Finite Group Databasebeta

← #28742004

Subgroups — #28742004

Order 15104.

738 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837839C211yesmaximal, characteristic
2C4×#14273750C211yesmaximal, characteristic, complemented
3C4×D8×D59C211yesmaximal, characteristic, complemented
4#21425973C211yesmaximal, characteristic, complemented
5#21425970C211yesmaximal, characteristic, complemented
6#21425958C211yesmaximal, characteristic, complemented
7SD32×Dic59C211yesmaximal, characteristic, complemented
8#21425945C211yesmaximal, characteristic, complemented
9#21425942C211yesmaximal, characteristic, complemented
10#21425940C211yesmaximal, characteristic, complemented
11#21425881C211yesmaximal, characteristic, complemented
12#21425810C211yesmaximal, characteristic, complemented
13#21425807C211yesmaximal, characteristic, complemented
14#21425689C211yesmaximal, characteristic, complemented
15C59×#2175C211yesmaximal, characteristic, complemented
16C2×#14273750C2211yescharacteristic
17C2×D8×D59C2211yescharacteristic
18C4×#11745689—21no—
19C4×D4×D59—21no—
20#16837839C418yescomplemented
21C2×#14273670C2211yescharacteristic
22C2×D944C2211yescharacteristic
23C2×#14273668C2211yescharacteristic
24C2×#14273666C2211yescharacteristic
25C2×#14273666C2211yescharacteristic
26Q16×Dic59C2211yescharacteristic, complemented
27C4×#11745671—21no—
28#16837487—21no—
29C4×D472C2211yescharacteristic
30C4×D472C2211yescharacteristic, complemented
31D8×Dic59C2211yescharacteristic
32#16837484C2211yescharacteristic
33#16837484C2211yescharacteristic, complemented
34C4×#11745669—21no—
35#16837482—21no—
36#16837439C2211yescharacteristic
37#107×D59C2211yescharacteristic
38#16837397—21no—
39#102×D59—21no—
40C4×C8×D59C2211yescharacteristic
41#16837230C2211yescharacteristic, complemented
42#16837229C2211yescharacteristic, complemented
43#16837227C2211yescharacteristic, complemented
44#16837206C2211yescharacteristic, complemented
45#16837204C2211yescharacteristic
46#16837204C2211yescharacteristic, complemented
47#16837203C2211yescharacteristic
48#16837202C2211yescharacteristic
49#16837202C2211yescharacteristic, complemented
50#16837178C2211yescharacteristic, complemented
51C16×Dic59C2211yescharacteristic, complemented
52C16×Dic59C2211yescharacteristic, complemented
53C2×SD32×C59C2211yescharacteristic
54C4×Q16×C59C2211yescharacteristic, complemented
55C4×D8×C59C2211yescharacteristic
56C59×#366C2211yescharacteristic, complemented
57C59×#358C2211yescharacteristic, complemented
58C59×#357C2211yescharacteristic
59C59×#346C2211yescharacteristic, complemented
60C2×#11745689—21no—
61C2×D4×D59—21no—
62C22×C4×D59—41no—
63#14273750—22no—
64#14273750C2×C414yescomplemented
65C2×#11745671—21no—
66C2×D472C2311yescharacteristic
67C2×D472C2311yescharacteristic
68C2×#11745669—21no—
