Finite Group Databasebeta

← #27541176

Subgroups — #27541176

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#20287104C211yesmaximal, characteristic, complemented
2#20286394C211yesmaximal, characteristic
3#20276953C211yesmaximal, characteristic, complemented
4#20276916C211yesmaximal, characteristic, complemented
5#20276896C211yesmaximal, characteristic, complemented
6#20276850C211yesmaximal, characteristic, complemented
7#20276830C211yesmaximal, characteristic, complemented
8#20276803C211yesmaximal, characteristic, complemented
9#20276779C211yesmaximal, characteristic, complemented
10#20276642C211yesmaximal, characteristic, complemented
11#20276580C211yesmaximal, characteristic, complemented
12#20276543C211yesmaximal, characteristic, complemented
13#20276427C211yesmaximal, characteristic, complemented
14#357×D53C211yesmaximal, characteristic, complemented
15C53×#2189C211yesmaximal, characteristic, complemented
16C2×D8×D53C2211yescharacteristic
17#16811331—21no—
18#16811261—21no—
19C2×#13133160C2211yescharacteristic
20C2×#13133157C2211yescharacteristic, complemented
21C2×C16×D53C2211yescharacteristic
22#16811027C2211yescharacteristic, complemented
23#16811008C2211yescharacteristic
24#16810957—21no—
25#16810956—21no—
26#16810916—21no—
27#16810861C2211yescharacteristic
28#16810861C2211yescharacteristic
29#16810859C2211yescharacteristic
30#16810840—21no—
31#16810772—21no—
32#16810768—21no—
33Q16×Dic53C2211yescharacteristic, complemented
34D8×Dic53C2211yescharacteristic
35C8×C22⋊D53C2211yescharacteristic
36C2×#13133151C2211yescharacteristic, complemented
37#107×D53C2211yescharacteristic
38#16810449C2211yescharacteristic, complemented
39#16810449C2211yescharacteristic, complemented
40#16810448C2211yescharacteristic, complemented
41#16810441C2211yescharacteristic, complemented
42#16810427C2211yescharacteristic, complemented
43#16810427C2211yescharacteristic, complemented
44#16810427C2211yescharacteristic, complemented
45#16810425C2211yescharacteristic
46#16810425C2211yescharacteristic
47#16810424C2211yescharacteristic
48#16810424C2211yescharacteristic
49#16810423C2211yescharacteristic
50#16810402C2211yescharacteristic
51#16810398C2211yescharacteristic, complemented
52C2×#135×C53C2211yescharacteristic
53C2×Q32×C53C2211yescharacteristic, complemented
54C2×SD32×C53C2211yescharacteristic
55C53×#467C2211yescharacteristic, complemented
56C53×#358C2211yescharacteristic, complemented
57C53×#357C2211yescharacteristic
58C53×C22⋊C16C2211yescharacteristic, complemented
59C2×D4×D53—21no—
60C2×#11719924—21no—
61C2×D424C2311yescharacteristic
62C2×#11719922—21no—
63D8×D53—22no—
64D8×D53—41no—
65C2×C8×D53C2311yescharacteristic
66#13133216—21no—
67#13133207—41no—
68#13133206—21no—
69#13133200—41no—
70#13133197—41no—
71#13133194—41no—
72#13133193—21no—
73#13133193—21no—
74#13133191—21no—
75#13133189—41no—
76#13133185—41no—
77Q8×Dic53—21no—
78D4×Dic53—21no—
79C4×C22⋊D53D411yescharacteristic
80#13133177—41no—
81C2×C4⋊Dic53—21no—
82C4⋊C4×D53—21no—
83C2×#11719921—41no—
84#13133160—41no—
85#13133157—41no—
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88C16×D53—41no—
89#13133151—21no—
90#13133151C2311yescharacteristic
91#13133151C2311yescharacteristic
92#13133151C2311yescharacteristic
93#13133150—21no—
94#13133150—21no—
95#13133146D411yescharacteristic
96#13133141—21no—
97#13133141—21no—
98#13133141—21no—
99#13133139—21no—
100#13133138—21no—
101C22⋊#88322D411yescharacteristic
102#13133129D411yescharacteristic
103C8×Dic53C2311yescharacteristic
104C2×#41×C53—21no—
105#135×C53—22no—
106#135×C53—41no—
107C2×Q16×C53C2311yescharacteristic
108C2×Q16×C53C2311yescharacteristic
109C2×SD16×C53—21no—
110C2×D8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22⋊Q8×C53—21no—
113Q32×C53—41no—
114SD32×C53—22no—
115C2×C16×C53C2311yescharacteristic
116C2×C16×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#103×C53—21no—
