2243—531no—
259#2236—531no—
260#2232—531no—
261#2224—531no—
262#2220—531no—
263#2216—531no—
264#2211—531no—
265#2189—531no—
266#2187—531no—
267#2175—531no—
268#2168—531no—
269#2139—531no—
270C2×#357—531no—
271C2×C53—22no—
272C2×C53—41no—
273C2×C53#329411yescharacteristic
274C2×C53C2×#51011yescharacteristic
275C2×C53#341511yescharacteristic
276D53—82no—
277D53—161no—
278C2×#135—531no—
279C22×D8—531no—
280#554—21no—
281#546—1061no—
282C2×Q32—531no—
283C2×SD32—531no—
284C2×SD32—531no—
285C2×M32—531no—
286C22×C16—531no—
287#500C2×D5311yescharacteristic, complemented
288#495—531no—
289#484—21no—
290#484—21no—
291#477—21no—
292#467—531no—
293#467C2×D5311yescharacteristic
294#467C2×D5311yescharacteristic, complemented
295#466—531no—
296#463—1061no—
297#450—1061no—
298#449—1061no—
299C4×Q16C2×D5311yescharacteristic, complemented
300C4×D8—531no—
301C8×D4C2×D5311yescharacteristic
302C2×#107C2×D5311yescharacteristic
303C2×#107C2×D5311yescharacteristic, complemented
304#366—531no—
305#366—531no—
306#366—531no—
307C4⋊C16—531no—
308#358—531no—
309#358—531no—
310#358—531no—
311#358—531no—
312#357—531no—
313#357—531no—
314#357—531no—
315#357—531no—
316#357—531no—
317#357—531no—
318C22⋊C16—531no—
319C22⋊C16—531no—
320#346—531no—
321C53#3388111yescharacteristic, complemented, Sylow
322C2×#41—1061no—
323C22×D4—1061no—
324#135—1062no—
325#135—2121no—
326C2×Q16—531no—
327C2×Q16C22×D5311yescharacteristic
328C2×SD16—1061no—
329C2×SD16—1061no—
330C2×D8—531no—
331C2×D8—531no—
332C2×D8—1062no—
333C2×D8—1061no—
334C2×D8—2121no—
335C22×C8C22×D5311yescharacteristic
336C22×C8C22×D5311yescharacteristic
337C4⋊Q8—21no—
338#125—41no—
339#124—1061no—
340#123—2121no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—41no—
344C22⋊Q8—41no—
345C22⋊Q8—1061no—
346#121—1061no—
347#121—2121no—
348#120—2121no—
349C4×Q8—21no—
350C4×D4—41no—
351C4×D4—1061no—
352C4×D4C22⋊D5311yescharacteristic
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C22⋊C4—2121no—
356Q32—2121no—
357SD32—1062no—
358SD32—2121no—
359M32—2121no—
360C2×C16—531no—
361C2×C16—531no—
362C2×C16—531no—
363C2×C16—531no—
364C2×C16—2121no—
365#107—21no—
366#107—21no—
367#107—21no—
368#107—531no—
369#107C22×D5311yescharacteristic
370#107C22×D5311yescharacteristic
371#107C22×D5311yescharacteristic
372C4⋊C8C22⋊D5311yescharacteristic
373#103—21no—
374#103—21no—
375#103—21no—
376#103—1061no—
377#102—1061no—
378#102—1061no—
379C22⋊C8C22⋊D5311yescharacteristic
380C22⋊C8C22⋊D5311yescharacteristic
381C4×C8C22×D5311yescharacteristic
382C24—2121no—
383#41—4241no—
384#41—2122no—
385C2×Q8—21no—
386C2×Q8—41no—
387C2×Q8—1061no—
388C2×D4—1061no—
389C2×D4—1061no—
390C2×D4—4241no—
391C2×D4—2122no—
392C2×D4—2121no—
393C2×D4—2121no—
394C2×D4—2121no—
395C2×D4#1171992211yescharacteristic
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—2121no—
399C22×C4—2121no—
400C22×C4C2×C22⋊D5311yescharacteristic
401C22×C4C2×C22⋊D5311yescharacteristic
402Q16—21no—
403Q16—21no—
404Q16—1061no—
405Q16—1061no—
406SD16—4241no—
407SD16—2122no—
408D8—532no—
409D8—1061no—
410D8—1061no—
411D8—1061no—
412D8—2122no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8C2×C22⋊D5311yescharacteristic
418C2×C8C2×C22⋊D5311yescharacteristic
419C2×C8C2×C22⋊D5311yescharacteristic
420C2×C8C23×D5311yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—1061no—
431C4⋊C4—2121no—
432C4⋊C4#1171992211yescharacteristic
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C22⋊C4—2121no—
440C22⋊C4#1171992211yescharacteristic
441C22⋊C4#1171992211yescharacteristic
442C42—41no—
443C42C2×C22⋊D5311yescharacteristic
444C16—1061no—
445C16—1061no—
446C16—1061no—
447C16—1061no—
448C23—8481no—
449C23—4242no—
450C23—4242no—
451C23—2121no—
452C23—2121no—
453C23C2×#1171992211yescharacteristic
454C23C2×#1171992211yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—161no—
458Q8—2121no—
459Q8—2121no—
460D4—22no—
461D4—8481no—
462D4—1062no—
463D4—4242no—
464D4—4242no—
465D4—2121no—
466D4—2121no—
467D4—2121no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—161no—
477C2×C4—161no—
478C2×C4—8481no—
479C2×C4—4242no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4C2×#1171992211yescharacteristic
483C2×C4C2×#1171992211yescharacteristic
484C2×C4C2×#1171992211yescharacteristic
485C2×C4C2×#1171992211yescharacteristic
486C2×C4C22×C22⋊D5311yescharacteristic
487C8—21no—
488C8—21no—
489C8—21no—
490C8#41×D5311yescharacteristic
491C8#1313326311yescharacteristic
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—8484no—
498C22—4241no—
499C22—4241no—
500C22—4241no—
501C22—4241no—
502C22—4241no—
503C22—2122no—
504C22#1681106412yes—
505C22C22×#1171992211yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—4241no—
515C4—4241no—
516C4#1681126111yescharacteristic
517C4#1681143111yescharacteristic
518C2—22no—
519C2—41no—
520C2—8481no—
521C2—4242no—
522C2#2028652211yescharacteristic
523C2C2×#1681106411yescharacteristic
524C2#2028785411yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3338C2
socleC22×C53C22×D8
prime radical#20286402C2
hypercenter#3338D53
hypocenterC53#33881
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3338

