Finite Group Databasebeta

← #27541119

Subgroups — #27541119

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
1#20287079C211yesmaximal, characteristic, complemented
2D8×C22⋊D53C211yesmaximal, characteristic
3#20276955C211yesmaximal, characteristic, complemented
4#20276905C211yesmaximal, characteristic, complemented
5#20276895C211yesmaximal, characteristic, complemented
6#20276874C211yesmaximal, characteristic, complemented
7#20276827C211yesmaximal, characteristic, complemented
8#20276811C211yesmaximal, characteristic, complemented
9#20276754C211yesmaximal, characteristic, complemented
10#20276645C211yesmaximal, characteristic, complemented
11#20276644C211yesmaximal, characteristic, complemented
12#20276576C211yesmaximal, characteristic, complemented
13#20276540C211yesmaximal, characteristic, complemented
14#20276429C211yesmaximal, characteristic, complemented
15C2×C53×#357C211yesmaximal, characteristic, complemented
16C2×D8×D53C2211yescharacteristic
17#16811308—21no—
18D4×C22⋊D53—21no—
19C2×#13133162C2211yescharacteristic, complemented
20C2×#13133161C2211yescharacteristic
21C2×#13133161C2211yescharacteristic, complemented
22C2×#13133156C2211yescharacteristic, complemented
23#16811024C2211yescharacteristic, complemented
24#16810998C2211yescharacteristic
25#16810966—21no—
26#16810953—21no—
27#16810929—21no—
28#16810865C2211yescharacteristic, complemented
29#16810864C2211yescharacteristic, complemented
30#16810859C2211yescharacteristic
31#16810856C2211yescharacteristic
32#16810816—21no—
33#16810757—21no—
34#16810755—21no—
35#16810717C2211yescharacteristic, complemented
36D8×Dic53C2211yescharacteristic
37C8×C22⋊D53C2211yescharacteristic
38#16810660C2211yescharacteristic, complemented
39#16810451C2211yescharacteristic, complemented
40#16810448C2211yescharacteristic, complemented
41#16810448C2211yescharacteristic, complemented
42#16810441C2211yescharacteristic, complemented
43#16810428C2211yescharacteristic, complemented
44#16810425C2211yescharacteristic, complemented
45#16810424C2211yescharacteristic
46#16810424C2211yescharacteristic, complemented
47#16810423C2211yescharacteristic, complemented
48#16810423C2211yescharacteristic, complemented
49C22⋊#11719908C2211yescharacteristic, complemented
50#16810402C2211yescharacteristic, complemented
51#16810398C2211yescharacteristic, complemented
52C22×D8×C53C2211yescharacteristic
53C22×C16×C53C2211yescharacteristic, complemented
54C2×#107×C53C2211yescharacteristic, complemented
55C53×#357C2211yescharacteristic
56C53×#357C2211yescharacteristic
57C53×#357C2211yescharacteristic, complemented
58C53×#357C2211yescharacteristic, complemented
59C22×C22⋊D53—41no—
60C2×D4×D53—21no—
61C2×Dic212C2311yescharacteristic
62C2×D424C2311yescharacteristic
63C2×#11719922—21no—
64C2×#11719922—21no—
65D8×D53—22no—
66D8×D53—41no—
67C2×C8×D53C2311yescharacteristic
68#13133213—21no—
69#13133212—41no—
70#13133208—21no—
71#13133207—41no—
72#13133200—41no—
73#13133197—21no—
74#13133196—21no—
75#13133191—21no—
76#13133189—41no—
77#13133188—21no—
78#13133186—41no—
79#13133185—41no—
80#13133182—21no—
81D4×Dic53—21no—
82C4×C22⋊D53D411yescharacteristic
83#13133176—41no—
84#13133175—21no—
85#13133162—41no—
86#13133161—22no—
87#13133161—41no—
88#13133156—41no—
89C2×#11719908C2311yescharacteristic
