Finite Group Databasebeta

← #28743161

Subgroups — #28743161

Order 15104.

738 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837836C211yesmaximal, characteristic, complemented
2C2×M32×D59C211yesmaximal, characteristic, complemented
3#21433675C211yesmaximal, characteristic, complemented
4#466×D59C211yesmaximal, characteristic, complemented
5#21426209C211yesmaximal, characteristic, complemented
6#21426207C211yesmaximal, characteristic, complemented
7#21426197C211yesmaximal, characteristic, complemented
8#21426193C211yesmaximal, characteristic, complemented
9#21426191C211yesmaximal, characteristic, complemented
10#21426181C211yesmaximal, characteristic, complemented
11#21426179C211yesmaximal, characteristic, complemented
12#21425883C211yesmaximal, characteristic, complemented
13#21425810C211yesmaximal, characteristic, complemented
14#21425805C211yesmaximal, characteristic, complemented
15C59×#2218C211yesmaximal, characteristic, complemented
16C2×#14273746C2211yescharacteristic
17C2×D8×D59C2211yescharacteristic
18C22×C8×D59C2211yescharacteristic
19#16837912—21no—
20#121×D59—21no—
21#16837836—24no—
22C2×#14273672C2211yescharacteristic, complemented
23C2×#14273671C2211yescharacteristic, complemented
24C2×#14273668C2211yescharacteristic, complemented
25C2×#14273667C2211yescharacteristic, complemented
26C2×#14273666C2211yescharacteristic
27C2×#14273666C2211yescharacteristic
28M32×D59—24no—
29C2×C16×D59C2211yescharacteristic, complemented
30C2×C16×D59C2211yescharacteristic, complemented
31#16837644C2211yescharacteristic, complemented
32#16837643C2211yescharacteristic, complemented
33#16837639C2211yescharacteristic
34#16837638C2211yescharacteristic
35#16837637—21no—
36#16837636—21no—
37#16837635C2211yescharacteristic
38#16837635C2211yescharacteristic, complemented
39#16837630—21no—
40#16837629—21no—
41#16837439C2211yescharacteristic
42#107×D59C2211yescharacteristic
43#16837395—21no—
44#102×D59—21no—
45#16837229C2211yescharacteristic, complemented
46#16837228C2211yescharacteristic, complemented
47#16837227C2211yescharacteristic, complemented
48#16837207C2211yescharacteristic, complemented
49#16837205C2211yescharacteristic, complemented
50#16837204C2211yescharacteristic, complemented
51#16837203C2211yescharacteristic, complemented
52#16837203C2211yescharacteristic, complemented
53#16837202C2211yescharacteristic, complemented
54C2×D16×C59C2211yescharacteristic, complemented
55C2×M32×C59C2211yescharacteristic, complemented
56C59×#466C2211yescharacteristic
57C59×#466C2211yescharacteristic, complemented
58C59×#367C2211yescharacteristic, complemented
59C59×#357C2211yescharacteristic, complemented
60C59×#357C2211yescharacteristic, complemented
61C2×#11745687—21no—
62C2×D4×D59—21no—
63C2×D4×D59—41no—
64C22×C4×D59D411yescharacteristic
65#14273746—22no—
66#14273746—22no—
67C2×Dic236C2311yescharacteristic
68C2×#11745670—21no—
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#14273709—41no—
81#14273707—21no—
82#14273706—21no—
83#14273702—21no—
84#14273701—21no—
85#14273700—41no—
86#14273699—41no—
87#14273698—21no—
88#14273698—21no—
89#14273695—41no—
90D4×Dic59—41no—
91(C22⋊D59)⋊C4—41no—
92#14273685—21no—
93C4⋊C4×D59—21no—
94C22⋊C4×D59—41no—
95#14273672—22no—
96#14273671—22no—
97#14273668—22no—
98#14273667—22no—
99#14273666—22no—
100#14273666D414yescomplemented
101C2×C59⋊C16C2311yescharacteristic
102C2×C59⋊C16C2311yescharacteristic
103C16×D59—22no—
104C16×D59—22no—
105#14273661C2311yescharacteristic
106#14273661C2311yescharacteristic
107#14273660—21no—
108#14273658—21no—
109#14273652—21no—
110#14273650—21no—
111#14273649—21no—
112#14273648—21no—
113#14273648—21no—
114#14273647—21no—
115C2×D8×C59C2311yescharacteristic
116C2×D8×C59C2311yescharacteristic
117C22×C8×C59C2311yescharacteristic
118#121×C59—21no—
119#121×C59—21no—
120D16×C59—22no—
121M32×C59—22no—
122C2×C16×C59C2311yescharacteristic
123C2×C16×C59C2311yescharacteristic
124#107×C59C2311yescharacteristic
125#107×C59C2311yescharacteristic
126#102×C59—21no—
127#102×C59—21no—
128C23×D59—41no—
129C23×D59D811yescharacteristic
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—41no—
133C2×D4×C59—41no—
