Finite Group Databasebeta

← #28745660

Subgroups — #28745660

Order 15104.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436444C211yesmaximal, characteristic, complemented
2#21426332C211yesmaximal, characteristic, complemented
3#21426291C211yesmaximal, characteristic, complemented
4#21426279C211yesmaximal, characteristic, complemented
5#21426243C211yesmaximal, characteristic, complemented
6#21426197C211yesmaximal, characteristic, complemented
7#21426194C211yesmaximal, characteristic, complemented
8#21426148C211yesmaximal, characteristic, complemented
9#21426019C211yesmaximal, characteristic, complemented
10#21426008C211yesmaximal, characteristic, complemented
11#21425970C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425808C211yesmaximal, characteristic, complemented
14C2×#16837203C211yesmaximal, characteristic, complemented
15C59×#3280C211yesmaximal, characteristic, complemented
16#16838056—21no—
17C2×#14273670C2211yescharacteristic, complemented
18C2×#14273668C2211yescharacteristic
19C2×#14273668C2211yescharacteristic
20C2×#14273667C2211yescharacteristic, complemented
21C22×C59⋊C16C2211yescharacteristic
22#16837807C2211yescharacteristic, complemented
23#16837716—21no—
24#16837716—21no—
25#16837683—21no—
26#16837638C2211yescharacteristic
27#16837638C2211yescharacteristic, complemented
28C4×D472C2211yescharacteristic, complemented
29C2×#14273661C2211yescharacteristic
30C2×#14273661C2211yescharacteristic, complemented
31#16837229C2211yescharacteristic, complemented
32#16837229C2211yescharacteristic, complemented
33#16837229C2211yescharacteristic, complemented
34C4⋊(C59⋊C16)C2211yescharacteristic, complemented
35#16837206C2211yescharacteristic, complemented
36#16837206C2211yescharacteristic, complemented
37#16837206C2211yescharacteristic, complemented
38#16837203C2211yescharacteristic
39#16837203C2211yescharacteristic
40#16837203C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44#16837202C2211yescharacteristic, complemented
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic, complemented
47#16837178C2211yescharacteristic, complemented
48C2×#135×C59C2211yescharacteristic
49C22×D8×C59C2211yescharacteristic
50C59×#546—21no—
51C59×#495C2211yescharacteristic
52C59×#467C2211yescharacteristic
53C59×#466C2211yescharacteristic
54C59×#463—21no—
55C59×#450—21no—
56C59×#449—21no—
57C4×D8×C59C2211yescharacteristic
58C8×D4×C59C2211yescharacteristic
59C2×#11745687—41no—
60C2×D472C2311yescharacteristic
61#14273727—21no—
62#14273711—41no—
63#14273711—41no—
64#14273707—41no—
65#14273701—21no—
66#14273701—21no—
67C4×D236—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70#14273670—41no—
71#14273668—22no—
72#14273668—41no—
73#14273667—41no—
74C2×C59⋊C16—41no—
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79#14273661—21no—
80#14273661—21no—
81#14273661—21no—
82#14273661C2311yescharacteristic
83#14273661C2311yescharacteristic
84#14273661C2311yescharacteristic
85#14273649—21no—
86#14273649—21no—
87#14273649—21no—
88C2×#41×C59—21no—
89C22×D4×C59—21no—
90#135×C59—22no—
91#135×C59—41no—
92C2×Q16×C59C2311yescharacteristic
93C2×SD16×C59—21no—
94C2×SD16×C59—21no—
95C2×D8×C59—22no—
96C2×D8×C59—21no—
97C2×D8×C59—41no—