90C2×#11719908C2311yescharacteristic
91#13133151C2311yescharacteristic
92#13133151C2311yescharacteristic
93#13133148—21no—
94#13133148—21no—
95#13133146D411yescharacteristic
96#13133142—21no—
97#13133140—21no—
98#13133140—21no—
99#13133139—21no—
100#13133138—21no—
101#13133138—21no—
102C22⋊#88322D411yescharacteristic
103#13133129D411yescharacteristic
104C8×Dic53C2311yescharacteristic
105C22×D4×C53—21no—
106C2×D8×C53—21no—
107C2×D8×C53—22no—
108C2×D8×C53—41no—
109C2×D8×C53C2311yescharacteristic
110C2×D8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C2×C4⋊C4×C53—21no—
113C2×C16×C53—41no—
114C2×C16×C53C2311yescharacteristic
115C2×C16×C53C2311yescharacteristic
116#107×C53—21no—
117#107×C53C2311yescharacteristic
118#107×C53C2311yescharacteristic
119C24×C53—41no—
120C23×D53—41no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—42no—
124C2×D4×C53—41no—
125C2×D4×C53—81no—
126C22×C4×C53—41no—
127C22×C4×C53C2×D411yescharacteristic
128C2×C22⋊D53—41no—
129C2×C22⋊D53—41no—
130C2×C22⋊D53—41no—
131C2×C22⋊D53—41no—
132C2×C22⋊D53—41no—
133C2×C22⋊D53—84no—
134C2×C22⋊D53D811yescharacteristic
135C22×Dic53—41no—
136D4×D53—42no—
137D4×D53—81no—
138C2×D212—21no—
139C2×C4×D53C2×D411yescharacteristic
140C2×Dic106—21no—
141D8×C53—21no—
142D8×C53—21no—
143D8×C53—21no—
144D8×C53—42no—
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—21no—
152C4⋊C4×C53—41no—
153C22⋊Dic53—41no—
154C22⋊Dic53—41no—
155C22⋊Dic53—41no—
156C22⋊Dic53D811yescharacteristic
157#11719922—42no—
158#11719922—81no—
159#11719921—41no—
160#11719921—41no—
161#11719921—41no—
162#11719921D811yescharacteristic
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—41no—
166C4⋊Dic53—41no—
167#11719919—41no—
168#11719919—41no—
169#11719919—41no—
170#11719919D811yescharacteristic
171C4×Dic53—41no—
172C4×Dic53—41no—
173C4×Dic53C2×D411yescharacteristic
174C2×#88322C2×D411yescharacteristic
175C2×#88322C2×D411yescharacteristic
176Dic212—21no—
177Dic212—21no—
178D424—21no—
179D424—21no—
180C8×D53—21no—
181C8×D53—21no—
182C16×C53—21no—
183C16×C53—21no—
184#11719908—21no—
185#11719908—21no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—82no—
189C23×C53—82no—
190C23×C53—161no—
191C23×C53C2×D811yescharacteristic
192C22×D53—41no—
193C22×D53—82no—
194C22×D53—82no—
195C22×D53—161no—
196C22×D53C2×D811yescharacteristic
197D4×C53—22no—
198D4×C53—41no—
199D4×C53—41no—
200D4×C53—41no—
201D4×C53—82no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—82no—
207C2×C4×C53C2×D811yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C22⋊D53—22no—
210C22⋊D53—82no—
211C22⋊D53—82no—
212C22⋊D53—82no—
213C22⋊D53—82no—
214C22⋊D53—161no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—82no—
219C2×Dic53C2×D811yescharacteristic
220C2×Dic53C2×D811yescharacteristic
221C2×Dic53C2×D811yescharacteristic
222D212—41no—
223D212—41no—
224C4×D53—21no—
225C4×D53—21no—
226Dic106—41no—
227Dic106—41no—
228C8×C53—21no—
229C8×C53C2×#4111yescharacteristic
230C8×C53H(5,𝔽2)11yescharacteristic
231#88322—21no—
232#88322—21no—
233#33877—531nomaximal, Sylow
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#50912yes—
244C22×C53C22×D811yescharacteristic
245C2×D53—21no—
246C2×D53—21no—
247C2×D53—21no—
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—81no—
251C2×D53—164no—