Complements

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

NH ≅ G/Ntype
2. #20276954C2semidirect
3. #20276917C2semidirect
4. #20276895C2semidirect
5. #20276849C2semidirect
6. #20276830C2semidirect
7. #20276804C2semidirect
8. #20276777C2semidirect
9. #20276653C2semidirect
10. #20276643C2semidirect
11. #20276577C2semidirect
12. #20276541C2semidirect
13. #20276425C2semidirect
14. C2×#16810423C2semidirect
15. C53×#3338C2semidirect
19. C2×#13133163C22semidirect
22. C2×#13133157C22semidirect
31. #16810448C22semidirect
32. #16810448C22semidirect
33. #16810448C22semidirect
34. C4⋊#11719908C22semidirect
35. #16810427C22semidirect
36. #16810426C22semidirect
37. #16810426C22semidirect
38. #16810426C22semidirect
46. C22⋊#11719908C22semidirect
47. #16810399C22semidirect
49. C53×#500C22semidirect
54. C53×#467C22semidirect
55. C4×Q16×C53C22semidirect
58. C2×#107×C53C22semidirect
256. #3338D53semidirect
287. #500C2×D53semidirect
294. #467C2×D53semidirect
299. C4×Q16C2×D53semidirect
303. C2×#107C2×D53semidirect
321. C53#33881semidirect

Maximal subgroups

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

subgroupindex
#202864022
#202769542
#202769172
#202768952
#202768492
#202768302
#202768042
#202767772
#202766532
#202766432
#202765772
#202765412
#202764252
C2×#168104232
C53×#33382
#3388153