Finite Group Databasebeta

← #28745727

Subgroups — #28745727

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#21436466C211yesmaximal, characteristic, complemented
2#21435771C211yesmaximal, characteristic
3#21426337C211yesmaximal, characteristic, complemented
4#21426286C211yesmaximal, characteristic, complemented
5#21426275C211yesmaximal, characteristic, complemented
6#21426254C211yesmaximal, characteristic, complemented
7#21426206C211yesmaximal, characteristic, complemented
8#21426189C211yesmaximal, characteristic, complemented
9#21426023C211yesmaximal, characteristic, complemented
10#21426023C211yesmaximal, characteristic, complemented
11#21425958C211yesmaximal, characteristic, complemented
12#21425920C211yesmaximal, characteristic, complemented
13#21425810C211yesmaximal, characteristic, complemented
14#357×D59C211yesmaximal, characteristic, complemented
15C59×#2207C211yesmaximal, characteristic, complemented
16C2×D8×D59C2211yescharacteristic
17C2×D8×D59C2211yescharacteristic
18#16838093—21no—
19#16838037—21no—
20C2×#14273670C2211yescharacteristic
21C2×#14273666C2211yescharacteristic, complemented
22C2×C16×D59C2211yescharacteristic, complemented
23#16837804C2211yescharacteristic, complemented
24#16837741—21no—
25#16837731—21no—
26#16837709—21no—
27#16837640C2211yescharacteristic, complemented
28#16837640C2211yescharacteristic, complemented
29#16837635C2211yescharacteristic
30#16837635C2211yescharacteristic
31#16837594—21no—
32#16837534—21no—
33#16837534—21no—
34C4×D472C2211yescharacteristic
35C8×D236C2211yescharacteristic
36#16837439C2211yescharacteristic, complemented
37#107×D59C2211yescharacteristic, complemented
38#16837230C2211yescharacteristic, complemented
39#16837228C2211yescharacteristic, complemented
40#16837227C2211yescharacteristic, complemented
41#16837206C2211yescharacteristic, complemented
42#16837204C2211yescharacteristic
43#16837204C2211yescharacteristic
44#16837203C2211yescharacteristic, complemented
45#16837203C2211yescharacteristic, complemented
46#16837203C2211yescharacteristic, complemented
47#16837203C2211yescharacteristic, complemented
48#16837202C2211yescharacteristic
49(C2×D59)⋊C16C2211yescharacteristic, complemented
50(C2×D59)⋊C16C2211yescharacteristic, complemented
51#16837178C2211yescharacteristic, complemented
52C2×SD32×C59C2211yescharacteristic, complemented
53C2×SD32×C59C2211yescharacteristic, complemented
54C59×#493C2211yescharacteristic
55C4×Q16×C59C2211yescharacteristic, complemented
56C59×C4⋊C16C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic, complemented
58C59×#357C2211yescharacteristic, complemented
59C2×D4×D59—21no—
60C2×D4×D59—21no—
61C2×D4×D59—41no—
62C2×D472C2311yescharacteristic
63C2×#11745669—21no—
64C2×#11745669—21no—
65D8×D59—22no—
66D8×D59—22no—
67D8×D59—41no—
68D8×D59—41no—
69C2×C8×D59C2311yescharacteristic
70C2×C8×D59C2311yescharacteristic
71#14273721—41no—
72#14273719—21no—
73#14273717—41no—
74#14273712—41no—
75#14273703—21no—
76#14273703—21no—
77#14273698—21no—
78#14273698—21no—
79#14273698—41no—
80#14273695—41no—
81#14273695—41no—
82C4×D236—21no—
83C4×D236D411yescharacteristic
84#14273685—21no—
85#14273684—41no—
86C4⋊C4×D59—21no—
87#14273670—22no—
88#14273666—41no—
89C2×C59⋊C16C2311yescharacteristic
90C2×C59⋊C16C2311yescharacteristic
91C16×D59—41no—