69D8×D59—22no—
70D8×D59C2×C414yescomplemented
71C2×C8×D59D411yescharacteristic
72C2×C8×D59C2311yescharacteristic
73Q8×Dic59—21no—
74C4×D236—21no—
75C4×D236—21no—
76C4×D236—41no—
77D4×Dic59—21no—
78#14273689—21no—
79#14273689—21no—
80#14273689—41no—
81C4×C22⋊D59—41no—
82(C22⋊D59)⋊C4—41no—
83#14273685—21no—
84#14273685—41no—
85C4⋊C4×D59—21no—
86C22⋊C4×D59—41no—
87C42×D59—41no—
88C42×D59D411yescharacteristic
89#14273670C2×C414yescomplemented
90D944C2×C414yescomplemented
91#14273668C2×C414yescomplemented
92#14273666—22no—
93#14273666C2×C414yescomplemented
94C2×C59⋊C16C2311yescharacteristic
95C2×C59⋊C16C2311yescharacteristic
96#14273661C2311yescharacteristic
97#14273661C2311yescharacteristic
98#14273660—21no—
99#14273658—21no—
100#14273651—21no—
101#14273649—21no—
102#14273649—21no—
103#14273648—21no—
104#14273647—21no—
105#14273647—21no—
106C4×C59⋊C8D411yescharacteristic
107C8×Dic59D411yescharacteristic
108C8×Dic59C2311yescharacteristic
109C2×Q16×C59C2311yescharacteristic
110C2×D8×C59C2311yescharacteristic
111C4×Q8×C59—21no—
112C4×D4×C59—21no—
113SD32×C59C2×C414yescomplemented
114C2×C16×C59C2311yescharacteristic
115C2×C16×C59C2311yescharacteristic
116#107×C59C2311yescharacteristic
117#107×C59C2311yescharacteristic
118#103×C59—21no—
119#102×C59—21no—
120C4×C8×C59C2311yescharacteristic
121C23×D59—41no—
122C2×Q8×C59—21no—
123C2×D4×C59—21no—
124C22×C4×C59—41no—
125C2×C22⋊D59—41no—
126C22×Dic59—41no—
127#11745689—24no—
128#11745689—42no—
129D4×D59—24no—
130D4×D59—42no—
131C2×D236—21no—
132C2×D236—21no—
133C2×D236—41no—
134C2×C4×D59—41no—
135C2×C4×D59—41no—
136C2×C4×D59—41no—
137C2×C4×D59—41no—
138C2×C4×D59—41no—
139C2×C4×D59—41no—
140C2×C4×D59—84no—
141C2×C4×D59D811yescharacteristic
142C2×C4×D59D811yescharacteristic
143C2×C4×D59C2×D411yescharacteristic
144Q16×C59—21no—
145Q16×C59C22×C412yescomplemented
146D8×C59—21no—
147D8×C59C22×C412yes—
148C2×C8×C59C2×D411yescharacteristic
149C2×C8×C59C2411yescharacteristic
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—41no—
153C22⋊C4×C59—41no—
154C42×C59—41no—
155C42×C59C2×D411yescharacteristic
156C22⋊Dic59—41no—
157#11745671—24no—
158#11745669—24no—
159(C2×D59)⋊C4—41no—
160(C2×D59)⋊C4—41no—
161(C2×D59)⋊C4—41no—
162(C2×D59)⋊C4—41no—
163(C2×D59)⋊C4—41no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—41no—
167Dic59⋊C4—41no—
168C4×Dic59—41no—
169C4×Dic59—41no—
170C4×Dic59D811yescharacteristic
171C4×Dic59D811yescharacteristic
172C4×Dic59C2×D411yescharacteristic
173C2×C59⋊C8C2×D411yescharacteristic
174C2×C59⋊C8C2×D411yescharacteristic
175D472—21no—
176D472—21no—
177D472C22×C412yes—
178D472C22×C412yescomplemented
179C8×D59—22no—
180C8×D59C22×C412yes—
181C8×D59#4112yes—
182C16×C59—21no—
183C16×C59C22×C412yescomplemented
184C59⋊C16—21no—
185C59⋊C16C22×C412yescomplemented
186C23×C59—41no—
