Finite Group Databasebeta

← #28745720

Subgroups — #28745720

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#21436465C211yesmaximal, characteristic, complemented
2#21426337C211yesmaximal, characteristic, complemented
3#21426284C211yesmaximal, characteristic, complemented
4#21426274C211yesmaximal, characteristic, complemented
5#21426253C211yesmaximal, characteristic, complemented
6#21426207C211yesmaximal, characteristic, complemented
7#21426190C211yesmaximal, characteristic, complemented
8#21426133C211yesmaximal, characteristic, complemented
9#21426025C211yesmaximal, characteristic, complemented
10#21426025C211yesmaximal, characteristic, complemented
11#21425960C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425812C211yesmaximal, characteristic, complemented
14C2×#16837203C211yesmaximal, characteristic, complemented
15D4×D8×C59C211yesmaximal, characteristic
16#16838092—21no—
17C2×#14273671C2211yescharacteristic
18C2×#14273671C2211yescharacteristic, complemented
19C2×#14273671C2211yescharacteristic, complemented
20C2×#14273667C2211yescharacteristic, complemented
21C22×C59⋊C16C2211yescharacteristic, complemented
22#16837804C2211yescharacteristic, complemented
23#16837744—21no—
24#16837733—21no—
25#16837710—21no—
26#16837643C2211yescharacteristic, complemented
27#16837643C2211yescharacteristic, complemented
28C4×Dic236C2211yescharacteristic, complemented
29#16837440C2211yescharacteristic, complemented
30C2×#14273661C2211yescharacteristic, complemented
31#16837229C2211yescharacteristic, complemented
32#16837227C2211yescharacteristic, complemented
33#16837227C2211yescharacteristic, complemented
34C4⋊(C59⋊C16)C2211yescharacteristic, complemented
35#16837205C2211yescharacteristic, complemented
36#16837203C2211yescharacteristic
37#16837203C2211yescharacteristic
38#16837203C2211yescharacteristic
39#16837203C2211yescharacteristic, complemented
40#16837203C2211yescharacteristic, complemented
41#16837203C2211yescharacteristic, complemented
42#16837203C2211yescharacteristic, complemented
43#16837203C2211yescharacteristic, complemented
44#16837203C2211yescharacteristic, complemented
45C22⋊(C59⋊C16)C2211yescharacteristic, complemented
46C22⋊(C59⋊C16)C2211yescharacteristic, complemented
47#16837178C2211yescharacteristic, complemented
48C22×D8×C59C2211yescharacteristic
49C22×D8×C59C2211yescharacteristic
50D42×C59—21no—
51C59×#493C2211yescharacteristic
52C59×#466C2211yescharacteristic
53C59×#466C2211yescharacteristic
54C59×#459—21no—
55C59×#447—21no—
56C59×#447—21no—
57C4×D8×C59C2211yescharacteristic
58C8×D4×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60#14273720—41no—
61#14273719—21no—
62#14273717—41no—
63#14273708—41no—
64#14273706—21no—
65#14273706—21no—
66C4×Dic118—21no—
67#14273686—21no—
68#14273683—41no—
69C2×C4⋊Dic59—21no—
70#14273671—22no—
71#14273671—41no—
72#14273671—41no—
73#14273667—41no—
74C2×C59⋊C16—41no—
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79#14273661—21no—
80#14273661C2311yescharacteristic
81#14273661C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273659—21no—
84#14273659—21no—
85#14273652—21no—
86#14273652—21no—
87#14273652—21no—
88C22×D4×C59—21no—
89C22×D4×C59—21no—
90C22×D4×C59—41no—
91C2×D8×C59—22no—
92C2×D8×C59—22no—
93C2×D8×C59—21no—
94C2×D8×C59—21no—
95C2×D8×C59—21no—
96C2×D8×C59—41no—
97C2×D8×C59—41no—
98C2×D8×C59C2311yescharacteristic
99C2×D8×C59C2311yescharacteristic
100C2×D8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102C22×C8×C59C2311yescharacteristic
103#127×C59—21no—
104#121×C59—21no—
105#121×C59—21no—
106#121×C59—41no—
107#120×C59—41no—
108#120×C59—41no—
109C4×D4×C59—21no—
110C4×D4×C59D411yescharacteristic
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#102×C59—21no—
