Finite Group Databasebeta

← #28743434

Subgroups — #28743434

Order 15104.

514 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434621C211yesmaximal, characteristic, complemented
2#21433008C211yesmaximal, characteristic
3#21426273C211yesmaximal, characteristic, complemented
4#21426272C211yesmaximal, characteristic, complemented
5#21426271C211yesmaximal, characteristic, complemented
6#21426270C211yesmaximal, characteristic, complemented
7#21426266C211yesmaximal, characteristic, complemented
8#21426265C211yesmaximal, characteristic, complemented
9#21426264C211yesmaximal, characteristic, complemented
10#21425867C211yesmaximal, characteristic, complemented
11#365×D59C211yesmaximal, characteristic, complemented
12#21425860C211yesmaximal, characteristic, complemented
13#21425830C211yesmaximal, characteristic
14#360×D59C211yesmaximal, characteristic
15C59×#2230C211yesmaximal, characteristic, complemented
16#135×D59C2211yescharacteristic
17#16838167—21no—
18#16838029—21no—
19M32×D59C2211yescharacteristic
20M32×D59C2211yescharacteristic
21#16837809—21no—
22#16837808—21no—
23#16837804C2211yescharacteristic, complemented
24#16837802C2211yescharacteristic, complemented
25#16837799C2211yescharacteristic, complemented
26#16837522—21no—
27#16837521—21no—
28#16837520C2211yescharacteristic
29#16837519C2211yescharacteristic
30#16837518C2211yescharacteristic
31#108×D59C2211yescharacteristic
32#16837441—21no—
33#16837437C2211yescharacteristic
34#106×D59C2211yescharacteristic
35#16837417—21no—
36#16837356C2211yescharacteristic
37#16837226C2211yescharacteristic, complemented
38#16837226C2211yescharacteristic, complemented
39#16837225C2211yescharacteristic, complemented
40#16837225C2211yescharacteristic, complemented
41#16837224C2211yescharacteristic, complemented
42#16837224C2211yescharacteristic, complemented
43#16837223C2211yescharacteristic, complemented
44#16837222C2211yescharacteristic, complemented
45#16837222C2211yescharacteristic, complemented
46#16837213C2211yescharacteristic
47#16837213C2211yescharacteristic
48#16837212C2211yescharacteristic
49#16837212C2211yescharacteristic
50#16837211C2211yescharacteristic
51#16837211C2211yescharacteristic
52C59×#498C2211yescharacteristic, complemented
53C59×#443C2211yescharacteristic
54C59×#365C2211yescharacteristic, complemented
55C59×#365C2211yescharacteristic, complemented
56C59×#364C2211yescharacteristic, complemented
57C59×#360C2211yescharacteristic
58C59×#360C2211yescharacteristic
59#41×D59—21no—
60#14273750Q811yescharacteristic, complemented
61#14273749C2311yescharacteristic
62#14273748—21no—
63#14273747—21no—
64#14273746Q811yescharacteristic, complemented
65Q16×D59Q811yescharacteristic, complemented
66SD16×D59—21no—
67D8×D59Q811yescharacteristic, complemented
68#14273735—21no—
69#14273735—41no—
70#14273734—21no—
71#14273733—21no—
72#14273733—41no—
73C2×#11745659—41no—
74C2×#11745659D411yescharacteristic
75M16×D59—21no—
76M16×D59—41no—
77C2×C8×D59C2311yescharacteristic
78#14273726—21no—
79#14273725—41no—
80#14273719—21no—
81#14273718—21no—
82#14273717—41no—
83#14273705—41no—
84#14273703—41no—
85#14273685—21no—
86#14273684D411yescharacteristic
87C4⋊C4×D59—21no—
88#14273667C2311yescharacteristic
89#14273667C2311yescharacteristic
90#14273666—21no—
91#14273666—21no—
92C16×D59—21no—
93C16×D59—21no—
94#14273663C2311yescharacteristic
95#14273662—21no—
96#14273661—21no—
97#14273660—21no—
98#14273660—21no—
99#14273659C2311yescharacteristic
100#14273659C2311yescharacteristic
101#14273658—21no—
102#14273658—21no—
103#14273655—21no—
104#14273654—21no—
105#14273653—21no—
106#14273638D411yescharacteristic
107#14273637D411yescharacteristic
108C8×Dic59C2311yescharacteristic
109#135×C59C2311yescharacteristic
110#131×C59—21no—
111C4⋊Q8×C59—21no—
112M32×C59C2311yescharacteristic
113M32×C59C2311yescharacteristic
114#108×C59C2311yescharacteristic
115#106×C59—21no—
116#106×C59C2311yescharacteristic
117#106×C59C2311yescharacteristic
118#104×C59—21no—
119C4×C8×C59C2311yescharacteristic