119C23×D53—41no—
120#41×C53—42no—
121#41×C53—81no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—41no—
126C22×C4×C53—41no—
127C22×C4×C53C2×D411yescharacteristic
128C2×C22⋊D53—41no—
129C2×C22⋊D53—41no—
130C2×C22⋊D53D811yescharacteristic
131C22×Dic53—41no—
132C22×Dic53—41no—
133D4×D53—42no—
134D4×D53—81no—
135C2×D212—21no—
136C2×C4×D53—41no—
137C2×C4×D53C2×D411yescharacteristic
138C2×Dic106—41no—
139Q16×C53—21no—
140Q16×C53—21no—
141Q16×C53—21no—
142Q16×C53—21no—
143SD16×C53—42no—
144D8×C53—21no—
145D8×C53#4112yes—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53C2×D411yescharacteristic
149C2×C8×C53C2411yescharacteristic
150C4⋊C4×C53—21no—
151C4⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C22⋊Dic53—41no—
154C22⋊Dic53D811yescharacteristic
155#11719924—81no—
156#11719922—42no—
157#11719921—41no—
158#11719921—41no—
159#11719921—41no—
160#11719921—41no—
161#11719921—41no—
162#11719921D811yescharacteristic
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—41no—
167C4⋊Dic53—41no—
168#11719919—41no—
169#11719919—41no—
170#11719919—41no—
171#11719919—41no—
172#11719919D811yescharacteristic
173C4×Dic53—41no—
174C4×Dic53C2×D411yescharacteristic
175C2×#88322C2×D411yescharacteristic
176C2×#88322C2×D411yescharacteristic
177D424—21no—
178D424—21no—
179C8×D53—21no—
180C8×D53—21no—
181C16×C53—21no—
182C16×C53—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53—41no—
186C23×C53C2×D811yescharacteristic
187C22×D53—41no—
188C22×D53—82no—
189C22×D53—82no—
190C22×D53—161no—
191C22×D53C2×D811yescharacteristic
192Q8×C53—41no—
193Q8×C53—41no—
194Q8×C53—41no—
195Q8×C53—41no—
196D4×C53—22no—
197D4×C53—41no—
198D4×C53—82no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—82no—
205C2×C4×C53C2×D811yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C22⋊D53—22no—
208C22⋊D53—82no—
209C22⋊D53—161no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—161no—
214C2×Dic53—161no—
215C2×Dic53C2×D811yescharacteristic
216C2×Dic53C2×D811yescharacteristic
217C2×Dic53C2×D811yescharacteristic
218D212—41no—
219D212—41no—
220C4×D53—21no—
221C4×D53—21no—
222C4×D53—161no—
223Dic106—161no—
224C8×C53—21no—
225C8×C53C2×#4111yescharacteristic
226C8×C53H(5,𝔽2)11yescharacteristic
227#88322—21no—
228#88322—21no—
229#33881—531nomaximal, Sylow
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—42no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53C22×D811yescharacteristic
237C2×D53—21no—
238C2×D53—21no—
239C2×D53—81no—
240C2×D53—81no—
241C2×D53—81no—
242C2×D53—164no—
243C2×D53#51012yes—
244C4×C53—21no—
245C4×C53—81no—
246C4×C53—81no—
247C4×C53—81no—
248C4×C53#54611yescharacteristic
249C4×C53#57711yescharacteristic
250Dic53—21no—
251Dic53—21no—
252Dic53—21no—
253Dic53—81no—
254Dic53—81no—
255Dic53—81no—
256#3338—531no—
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#2189D5311yescharacteristic, complemented
266#2187—531no—
267#2175—531no—
268#2168—531no—
269#2139—531no—
270C2×#357—531no—
271C2×C53—41no—
272C2×C53—82no—
273C2×C53#329411yescharacteristic
274C2×C53C2×#51011yescharacteristic
275C2×C53#341511yescharacteristic
276D53—22no—
277D53—161no—
278C2×#135C2×D5311yescharacteristic
279C22×D8—531no—
280#554—1061no—
281#546—1061no—
282C2×Q32C2×D5311yescharacteristic, complemented
283C2×SD32—531no—
284C2×SD32C2×D5311yescharacteristic
285C2×M32—531no—
286C22×C16—531no—
287#500—531no—
288#495—531no—
289#484—1061no—
290#484—1061no—
291#477—1061no—
292#467—531no—
293#467—531no—
294#467C2×D5311yescharacteristic, complemented
295#466—531no—
296#463—1061no—
297#450—1061no—
298#449—1061no—
299C4×Q16—531no—
300C4×D8—531no—
301C8×D4—531no—
302C2×#107—531no—
303C2×#107—531no—
304#366—531no—
305#366—531no—
306#366—531no—
307C4⋊C16—531no—
308#358—531no—
309#358—531no—
310#358—531no—
311#358C2×D5311yescharacteristic, complemented
312#357—531no—
313#357—531no—
314#357—531no—
315#357—531no—
316#357—531no—