134C22×C4×C59C2×D411yescharacteristic
135C2×C22⋊D59—41no—
136C2×C22⋊D59—41no—
137C2×C22⋊D59—41no—
138C2×C22⋊D59—41no—
139C2×C22⋊D59—41no—
140C22×Dic59—41no—
141C22×Dic59D811yescharacteristic
142#11745687—42no—
143#11745687—42no—
144D4×D59—42no—
145D4×D59—42no—
146D4×D59—84no—
147C2×D236—21no—
148C2×D236—41no—
149C2×C4×D59—22no—
150C2×C4×D59—22no—
151C2×C4×D59—41no—
152C2×C4×D59D811yescharacteristic
153C2×C4×D59D811yescharacteristic
154C2×C4×D59C2×D411yescharacteristic
155C2×C4×D59C2×D411yescharacteristic
156C2×Dic118—21no—
157D8×C59—21no—
158D8×C59—21no—
159D8×C59—21no—
160D8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59C2×D411yescharacteristic
164C2×C8×C59C2411yescharacteristic
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C22⋊C4×C59—41no—
168C22⋊C4×C59—41no—
169C22⋊Dic59—41no—
170C22⋊Dic59—41no—
171C22⋊Dic59—41no—
172#11745670—42no—
173#11745669—42no—
174(C2×D59)⋊C4—41no—
175(C2×D59)⋊C4—41no—
176(C2×D59)⋊C4—41no—
177(C2×D59)⋊C4—41no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—41no—
181Dic59⋊C4—41no—
182Dic59⋊C4—41no—
183C4×Dic59—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×C59—41no—
203C23×C59C2×D811yescharacteristic
204C22×D59—22no—
205C22×D59—22no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—82no—
209C22×D59—82no—
210C22×D59—82no—
211C22×D59C2×D811yescharacteristic
212C22×D59C2×D811yescharacteristic
213D4×C59—41no—
214D4×C59—41no—
215D4×C59—41no—
216D4×C59—41no—
217D4×C59—82no—
218D4×C59—82no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59C2×D811yescharacteristic
224C2×C4×C59C22×D411yescharacteristic
225C22⋊D59—82no—
226C22⋊D59—82no—
227C22⋊D59—82no—
228C22⋊D59—82no—
229C22⋊D59—82no—
230C2×Dic59—22no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—82no—
235C2×Dic59C2×D811yescharacteristic
236C2×Dic59C2×D811yescharacteristic
237D236—41no—
238D236—41no—
239D236—82no—
240C4×D59—22no—
241C4×D59—21no—
242C4×D59—21no—
243C4×D59—22no—
244C4×D59—82no—
245C4×D59#12112yes—
246C4×D59#13512yes—
247Dic118—41no—
248Dic118—41no—
249C8×C59—21no—
250C8×C59C22×D411yescharacteristic
251C8×C59C2×#4111yescharacteristic
252C59⋊C8—21no—
253C59⋊C8#12112yes—
254#33605—591nomaximal, Sylow
255C22×C59—21no—
256C22×C59—21no—
257C22×C59—21no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59C22×D811yescharacteristic
265C2×D59—21no—
266C2×D59—21no—
267C2×D59—21no—
268C2×D59—21no—
269C2×D59—44no—
270C2×D59—81no—
271C2×D59—81no—
272C2×D59—81no—
273C2×D59—164no—
274C2×D59#46612yes—
275C2×D59#50912yescomplemented
276C4×C59—21no—
277C4×C59—81no—
278C4×C59—81no—
279C4×C59C2×#12111yescharacteristic
280C4×C59C2×#13511yescharacteristic
281Dic59—21no—
282Dic59—81no—
283Dic59—81no—
284Dic59—81no—
285Dic59#46612yes—
286C2×#509—591no—
287C22×M32—591no—
288#3050—591no—
289C2×#466—591no—
290#2220—591no—
291#2220—591no—
292#2218—591no—
293#2218D5911yescharacteristic, complemented
294#2217—591no—
295#2217—591no—
296#2215—591no—
297#2215—591no—
298#2159—591no—
299#2140—591no—
300#2139—591no—
301C2×C59—41no—
302C2×C59—161no—
303C2×C59—161no—
304C2×C59C2×#46611yescharacteristic
305C2×C59C2×#50911yescharacteristic
306C2×C59#341411yescharacteristic
307D59—22no—
308D59—22no—
309D59—161no—
310C2×#135—591no—
311C22×D8—591no—
312C23×C8—591no—
313#525—1181no—
314C2×#121—1181no—
315#509—1184no—
316C2×SD32—591no—
317C2×SD32—591no—
318C2×D16—591no—
319C2×D16C2×D5911yescharacteristic, complemented
320C2×M32—591no—
321C2×M32—591no—
322C2×M32—591no—
323C2×M32—1184no—
324C2×M32C2×D5911yescharacteristic, complemented
325C22×C16—591no—
326C22×C16—591no—
327#467—591no—
328#467—591no—
329#466—591no—
330#466—591no—
331#466—591no—
332#466—591no—
333#466—1181no—
334#466—1181no—
335#466C2×D5911yescharacteristic
336#466C2×D5911yescharacteristic, complemented
337#465—1181no—
338#465—1181no—
339#427—591no—
340C2×#107—591no—
341#416—1181no—
342C2×#102—1181no—
343#367—591no—
344#367C2×D5911yescharacteristic, complemented
345#366—591no—
346#366—591no—
347#358—591no—
348#358—591no—
349#357—591no—
350#357—591no—
351#357—591no—
352#357—591no—
353#357C2×D5911yescharacteristic, complemented
354#357C2×D5911yescharacteristic, complemented
355C59#3360511yescharacteristic, complemented, Sylow