92#14273661C2311yescharacteristic
93#14273661C2311yescharacteristic
94#14273661C2311yescharacteristic
95#14273660—21no—
96#14273659—21no—
97#14273658—21no—
98C4⋊(C59⋊C8)D411yescharacteristic
99#14273651—21no—
100#14273651—21no—
101#14273649—21no—
102#14273648—21no—
103#14273648—21no—
104(C2×D59)⋊C8D411yescharacteristic
105(C2×D59)⋊C8D411yescharacteristic
106C2×Q16×C59C2311yescharacteristic
107C2×D8×C59—21no—
108C2×D8×C59C2311yescharacteristic
109C2×D8×C59C2311yescharacteristic
110#127×C59—21no—
111C4×Q8×C59—21no—
112SD32×C59—41no—
113SD32×C59—41no—
114C2×C16×C59C2311yescharacteristic
115C2×C16×C59C2311yescharacteristic
116#107×C59C2311yescharacteristic
117#103×C59—21no—
118C4×C8×C59C2311yescharacteristic
119C23×D59—41no—
120C23×D59—41no—
121C2×Q8×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—41no—
125C2×D4×C59—41no—
126C2×C22⋊D59—41no—
127C2×C22⋊D59—41no—
128C2×C22⋊D59—41no—
129C2×C22⋊D59—41no—
130D4×D59—42no—
131D4×D59—42no—
132D4×D59—81no—
133D4×D59—81no—
134D4×D59—84no—
135C2×D236—21no—
136C2×D236D811yescharacteristic
137C2×C4×D59—41no—
138C2×C4×D59—41no—
139C2×C4×D59C2×D411yescharacteristic
140C2×C4×D59C2×D411yescharacteristic
141Q16×C59—21no—
142Q16×C59—21no—
143D8×C59—21no—
144D8×C59—21no—
145D8×C59—21no—
146D8×C59—21no—
147D8×C59—81no—
148C2×C8×C59C2×D411yescharacteristic
149C2×C8×C59C2411yescharacteristic
150C4⋊C4×C59—21no—
151C4⋊C4×C59—41no—
152C42×C59—41no—
153C42×C59C2×D411yescharacteristic
154C22⋊Dic59—41no—
155C22⋊Dic59—41no—
156#11745669—42no—
157#11745669—42no—
158(C2×D59)⋊C4—41no—
159(C2×D59)⋊C4—41no—
160(C2×D59)⋊C4—41no—
161(C2×D59)⋊C4—41no—
162(C2×D59)⋊C4D811yescharacteristic
163(C2×D59)⋊C4D811yescharacteristic
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59D811yescharacteristic
168Dic59⋊C4—41no—
169Dic59⋊C4—41no—
170Dic59⋊C4—41no—
171Dic59⋊C4—41no—
172Dic59⋊C4—41no—
173C4×Dic59—41no—
174C2×C59⋊C8C2×D411yescharacteristic
175C2×C59⋊C8C2×D411yescharacteristic
176D472—21no—
177D472#4112yes—
178C8×D59—21no—
179C8×D59—21no—
180C8×D59—21no—
181C8×D59—21no—
182C16×C59—21no—
183C16×C59—21no—
184C59⋊C16—21no—
185C59⋊C16—21no—
186C23×C59—41no—
187C23×C59—41no—
188C22×D59—41no—
189C22×D59—82no—
190C22×D59—82no—
191C22×D59—82no—
192C22×D59—82no—
193C22×D59—161no—
194C22×D59—161no—
195C22×D59C2×D811yescharacteristic
196C22×D59C2×D811yescharacteristic
197Q8×C59—41no—
198Q8×C59—41no—
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—41no—
202D4×C59—41no—
203D4×C59—82no—
204D4×C59—161no—
205C2×C4×C59—41no—
206C2×C4×C59—41no—
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C2×D811yescharacteristic
209C2×C4×C59C22×D411yescharacteristic
210C22⋊D59—82no—
211C22⋊D59—82no—
212C22⋊D59—82no—
213C22⋊D59—82no—
214C2×Dic59—41no—
215C2×Dic59—41no—
216C2×Dic59—41no—
217C2×Dic59C2×D811yescharacteristic
218C2×Dic59C2×D811yescharacteristic
219D236—22no—
220D236—22no—
221D236—41no—
222C4×D59—21no—
223C4×D59—21no—
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—81no—
235C22×C59—81no—
236C22×C59—81no—
237C22×C59—81no—
238C22×C59—81no—
239C22×C59—81no—
240C22×C59C22×D811yescharacteristic
241C2×D59—21no—
242C2×D59—21no—
243C2×D59—21no—
244C2×D59—21no—
245C2×D59—21no—
246C2×D59—42no—
247C2×D59—81no—
248C2×D59—81no—
249C2×D59—164no—
250C2×D59—164no—
251C2×D59#50912yes—
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—81no—