114#102×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C24×C59—41no—
120C24×C59—41no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—42no—
125C2×D4×C59—42no—
126C2×D4×C59—41no—
127C2×D4×C59—41no—
128C2×D4×C59—41no—
129C2×D4×C59—41no—
130C2×D4×C59—41no—
131C2×D4×C59—41no—
132C2×D4×C59—81no—
133C2×D4×C59—81no—
134C2×D4×C59—84no—
135C2×D4×C59D811yescharacteristic
136C22×C4×C59—41no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×Dic59—41no—
140C2×Dic118—21no—
141D8×C59—21no—
142D8×C59—21no—
143D8×C59—21no—
144D8×C59—21no—
145D8×C59—21no—
146D8×C59—42no—
147D8×C59—42no—
148D8×C59—81no—
149D8×C59#4112yes—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59C2×D411yescharacteristic
155C2×C8×C59C2×D411yescharacteristic
156C2×C8×C59C2×D411yescharacteristic
157C2×C8×C59C2411yescharacteristic
158C4⋊C4×C59—21no—
159C4⋊C4×C59D811yescharacteristic
160C22⋊C4×C59—41no—
161C22⋊C4×C59—41no—
162C22⋊C4×C59—41no—
163C22⋊C4×C59D811yescharacteristic
164C22⋊C4×C59D811yescharacteristic
165C42×C59C2×D411yescharacteristic
166C22⋊Dic59—41no—
167C22⋊Dic59—41no—
168C22⋊Dic59—41no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—41no—
173Dic59⋊C4—41no—
174Dic59⋊C4—41no—
175Dic59⋊C4—41no—
176Dic59⋊C4—41no—
177Dic59⋊C4—41no—
178C4×Dic59—41no—
179C4×Dic59—41no—
180Dic236—21no—
181Dic236—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C59⋊C16—21no—
186C23×C59—41no—
187C23×C59—41no—
188C23×C59—41no—
189C23×C59—82no—
190C23×C59—82no—
191C23×C59—82no—
192C23×C59—82no—
193C23×C59—161no—
194C23×C59—161no—
195C23×C59C2×D811yescharacteristic
196C23×C59C2×D811yescharacteristic
197D4×C59—22no—
198D4×C59—22no—
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—41no—
202D4×C59—41no—
203D4×C59—41no—
204D4×C59—82no—
205D4×C59—82no—
206D4×C59—82no—
207D4×C59—82no—
208D4×C59—82no—
209D4×C59—161no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—41no—
215C2×C4×C59—82no—
216C2×C4×C59C2×D811yescharacteristic
217C2×C4×C59C2×D811yescharacteristic
218C2×C4×C59C2×D811yescharacteristic
219C2×C4×C59C2×D811yescharacteristic
220C2×C4×C59C22×D411yescharacteristic
221C2×Dic59—41no—
222C2×Dic59—41no—
223C2×Dic59—41no—
224C2×Dic59—41no—
225C2×Dic59—82no—
226Dic118—41no—
227Dic118—41no—
228C8×C59—21no—
229C8×C59—21no—
230C8×C59—21no—
231C8×C59C2×#4111yescharacteristic
232C8×C59H(5,𝔽2)11yescharacteristic
233#33877—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—81no—
246C22×C59—81no—
247C22×C59—81no—
248C22×C59—164no—
249C22×C59—164no—
250C22×C59#50912yes—
251C22×C59C22×D811yescharacteristic
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—21no—
255C4×C59—21no—
256C4×C59—81no—
257C4×C59#54611yescharacteristic
258C4×C59#57711yescharacteristic
259Dic59—81no—
260Dic59—81no—
261Dic59—81no—
262Dic59—81no—
263#3336—591no—
264D4×D8D5911yescharacteristic
265#2244—591no—
266#2234—591no—
267#2232—591no—
268#2228—591no—
269#2220—591no—
270#2217—591no—
271#2207—591no—
272#2187—591no—
273#2187—591no—
274#2175—591no—
275#2168—591no—
276#2140—591no—
277C2×#357—591no—
278C2×C59—22no—
279C2×C59—41no—
280C2×C59—82no—
281C2×C59—161no—
282C2×C59—161no—
283C2×C59#329411yescharacteristic
284C2×C59C2×#50911yescharacteristic
285C2×C59#341611yescharacteristic
286C22×D8C2×D5911yescharacteristic
287C22×D8C2×D5911yescharacteristic
288#552—1181no—
289D42—21no—
290C2×SD32—591no—
291C2×SD32—591no—
292C2×SD32—591no—
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#466C2×D5911yescharacteristic
303#466C2×D5911yescharacteristic
304#459—21no—
305#447—21no—
306#447—21no—
307C4×Q16—591no—
308C4×D8C2×D5911yescharacteristic
309C8×D4C2×D5911yescharacteristic
310#427—591no—
311C2×#107—591no—
312#367—591no—
313#366—591no—
314#366—591no—
315C4⋊C16—591no—
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#357—591no—
325#357—591no—
326C22⋊C16—591no—
327C22⋊C16—591no—