356C2×#41—1181no—
357C22×D4—1181no—
358C22×D4—2361no—
359C23×C4—591no—
360#135—1182no—
361#135—1182no—
362C2×Q16—591no—
363C2×SD16—1181no—
364C2×D8—591no—
365C2×D8—1182no—
366C2×D8—1182no—
367C2×D8—1181no—
368C2×D8C22×D5911yescharacteristic
369C2×D8C22×D5911yescharacteristic
370C22×C8—591no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8—591no—
374C22×C8—591no—
375C22×C8—1182no—
376C22×C8—1182no—
377C22×C8C22×D5911yescharacteristic
378#123—2361no—
379C22⋊Q8—1181no—
380C22⋊Q8—1181no—
381#121—21no—
382#121—21no—
383#121—1181no—
384#121—1181no—
385#121—1181no—
386#121—1181no—
387#121—2361no—
388#121—2361no—
389#120—2361no—
390C4×D4—2361no—
391C4×D4—2361no—
392#117—1181no—
393C2×C4⋊C4—1181no—
394C2×C22⋊C4—2361no—
395SD32—1182no—
396SD32—1182no—
397D16—22no—
398D16—1182no—
399M32—22no—
400M32—594no—
401M32—1182no—
402M32—1182no—
403C2×C16—591no—
404C2×C16—591no—
405C2×C16—1182no—
406C2×C16—1182no—
407C2×C16C22×D5911yescharacteristic
408C2×C16C22×D5911yescharacteristic
409#107—591no—
410#107—591no—
411#107—1181no—
412#107C22×D5911yescharacteristic
413#107C22×D5911yescharacteristic
414#106—1181no—
415#103—1181no—
416#103—1181no—
417#102—21no—
418#102—21no—
419#102—1181no—
420#102—1181no—
421#102—1181no—
422#102—1181no—
423C24—591no—
424C24—2361no—
425#41—2362no—
426#41—2362no—
427C2×Q8—1181no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—1181no—
433C2×D4—4724no—
434C2×D4—2361no—
435C2×D4—2362no—
436C2×D4—2362no—
437C2×D4—2361no—
438C2×D4—2361no—
439C2×D4—2361no—
440C2×D4—2361no—
441C2×D4—2361no—
442C22×C4—591no—
443C22×C4—591no—
444C22×C4—591no—
445C22×C4—591no—
446C22×C4—591no—
447C22×C4—1182no—
448C22×C4—1182no—
449C22×C4—2361no—
450C22×C4—2361no—
451C22×C4D4×D5911yescharacteristic
452Q16—1181no—
453Q16—1181no—
454SD16—2362no—
455D8—21no—
456D8—21no—
457D8—21no—
458D8—21no—
459D8—1181no—
460D8—1181no—
461D8—2362no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—591no—
465C2×C8—592no—
466C2×C8—592no—
467C2×C8—591no—
468C2×C8—1182no—
469C2×C8—1182no—
470C2×C8—1182no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8D4×D5911yescharacteristic
474C2×C8C23×D5911yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—2361no—
480C4⋊C4—2361no—
481C4⋊C4—2361no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—2361no—
485C22⋊C4—2361no—
486C22⋊C4—2361no—
487C22⋊C4—2361no—
488C22⋊C4—2361no—
489C22⋊C4—2361no—
490C22⋊C4—2361no—
491C42—2361no—
492C16—21no—
493C16—592no—
494C16—1181no—
495C16D4×D5912yescomplemented
496C23—41no—
497C23—41no—
498C23—591no—
499C23—591no—
500C23—1182no—
501C23—1182no—
502C23—4722no—
503C23—4722no—
504C23—4722no—
505C23—2361no—
506C23—2361no—
507C23D8×D5911yescharacteristic
508Q8—2361no—
509Q8—2361no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—82no—
515D4—82no—
516D4—4722no—
517D4—4722no—
518D4—4722no—
519D4—4722no—
520D4—4722no—
521D4—4722no—
522D4—2361no—
523D4—2361no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—591no—
529C2×C4—592no—
530C2×C4—592no—
531C2×C4—591no—
532C2×C4—1182no—
533C2×C4—1182no—
534C2×C4—1182no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—4722no—
538C2×C4—4722no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4D8×D5911yescharacteristic
543C2×C4C2×D4×D5911yescharacteristic
544C8—21no—
545C8—592no—
546C8—1181no—
547C8C2×D4×D5911yescharacteristic
548C8#41×D5911yescharacteristic
549C22—21no—
550C22—21no—
551C22—21no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—592no—
559C22—592no—
560C22—1181no—
561C22—1181no—
562C22—1181no—
563C22—1181no—
564C22—9444no—
565C22—4721no—
566C22—4721no—
567C22—4721no—
568C22—2364no—
569C22C2×D8×D5911yescharacteristic
570C4—21no—
571C4—81no—
572C4—81no—
573C4—592no—
574C4—1181no—
575C4—4721no—
576C4—4721no—
577C4—4721no—
578C4#121×D5911yescharacteristic
579C4#135×D5911yescharacteristic
580C2—41no—
581C2—161no—
582C2—161no—
583C2—1182no—
584C2—1182no—
585C2—9441no—
586C2#466×D5911yescharacteristic
587C2#509×D5911yescharacteristic
588C2#2143722811yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2218C2
socleC22×C59C22×D8
hypercenter#2218D59
hypocenterC59#33605
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2218

