Finite Group Databasebeta

← #28745715

Subgroups — #28745715

Order 15104.

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#21436459C211yesmaximal, characteristic, complemented
2#21435770C211yesmaximal, characteristic
3#21426336C211yesmaximal, characteristic, complemented
4#21426284C211yesmaximal, characteristic, complemented
5#21426278C211yesmaximal, characteristic, complemented
6#21426256C211yesmaximal, characteristic, complemented
7#21426195C211yesmaximal, characteristic, complemented
8#21426134C211yesmaximal, characteristic, complemented
9#21426024C211yesmaximal, characteristic, complemented
10#21426023C211yesmaximal, characteristic, complemented
11#21425963C211yesmaximal, characteristic, complemented
12#21425919C211yesmaximal, characteristic, complemented
13#21425813C211yesmaximal, characteristic, complemented
14#357×D59C211yesmaximal, characteristic, complemented
15C59×#2220C211yesmaximal, characteristic, complemented
16C22×D472C2211yescharacteristic
17C2×D8×D59C2211yescharacteristic
18#16838086—21no—
19#16838036—21no—
20C2×#14273670C2211yescharacteristic
21C2×#14273670C2211yescharacteristic, complemented
22C2×C16×D59C2211yescharacteristic, complemented
23#16837803C2211yescharacteristic, complemented
24#16837777C2211yescharacteristic
25#16837740—21no—
26#16837731—21no—
27#16837707—21no—
28#16837640C2211yescharacteristic, complemented
29#16837639C2211yescharacteristic
30#16837596—21no—
31#16837537—21no—
32#16837535—21no—
33Q16×Dic59C2211yescharacteristic, complemented
34#16837484C2211yescharacteristic
35C8×C22⋊D59C2211yescharacteristic
36#16837440C2211yescharacteristic, complemented
37#107×D59C2211yescharacteristic, complemented
38#16837229C2211yescharacteristic, complemented
39#16837227C2211yescharacteristic, complemented
40Dic59⋊C16C2211yescharacteristic, complemented
41#16837204C2211yescharacteristic
42#16837204C2211yescharacteristic
43#16837204C2211yescharacteristic, complemented
44#16837204C2211yescharacteristic, complemented
45#16837203C2211yescharacteristic, complemented
46#16837203C2211yescharacteristic, complemented
47#16837202C2211yescharacteristic
48#16837202C2211yescharacteristic, complemented
49C22⋊(C59⋊C16)C2211yescharacteristic, complemented
50(C2×D59)⋊C16C2211yescharacteristic, complemented
51#16837177C2211yescharacteristic, complemented
52C2×SD32×C59C2211yescharacteristic, complemented
53C2×M32×C59C2211yescharacteristic, complemented
54C59×#467C2211yescharacteristic, complemented
55C59×#466C2211yescharacteristic
56C59×#366C2211yescharacteristic, complemented
57C59×#358C2211yescharacteristic, complemented
58C59×#357C2211yescharacteristic, complemented
59C22×D236—21no—
60C2×D4×D59—21no—
61C2×D4×D59—41no—
62C2×D472—22no—
63C2×D472—21no—
64C2×D472—41no—
65C2×D472C2311yescharacteristic
66C2×D472C2311yescharacteristic
67C2×#11745669—21no—
68C2×#11745669—21no—
69D8×D59—22no—
70D8×D59—41no—
71C2×C8×D59C2311yescharacteristic
72#14273723—21no—
73#14273720—41no—
74#14273718—21no—
75#14273717—41no—
76#14273709—41no—
77#14273703—21no—
78#14273702—21no—
79#14273700—41no—
80#14273697—41no—
81#14273695—41no—
82Q8×Dic59—21no—
83#14273689—21no—
84C4×C22⋊D59D411yescharacteristic
85#14273686—21no—
86#14273685—41no—
87C4⋊C4×D59—21no—
88#14273670—22no—
89#14273670—41no—
90C2×C59⋊C16C2311yescharacteristic
91C2×C59⋊C16C2311yescharacteristic
92C16×D59—41no—
93#14273661C2311yescharacteristic
94#14273661C2311yescharacteristic
95#14273660—21no—
96#14273659—21no—
97#14273658—21no—
98Dic59⋊C8D411yescharacteristic
99#14273651—21no—
100#14273651—21no—
101#14273647—21no—
102#14273647—21no—
103C22⋊(C59⋊C8)D411yescharacteristic
104(C2×D59)⋊C8D411yescharacteristic
105C8×Dic59C2311yescharacteristic
106C2×Q16×C59C2311yescharacteristic
107C2×D8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22⋊Q8×C59—21no—
110#121×C59—21no—
111SD32×C59—41no—
112M32×C59—41no—
113C2×C16×C59C2311yescharacteristic
114C2×C16×C59C2311yescharacteristic
115#107×C59C2311yescharacteristic
116#107×C59C2311yescharacteristic
117#103×C59—21no—
118#102×C59—21no—
119C23×D59—41no—