328#346—591no—
329C59#3387711yescharacteristic, complemented, Sylow
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—41no—
333C2×Q16—591no—
334C2×D8—22no—
335C2×D8—22no—
336C2×D8—21no—
337C2×D8—21no—
338C2×D8—21no—
339C2×D8—41no—
340C2×D8—41no—
341C2×D8C22×D5911yescharacteristic
342C2×D8C22×D5911yescharacteristic
343C2×D8C22×D5911yescharacteristic
344C22×C8C22×D5911yescharacteristic
345C22×C8C22×D5911yescharacteristic
346#127—21no—
347#126—2361no—
348#125—1181no—
349#125—2361no—
350#123—2361no—
351C22⋊Q8—1181no—
352C22⋊Q8—1181no—
353#121—21no—
354#121—21no—
355#121—41no—
356#120—41no—
357#120—41no—
358C4×Q8—1181no—
359C4×D4—21no—
360C4×D4C22⋊D5911yescharacteristic
361#117—1181no—
362#117—2361no—
363C2×C4⋊C4—1181no—
364SD32—1182no—
365SD32—2361no—
366SD32—2361no—
367M32—2361no—
368C2×C16—591no—
369C2×C16—591no—
370C2×C16—591no—
371C2×C16—591no—
372C2×C16—2361no—
373#107—591no—
374#107—591no—
375#107—591no—
376#107—1181no—
377#107C22×D5911yescharacteristic
378#106—1181no—
379#106—1181no—
380C4⋊C8C22⋊D5911yescharacteristic
381#103—1181no—
382#103—1181no—
383#103—1181no—
384#102—21no—
385#102—21no—
386#102—21no—
387C22⋊C8C22⋊D5911yescharacteristic
388C22⋊C8C22⋊D5911yescharacteristic
389C4×C8C22×D5911yescharacteristic
390C24—41no—
391C24—41no—
392C2×Q8—1181no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—84no—
407C2×D4#1174566911yescharacteristic
408C22×C4—41no—
409C22×C4—2361no—
410C22×C4C2×C22⋊D5911yescharacteristic
411C22×C4C2×C22⋊D5911yescharacteristic
412Q16—1181no—
413Q16—1181no—
414D8—21no—
415D8—21no—
416D8—21no—
417D8—21no—
418D8—21no—
419D8—42no—
420D8—42no—
421D8—81no—
422D8#1174568712yes—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×C22⋊D5911yescharacteristic
428C2×C8C2×C22⋊D5911yescharacteristic
429C2×C8C2×C22⋊D5911yescharacteristic
430C2×C8C23×D5911yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—1181no—
433C4⋊C4—1181no—
434C4⋊C4—1181no—
435C4⋊C4—2361no—
436C4⋊C4—2361no—
437C4⋊C4—2361no—
438C4⋊C4—2361no—
439C4⋊C4—2361no—
440C4⋊C4—2361no—
441C4⋊C4#1174566911yescharacteristic
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—2361no—
446C22⋊C4—2361no—
447C22⋊C4—2361no—
448C22⋊C4#1174566911yescharacteristic
449C22⋊C4#1174566911yescharacteristic
450C42—2361no—
451C42—2361no—
452C42C2×C22⋊D5911yescharacteristic
453C16—1181no—
454C16—1181no—
455C16—1181no—
456C16—1181no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—161no—
465C23—161no—
466C23C2×#1174566911yescharacteristic
467C23C2×#1174566911yescharacteristic
468Q8—2361no—
469Q8—2361no—
470D4—22no—
471D4—22no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—82no—
478D4—82no—
479D4—82no—
480D4—82no—
481D4—82no—
482D4—161no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—82no—
489C2×C4—4722no—
490C2×C4—2361no—
491C2×C4—2361no—
492C2×C4—2361no—
493C2×C4—2361no—
494C2×C4C2×#1174566911yescharacteristic
495C2×C4C2×#1174566911yescharacteristic
496C2×C4C2×#1174566911yescharacteristic
497C2×C4C2×#1174566911yescharacteristic
498C2×C4C22×C22⋊D5911yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8C2×#1174568711yescharacteristic
503C8#1427377211yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—164no—
519C22—164no—
520C22#1683783812yes—
521C22C22×#1174566911yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—81no—
527C4—4721no—
528C4—4721no—
529C4—4721no—
530C4—4721no—
531C4#1683804611yescharacteristic
532C4#1683821311yescharacteristic
533C2—22no—
534C2—41no—
535C2—82no—
536C2—161no—
537C2—161no—
538C2#2143590911yescharacteristic
539C2C2×#1683783811yescharacteristic
540C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
Fitting, prime radicalD4×D8×C59C2
socleC22×C59C22×D8
hypercenterD4×D8D59
hypocenterC59#33877
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×D8