187C22×D59—41no—
188C22×D59—41no—
189C22×D59—44no—
190C22×D59—82no—
191C22×D59—82no—
192C22×D59C2×D811yescharacteristic
193Q8×C59—22no—
194Q8×C59—41no—
195D4×C59—22no—
196D4×C59—41no—
197C2×C4×C59—41no—
198C2×C4×C59—41no—
199C2×C4×C59—41no—
200C2×C4×C59—82no—
201C2×C4×C59C2×D811yescharacteristic
202C2×C4×C59C2×D811yescharacteristic
203C2×C4×C59C22×D411yescharacteristic
204C22⋊D59—44no—
205C2×Dic59—41no—
206C2×Dic59—41no—
207C2×Dic59—82no—
208C2×Dic59C2×D811yescharacteristic
209C2×Dic59C2×D811yescharacteristic
210C2×Dic59C2×D811yescharacteristic
211D236—22no—
212D236—22no—
213D236—44no—
214D236—41no—
215D236—41no—
216C4×D59—22no—
217C4×D59—41no—
218C4×D59—44no—
219C4×D59—82no—
220C4×D59—82no—
221C4×D59—82no—
222C4×D59—82no—
223C4×D59—82no—
224C4×D59C4×D412yes—
225C4×D59#13512yes—
226C8×C59—21no—
227C8×C59C23×C411yescharacteristic
228C8×C59C2×#4111yescharacteristic
229C59⋊C8—21no—
230C59⋊C8C4×D412yes—
231C4×#509—591nomaximal, Sylow
232C22×C59—42no—
233C22×C59—81no—
234C22×C59—81no—
235C22×C59C22×D811yescharacteristic
236C2×D59—21no—
237C2×D59—21no—
238C2×D59—42no—
239C2×D59—42no—
240C2×D59—88no—
241C2×D59—81no—
242C2×D59—81no—
243C2×D59—81no—
244C2×D59—81no—
245C2×D59C4×D812yes—
246C4×C59—21no—
247C4×C59—21no—
248C4×C59—42no—
249C4×C59—81no—
250C4×C59—81no—
251C4×C59C2×C4×D411yescharacteristic
252C4×C59C2×#13511yescharacteristic
253Dic59—21no—
254Dic59—81no—
255Dic59—81no—
256Dic59C4×D812yes—
257Dic59#50912yescomplemented
258C2×#509—591no—
259C4×#135—591no—
260C2×C4×D8—591no—
261#2177—591no—
262#2177—591no—
263#2175—591no—
264#2175D5911yescharacteristic, complemented
265C4×SD32—591no—
266C4×SD32—591no—
267C4×D16—591no—
268C4×D16—591no—
269#2159—591no—
270#2140—591no—
271#2139—591no—
272C4×M32—591no—
273C2×C59—82no—
274C2×C59C2×C4×D811yescharacteristic
275C2×C59C2×#50911yescharacteristic
276C2×C59#341411yescharacteristic
277D59—22no—
278D59—82no—
279D59—82no—
280C2×#135—591no—
281C22×D8—591no—
282C4×#41—1181no—
283C2×C4×D4—1181no—
284#509—598no—
285C2×SD32—591no—
286C2×SD32C2×D5911yescharacteristic
287C2×D16—591no—
288C2×D16—591no—
289C2×M32—591no—
290C2×M32—591no—
291C4×Q16—591no—
292C4×Q16C2×D5911yescharacteristic, complemented
293C4×SD16—1181no—
294C4×SD16—1181no—
295C4×D8—591no—
296C4×D8—591no—
297C4×D8—591no—
298C4×D8—591no—
299C4×D8—591no—
300C4×D8—1181no—
301C4×D8—1181no—
302C4×D8C2×D5911yescharacteristic
303#427—591no—
304C2×#107—591no—
305#416—1181no—
306C2×#102—1181no—
307C2×C4×C8—591no—
308#367—591no—
309#367—591no—
310#366—591no—
311#366C2×D5911yescharacteristic, complemented
312#358—591no—
313#358C2×D5911yescharacteristic, complemented
314#357—591no—
315#357—591no—
316#357—591no—
317#357—591no—
318#357—591no—
319#357C2×D5911yescharacteristic
320#346—591no—
321#346C2×D5911yescharacteristic, complemented
