Finite Group Databasebeta

← D59×#2220 #28743143

Subgroups — D59×#2220 #28743143

Order 15104.

734 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD32×D59C211yesmaximal, characteristic, complemented
2C2×M32×D59C211yesmaximal, characteristic, complemented
3#467×D59C211yesmaximal, characteristic, complemented
4#466×D59C211yesmaximal, characteristic
5#21426212C211yesmaximal, characteristic, complemented
6#21426211C211yesmaximal, characteristic, complemented
7#21426210C211yesmaximal, characteristic, complemented
8#21426209C211yesmaximal, characteristic, complemented
9#21426208C211yesmaximal, characteristic, complemented
10#21426207C211yesmaximal, characteristic, complemented
11#21426206C211yesmaximal, characteristic, complemented
12#366×D59C211yesmaximal, characteristic, complemented
13#358×D59C211yesmaximal, characteristic, complemented
14#357×D59C211yesmaximal, characteristic, complemented
15C59×#2220C211yesmaximal, characteristic, complemented
16C2×Q16×D59C2211yescharacteristic
17C2×D8×D59C2211yescharacteristic
18C22×C8×D59C2211yescharacteristic
19C22⋊Q8×D59—21no—
20#121×D59—21no—
21C2×#14273672C2211yescharacteristic, complemented
22C2×#14273671C2211yescharacteristic, complemented
23C2×#14273670C2211yescharacteristic, complemented
24SD32×D59—24no—
25C2×#14273667C2211yescharacteristic, complemented
26C2×#14273666C2211yescharacteristic, complemented
27C2×#14273666C2211yescharacteristic, complemented
28M32×D59—24no—
29C2×C16×D59C2211yescharacteristic
30C2×C16×D59C2211yescharacteristic
31#16837644C2211yescharacteristic, complemented
32#16837643C2211yescharacteristic, complemented
33#16837642—21no—
34#16837641—21no—
35#16837640C2211yescharacteristic, complemented
36#16837639C2211yescharacteristic
37#16837638C2211yescharacteristic
38#16837637—21no—
39#16837636—21no—
40#16837635C2211yescharacteristic
41#107×D59C2211yescharacteristic
42#107×D59C2211yescharacteristic
43#103×D59—21no—
44#102×D59—21no—
45#16837228C2211yescharacteristic, complemented
46#16837227C2211yescharacteristic, complemented
47#16837227C2211yescharacteristic, complemented
48#16837207C2211yescharacteristic, complemented
49#16837206C2211yescharacteristic, complemented
50#16837205C2211yescharacteristic, complemented
51#16837204C2211yescharacteristic, complemented
52#16837203C2211yescharacteristic, complemented
53#16837202C2211yescharacteristic, complemented
54C2×SD32×C59C2211yescharacteristic, complemented
55C2×M32×C59C2211yescharacteristic, complemented
56C59×#467C2211yescharacteristic, complemented
57C59×#466C2211yescharacteristic
58C59×#366C2211yescharacteristic, complemented
59C59×#358C2211yescharacteristic, complemented
60C59×#357C2211yescharacteristic, complemented
61C2×Q8×D59—21no—
62C2×D4×D59—21no—
63C2×D4×D59—41no—
64C22×C4×D59D411yescharacteristic
65C2×Dic236C2311yescharacteristic
66C2×#11745672—21no—
67Q16×D59—22no—
68Q16×D59—22no—
69C2×D472C2311yescharacteristic
70C2×#11745669—21no—
71D8×D59—22no—