120C23×D59—41no—
121C2×Q8×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—41no—
124C22×C4×C59C2×D411yescharacteristic
125C2×C22⋊D59—41no—
126C2×C22⋊D59—41no—
127C2×C22⋊D59—41no—
128C2×C22⋊D59D811yescharacteristic
129D4×D59—42no—
130D4×D59—81no—
131D4×D59—84no—
132C2×D236—21no—
133C2×D236—21no—
134C2×D236—42no—
135C2×D236—41no—
136C2×D236—41no—
137C2×D236—81no—
138C2×C4×D59—41no—
139C2×C4×D59—41no—
140C2×C4×D59C2×D411yescharacteristic
141Q16×C59—21no—
142Q16×C59—21no—
143D8×C59—21no—
144D8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59C2×D411yescharacteristic
148C2×C8×C59C2411yescharacteristic
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—41no—
152C22⋊C4×C59—41no—
153C22⋊C4×C59—41no—
154C22⋊Dic59—41no—
155C22⋊Dic59D811yescharacteristic
156#11745669—42no—
157#11745669—81no—
158(C2×D59)⋊C4—41no—
159(C2×D59)⋊C4—41no—
160(C2×D59)⋊C4—41no—
161(C2×D59)⋊C4D811yescharacteristic
162C4⋊Dic59—21no—
163C4⋊Dic59—21no—
164C4⋊Dic59—41no—
165C4⋊Dic59—41no—
166Dic59⋊C4—41no—
167Dic59⋊C4—41no—
168Dic59⋊C4—41no—
169Dic59⋊C4D811yescharacteristic
170C4×Dic59—41no—
171C4×Dic59—41no—
172C4×Dic59C2×D411yescharacteristic
173C2×C59⋊C8C2×D411yescharacteristic
174C2×C59⋊C8C2×D411yescharacteristic
175D472—21no—
176D472—21no—
177D472—21no—
178D472—42no—
179D472#4112yes—
180C8×D59—21no—
181C8×D59—21no—
182C16×C59—21no—
183C16×C59—21no—
184C59⋊C16—21no—
185C59⋊C16—21no—
186C23×C59—41no—
187C23×C59C2×D811yescharacteristic
188C22×D59—41no—
189C22×D59—41no—
190C22×D59—82no—
191C22×D59—82no—
192C22×D59—82no—
193C22×D59—82no—
194C22×D59—161no—
195C22×D59—161no—
196C22×D59C2×D811yescharacteristic
197Q8×C59—41no—
198Q8×C59—41no—
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—82no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—41no—
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C22×D411yescharacteristic
209C22⋊D59—22no—
210C22⋊D59—82no—
211C22⋊D59—82no—
212C22⋊D59—161no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215C2×Dic59C2×D811yescharacteristic
216C2×Dic59C2×D811yescharacteristic
217C2×Dic59C2×D811yescharacteristic
218D236—22no—
219D236—41no—
220D236—41no—
221D236—41no—
222D236—82no—
223D236—82no—
224C4×D59—21no—
225C4×D59—21no—
226C4×D59—82no—
227C4×D59—82no—
228C8×C59—21no—
229C8×C59C2×#4111yescharacteristic
230C8×C59H(5,𝔽2)11yescharacteristic
231C59⋊C8—21no—
232C59⋊C8—21no—
233#33877—591nomaximal, Sylow
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—21no—
237C22×C59—81no—
238C22×C59—81no—
239C22×C59—81no—
240C22×C59C22×D811yescharacteristic
241C2×D59—21no—
242C2×D59—21no—
243C2×D59—42no—
244C2×D59—81no—
245C2×D59—81no—
246C2×D59—81no—
247C2×D59—81no—
248C2×D59—81no—
249C2×D59—164no—
250C2×D59—164no—
251C2×D59#50912yes—
252C4×C59—21no—
253C4×C59—81no—
254C4×C59—81no—
255C4×C59—81no—
256C4×C59#54611yescharacteristic
257C4×C59#57711yescharacteristic
258Dic59—21no—
259Dic59—21no—
260Dic59—21no—
261Dic59—81no—
262Dic59—81no—
263#3336—591no—
264D4×D8—591no—
265#2244—591no—
266#2234—591no—
267#2232—591no—
268#2228—591no—
269#2220D5911yescharacteristic, complemented
270#2217—591no—
271#2207—591no—
272#2187—591no—
273#2187—591no—
274#2175—591no—
275#2168—591no—
276#2140—591no—
277C2×#357—591no—
278C2×C59—41no—
279C2×C59—161no—
280C2×C59#329411yescharacteristic
281C2×C59C2×#50911yescharacteristic
282C2×C59#341611yescharacteristic
283D59—22no—
284D59—82no—
285D59—161no—
286C22×D8—591no—
287C22×D8—591no—
288#552—1181no—
289D42—1181no—
290C2×SD32—591no—
291C2×SD32—591no—
292C2×SD32C2×D5911yescharacteristic, complemented
293C2×M32C2×D5911yescharacteristic, complemented
294C22×C16—591no—
295#499—591no—
296#493—591no—
297#485—1181no—
298#483—1181no—
299#479—1181no—
300#467—591no—
301#467C2×D5911yescharacteristic, complemented
302#466—591no—
303#466C2×D5911yescharacteristic
304#459—1181no—
305#447—1181no—
306#447—1181no—
307C4×Q16—591no—
308C4×D8—591no—
309C8×D4—591no—
310#427—591no—
311C2×#107—591no—
312#367—591no—
313#366—591no—
314#366C2×D5911yescharacteristic, complemented
315C4⋊C16—591no—
316#358C2×D5911yescharacteristic, complemented