255C4×C59—81no—
256C4×C59#54611yescharacteristic
257C4×C59#57711yescharacteristic
258Dic59—21no—
259Dic59—21no—
260Dic59—81no—
261Dic59—81no—
262Dic59—81no—
263#3336—591no—
264D4×D8—591no—
265#2244—591no—
266#2234—591no—
267#2232—591no—
268#2228—591no—
269#2220—591no—
270#2217—591no—
271#2207D5911yescharacteristic, complemented
272#2187—591no—
273#2187—591no—
274#2175—591no—
275#2168—591no—
276#2140—591no—
277C2×#357—591no—
278C2×C59—161no—
279C2×C59—161no—
280C2×C59#329411yescharacteristic
281C2×C59C2×#50911yescharacteristic
282C2×C59#341611yescharacteristic
283D59—22no—
284D59—41no—
285D59—82no—
286C22×D8—591no—
287C22×D8—591no—
288#552—1181no—
289D42—1181no—
290C2×SD32—591no—
291C2×SD32C2×D5911yescharacteristic, complemented
292C2×SD32C2×D5911yescharacteristic, complemented
293C2×M32—591no—
294C22×C16—591no—
295#499—591no—
296#493C2×D5911yescharacteristic
297#485—1181no—
298#483—1181no—
299#479—1181no—
300#467—591no—
301#467—591no—
302#466—591no—
303#466—591no—
304#459—1181no—
305#447—1181no—
306#447—1181no—
307C4×Q16C2×D5911yescharacteristic, complemented
308C4×D8—591no—
309C8×D4—591no—
310#427—591no—
311C2×#107—591no—
312#367—591no—
313#366—591no—
314#366—591no—
315C4⋊C16C2×D5911yescharacteristic, complemented
316#358—591no—
317#357—591no—
318#357—591no—
319#357—591no—
320#357—591no—
321#357—591no—
322#357—591no—
323#357—591no—
324#357C2×D5911yescharacteristic, complemented
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—21no—
335C2×D8—591no—
336C2×D8—1181no—
337C2×D8—1181no—
338C2×D8—1182no—
339C2×D8—1182no—
340C2×D8—2361no—
341C2×D8—2361no—
342C2×D8C22×D5911yescharacteristic
343C2×D8C22×D5911yescharacteristic
344C22×C8—591no—
345C22×C8—591no—
346#127—21no—
347#126—2361no—
348#125—1181no—
349#125—2361no—
350#123—2361no—
351C22⋊Q8—1181no—
352C22⋊Q8—1181no—
353#121—1181no—
354#121—1181no—
355#121—2361no—
356#120—2361no—
357#120—2361no—
358C4×Q8—21no—
359C4×D4—591no—
360C4×D4—1181no—
361#117—1181no—
362#117—2361no—
363C2×C4⋊C4—1181no—
364SD32—41no—
365SD32—41no—
366SD32—1182no—
367M32—2361no—
368C2×C16—591no—
369C2×C16—591no—
370C2×C16—2361no—
371C2×C16C22×D5911yescharacteristic
372C2×C16C22×D5911yescharacteristic
373#107—591no—
374#107—591no—
375#107—591no—
376#107—1181no—
377#107C22×D5911yescharacteristic
378#106—1181no—
379#106—1181no—
380C4⋊C8—591no—
381#103—21no—
382#103—1181no—
383#103—1181no—
384#102—1181no—
385#102—1181no—
386#102—1181no—
387C22⋊C8—591no—
388C22⋊C8—591no—
389C4×C8C22×D5911yescharacteristic
390C24—2361no—
391C24—2361no—
392C2×Q8—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—591no—
398C2×D4—1181no—
399C2×D4—4721no—
400C2×D4—4721no—
401C2×D4—4724no—
402C2×D4—2361no—
403C2×D4—2361no—
404C2×D4—2362no—
405C2×D4—2362no—
406C2×D4—2361no—
407C2×D4—2361no—
408C22×C4—591no—
409C22×C4—591no—
410C22×C4—2361no—
411C22×C4—2361no—
412Q16—21no—
413Q16—21no—
414D8—21no—
415D8—21no—
416D8—21no—
417D8—21no—
418D8—81no—
419D8—592no—
420D8—1181no—
421D8—2362no—
422D8—2362no—
423C2×C8—591no—
424C2×C8—591no—
425C2×C8—1181no—
426C2×C8—1181no—
427C2×C8—1181no—
428C2×C8—1181no—
429C2×C8D4×D5911yescharacteristic
430C2×C8C23×D5911yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—41no—
433C4⋊C4—591no—
434C4⋊C4—1181no—
435C4⋊C4—1181no—
436C4⋊C4—1181no—
437C4⋊C4—2361no—