72D8×D59—22no—
73C22×C59⋊C8D411yescharacteristic
74C2×C8×D59—22no—
75C2×C8×D59—22no—
76C2×C8×D59D411yescharacteristic
77C2×C8×D59D411yescharacteristic
78C2×C8×D59C2311yescharacteristic
79C2×C8×D59C2311yescharacteristic
80#14273707—21no—
81#14273706—21no—
82#14273705—41no—
83#14273704—41no—
84#14273703—21no—
85#14273702—21no—
86#14273701—21no—
87#14273700—41no—
88#14273699—41no—
89#14273698—21no—
90C4⋊C4×D59—21no—
91C4⋊C4×D59—21no—
92C4⋊C4×D59—41no—
93C22⋊C4×D59—41no—
94C22⋊C4×D59—41no—
95#14273672—22no—
96#14273671—22no—
97#14273670—22no—
98#14273667—22no—
99#14273666—22no—
100#14273666—22no—
101C2×C59⋊C16C2311yescharacteristic
102C2×C59⋊C16C2311yescharacteristic
103C16×D59—22no—
104C16×D59D414yescomplemented
105#14273661C2311yescharacteristic
106#14273661C2311yescharacteristic
107#14273660—21no—
108#14273660—21no—
109#14273652—21no—
110#14273651—21no—
111#14273650—21no—
112#14273649—21no—
113#14273648—21no—
114#14273647—21no—
115C2×Q16×C59C2311yescharacteristic
116C2×D8×C59C2311yescharacteristic
117C22×C8×C59C2311yescharacteristic
118C22⋊Q8×C59—21no—
119#121×C59—21no—
120SD32×C59—22no—
121M32×C59—22no—
122C2×C16×C59C2311yescharacteristic
123C2×C16×C59C2311yescharacteristic
124#107×C59C2311yescharacteristic
125#107×C59C2311yescharacteristic
126#103×C59—21no—
127#102×C59—21no—
128C23×D59—41no—
129C23×D59D811yescharacteristic
130C2×Q8×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—41no—
133C22×C4×C59C2×D411yescharacteristic
134C2×C22⋊D59—41no—
135C2×C22⋊D59—41no—
136C2×C22⋊D59—41no—
137C22×Dic59D811yescharacteristic
138Q8×D59—42no—
139Q8×D59—42no—
140D4×D59—42no—
141D4×D59—42no—
142D4×D59—84no—
143C2×D236—21no—
144C2×D236—41no—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—41no—
148C2×C4×D59—41no—
149C2×C4×D59—41no—
150C2×C4×D59D811yescharacteristic
151C2×C4×D59D811yescharacteristic
152C2×C4×D59C2×D411yescharacteristic
153C2×C4×D59C2×D411yescharacteristic
154C2×Dic118—21no—
155C2×Dic118—41no—
156Q16×C59—21no—
157Q16×C59—21no—
158D8×C59—21no—
159D8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C2×D411yescharacteristic
163C2×C8×C59C2411yescharacteristic
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—41no—
167C22⋊C4×C59—41no—
168C22⋊C4×C59—41no—
169C22⋊Dic59—41no—
170C22⋊Dic59—41no—
171#11745672—42no—
172#11745669—42no—
173(C2×D59)⋊C4—41no—
174(C2×D59)⋊C4—41no—
175(C2×D59)⋊C4—41no—
176(C2×D59)⋊C4—41no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—41no—
180Dic59⋊C4—41no—
181Dic59⋊C4—41no—
182Dic59⋊C4—41no—
183Dic59⋊C4—41no—
184C2×C59⋊C8—22no—
185C2×C59⋊C8C2×D411yescharacteristic
186C2×C59⋊C8C2×D411yescharacteristic
187Dic236—21no—
188Dic236—21no—
189D472—21no—
190D472—21no—
191C8×D59—22no—
192C8×D59—21no—
193C8×D59—21no—
194C8×D59—22no—
195C8×D59C2×D412yes—
196C8×D59#4112yes—
197C16×C59—21no—
198C16×C59C2×D412yescomplemented
199C59⋊C16—21no—
200C59⋊C16C2×D412yescomplemented
201C23×C59—41no—
202C23×C59C2×D811yescharacteristic
203C22×D59—22no—
204C22×D59—22no—