317#357—591no—
318#357—591no—
319#357—591no—
320#357—591no—
321#357—591no—
322#357—591no—
323#357—591no—
324#357—591no—
325#357C2×D5911yescharacteristic, complemented
326C22⋊C16—591no—
327C22⋊C16—591no—
328#346—591no—
329C59#3387711yescharacteristic, complemented, Sylow
330C22×D4—1181no—
331C22×D4—1181no—
332C22×D4—2361no—
333C2×Q16C22×D5911yescharacteristic
334C2×D8—591no—
335C2×D8—591no—
336C2×D8—1182no—
337C2×D8—1181no—
338C2×D8—1181no—
339C2×D8—1182no—
340C2×D8—1181no—
341C2×D8—2361no—
342C2×D8—2361no—
343C2×D8C22×D5911yescharacteristic
344C22×C8—591no—
345C22×C8C22×D5911yescharacteristic
346#127—1181no—
347#126—2361no—
348#125—1181no—
349#125—2361no—
350#123—2361no—
351C22⋊Q8—21no—
352C22⋊Q8—1181no—
353#121—21no—
354#121—1181no—
355#121—2361no—
356#120—2361no—
357#120—2361no—
358C4×Q8—1181no—
359C4×D4—591no—
360C4×D4—1181no—
361#117—1181no—
362#117—2361no—
363C2×C4⋊C4—1181no—
364SD32—41no—
365SD32—1182no—
366SD32—2361no—
367M32—41no—
368C2×C16—591no—
369C2×C16—591no—
370C2×C16—2361no—
371C2×C16C22×D5911yescharacteristic
372C2×C16C22×D5911yescharacteristic
373#107—591no—
374#107—591no—
375#107—1181no—
376#107C22×D5911yescharacteristic
377#107C22×D5911yescharacteristic
378#106—1181no—
379#106—1181no—
380C4⋊C8—591no—
381#103—21no—
382#103—1181no—
383#103—1181no—
384#102—21no—
385#102—1181no—
386#102—1181no—
387C22⋊C8—591no—
388C22⋊C8—591no—
389C4×C8—591no—
390C24—2361no—
391C24—2361no—
392C2×Q8—21no—
393C2×D4—21no—
394C2×D4—41no—
395C2×D4—591no—
396C2×D4—1181no—
397C2×D4—1181no—
398C2×D4—4721no—
399C2×D4—4721no—
400C2×D4—4724no—
401C2×D4—2362no—
402C2×D4—2361no—
403C2×D4—2361no—
404C2×D4—2361no—
405C2×D4—2361no—
406C2×D4—2362no—
407C2×D4—2361no—
408C22×C4—591no—
409C22×C4—2361no—
410C22×C4—2361no—
411C22×C4D4×D5911yescharacteristic
412Q16—21no—
413Q16—21no—
414D8—21no—
415D8—21no—
416D8—592no—
417D8—1181no—
418D8—1181no—
419D8—1181no—
420D8—4721no—
421D8—2362no—
422D8—2362no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—591no—
426C2×C8—591no—
427C2×C8—1181no—
428C2×C8—1181no—
429C2×C8D4×D5911yescharacteristic
430C2×C8C23×D5911yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—591no—
435C4⋊C4—1181no—
436C4⋊C4—1181no—
437C4⋊C4—2361no—
438C4⋊C4—2361no—
439C4⋊C4—2361no—
440C4⋊C4—2361no—
441C4⋊C4—2361no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—591no—
445C22⋊C4—591no—
446C22⋊C4—2361no—
447C22⋊C4—2361no—
448C22⋊C4—2361no—
449C22⋊C4—2361no—
450C42—591no—
451C42—2361no—
452C42—2361no—
453C16—21no—
454C16—21no—
455C16—1181no—
456C16—1181no—
457C23—41no—
458C23—591no—
459C23—9441no—
460C23—9441no—
461C23—4722no—
462C23—4722no—
463C23—4722no—
464C23—4722no—
465C23—2361no—
466C23—2361no—
467C23D8×D5911yescharacteristic
468Q8—41no—
469Q8—41no—
470D4—41no—
471D4—41no—
472D4—82no—
473D4—1182no—
474D4—1182no—
475D4—9441no—
476D4—4722no—
477D4—4722no—
478D4—4722no—
479D4—4722no—
480D4—2361no—
481D4—2361no—
482D4—2361no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—591no—
489C2×C4—591no—
490C2×C4—591no—
491C2×C4—1181no—
492C2×C4—1181no—
493C2×C4—4722no—
494C2×C4—4722no—
495C2×C4—2361no—
496C2×C4—2361no—
497C2×C4D8×D5911yescharacteristic
498C2×C4C2×D4×D5911yescharacteristic
499C8—21no—
500C8—1181no—
501C8—1181no—
502C8#41×D5911yescharacteristic
503C8#1427377211yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—592no—
511C22—1181no—
512C22—1181no—
513C22—9444no—
514C22—9444no—
515C22—4721no—
516C22—4721no—
517C22—4721no—
518C22—4721no—
519C22—4721no—
520C22—2362no—
521C22C2×D8×D5911yescharacteristic
522C4—21no—
523C4—81no—
524C4—81no—
525C4—81no—
526C4—1181no—
527C4—1181no—
528C4—1181no—
529C4—4721no—
530C4—4721no—
531C4#1683804111yescharacteristic
532C4#1683820911yescharacteristic
533C2—41no—
534C2—161no—
535C2—1182no—
536C2—9441no—
537C2—4722no—
538C2#2143590411yescharacteristic
539C2#509×D5911yescharacteristic
540C2#2143724111yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2220C2
socleC22×C59C22×D8
prime radical#21435770C2
hypercenter#2220D59
hypocenterC59#33877
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2220