120#41×C59—21no—
121C2×Q8×C59—41no—
122#11745689—21no—
123Q8×D59—21no—
124#11745687—21no—
125#11745687—41no—
126D4×D59—21no—
127D4×D59—41no—
128#11745685—21no—
129#11745685—41no—
130C2×C4×D59—41no—
131C2×C4×D59—41no—
132C2×C4×D59C2×D411yescharacteristic
133C2×Dic118—41no—
134Q16×C59C2×Q811yescharacteristic
135SD16×C59—21no—
136D8×C59C2×Q811yescharacteristic
137M16×C59—21no—
138M16×C59—41no—
139C2×C8×C59—41no—
140C2×C8×C59C2×D411yescharacteristic
141C2×C8×C59C2411yescharacteristic
142C4⋊C4×C59—21no—
143C4⋊C4×C59—21no—
144C4⋊C4×C59—41no—
145C42×C59C2×D411yescharacteristic
146#11745672—21no—
147#11745671—21no—
148#11745670—21no—
149#11745669—21no—
150(C2×D59)⋊C4—21no—
151(C2×D59)⋊C4—41no—
152C4⋊Dic59—21no—
153C4⋊Dic59—21no—
154C4⋊Dic59—41no—
155Dic59⋊C4—21no—
156Dic59⋊C4—41no—
157Dic59⋊C4—41no—
158Dic59⋊C4—41no—
159Dic59⋊C4—41no—
160Dic59⋊C4—41no—
161C4×Dic59—41no—
162C4×Dic59C2×D411yescharacteristic
163#11745664—21no—
164#11745664—41no—
165C2×C59⋊C8—41no—
166C2×C59⋊C8C2×D411yescharacteristic
167C2×C59⋊C8C2×D411yescharacteristic
168Dic236C2×Q811yescharacteristic
169D472C2×Q811yescharacteristic
170#11745660—21no—
171#11745659—41no—
172#11745659—41no—
173#11745659—41no—
174#11745659SD1611yescharacteristic
175#11745659SD1611yescharacteristic
176#11745659SD1611yescharacteristic
177#11745659SD1611yescharacteristic
178C8×D59—41no—
179C8×D59—41no—
180C8×D59—41no—
181C8×D59C2×Q811yescharacteristic
182C8×D59C2×Q811yescharacteristic
183C8×D59#4111yescharacteristic
184C8×D59#4111yescharacteristic
185C16×C59—21no—
186C16×C59—21no—
187C59⋊C16—21no—
188C59⋊C16—21no—
189C22×D59—41no—
190C22×D59AGL(1,ℤ/8)11yescharacteristic
191Q8×C59—21no—
192Q8×C59—161no—
193D4×C59—21no—
194D4×C59—41no—
195C2×C4×C59—21no—
196C2×C4×C59—41no—
197C2×C4×C59—41no—
198C2×C4×C59—41no—
199C2×C4×C59C22×D411yescharacteristic
200C22⋊D59—41no—
201C22⋊D59—41no—
202C22⋊D59—41no—
203C2×Dic59—21no—
204C2×Dic59—41no—
205C2×Dic59—41no—
206C2×Dic59—41no—
207C2×Dic59AGL(1,ℤ/8)11yescharacteristic
208D236—21no—
209D236—41no—
210C4×D59—41no—
211C4×D59—41no—
212C4×D59—41no—
213C4×D59—161no—
214C4×D59C22⋊Q811yescharacteristic
215C4×D59C22⋊Q811yescharacteristic
216C4×D59C2×SD1611yescharacteristic
217C4×D59C2×SD1611yescharacteristic
218Dic118—21no—
219Dic118—41no—
220Dic118—161no—
221C8×C59—41no—
222C8×C59C2×SD1611yescharacteristic
223C8×C59C2×SD1611yescharacteristic
224C8×C59C22×Q811yescharacteristic
225C8×C59C2×#4111yescharacteristic
226C59⋊C8—41no—
227C59⋊C8C22⋊Q811yescharacteristic
228C59⋊C8C22⋊Q811yescharacteristic
229C59⋊C8C2×SD1611yescharacteristic
230C59⋊C8C2×SD1611yescharacteristic
231#33638—591nomaximal, Sylow
232C22×C59—41no—
233C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
234C2×D59—41no—
235C2×D59—41no—
236C2×D59—161no—
237C2×D59#47611yescharacteristic
238C2×D59#47611yescharacteristic
239C4×C59—41no—
240C4×C59—41no—
241C4×C59—81no—
242C4×C59—81no—
243C4×C59C2×C22⋊Q811yescharacteristic
244C4×C59C22×SD1611yescharacteristic
245Dic59—41no—
246Dic59—41no—
247Dic59—81no—
248Dic59—81no—
249Dic59#47611yescharacteristic
250Dic59#47611yescharacteristic
251#3163—591no—
252#2957—591no—
253#2231—591no—
254#2231—591no—
255#2231—591no—
256#2231—591no—
257#2230—591no—
258#2230—591no—
259#2230—591no—
260#2230D5911yescharacteristic, complemented
261#2155—591no—
262C2×#365—591no—
263#2153—591no—
264#2145—591no—
265C2×#360—591no—
266C2×C59—21no—
267C2×C59—81no—
268C2×C59C2×#47611yescharacteristic
269D59—21no—
270D59—21no—
271D59—81no—
272C2×#135—591no—
273#568—1181no—
274#544—1181no—
275C2×M32—591no—
276C2×M32—591no—
277#501—1181no—
278#501—1181no—
279#499—591no—
280#499—591no—
281#498—591no—
282#498C2×D5911yescharacteristic, complemented
283#444—1181no—
284#444—1181no—
285#443—591no—
286#443—591no—
287#443—591no—
288#443C2×D5911yescharacteristic
289C2×#108—591no—
290#428—1181no—
291#427—591no—
292C2×#106—591no—
293#421—1181no—
294#405—591no—
295#365—591no—
296#365—591no—
297#365—591no—
298#365—591no—
299#365—591no—
300#365—591no—
301#365C2×D5911yescharacteristic, complemented
