Finite Group Databasebeta

← #27541185

Subgroups — #27541185

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#20287108C211yesmaximal, characteristic, complemented
2#20286402C211yesmaximal, characteristic
3#20276952C211yesmaximal, characteristic, complemented
4#20276918C211yesmaximal, characteristic, complemented
5#20276896C211yesmaximal, characteristic, complemented
6#20276853C211yesmaximal, characteristic, complemented
7#20276828C211yesmaximal, characteristic, complemented
8#20276806C211yesmaximal, characteristic, complemented
9#20276778C211yesmaximal, characteristic, complemented
10#20276657C211yesmaximal, characteristic, complemented
11#20276646C211yesmaximal, characteristic, complemented
12#20276581C211yesmaximal, characteristic, complemented
13#20276428C211yesmaximal, characteristic, complemented
14C2×#16810425C211yesmaximal, characteristic, complemented
15C53×#2168C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C22×D424C2211yescharacteristic
18#16811330—21no—
19#16811269—21no—
20C2×Dic424C2211yescharacteristic, complemented
21C2×#13133162C2211yescharacteristic
22C2×#13133162C2211yescharacteristic
23#16811028C2211yescharacteristic, complemented
24#16811007C2211yescharacteristic
25#16810959—21no—
26#16810959—21no—
27#16810919—21no—
28#16810864C2211yescharacteristic
29#16810864C2211yescharacteristic
30#16810864C2211yescharacteristic, complemented
31#16810859C2211yescharacteristic
32#16810839—21no—
33#16810776—21no—
34#16810770—21no—
35C4×Dic212C2211yescharacteristic, complemented
36C4×D424C2211yescharacteristic
37C2×#13133151C2211yescharacteristic
38C2×#13133151C2211yescharacteristic, complemented
39#16810449C2211yescharacteristic, complemented
40#16810449C2211yescharacteristic, complemented
41#16810449C2211yescharacteristic, complemented
42#16810428C2211yescharacteristic, complemented
43#16810428C2211yescharacteristic, complemented
44#16810428C2211yescharacteristic, complemented
45#16810428C2211yescharacteristic, complemented
46#16810425C2211yescharacteristic
47#16810425C2211yescharacteristic
48#16810425C2211yescharacteristic
49#16810425C2211yescharacteristic
50#16810425C2211yescharacteristic
51#16810425C2211yescharacteristic
52C2×M32×C53C2211yescharacteristic, complemented
53C22×C16×C53C2211yescharacteristic
54C8×D4×C53C2211yescharacteristic
55C53×C4⋊C16C2211yescharacteristic, complemented
56C53×C22⋊C16C2211yescharacteristic
57C53×C22⋊C16C2211yescharacteristic, complemented
58C53×#346C2211yescharacteristic, complemented
59C2×#11719945—21no—
60C22×D212—21no—
61#13133239—22no—
62#13133239—41no—
63C2×Dic212C2311yescharacteristic
64C2×Dic212C2311yescharacteristic
65C2×#11719913—21no—
66C2×#11719913—21no—
67C2×D424—21no—
68C2×D424—22no—
69C2×D424—41no—
70C2×D424C2311yescharacteristic
71C2×D424C2311yescharacteristic
72#13133217—21no—
73#13133208—41no—
74#13133205—21no—
75#13133202—41no—
76#13133196—21no—
77#13133196—21no—
78#13133196—21no—
79#13133194—41no—
80#13133194—41no—
81#13133191—21no—
82#13133190—41no—
83#13133185—41no—
84C4×Dic106—21no—
85C4×D212—21no—
86D4×Dic53—41no—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53—21no—
89C22⋊C4×D53—41no—
90Dic424—41no—
91#13133162—22no—
92#13133162—41no—
93#13133151—21no—
94#13133151—21no—
95#13133151—21no—
96#13133151C2311yescharacteristic
97#13133151C2311yescharacteristic
98#13133151C2311yescharacteristic
99#13133151C2311yescharacteristic
100#13133142—21no—
101#13133142—21no—
102#13133142—21no—
103#13133142—21no—
104#13133139—21no—
105#13133139—21no—
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C4×D4×C53D411yescharacteristic
109M32×C53—41no—
110C2×C16×C53—41no—
111C2×C16×C53C2311yescharacteristic
112C2×C16×C53C2311yescharacteristic
113C2×C16×C53C2311yescharacteristic
114C2×C16×C53C2311yescharacteristic