Complements

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

NH ≅ G/Ntype
1. #21436465C2semidirect
2. #21426337C2semidirect
3. #21426284C2semidirect
4. #21426274C2semidirect
5. #21426253C2semidirect
6. #21426207C2semidirect
7. #21426190C2semidirect
8. #21426133C2semidirect
9. #21426025C2semidirect
10. #21426025C2semidirect
11. #21425960C2semidirect
12. #21425922C2semidirect
13. #21425812C2semidirect
14. C2×#16837203C2semidirect
18. C2×#14273671C22semidirect
19. C2×#14273671C22semidirect
20. C2×#14273667C22semidirect
21. C22×C59⋊C16C22semidirect
22. #16837804C22semidirect
26. #16837643C22semidirect
27. #16837643C22semidirect
28. C4×Dic236C22semidirect
29. #16837440C22semidirect
30. C2×#14273661C22semidirect
31. #16837229C22semidirect
32. #16837227C22semidirect
33. #16837227C22semidirect
34. C4⋊(C59⋊C16)C22semidirect
35. #16837205C22semidirect
39. #16837203C22semidirect
40. #16837203C22semidirect
41. #16837203C22semidirect
42. #16837203C22semidirect
43. #16837203C22semidirect
44. #16837203C22semidirect
45. C22⋊(C59⋊C16)C22semidirect
46. C22⋊(C59⋊C16)C22semidirect
47. #16837178C22semidirect
329. C59#33877semidirect

Maximal subgroups

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

subgroupindex
#214364652
#214263372
#214262842
#214262742
#214262532
#214262072
#214261902
#214261332
#214260252
#214259602
#214259222
#214258122
C2×#168372032
D4×D8×C592
#3387759