205C22×D59—22no—
206C22×D59—41no—
207C22×D59—82no—
208C22×D59—82no—
209C22×D59—82no—
210C22×D59C2×D811yescharacteristic
211C22×D59C2×D811yescharacteristic
212Q8×C59—41no—
213Q8×C59—41no—
214D4×C59—41no—
215D4×C59—41no—
216D4×C59—82no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59C2×D811yescharacteristic
223C2×C4×C59C22×D411yescharacteristic
224C22⋊D59—82no—
225C22⋊D59—82no—
226C22⋊D59—82no—
227C2×Dic59—22no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231C2×Dic59C2×D811yescharacteristic
232C2×Dic59C2×D811yescharacteristic
233D236—41no—
234D236—41no—
235D236—82no—
236C4×D59—22no—
237C4×D59—21no—
238C4×D59—21no—
239C4×D59—22no—
240C4×D59—82no—
241C4×D59—82no—
242C4×D59—82no—
243C4×D59#12112yes—
244C4×D59#13512yes—
245Dic118—41no—
246Dic118—41no—
247Dic118—82no—
248C8×C59—21no—
249C8×C59C22×D411yescharacteristic
250C8×C59C2×#4111yescharacteristic
251C59⋊C8—21no—
252C59⋊C8#12112yes—
253C2×#2220—591nomaximal, Sylow
254C22×C59—21no—
255C22×C59—21no—
256C22×C59—21no—
257C22×C59—81no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59C22×D811yescharacteristic
261C2×D59—21no—
262C2×D59—21no—
263C2×D59—21no—
264C2×D59—44no—
265C2×D59—81no—
266C2×D59—81no—
267C2×D59—81no—
268C2×D59—164no—
269C2×D59#46612yes—
270C2×D59#50912yes—
271C2×D59#51012yes—
272C4×C59—21no—
273C4×C59—81no—
274C4×C59—81no—
275C4×C59—81no—
276C4×C59C2×#12111yescharacteristic
277C4×C59C2×#13511yescharacteristic
278Dic59—21no—
279Dic59—81no—
280Dic59—81no—
281Dic59—81no—
282Dic59#46612yes—
283C22×SD32—591no—
284C22×M32—591no—
285C2×#467—591no—
286C2×#466—591no—
287#2220—591no—
288#2220—591no—
289#2220—591no—
290#2220—591no—
291#2220—591no—
292#2220—591no—
293#2220—591no—
294#2220D5911yescharacteristic, direct product factor
295C2×#366—591no—
296C2×#358—591no—
297C2×#357—591no—
298C2×C59—41no—
299C2×C59—161no—
300C2×C59C2×#46611yescharacteristic
301C2×C59C2×#50911yescharacteristic
302C2×C59C2×#51011yescharacteristic
303D59—41no—
304D59—161no—
305D59#222014yesdirect product factor
306C22×Q16—591no—
307C22×D8—591no—
308C23×C8—591no—
309C2×C22⋊Q8—1181no—
310C2×#121—1181no—
311C2×SD32—591no—
312C2×SD32—591no—
313C2×SD32—591no—
314C2×SD32—1184no—
315C2×SD32C2×D5911yescharacteristic, complemented
316C2×M32—591no—
317C2×M32—591no—
318C2×M32—591no—
319C2×M32—1184no—
320C2×M32C2×D5911yescharacteristic, complemented
321C22×C16—591no—
322C22×C16—591no—
323#467—591no—
324#467—591no—
325#467—591no—
326#467—1181no—
327#467—1181no—
328#467C2×D5911yescharacteristic, complemented
329#466—591no—
330#466—591no—
331#466—591no—
332#466—1181no—
333#466—1181no—
334#466C2×D5911yescharacteristic
335C2×#107—591no—
336C2×#107—591no—
337C2×#103—1181no—
338C2×#102—1181no—
339#366—591no—
340#366—591no—
341#366—591no—
342#366C2×D5911yescharacteristic, complemented
343#358—591no—
344#358—591no—
345#358—591no—
346#358C2×D5911yescharacteristic, complemented
347#357—591no—
348#357—591no—
349#357—591no—
350#357C2×D5911yescharacteristic, complemented