98C2×D8×C59C2311yescharacteristic
99C2×D8×C59C2311yescharacteristic
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102#124×C59—21no—
103#123×C59—41no—
104C22⋊Q8×C59—21no—
105#121×C59—21no—
106#121×C59—41no—
107#120×C59—41no—
108C4×D4×C59—21no—
109C4×D4×C59D411yescharacteristic
110C2×C22⋊C4×C59—41no—
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#103×C59—21no—
114#102×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C24×C59—41no—
120#41×C59—42no—
121#41×C59—81no—
122C2×Q8×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—42no—
126C2×D4×C59—41no—
127C2×D4×C59—41no—
128C2×D4×C59—41no—
129C2×D4×C59—81no—
130C2×D4×C59D811yescharacteristic
131C22×C4×C59—41no—
132C22×C4×C59—41no—
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C2×C22⋊D59—41no—
136C2×C22⋊D59—41no—
137C22×Dic59—41no—
138C22×Dic59—41no—
139#11745687—84no—
140C2×D236—21no—
141C2×C4×D59—41no—
142C2×Dic118—41no—
143Q16×C59—21no—
144Q16×C59—21no—
145SD16×C59—42no—
146SD16×C59—81no—
147D8×C59—21no—
148D8×C59—21no—
149D8×C59—21no—
150D8×C59—42no—
151D8×C59#4112yescomplemented
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59C2×D411yescharacteristic
157C2×C8×C59C2×D411yescharacteristic
158C2×C8×C59C2×D411yescharacteristic
159C2×C8×C59C2411yescharacteristic
160C4⋊C4×C59—21no—
161C4⋊C4×C59—41no—
162C4⋊C4×C59D811yescharacteristic
163C22⋊C4×C59—41no—
164C22⋊C4×C59—41no—
165C22⋊C4×C59—41no—
166C22⋊C4×C59—41no—
167C22⋊C4×C59—41no—
168C22⋊C4×C59D811yescharacteristic
169C22⋊C4×C59D811yescharacteristic
170C42×C59C2×D411yescharacteristic
171(C2×D59)⋊C4—41no—
172(C2×D59)⋊C4—41no—
173(C2×D59)⋊C4—41no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—41no—
178C4⋊Dic59—41no—
179C4⋊Dic59—41no—
180D472—21no—
181D472—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C59⋊C16—21no—
186C23×C59—41no—
187C23×C59—41no—
188C23×C59—82no—
189C23×C59—82no—
190C23×C59—161no—
191C23×C59C2×D811yescharacteristic
192C23×C59C2×D811yescharacteristic
193C22×D59—41no—
194Q8×C59—41no—
195Q8×C59—41no—
196D4×C59—22no—
197D4×C59—22no—
198D4×C59—41no—
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—82no—
202D4×C59—82no—
203D4×C59—161no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—82no—
211C2×C4×C59—161no—
212C2×C4×C59C2×D811yescharacteristic
213C2×C4×C59C2×D811yescharacteristic
214C2×C4×C59C2×D811yescharacteristic
215C2×C4×C59C2×D811yescharacteristic
216C2×C4×C59C22×D411yescharacteristic
217C22⋊D59—82no—
218C22⋊D59—82no—
219C2×Dic59—41no—
220C2×Dic59—41no—
221C2×Dic59—41no—
222C2×Dic59—82no—
223C2×Dic59—82no—
224D236—41no—
225D236—41no—
226C4×D59—82no—
227Dic118—82no—
228C8×C59—21no—
229C8×C59—21no—
230C8×C59—21no—
231C8×C59C2×#4111yescharacteristic
232C8×C59H(5,𝔽2)11yescharacteristic
233#33873—591nomaximal, Sylow
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—21no—
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—42no—
240C22×C59—81no—
241C22×C59—81no—
242C22×C59—81no—
243C22×C59—81no—
244C22×C59—81no—
245C22×C59—164no—
246C22×C59#50912yes—
247C22×C59C22×D811yescharacteristic
248C2×D59—81no—
249C2×D59—81no—
250C2×D59—81no—
251C4×C59—21no—
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—21no—
255C4×C59—81no—
256C4×C59—81no—
257C4×C59#54611yescharacteristic