322C4×C16—591no—
323C4×C16—591no—
324C59C4×#50911yescharacteristic, complemented, Sylow
325C2×#41—1181no—
326C22×D4—1181no—
327C23×C4—2361no—
328#135—594no—
329#135—1182no—
330C2×Q16C22×D5911yescharacteristic
331C2×SD16—1181no—
332C2×D8—594no—
333C2×D8—591no—
334C2×D8—591no—
335C2×D8—1182no—
336C2×D8—1181no—
337C2×D8C22×D5911yescharacteristic
338C22×C8—591no—
339C22×C8—591no—
340C4×Q8—21no—
341C4×Q8—1181no—
342C4×D4—21no—
343C4×D4—1181no—
344C4×D4—1181no—
345C4×D4—1181no—
346C4×D4—1181no—
347C4×D4—1181no—
348C4×D4—2361no—
349C4×D4—2361no—
350C4×D4—2361no—
351C4×D4—2361no—
352#117—1181no—
353#117—2361no—
354C2×C4⋊C4—1181no—
355C2×C22⋊C4—2361no—
356C2×C42—591no—
357C2×C42—2361no—
358SD32—594no—
359SD32C4×D5914yescomplemented
360D16—594no—
361D16—594no—
362M32—594no—
363M32—1182no—
364C2×C16—591no—
365C2×C16—591no—
366C2×C16C22×D5911yescharacteristic
367C2×C16C22×D5911yescharacteristic
368#107—591no—
369#107—591no—
370#107—1181no—
371#107C22×D5911yescharacteristic
372#107C22×D5911yescharacteristic
373#106—1181no—
374#103—21no—
375#103—1181no—
376#102—21no—
377#102—1181no—
378#102—1181no—
379#102—1181no—
380#102—1181no—
381#102—1181no—
382C4×C8—591no—
383C4×C8—591no—
384C4×C8—591no—
385C4×C8C22×D5911yescharacteristic
386C24—2361no—
387#41—1184no—
388#41—2362no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—1184no—
392C2×D4—1181no—
393C2×D4—1181no—
394C2×D4—2361no—
395C2×D4—2362no—
396C2×D4—2361no—
397C22×C4—41no—
398C22×C4—591no—
399C22×C4—591no—
400C22×C4—591no—
401C22×C4—4724no—
402C22×C4—2361no—
403C22×C4—2361no—
404C22×C4—2361no—
405C22×C4—2361no—
406C22×C4—2361no—
407C22×C4—2361no—
408C22×C4—2361no—
409Q16—21no—
410Q16C2×C4×D5912yescomplemented
411SD16—1184no—
412D8—21no—
413D8—592no—
414D8—592no—
415D8—1184no—
416D8—1181no—
417D8—1181no—
418D8C2×C4×D5912yes—
419C2×C8—591no—
420C2×C8—592no—
421C2×C8—592no—
422C2×C8—591no—
423C2×C8—1182no—
424C2×C8D4×D5911yescharacteristic
425C2×C8C23×D5911yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—1181no—
430C4⋊C4—1181no—
431C4⋊C4—2361no—
432C4⋊C4—2361no—
433C22⋊C4—41no—
434C22⋊C4—2361no—
435C22⋊C4—2361no—
436C22⋊C4—2361no—
437C22⋊C4—2361no—
438C22⋊C4—2361no—
439C22⋊C4—2361no—
440C42—41no—
441C42—591no—
442C42—591no—
443C42—591no—
444C42—2361no—
445C42—2361no—
446C42D4×D5911yescharacteristic
447C16—21no—
448C16—592no—
449C16—1181no—
450C16C2×C4×D5912yescomplemented
451C23—41no—
452C23—591no—
453C23—4722no—
454C23—4722no—
455C23—2364no—
456C23—2361no—
457C23—2361no—
458Q8—22no—
459Q8—41no—
460D4—22no—
461D4—41no—
462D4—1182no—
463D4—1182no—
464D4—2364no—
465D4—2364no—
466D4—2361no—
467D4—2361no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—591no—
473C2×C4—591no—
474C2×C4—592no—
475C2×C4—592no—
476C2×C4—591no—
477C2×C4—1182no—
478C2×C4—4722no—
479C2×C4—4722no—