252C4×C53—21no—
253C4×C53—81no—
254C4×C53—81no—
255C4×C53#54611yescharacteristic
256C4×C53#57711yescharacteristic
257Dic53—21no—
258Dic53—21no—
259Dic53—21no—
260Dic53—81no—
261Dic53—81no—
262Dic53—81no—
263#3336—531no—
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×#357D5311yescharacteristic, complemented
278C2×C53—22no—
279C2×C53—82no—
280C2×C53—161no—
281C2×C53#329411yescharacteristic
282C2×C53C2×#50911yescharacteristic
283C2×C53#341611yescharacteristic
284D53—41no—
285D53—161no—
286C22×D8—531no—
287C22×D8C2×D5311yescharacteristic
288#552—1061no—
289D42—1061no—
290C2×SD32—531no—
291C2×SD32—531no—
292C2×SD32—531no—
293C2×M32—531no—
294C22×C16C2×D5311yescharacteristic, complemented
295#499—531no—
296#493—531no—
297#485—1061no—
298#483—1061no—
299#479—1061no—
300#467—531no—
301#467—531no—
302#466—531no—
303#466—531no—
304#459—1061no—
305#447—1061no—
306#447—1061no—
307C4×Q16—531no—
308C4×D8—531no—
309C8×D4—531no—
310#427—531no—
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#357C2×D5311yescharacteristic
323#357C2×D5311yescharacteristic
324#357C2×D5311yescharacteristic, complemented
325#357C2×D5311yescharacteristic, complemented
326C22⋊C16—531no—
327C22⋊C16—531no—
328#346—531no—
329C53#3387711yescharacteristic, complemented, Sylow
330C22×D4—21no—
331C22×D4—1061no—
332C22×D4—2121no—
333C2×Q16—531no—
334C2×D8—22no—
335C2×D8—21no—
336C2×D8—41no—
337C2×D8—531no—
338C2×D8—1061no—
339C2×D8—1061no—
340C2×D8—1062no—
341C2×D8—2121no—
342C2×D8C22×D5311yescharacteristic
343C2×D8C22×D5311yescharacteristic
344C22×C8—531no—
345C22×C8C22×D5311yescharacteristic
346#127—1061no—
347#126—2121no—
348#125—1061no—
349#125—2121no—
350#123—2121no—
351C22⋊Q8—1061no—
352C22⋊Q8—1061no—
353#121—1061no—
354#121—1061no—
355#121—2121no—
356#120—2121no—
357#120—2121no—
358C4×Q8—1061no—
359C4×D4—531no—
360C4×D4—1061no—
361#117—1061no—
362#117—2121no—
363C2×C4⋊C4—21no—
364SD32—1062no—
365SD32—2121no—
366SD32—2121no—
367M32—2121no—
368C2×C16—41no—
369C2×C16—531no—
370C2×C16—531no—
371C2×C16C22×D5311yescharacteristic
372C2×C16C22×D5311yescharacteristic
373#107—21no—
374#107—531no—
375#107—531no—
376#107C22×D5311yescharacteristic
377#107C22×D5311yescharacteristic
378#106—1061no—
379#106—1061no—
380C4⋊C8—531no—
381#103—1061no—
382#103—1061no—
383#103—1061no—
384#102—1061no—
385#102—1061no—
386#102—1061no—
387C22⋊C8—531no—
388C22⋊C8—531no—
389C4×C8—531no—
390C24—41no—
391C24—2121no—
392C2×Q8—1061no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—42no—
396C2×D4—41no—
397C2×D4—81no—
398C2×D4—531no—
399C2×D4—1061no—
400C2×D4—4241no—
401C2×D4—4244no—
402C2×D4—2121no—
403C2×D4—2121no—
404C2×D4—2121no—
405C2×D4—2122no—
406C2×D4—2121no—
407C2×D4—2121no—
408C22×C4—41no—
409C22×C4—531no—
410C22×C4—2121no—
411C22×C4D4×D5311yescharacteristic
412Q16—1061no—
413Q16—1061no—
414D8—21no—
415D8—21no—
416D8—21no—
417D8—42no—
418D8—1061no—
419D8—1061no—
420D8—4241no—
421D8—2122no—
422D8#1171994512yes—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—531no—
426C2×C8—531no—
427C2×C8—1061no—
428C2×C8—1061no—
429C2×C8D4×D5311yescharacteristic
430C2×C8C23×D5311yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—531no—
435C4⋊C4—1061no—
436C4⋊C4—1061no—
437C4⋊C4—2121no—
438C4⋊C4—2121no—
439C4⋊C4—2121no—
440C4⋊C4—2121no—
441C4⋊C4—2121no—