302#365C2×D5911yescharacteristic, complemented
303#364—591no—
304#364—591no—
305#364—591no—
306#364C2×D5911yescharacteristic, complemented
307#360—591no—
308#360—591no—
309#360—591no—
310#360—591no—
311#360—591no—
312#360—591no—
313#360C2×D5911yescharacteristic
314#360C2×D5911yescharacteristic
315C59#3363811yescharacteristic, complemented, Sylow
316C2×#41—1181no—
317#135—591no—
318#135—591no—
319#135—591no—
320#135—1181no—
321#135—1181no—
322#135C22×D5911yescharacteristic
323C2×Q16—591no—
324C2×SD16—1181no—
325C2×D8—591no—
326#131—21no—
327#131—1181no—
328#131—1181no—
329#131—1181no—
330#131—2361no—
331#131—2361no—
332C2×M16—591no—
333C2×M16—1181no—
334C2×M16—2361no—
335C2×M16—2361no—
336C22×C8—591no—
337C4⋊Q8—21no—
338C4⋊Q8—1181no—
339C4⋊Q8—2361no—
340#125—1181no—
341#125—1181no—
342#125—2361no—
343C22⋊Q8—2361no—
344C22⋊Q8—2361no—
345#117—591no—
346#117—1181no—
347C2×C4⋊C4—1181no—
348M32—591no—
349M32—591no—
350M32—1181no—
351M32—1181no—
352M32C22×D5911yescharacteristic
353M32C22×D5911yescharacteristic
354C2×C16—1181no—
355C2×C16—1181no—
356#108—591no—
357#108—1181no—
358#108C22×D5911yescharacteristic
359#107—1181no—
360#107—1181no—
361#107—1181no—
362#106—21no—
363#106—591no—
364#106—591no—
365#106—1181no—
366#106—1181no—
367#106C22×D5911yescharacteristic
368#106C22×D5911yescharacteristic
369#104—21no—
370#104—1181no—
371#104—1181no—
372#104—1181no—
373#97—591no—
374#97—591no—
375C4×C8—591no—
376C4×C8C22×D5911yescharacteristic
377#41—21no—
378#41—1181no—
379#41—1181no—
380#41—1181no—
381#41—2361no—
382#41—2361no—
383C2×Q8—41no—
384C2×Q8—1181no—
385C2×Q8—2361no—
386C2×D4—1181no—
387C2×D4—2361no—
388C22×C4—591no—
389C22×C4—2361no—
390C22×C4—2361no—
391Q16—591no—
392Q16—1181no—
393Q16Q8×D5911yescharacteristic
394SD16—21no—
395SD16—1181no—
396SD16—1181no—
397SD16—1181no—
398D8—591no—
399D8—1181no—
400D8Q8×D5911yescharacteristic
401M16—21no—
402M16—41no—
403M16—591no—
404M16—591no—
405M16—591no—
406M16—591no—
407M16—1181no—
408M16—2361no—
409M16—2361no—
410M16—2361no—
411M16—2361no—
412C2×C8—41no—
413C2×C8—591no—
414C2×C8—591no—
415C2×C8—591no—
416C2×C8—591no—
417C2×C8—591no—
418C2×C8—591no—
419C2×C8—2361no—
420C2×C8—2361no—
421C2×C8—2361no—
422C2×C8—2361no—
423C2×C8D4×D5911yescharacteristic
424C2×C8C23×D5911yescharacteristic
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—1181no—
429C4⋊C4—1181no—
430C4⋊C4—1181no—
431C4⋊C4—2361no—
432C4⋊C4—2361no—
433C4⋊C4—2361no—
434C4⋊C4—2361no—
435C4⋊C4—2361no—
436C4⋊C4—2361no—
437C22⋊C4—1181no—
438C22⋊C4—2361no—
439C42—591no—
440C42—2361no—
441C42D4×D5911yescharacteristic
442C16—21no—
443C16—21no—
444C16—1181no—
445C16—1181no—
446C23—591no—
447C23—2361no—
448Q8—21no—
449Q8—161no—
450Q8—1181no—
451Q8—9441no—
452Q8—2361no—
453D4—21no—
454D4—41no—
455D4—1181no—
456D4—2361no—
457D4—2361no—
458D4—2361no—
459D4—2361no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—591no—
465C2×C4—591no—
466C2×C4—591no—
467C2×C4—591no—
468C2×C4—591no—
469C2×C4—1181no—
470C2×C4—9441no—
471C2×C4—2361no—
472C2×C4—2361no—
473C2×C4—2361no—
474C2×C4—2361no—
475C2×C4—2361no—
476C2×C4—2361no—
477C2×C4C2×D4×D5911yescharacteristic
478C8—41no—
479C8—591no—
480C8—591no—
481C8—591no—
482C8—591no—
483C8—2361no—
484C8SD16×D5911yescharacteristic
485C8SD16×D5911yescharacteristic
486C8C2×Q8×D5911yescharacteristic
487C8#41×D5911yescharacteristic
488C22—41no—
489C22—591no—
490C22—591no—
491C22—9441no—
492C22—2361no—
493C22—2361no—
494C22AGL(1,ℤ/8)×D5911yescharacteristic
495C4—41no—
496C4—41no—
497C4—81no—
498C4—81no—
499C4—591no—
500C4—591no—
501C4—4721no—
502C4—4721no—
503C4—2361no—
504C4—2361no—
505C4C22⋊Q8×D5911yescharacteristic
506C4C2×SD16×D5911yescharacteristic
507C2—21no—
508C2—81no—
509C2—1181no—
510C2—1181no—
511C2—4721no—
512C2#476×D5911yescharacteristic
subgroupG/H
centerC2#476×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2230C2
socleC2×C59C2×#476
prime radical#135×D59C22
hypercenter#2230D59
hypocenterC59#33638
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2230