Complements

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

NH ≅ G/Ntype
1. C2×#16837836C2semidirect
2. C2×M32×D59C2semidirect
3. #21433675C2semidirect
4. #466×D59C2semidirect
5. #21426209C2semidirect
6. #21426207C2semidirect
7. #21426197C2semidirect
8. #21426193C2semidirect
9. #21426191C2semidirect
10. #21426181C2semidirect
11. #21426179C2semidirect
12. #21425883C2semidirect
13. #21425810C2semidirect
14. #21425805C2semidirect
15. C59×#2218C2semidirect
22. C2×#14273672C22semidirect
23. C2×#14273671C22semidirect
24. C2×#14273668C22semidirect
25. C2×#14273667C22semidirect
29. C2×C16×D59C22semidirect
30. C2×C16×D59C22semidirect
31. #16837644C22semidirect
32. #16837643C22semidirect
38. #16837635C22semidirect
45. #16837229C22semidirect
46. #16837228C22semidirect
47. #16837227C22semidirect
48. #16837207C22semidirect
49. #16837205C22semidirect
50. #16837204C22semidirect
51. #16837203C22semidirect
52. #16837203C22semidirect
53. #16837202C22semidirect
54. C2×D16×C59C22semidirect
55. C2×M32×C59C22semidirect
57. C59×#466C22semidirect
58. C59×#367C22semidirect
59. C59×#357C22semidirect
60. C59×#357C22semidirect
100. #14273666D4semidirect
198. C16×C59C2×D4semidirect
200. C59⋊C16C2×D4semidirect
275. C2×D59#509semidirect
293. #2218D59semidirect
319. C2×D16C2×D59semidirect
324. C2×M32C2×D59semidirect
336. #466C2×D59semidirect
344. #367C2×D59semidirect
353. #357C2×D59semidirect
354. #357C2×D59semidirect
355. C59#33605semidirect
495. C16D4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378362
C2×M32×D592
#214336752
#466×D592
#214262092
#214262072
#214261972
#214261932
#214261912
#214261812
#214261792
#214258832
#214258102
#214258052
C59×#22182
#3360559