115C4⋊C8×C53D411yescharacteristic
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C23×D53—41no—
120C2×D4×C53D811yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C2×C22⋊D53—41no—
124C2×C22⋊D53—41no—
125C22×Dic53—41no—
126C22×Dic53—41no—
127#11719945—42no—
128#11719945—81no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—42no—
132C2×D212—41no—
133C2×D212—81no—
134C2×C4×D53—41no—
135C2×C4×D53—41no—
136C2×Dic106—21no—
137C2×Dic106—21no—
138C2×Dic106—41no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53C2×D411yescharacteristic
144C2×C8×C53C2×D411yescharacteristic
145C2×C8×C53C2×D411yescharacteristic
146C2×C8×C53C2411yescharacteristic
147C4⋊C4×C53D811yescharacteristic
148C22⋊C4×C53D811yescharacteristic
149C22⋊C4×C53D811yescharacteristic
150C42×C53C2×D411yescharacteristic
151C22⋊Dic53—41no—
152C22⋊Dic53—41no—
153C22⋊Dic53—41no—
154#11719921—41no—
155#11719921—41no—
156#11719921—41no—
157#11719921—41no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—41no—
163C4⋊Dic53—41no—
164C4⋊Dic53—41no—
165#11719919—41no—
166#11719919—41no—
167#11719919—41no—
168#11719919—41no—
169C4×Dic53—41no—
170Dic212—21no—
171Dic212—21no—
172Dic212—21no—
173Dic212—21no—
174D424—21no—
175D424—21no—
176D424—21no—
177D424—42no—
178D424#4112yes—
179#11719913—42no—
180#11719913—81no—
181C16×C53—21no—
182C16×C53—21no—
183C16×C53—21no—
184C16×C53—21no—
185C23×C53C2×D811yescharacteristic
186C23×C53C2×D811yescharacteristic
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—82no—
190C22×D53—82no—
191C22×D53—161no—
192D4×C53—22no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C22⋊D53—82no—
203C22⋊D53—161no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—161no—
211C2×Dic53—161no—
212D212—22no—
213D212—41no—
214D212—41no—
215D212—41no—
216D212—82no—
217C4×D53—82no—
218C4×D53—161no—
219Dic106—41no—
220Dic106—41no—
221Dic106—41no—
222Dic106—41no—
223Dic106—161no—
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#54611yescharacteristic
249C4×C53#57711yescharacteristic
250Dic53—81no—
251Dic53—81no—
252Dic53—81no—
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#2189—531no—
266#2187—531no—
267#2175—531no—
268#2168D5311yescharacteristic, complemented
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—1061no—
281#546—1061no—
282C2×Q32—531no—
283C2×SD32—531no—
284C2×SD32—531no—
285C2×M32C2×D5311yescharacteristic, complemented
286C22×C16C2×D5311yescharacteristic
287#500—531no—
288#495—531no—
289#484—1061no—
290#484—1061no—
291#477—1061no—
292#467—531no—
293#467—531no—
294#467—531no—
295#466—531no—
296#463—1061no—
297#450—1061no—
298#449—1061no—
299C4×Q16—531no—
300C4×D8—531no—
301C8×D4C2×D5311yescharacteristic
302C2×#107—531no—
303C2×#107—531no—
304#366—531no—
305#366—531no—
306#366—531no—
307C4⋊C16C2×D5311yescharacteristic, complemented
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⋊C16C2×D5311yescharacteristic
319C22⋊C16C2×D5311yescharacteristic, complemented
320#346C2×D5311yescharacteristic, complemented
321C53#3388111yescharacteristic, complemented, Sylow
322C2×#41—1061no—
323C22×D4—1061no—
324#135—1062no—
325#135—2121no—
326C2×Q16—531no—
327C2×Q16—531no—
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—1061no—
338#125—2121no—
339#124—1061no—
340#123—2121no—
341C22⋊Q8—1061no—
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—1061no—
351C4×D4—2121no—
352C4×D4D21211yescharacteristic
353C2×C4⋊C4—1061no—
354C2×C4⋊C4—1061no—
355C2×C22⋊C4—2121no—
356Q32—2121no—
357SD32—1062no—
358SD32—2121no—
359M32—41no—
360C2×C16—41no—
361C2×C16C22×D5311yescharacteristic
362C2×C16C22×D5311yescharacteristic
363C2×C16C22×D5311yescharacteristic
364C2×C16C22×D5311yescharacteristic
365#107—531no—
366#107—531no—
367#107—531no—
368#107—531no—
369#107—1061no—
370#107—1061no—
371#107—1061no—
372C4⋊C8D21211yescharacteristic