258C4×C59#57711yescharacteristic
259Dic59—81no—
260Dic59—81no—
261Dic59—81no—
262#3334—591no—
263#3280D5911yescharacteristic, complemented
264#2243—591no—
265#2235—591no—
266#2233—591no—
267#2226—591no—
268#2218—591no—
269#2217—591no—
270#2210—591no—
271#2186—591no—
272#2184—591no—
273#2177—591no—
274#2168—591no—
275#2139—591no—
276C2×#357—591no—
277C2×C59—22no—
278C2×C59—41no—
279C2×C59—82no—
280C2×C59—161no—
281C2×C59#329411yescharacteristic
282C2×C59C2×#50911yescharacteristic
283C2×C59#341611yescharacteristic
284D59—161no—
285C2×#135C2×D5911yescharacteristic
286C22×D8C2×D5911yescharacteristic
287#547—1181no—
288#546—21no—
289C2×SD32—591no—
290C2×D16—591no—
291C2×D16—591no—
292C2×M32—591no—
293C22×C16—591no—
294#500—591no—
295#495C2×D5911yescharacteristic
296#480—1181no—
297#480—1181no—
298#474—1181no—
299#467C2×D5911yescharacteristic
300#466—591no—
301#466—591no—
302#466C2×D5911yescharacteristic
303#463—21no—
304#450—21no—
305#449—21no—
306C4×D8—591no—
307C4×D8C2×D5911yescharacteristic
308C8×D4C2×D5911yescharacteristic
309C2×#107—591no—
310C2×#107—591no—
311#367—591no—
312#367—591no—
313#367—591no—
314C4⋊C16—591no—
315#358—591no—
316#358—591no—
317#358—591no—
318#357—591no—
319#357—591no—
320#357—591no—
321#357—591no—
322#357—591no—
323#357—591no—
324#357—591no—
325C22⋊C16—591no—
326C22⋊C16—591no—
327#346—591no—
328C59#3387311yescharacteristic, complemented, Sylow
329C2×#41—21no—
330C2×#41—2361no—
331C22×D4—21no—
332#135—22no—
333#135—41no—
334C2×Q16C22×D5911yescharacteristic
335C2×SD16—21no—
336C2×SD16—21no—
337C2×D8—22no—
338C2×D8—21no—
339C2×D8—41no—
340C2×D8—591no—
341C2×D8C22×D5911yescharacteristic
342C2×D8C22×D5911yescharacteristic
343C22×C8C22×D5911yescharacteristic
344C22×C8C22×D5911yescharacteristic
345C4⋊Q8—1181no—
346#124—21no—
347#123—41no—
348#123—2361no—
349#123—2361no—
350C22⋊Q8—21no—
351C22⋊Q8—2361no—
352#121—21no—
353#121—41no—
354#121—1181no—
355#121—1181no—
356#120—41no—
357C4×D4—21no—
358C4×D4—1181no—
359C4×D4C22⋊D5911yescharacteristic
360C2×C4⋊C4—1181no—
361C2×C4⋊C4—1181no—
362C2×C22⋊C4—41no—
363SD32—2361no—
364D16—1182no—
365D16—2361no—
366M32—2361no—
367C2×C16—591no—
368C2×C16—591no—
369C2×C16—591no—
370C2×C16—591no—
371C2×C16—2361no—
372#107—591no—
373#107—591no—
374#107—591no—
375#107—1181no—
376#107—1181no—
377#107—1181no—
378#107C22×D5911yescharacteristic
379C4⋊C8C22⋊D5911yescharacteristic
380#103—21no—
381#102—21no—
382#102—21no—
383#102—1181no—
384#102—1181no—
385#102—1181no—
386C22⋊C8C22⋊D5911yescharacteristic
387C22⋊C8C22⋊D5911yescharacteristic
388C4×C8C22×D5911yescharacteristic
389C24—41no—
390#41—42no—
391#41—81no—
392#41—4724no—
393C2×Q8—21no—
394C2×Q8—2361no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—81no—
402C2×D4—1181no—
403C2×D4—2361no—
404C2×D4—2361no—
405C2×D4#1174566911yescharacteristic
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—2361no—
409C22×C4—2361no—
410C22×C4—2361no—
411C22×C4C2×C22⋊D5911yescharacteristic
412C22×C4C2×C22⋊D5911yescharacteristic
413Q16—21no—
414Q16—21no—
415SD16—42no—
416SD16—81no—
417D8—21no—
418D8—21no—
419D8—21no—
420D8—42no—
421D8—1181no—
422D8—1181no—
423D8#1174568712yescomplemented
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×C22⋊D5911yescharacteristic