Complements

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

NH ≅ G/Ntype
1. #21436459C2semidirect
3. #21426336C2semidirect
4. #21426284C2semidirect
5. #21426278C2semidirect
6. #21426256C2semidirect
7. #21426195C2semidirect
8. #21426134C2semidirect
9. #21426024C2semidirect
10. #21426023C2semidirect
11. #21425963C2semidirect
12. #21425919C2semidirect
13. #21425813C2semidirect
14. #357×D59C2semidirect
15. C59×#2220C2semidirect
21. C2×#14273670C22semidirect
22. C2×C16×D59C22semidirect
23. #16837803C22semidirect
28. #16837640C22semidirect
33. Q16×Dic59C22semidirect
36. #16837440C22semidirect
37. #107×D59C22semidirect
38. #16837229C22semidirect
39. #16837227C22semidirect
40. Dic59⋊C16C22semidirect
43. #16837204C22semidirect
44. #16837204C22semidirect
45. #16837203C22semidirect
46. #16837203C22semidirect
48. #16837202C22semidirect
49. C22⋊(C59⋊C16)C22semidirect
50. (C2×D59)⋊C16C22semidirect
51. #16837177C22semidirect
52. C2×SD32×C59C22semidirect
53. C2×M32×C59C22semidirect
54. C59×#467C22semidirect
56. C59×#366C22semidirect
57. C59×#358C22semidirect
58. C59×#357C22semidirect
269. #2220D59semidirect
292. C2×SD32C2×D59semidirect
293. C2×M32C2×D59semidirect
301. #467C2×D59semidirect
314. #366C2×D59semidirect
316. #358C2×D59semidirect
325. #357C2×D59semidirect
329. C59#33877semidirect

Maximal subgroups

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

subgroupindex
#214364592
#214357702
#214263362
#214262842
#214262782
#214262562
#214261952
#214261342
#214260242
#214260232
#214259632
#214259192
#214258132
#357×D592
C59×#22202
#3387759