351C59C2×#222011yescharacteristic, complemented, Sylow
352C22×Q8—1181no—
353C22×D4—1181no—
354C22×D4—2361no—
355C23×C4—591no—
356C2×Q16—591no—
357C2×Q16—1182no—
358C2×Q16—1182no—
359C2×Q16—1181no—
360C2×Q16C22×D5911yescharacteristic
361C2×D8—591no—
362C2×D8—1182no—
363C2×D8—1182no—
364C2×D8—1181no—
365C2×D8C22×D5911yescharacteristic
366C22×C8—591no—
367C22×C8—591no—
368C22×C8—591no—
369C22×C8—591no—
370C22×C8—591no—
371C22×C8—1182no—
372C22×C8—1182no—
373C22×C8C22×D5911yescharacteristic
374C22⋊Q8—21no—
375C22⋊Q8—1181no—
376C22⋊Q8—1181no—
377C22⋊Q8—1181no—
378C22⋊Q8—2361no—
379C22⋊Q8—2361no—
380#121—21no—
381#121—1181no—
382#121—1181no—
383#121—1181no—
384#121—2361no—
385#121—2361no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—2361no—
389C2×C22⋊C4—2361no—
390C2×C22⋊C4—2361no—
391SD32—22no—
392SD32—1182no—
393SD32—1182no—
394SD32—1182no—
395M32—22no—
396M32—1182no—
397M32—1182no—
398M32—1182no—
399C2×C16—591no—
400C2×C16—594no—
401C2×C16—591no—
402C2×C16—1182no—
403C2×C16C22×D5911yescharacteristic
404C2×C16C22×D5911yescharacteristic
405#107—591no—
406#107—591no—
407#107—1181no—
408#107—1181no—
409#107C22×D5911yescharacteristic
410#107C22×D5911yescharacteristic
411#103—21no—
412#103—1181no—
413#103—1181no—
414#103—1181no—
415#102—21no—
416#102—1181no—
417#102—1181no—
418#102—1181no—
419C24—591no—
420C24—2361no—
421C2×Q8—21no—
422C2×Q8—1181no—
423C2×Q8—2361no—
424C2×Q8—2362no—
425C2×Q8—2362no—
426C2×D4—21no—
427C2×D4—41no—
428C2×D4—1181no—
429C2×D4—4724no—
430C2×D4—2361no—
431C2×D4—2362no—
432C2×D4—2362no—
433C2×D4—2361no—
434C2×D4—2361no—
435C2×D4—2361no—
436C22×C4—591no—
437C22×C4—591no—
438C22×C4—591no—
439C22×C4—591no—
440C22×C4—591no—
441C22×C4—1182no—
442C22×C4—1182no—
443C22×C4—2361no—
444C22×C4—2361no—
445C22×C4—2361no—
446C22×C4D4×D5911yescharacteristic
447Q16—21no—
448Q16—21no—
449Q16—1181no—
450Q16—1181no—
451Q16—2362no—
452D8—21no—
453D8—21no—
454D8—1181no—
455D8—1181no—
456D8—2362no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—591no—
460C2×C8—592no—
461C2×C8—592no—
462C2×C8—591no—
463C2×C8—1182no—
464C2×C8—1182no—
465C2×C8—1182no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8D4×D5911yescharacteristic
469C2×C8C23×D5911yescharacteristic
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—41no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—2361no—
476C4⋊C4—2361no—
477C4⋊C4—2361no—
478C4⋊C4—2361no—
479C4⋊C4—2361no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—2361no—
483C22⋊C4—2361no—
484C22⋊C4—2361no—
485C22⋊C4—2361no—
486C22⋊C4—2361no—
487C22⋊C4—2361no—
488C16—21no—
489C16—592no—
490C16—1181no—
491C16D4×D5912yescomplemented
492C23—41no—
493C23—591no—
494C23—591no—
495C23—1182no—
496C23—1182no—
497C23—1182no—
498C23—4722no—
499C23—4722no—
500C23—4722no—
501C23—2361no—
502C23D8×D5911yescharacteristic
503Q8—41no—
504Q8—41no—
505Q8—4722no—
506Q8—2361no—
507Q8—2361no—
508D4—41no—
509D4—41no—
510D4—82no—
511D4—4722no—
512D4—4722no—