429C2×C8C2×C22⋊D5911yescharacteristic
430C2×C8C2×C22⋊D5911yescharacteristic
431C2×C8C23×D5911yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—1181no—
435C4⋊C4—1181no—
436C4⋊C4—1181no—
437C4⋊C4—2361no—
438C4⋊C4—2361no—
439C4⋊C4—2361no—
440C4⋊C4#1174566911yescharacteristic
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—2361no—
447C22⋊C4—2361no—
448C22⋊C4—2361no—
449C22⋊C4#1174566911yescharacteristic
450C22⋊C4#1174566911yescharacteristic
451C42C2×C22⋊D5911yescharacteristic
452C16—1181no—
453C16—1181no—
454C16—1181no—
455C16—1181no—
456C23—41no—
457C23—41no—
458C23—82no—
459C23—82no—
460C23—161no—
461C23—2361no—
462C23C2×#1174566911yescharacteristic
463C23C2×#1174566911yescharacteristic
464Q8—41no—
465Q8—41no—
466Q8—4722no—
467D4—22no—
468D4—22no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—82no—
473D4—82no—
474D4—161no—
475D4—4722no—
476D4—4722no—
477D4—2361no—
478D4—2361no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4—161no—
487C2×C4—4722no—
488C2×C4—4722no—
489C2×C4—4722no—
490C2×C4—2361no—
491C2×C4—2361no—
492C2×C4—2361no—
493C2×C4C2×#1174566911yescharacteristic
494C2×C4C2×#1174566911yescharacteristic
495C2×C4C2×#1174566911yescharacteristic
496C2×C4C2×#1174566911yescharacteristic
497C2×C4C22×C22⋊D5911yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8C2×#1174568711yescharacteristic
502C8#1427377211yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—164no—
515C22—4721no—
516C22—4721no—
517C22—4721no—
518C22#1683783712yes—
519C22C22×#1174566911yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—4721no—
527C4—4721no—
528C4—4721no—
529C4#1683804611yescharacteristic
530C4#1683821311yescharacteristic
531C2—22no—
532C2—41no—
533C2—82no—
534C2—161no—
535C2—9441no—
536C2#2143590911yescharacteristic
537C2C2×#1683783711yescharacteristic
538C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3280C2
socleC22×C59C22×D8
hypercenter#3280D59
hypocenterC59#33873
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #21436444C2semidirect
2. #21426332C2semidirect
3. #21426291C2semidirect
4. #21426279C2semidirect
5. #21426243C2semidirect
6. #21426197C2semidirect
7. #21426194C2semidirect
8. #21426148C2semidirect
9. #21426019C2semidirect
10. #21426008C2semidirect
11. #21425970C2semidirect
12. #21425922C2semidirect
13. #21425808C2semidirect
14. C2×#16837203C2semidirect
15. C59×#3280C2semidirect
17. C2×#14273670C22semidirect
20. C2×#14273667C22semidirect
22. #16837807C22semidirect
27. #16837638C22semidirect
28. C4×D472C22semidirect
30. C2×#14273661C22semidirect
31. #16837229C22semidirect
32. #16837229C22semidirect
33. #16837229C22semidirect
34. C4⋊(C59⋊C16)C22semidirect
35. #16837206C22semidirect
36. #16837206C22semidirect
37. #16837206C22semidirect
44. #16837202C22semidirect
46. C22⋊(C59⋊C16)C22semidirect
47. #16837178C22semidirect
151. D8×C59#41semidirect
263. #3280D59semidirect
328. C59#33873semidirect
423. D8#11745687semidirect

Maximal subgroups

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

subgroupindex
#214364442
#214263322
#214262912
#214262792
#214262432
#214261972
#214261942
#214261482
#214260192
#214260082
#214259702
#214259222
#214258082
C2×#168372032
C59×#32802
#3387359