480C2×C4—4722no—
481C2×C4—4722no—
482C2×C4—4722no—
483C2×C4—4722no—
484C2×C4—2361no—
485C2×C4—2364no—
486C2×C4—2361no—
487C2×C4—2361no—
488C2×C4D8×D5911yescharacteristic
489C2×C4D8×D5911yescharacteristic
490C2×C4C2×D4×D5911yescharacteristic
491C8—21no—
492C8—592no—
493C8—1181no—
494C8C22×C4×D5911yescharacteristic
495C8#41×D5911yescharacteristic
496C22—42no—
497C22—81no—
498C22—81no—
499C22—592no—
500C22—1181no—
501C22—1181no—
502C22—4728no—
503C22—4721no—
504C22—4721no—
505C22—4721no—
506C22—4721no—
507C22—2362no—
508C22—2362no—
509C22C2×D8×D5911yescharacteristic
510C4—21no—
511C4—21no—
512C4—42no—
513C4—81no—
514C4—81no—
515C4—592no—
516C4—592no—
517C4—1181no—
518C4—4721no—
519C4—4721no—
520C4C4×D4×D5911yescharacteristic
521C4#135×D5911yescharacteristic
522C2—82no—
523C2—1182no—
524C2—4722no—
525C2—4722no—
526C2C4×D8×D5911yescharacteristic
527C2#509×D5911yescharacteristic
528C2#2143723111yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutatorC8×C59C23×C4
FrattiniC2×C8C23×D59
FittingC59×#2175C2
socleC22×C59C22×D8
prime radicalC2×#16837839C2
cosocleC2×C8×C59C24
hypercenter#2175D59
hypocenterC59C4×#509
LCSG ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. C4×#14273750C2semidirect
3. C4×D8×D59C2semidirect
4. #21425973C2semidirect
5. #21425970C2semidirect
6. #21425958C2semidirect
7. SD32×Dic59C2semidirect
8. #21425945C2semidirect
9. #21425942C2semidirect
10. #21425940C2semidirect
11. #21425881C2semidirect
12. #21425810C2semidirect
13. #21425807C2semidirect
14. #21425689C2semidirect
15. C59×#2175C2semidirect
20. #16837839C4semidirect
26. Q16×Dic59C22semidirect
30. C4×D472C22semidirect
33. #16837484C22semidirect
41. #16837230C22semidirect
42. #16837229C22semidirect
43. #16837227C22semidirect
44. #16837206C22semidirect
46. #16837204C22semidirect
49. #16837202C22semidirect
50. #16837178C22semidirect
51. C16×Dic59C22semidirect
52. C16×Dic59C22semidirect
54. C4×Q16×C59C22semidirect
56. C59×#366C22semidirect
57. C59×#358C22semidirect
59. C59×#346C22semidirect
64. #14273750C2×C4semidirect
70. D8×D59C2×C4semidirect
89. #14273670C2×C4semidirect
90. D944C2×C4semidirect
91. #14273668C2×C4semidirect
93. #14273666C2×C4semidirect
113. SD32×C59C2×C4semidirect
145. Q16×C59C22×C4semidirect
178. D472C22×C4semidirect
183. C16×C59C22×C4semidirect
185. C59⋊C16C22×C4semidirect
257. Dic59#509semidirect
264. #2175D59semidirect
292. C4×Q16C2×D59semidirect
311. #366C2×D59semidirect
313. #358C2×D59semidirect
321. #346C2×D59semidirect
324. C59C4×#509semidirect
359. SD32C4×D59semidirect
410. Q16C2×C4×D59semidirect
450. C16C2×C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378392
C4×#142737502
C4×D8×D592
#214259732
#214259702
#214259582
SD32×Dic592
#214259452
#214259422
#214259402
#214258812
#214258102
#214258072
#214256892
C59×#21752
C4×#50959