442C22⋊C4—531no—
443C22⋊C4—531no—
444C22⋊C4—2121no—
445C22⋊C4—2121no—
446C22⋊C4—2121no—
447C22⋊C4—2121no—
448C22⋊C4—2121no—
449C22⋊C4—2121no—
450C42—531no—
451C42—2121no—
452C42—2121no—
453C16—21no—
454C16—21no—
455C16—1061no—
456C16—1061no—
457C23—41no—
458C23—41no—
459C23—82no—
460C23—82no—
461C23—161no—
462C23—531no—
463C23—8481no—
464C23—4242no—
465C23—4242no—
466C23—2121no—
467C23D8×D5311yescharacteristic
468Q8—2121no—
469Q8—2121no—
470D4—22no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—82no—
475D4—8481no—
476D4—1062no—
477D4—4242no—
478D4—4242no—
479D4—4242no—
480D4—4242no—
481D4—2121no—
482D4—2121no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—82no—
488C2×C4—531no—
489C2×C4—531no—
490C2×C4—531no—
491C2×C4—1061no—
492C2×C4—1061no—
493C2×C4—4242no—
494C2×C4—2121no—
495C2×C4—2121no—
496C2×C4—2121no—
497C2×C4D8×D5311yescharacteristic
498C2×C4C2×D4×D5311yescharacteristic
499C8—21no—
500C8—1061no—
501C8—1061no—
502C8C2×#1171994511yescharacteristic
503C8#1313326211yescharacteristic
504C22—21no—
505C22—21no—
506C22—42no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—164no—
513C22—8484no—
514C22—1061no—
515C22—1061no—
516C22—1061no—
517C22—4241no—
518C22—4241no—
519C22—4241no—
520C22#1681105712yes—
521C22C2×D8×D5311yescharacteristic
522C4—21no—
523C4—81no—
524C4—81no—
525C4—1061no—
526C4—1061no—
527C4—1061no—
528C4—4241no—
529C4—4241no—
530C4—4241no—
531C4#1681126311yescharacteristic
532C4#1681142911yescharacteristic
533C2—22no—
534C2—82no—
535C2—161no—
536C2—8481no—
537C2—2121no—
538C2#2028652411yescharacteristic
539C2C2×#1681105711yescharacteristic
540C2#2028785711yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#357C2
socleC22×C53C22×D8
prime radicalD8×C22⋊D53C2
hypercenterC2×#357D53
hypocenterC53#33877
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. #20287079C2semidirect
3. #20276955C2semidirect
4. #20276905C2semidirect
5. #20276895C2semidirect
6. #20276874C2semidirect
7. #20276827C2semidirect
8. #20276811C2semidirect
9. #20276754C2semidirect
10. #20276645C2semidirect
11. #20276644C2semidirect
12. #20276576C2semidirect
13. #20276540C2semidirect
14. #20276429C2semidirect
15. C2×C53×#357C2semidirect
19. C2×#13133162C22semidirect
21. C2×#13133161C22semidirect
22. C2×#13133156C22semidirect
23. #16811024C22semidirect
28. #16810865C22semidirect
29. #16810864C22semidirect
35. #16810717C22semidirect
38. #16810660C22semidirect
39. #16810451C22semidirect
40. #16810448C22semidirect
41. #16810448C22semidirect
42. #16810441C22semidirect
43. #16810428C22semidirect
44. #16810425C22semidirect
46. #16810424C22semidirect
47. #16810423C22semidirect
48. #16810423C22semidirect
49. C22⋊#11719908C22semidirect
50. #16810402C22semidirect
51. #16810398C22semidirect
53. C22×C16×C53C22semidirect
54. C2×#107×C53C22semidirect
57. C53×#357C22semidirect
58. C53×#357C22semidirect
277. C2×#357D53semidirect
294. C22×C16C2×D53semidirect
311. C2×#107C2×D53semidirect
324. #357C2×D53semidirect
325. #357C2×D53semidirect
329. C53#33877semidirect

Maximal subgroups

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

subgroupindex
#202870792
D8×C22⋊D532
#202769552
#202769052
#202768952
#202768742
#202768272
#202768112
#202767542
#202766452
#202766442
#202765762
#202765402
#202764292
C2×C53×#3572
#3387753