317#357C2×D5311yescharacteristic
318C22⋊C16—531no—
319C22⋊C16C2×D5311yescharacteristic, complemented
320#346—531no—
321C53#3388111yescharacteristic, complemented, Sylow
322C2×#41—21no—
323C22×D4—1061no—
324#135—22no—
325#135—41no—
326C2×Q16C22×D5311yescharacteristic
327C2×Q16C22×D5311yescharacteristic
328C2×SD16—21no—
329C2×SD16—1061no—
330C2×D8—531no—
331C2×D8—1061no—
332C2×D8—1062no—
333C2×D8—2121no—
334C2×D8C22×D5311yescharacteristic
335C22×C8—531no—
336C22×C8C22×D5311yescharacteristic
337C4⋊Q8—1061no—
338#125—2121no—
339#124—1061no—
340#123—2121no—
341C22⋊Q8—21no—
342C22⋊Q8—1061no—
343C22⋊Q8—1061no—
344C22⋊Q8—2121no—
345C22⋊Q8—2121no—
346#121—1061no—
347#121—2121no—
348#120—2121no—
349C4×Q8—1061no—
350C4×D4—531no—
351C4×D4—1061no—
352C4×D4—2121no—
353C2×C4⋊C4—1061no—
354C2×C4⋊C4—1061no—
355C2×C22⋊C4—2121no—
356Q32—41no—
357SD32—22no—
358SD32—2121no—
359M32—2121no—
360C2×C16—531no—
361C2×C16—531no—
362C2×C16—2121no—
363C2×C16C22×D5311yescharacteristic
364C2×C16C22×D5311yescharacteristic
365#107—531no—
366#107—531no—
367#107—531no—
368#107—1061no—
369#107—1061no—
370#107—1061no—
371#107C22×D5311yescharacteristic
372C4⋊C8—531no—
373#103—21no—
374#103—1061no—
375#103—1061no—
376#103—1061no—
377#102—1061no—
378#102—1061no—
379C22⋊C8—531no—
380C22⋊C8—531no—
381C4×C8—531no—
382C24—2121no—
383#41—42no—
384#41—81no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—2121no—
388C2×D4—21no—
389C2×D4—41no—
390C2×D4—531no—
391C2×D4—1061no—
392C2×D4—4241no—
393C2×D4—2121no—
394C2×D4—2122no—
395C2×D4—2121no—
396C22×C4—41no—
397C22×C4—531no—
398C22×C4—2121no—
399C22×C4—2121no—
400C22×C4—2121no—
401C22×C4D4×D5311yescharacteristic
402Q16—21no—
403Q16—21no—
404Q16—21no—
405Q16—21no—
406SD16—42no—
407SD16—4241no—
408D8—21no—
409D8—1061no—
410D8—1061no—
411D8—2122no—
412D8#1171994712yes—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—531no—
416C2×C8—531no—
417C2×C8—1061no—
418C2×C8—1061no—
419C2×C8D4×D5311yescharacteristic
420C2×C8C23×D5311yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—531no—
424C4⋊C4—1061no—
425C4⋊C4—1061no—
426C4⋊C4—1061no—
427C4⋊C4—2121no—
428C4⋊C4—2121no—
429C4⋊C4—2121no—
430C4⋊C4—2121no—
431C4⋊C4—2121no—
432C4⋊C4—2121no—
433C22⋊C4—41no—
434C22⋊C4—531no—
435C22⋊C4—531no—
436C22⋊C4—2121no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C22⋊C4—2121no—
440C22⋊C4—2121no—
441C22⋊C4—2121no—
442C42—531no—
443C42—2121no—
444C16—21no—
445C16—21no—
446C16—1061no—
447C16—1061no—
448C23—41no—
449C23—531no—
450C23—8481no—
451C23—4242no—
452C23—4242no—
453C23—2121no—
454C23D8×D5311yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—8481no—
460D4—22no—
461D4—41no—
462D4—82no—
463D4—8481no—
464D4—1062no—
465D4—4242no—
466D4—2121no—
467D4—2121no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—531no—
475C2×C4—531no—
476C2×C4—531no—
477C2×C4—8481no—
478C2×C4—8481no—
479C2×C4—8481no—
480C2×C4—1061no—
481C2×C4—1061no—
482C2×C4—2121no—
483C2×C4—2121no—
484C2×C4—2121no—
485C2×C4D8×D5311yescharacteristic
486C2×C4C2×D4×D5311yescharacteristic
487C8—21no—
488C8—1061no—
489C8—1061no—
490C8C2×#1171994711yescharacteristic
491C8#1313326211yescharacteristic
492C22—21no—
493C22—21no—
494C22—21no—
495C22—42no—
496C22—81no—
497C22—81no—
498C22—532no—
499C22—8484no—
500C22—1061no—
501C22—1061no—
502C22—4241no—
503C22—4241no—
504C22—4241no—
505C22C2×D8×D5311yescharacteristic
506C4—21no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—1061no—
511C4—1061no—
512C4—1061no—
513C4—4241no—
514C4—4241no—
515C4—4241no—
516C4#1681126011yescharacteristic
517C4#1681143311yescharacteristic
518C2—41no—
519C2—82no—
520C2—8481no—
521C2—1062no—
522C2#2028652111yescharacteristic
523C2#510×D5311yescharacteristic
524C2#2028785311yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2189C2
socleC22×C53C22×D8
prime radical#20286394C2
hypercenter#2189D53
hypocenterC53#33881
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2189