Complements

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

NH ≅ G/Ntype
1. #21434621C2semidirect
3. #21426273C2semidirect
4. #21426272C2semidirect
5. #21426271C2semidirect
6. #21426270C2semidirect
7. #21426266C2semidirect
8. #21426265C2semidirect
9. #21426264C2semidirect
10. #21425867C2semidirect
11. #365×D59C2semidirect
12. #21425860C2semidirect
15. C59×#2230C2semidirect
23. #16837804C22semidirect
24. #16837802C22semidirect
25. #16837799C22semidirect
37. #16837226C22semidirect
38. #16837226C22semidirect
39. #16837225C22semidirect
40. #16837225C22semidirect
41. #16837224C22semidirect
42. #16837224C22semidirect
43. #16837223C22semidirect
44. #16837222C22semidirect
45. #16837222C22semidirect
52. C59×#498C22semidirect
54. C59×#365C22semidirect
55. C59×#365C22semidirect
56. C59×#364C22semidirect
60. #14273750Q8semidirect
64. #14273746Q8semidirect
65. Q16×D59Q8semidirect
67. D8×D59Q8semidirect
260. #2230D59semidirect
282. #498C2×D59semidirect
301. #365C2×D59semidirect
302. #365C2×D59semidirect
306. #364C2×D59semidirect
315. C59#33638semidirect

Maximal subgroups

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

subgroupindex
#214346212
#214330082
#214262732
#214262722
#214262712
#214262702
#214262662
#214262652
#214262642
#214258672
#365×D592
#214258602
#214258302
#360×D592
C59×#22302
#3363859