513D4—4722no—
514D4—4722no—
515D4—2361no—
516D4—2361no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—591no—
523C2×C4—592no—
524C2×C4—592no—
525C2×C4—591no—
526C2×C4—1182no—
527C2×C4—1182no—
528C2×C4—1182no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—4722no—
532C2×C4—4722no—
533C2×C4—4722no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4D8×D5911yescharacteristic
538C2×C4C2×D4×D5911yescharacteristic
539C8—21no—
540C8—592no—
541C8—1181no—
542C8C2×D4×D5911yescharacteristic
543C8#41×D5911yescharacteristic
544C22—21no—
545C22—21no—
546C22—21no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—592no—
551C22—592no—
552C22—592no—
553C22—1181no—
554C22—1181no—
555C22—1181no—
556C22—9444no—
557C22—4721no—
558C22—4721no—
559C22—4721no—
560C22—2364no—
561C22C2×D8×D5911yescharacteristic
562C4—21no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—592no—
567C4—1181no—
568C4—4721no—
569C4—4721no—
570C4—4721no—
571C4#121×D5911yescharacteristic
572C4#135×D5911yescharacteristic
573C2—41no—
574C2—161no—
575C2—594no—
576C2—9441no—
577C2—2361no—
578C2#466×D5911yescharacteristic
579C2#509×D5911yescharacteristic
580C2#510×D5911yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2220C2
socleC22×C59C22×D8
prime radical#466×D59C2
hypercenter#2220D59
hypocenterC59C2×#2220
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. C2×SD32×D59C2semidirect
2. C2×M32×D59C2semidirect
3. #467×D59C2semidirect
5. #21426212C2semidirect
6. #21426211C2semidirect
7. #21426210C2semidirect
8. #21426209C2semidirect
9. #21426208C2semidirect
10. #21426207C2semidirect
11. #21426206C2semidirect
12. #366×D59C2semidirect
13. #358×D59C2semidirect
14. #357×D59C2semidirect
15. C59×#2220C2semidirect
21. C2×#14273672C22semidirect
22. C2×#14273671C22semidirect
23. C2×#14273670C22semidirect
25. C2×#14273667C22semidirect
26. C2×#14273666C22semidirect
27. C2×#14273666C22semidirect
31. #16837644C22semidirect
32. #16837643C22semidirect
35. #16837640C22semidirect
45. #16837228C22semidirect
46. #16837227C22semidirect
47. #16837227C22semidirect
48. #16837207C22semidirect
49. #16837206C22semidirect
50. #16837205C22semidirect
51. #16837204C22semidirect
52. #16837203C22semidirect
53. #16837202C22semidirect
54. C2×SD32×C59C22semidirect
55. C2×M32×C59C22semidirect
56. C59×#467C22semidirect
58. C59×#366C22semidirect
59. C59×#358C22semidirect
60. C59×#357C22semidirect
104. C16×D59D4semidirect
198. C16×C59C2×D4semidirect
200. C59⋊C16C2×D4semidirect
294. #2220D59direct (N × H)
305. D59#2220direct (N × H)
315. C2×SD32C2×D59semidirect
320. C2×M32C2×D59semidirect
328. #467C2×D59semidirect
342. #366C2×D59semidirect
346. #358C2×D59semidirect
350. #357C2×D59semidirect
351. C59C2×#2220semidirect
491. C16D4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×SD32×D592
C2×M32×D592
#467×D592
#466×D592
#214262122
#214262112
#214262102
#214262092
#214262082
#214262072
#214262062
#366×D592
#358×D592
#357×D592
C59×#22202
C2×#222059