Complements

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

NH ≅ G/Ntype
1. #20287104C2semidirect
3. #20276953C2semidirect
4. #20276916C2semidirect
5. #20276896C2semidirect
6. #20276850C2semidirect
7. #20276830C2semidirect
8. #20276803C2semidirect
9. #20276779C2semidirect
10. #20276642C2semidirect
11. #20276580C2semidirect
12. #20276543C2semidirect
13. #20276427C2semidirect
14. #357×D53C2semidirect
15. C53×#2189C2semidirect
20. C2×#13133157C22semidirect
22. #16811027C22semidirect
33. Q16×Dic53C22semidirect
36. C2×#13133151C22semidirect
38. #16810449C22semidirect
39. #16810449C22semidirect
40. #16810448C22semidirect
41. #16810441C22semidirect
42. #16810427C22semidirect
43. #16810427C22semidirect
44. #16810427C22semidirect
51. #16810398C22semidirect
53. C2×Q32×C53C22semidirect
55. C53×#467C22semidirect
56. C53×#358C22semidirect
58. C53×C22⋊C16C22semidirect
265. #2189D53semidirect
282. C2×Q32C2×D53semidirect
294. #467C2×D53semidirect
311. #358C2×D53semidirect
319. C22⋊C16C2×D53semidirect
321. C53#33881semidirect

Maximal subgroups

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

subgroupindex
#202871042
#202863942
#202769532
#202769162
#202768962
#202768502
#202768302
#202768032
#202767792
#202766422
#202765802
#202765432
#202764272
#357×D532
C53×#21892
#3388153