373#103—1061no—
374#103—1061no—
375#103—1061no—
376#103—1061no—
377#102—1061no—
378#102—1061no—
379C22⋊C8D21211yescharacteristic
380C22⋊C8D21211yescharacteristic
381C4×C8C22×D5311yescharacteristic
382C24—2121no—
383#41—4241no—
384#41—2122no—
385C2×Q8—1061no—
386C2×Q8—1061no—
387C2×Q8—2121no—
388C2×D4—1061no—
389C2×D4—1061no—
390C2×D4—4241no—
391C2×D4—2122no—
392C2×D4—2121no—
393C2×D4—2121no—
394C2×D4—2121no—
395C2×D4D42411yescharacteristic
396C22×C4—2121no—
397C22×C4—2121no—
398C22×C4—2121no—
399C22×C4—2121no—
400C22×C4C2×D21211yescharacteristic
401C22×C4C2×D21211yescharacteristic
402Q16—1061no—
403Q16—1061no—
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×D21211yescharacteristic
418C2×C8C2×D21211yescharacteristic
419C2×C8C2×D21211yescharacteristic
420C2×C8C23×D5311yescharacteristic
421C4⋊C4—1061no—
422C4⋊C4—1061no—
423C4⋊C4—1061no—
424C4⋊C4—1061no—
425C4⋊C4—2121no—
426C4⋊C4—2121no—
427C4⋊C4—2121no—
428C4⋊C4—2121no—
429C4⋊C4—2121no—
430C4⋊C4—2121no—
431C4⋊C4—2121no—
432C4⋊C4D42411yescharacteristic
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C22⋊C4—2121no—
440C22⋊C4D42411yescharacteristic
441C22⋊C4D42411yescharacteristic
442C42—2121no—
443C42C2×D21211yescharacteristic
444C16—21no—
445C16—21no—
446C16—21no—
447C16—21no—
448C23—8481no—
449C23—4242no—
450C23—4242no—
451C23—2121no—
452C23—2121no—
453C23C2×D42411yescharacteristic
454C23C2×D42411yescharacteristic
455Q8—8481no—
456Q8—2121no—
457Q8—2121no—
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—8481no—
473C2×C4—8481no—
474C2×C4—8481no—
475C2×C4—4242no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4C2×D42411yescharacteristic
483C2×C4C2×D42411yescharacteristic
484C2×C4C2×D42411yescharacteristic
485C2×C4C2×D42411yescharacteristic
486C2×C4C22×D21211yescharacteristic
487C8—21no—
488C8—21no—
489C8—21no—
490C8#41×D5311yescharacteristic
491C8#1313326211yescharacteristic
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#1681106612yes—
505C22C22×D42411yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—4241no—
511C4—4241no—
512C4—4241no—
513C4—4241no—
514C4—4241no—
515C4—4241no—
516C4#1681126811yescharacteristic
517C4#1681143511yescharacteristic
518C2—22no—
519C2—41no—
520C2—8481no—
521C2—4242no—
522C2#2028652911yescharacteristic
523C2C2×#1681106611yescharacteristic
524C2#2028785611yescharacteristic
subgroupG/H
centerC22C22×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2168C2
socleC22×C53C22×D8
prime radical#20286402C2
hypercenter#2168D53
hypocenterC53#33881
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. #20287108C2semidirect
3. #20276952C2semidirect
4. #20276918C2semidirect
5. #20276896C2semidirect
6. #20276853C2semidirect
7. #20276828C2semidirect
8. #20276806C2semidirect
9. #20276778C2semidirect
10. #20276657C2semidirect
11. #20276646C2semidirect
12. #20276581C2semidirect
13. #20276428C2semidirect
14. C2×#16810425C2semidirect
15. C53×#2168C2semidirect
20. C2×Dic424C22semidirect
23. #16811028C22semidirect
30. #16810864C22semidirect
35. C4×Dic212C22semidirect
38. C2×#13133151C22semidirect
39. #16810449C22semidirect
40. #16810449C22semidirect
41. #16810449C22semidirect
42. #16810428C22semidirect
43. #16810428C22semidirect
44. #16810428C22semidirect
45. #16810428C22semidirect
52. C2×M32×C53C22semidirect
55. C53×C4⋊C16C22semidirect
57. C53×C22⋊C16C22semidirect
58. C53×#346C22semidirect
268. #2168D53semidirect
285. C2×M32C2×D53semidirect
307. C4⋊C16C2×D53semidirect
319. C22⋊C16C2×D53semidirect
320. #346C2×D53semidirect
321. C53#33881semidirect

Maximal subgroups

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

subgroupindex
#202871082
#202864022
#202769522
#202769182
#202768962
#202768532
#202768282
#202768062
#202767782
#202766572
#202766462
#202765812
#202764282
C2×#168104252
C53×#21682
#3388153