438C4⋊C4—2361no—
439C4⋊C4—2361no—
440C4⋊C4—2361no—
441C4⋊C4—2361no—
442C22⋊C4—591no—
443C22⋊C4—591no—
444C22⋊C4—2361no—
445C22⋊C4—2361no—
446C22⋊C4—2361no—
447C22⋊C4—2361no—
448C22⋊C4—2361no—
449C22⋊C4—2361no—
450C42—41no—
451C42—2361no—
452C42D4×D5911yescharacteristic
453C16—21no—
454C16—21no—
455C16—1181no—
456C16—1181no—
457C23—41no—
458C23—41no—
459C23—591no—
460C23—591no—
461C23—9441no—
462C23—9441no—
463C23—4722no—
464C23—4722no—
465C23—4722no—
466C23—4722no—
467C23—2361no—
468Q8—41no—
469Q8—41no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—82no—
475D4—161no—
476D4—1182no—
477D4—1182no—
478D4—4722no—
479D4—4722no—
480D4—4722no—
481D4—4722no—
482D4—2361no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—591no—
486C2×C4—591no—
487C2×C4—1181no—
488C2×C4—1181no—
489C2×C4—1181no—
490C2×C4—1181no—
491C2×C4—4722no—
492C2×C4—4722no—
493C2×C4—2361no—
494C2×C4—2361no—
495C2×C4—2361no—
496C2×C4D8×D5911yescharacteristic
497C2×C4D8×D5911yescharacteristic
498C2×C4C2×D4×D5911yescharacteristic
499C8—21no—
500C8—1181no—
501C8—1181no—
502C8#41×D5911yescharacteristic
503C8#1427377211yescharacteristic
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—592no—
511C22—1181no—
512C22—1181no—
513C22—1181no—
514C22—1181no—
515C22—1181no—
516C22—9444no—
517C22—9444no—
518C22—4721no—
519C22—4721no—
520C22—2362no—
521C22C2×D8×D5911yescharacteristic
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—1181no—
527C4—1181no—
528C4—4721no—
529C4—4721no—
530C4—4721no—
531C4#1683804911yescharacteristic
532C4#1683821211yescharacteristic
533C2—161no—
534C2—161no—
535C2—1182no—
536C2—4722no—
537C2—2361no—
538C2#2143591211yescharacteristic
539C2#509×D5911yescharacteristic
540C2#2143723811yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2207C2
socleC22×C59C22×D8
prime radical#21435771C2
hypercenter#2207D59
hypocenterC59#33877
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #21436466C2semidirect
3. #21426337C2semidirect
4. #21426286C2semidirect
5. #21426275C2semidirect
6. #21426254C2semidirect
7. #21426206C2semidirect
8. #21426189C2semidirect
9. #21426023C2semidirect
10. #21426023C2semidirect
11. #21425958C2semidirect
12. #21425920C2semidirect
13. #21425810C2semidirect
14. #357×D59C2semidirect
15. C59×#2207C2semidirect
21. C2×#14273666C22semidirect
22. C2×C16×D59C22semidirect
23. #16837804C22semidirect
27. #16837640C22semidirect
28. #16837640C22semidirect
36. #16837439C22semidirect
37. #107×D59C22semidirect
38. #16837230C22semidirect
39. #16837228C22semidirect
40. #16837227C22semidirect
41. #16837206C22semidirect
44. #16837203C22semidirect
45. #16837203C22semidirect
46. #16837203C22semidirect
47. #16837203C22semidirect
49. (C2×D59)⋊C16C22semidirect
50. (C2×D59)⋊C16C22semidirect
51. #16837178C22semidirect
52. C2×SD32×C59C22semidirect
53. C2×SD32×C59C22semidirect
55. C4×Q16×C59C22semidirect
56. C59×C4⋊C16C22semidirect
57. C59×#357C22semidirect
58. C59×#357C22semidirect
271. #2207D59semidirect
291. C2×SD32C2×D59semidirect
292. C2×SD32C2×D59semidirect
307. C4×Q16C2×D59semidirect
315. C4⋊C16C2×D59semidirect
324. #357C2×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
#214364662
#214357712
#214263372
#214262862
#214262752
#214262542
#214262062
#214261892
#214260232
#214259582
#214259202
#214258102
#357×D592
C59×#22072
#3387759