Finite Group Databasebeta

← #28745178

Subgroups — #28745178

Order 15104.

686 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837843C211yesmaximal, characteristic, complemented
2C22×#14273670C211yesmaximal, characteristic, complemented
3#21435783C211yesmaximal, characteristic
4#21426365C211yesmaximal, characteristic, complemented
5#21426211C211yesmaximal, characteristic, complemented
6#21426185C211yesmaximal, characteristic, complemented
7#21426158C211yesmaximal, characteristic, complemented
8#21426138C211yesmaximal, characteristic, complemented
9#21426034C211yesmaximal, characteristic, complemented
10#21426030C211yesmaximal, characteristic, complemented
11#21426024C211yesmaximal, characteristic, complemented
12#21426013C211yesmaximal, characteristic, complemented
13#21425958C211yesmaximal, characteristic, complemented
14#21425922C211yesmaximal, characteristic, complemented
15D4×Q16×C59C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×D472C2211yescharacteristic
18#16838048—21no—
19#16837843—24no—
20C2×#14273673C2211yescharacteristic, complemented
21C2×#14273673C2211yescharacteristic, complemented
22C2×#14273673C2211yescharacteristic, complemented
23C2×#14273670—24no—
24C2×#14273670C2211yescharacteristic
25C2×#14273670C2211yescharacteristic
26C2×#14273670C2211yescharacteristic
27C2×#14273670C2211yescharacteristic
28C2×#14273670C2211yescharacteristic
29C2×#14273670C2211yescharacteristic
30C2×#14273667C2211yescharacteristic, complemented
31C22×C59⋊C16C2211yescharacteristic
32#16837786C2211yescharacteristic
33#16837643C2211yescharacteristic
34#16837638C2211yescharacteristic
35#16837618—21no—
36#16837555—21no—
37#16837549—21no—
38C4×D472C2211yescharacteristic
39#16837227C2211yescharacteristic, complemented
40C4⋊(C59⋊C16)C2211yescharacteristic, complemented
41#16837206C2211yescharacteristic, complemented
42#16837206C2211yescharacteristic, complemented
43#16837206C2211yescharacteristic, complemented
44#16837205C2211yescharacteristic, complemented
45#16837202C2211yescharacteristic
46#16837202C2211yescharacteristic
47C22⋊(C59⋊C16)C2211yescharacteristic
48C22⋊(C59⋊C16)C2211yescharacteristic, complemented
49#16837178C2211yescharacteristic, complemented
50C22×Q16×C59C2211yescharacteristic
51C22×Q16×C59C2211yescharacteristic, complemented
52D4×Q8×C59—21no—
53C59×#494C2211yescharacteristic, complemented
54C59×#467C2211yescharacteristic
55C59×#467C2211yescharacteristic, complemented
56C59×#462—21no—
57C59×#451—21no—
58C59×#451—21no—
59C4×Q16×C59C2211yescharacteristic, complemented
60C8×D4×C59C2211yescharacteristic
61C2×#11745685—21no—
62C22×D236—21no—
63#14273749—22no—
64#14273749—41no—
65C2×Dic236C2311yescharacteristic
66C2×#11745660—21no—
67C2×#11745660—21no—
68C2×D472—22no—
69C2×D472—21no—
70C2×D472—41no—
71C2×D472C2311yescharacteristic
72C2×D472C2311yescharacteristic
73#14273715—21no—
74#14273712—41no—
75#14273706—21no—
76#14273701—21no—
77#14273700—41no—
78#14273695—41no—
79C4×D236—21no—
80C22⋊C4×D59—41no—
81#14273673—22no—
82#14273673—22no—
83#14273673—41no—
84#14273670—22no—
85#14273670—22no—
86#14273670—22no—
87#14273670—22no—
88#14273670—41no—
89#14273670D414yescomplemented
90#14273667—22no—
91C2×C59⋊C16—22no—
92C2×C59⋊C16C2311yescharacteristic
93C2×C59⋊C16C2311yescharacteristic
94C2×C59⋊C16C2311yescharacteristic
95C2×C59⋊C16C2311yescharacteristic
96#14273661C2311yescharacteristic
97#14273652—21no—
98#14273649—21no—
99#14273649—21no—
100C22×Q8×C59—21no—
101C22×Q8×C59—21no—
102C2×Q16×C59—22no—
103C2×Q16×C59—22no—
104C2×Q16×C59—21no—
105C2×Q16×C59—21no—
106C2×Q16×C59—21no—
107C2×Q16×C59—41no—
108C2×Q16×C59—41no—
109C2×Q16×C59C2311yescharacteristic
110C2×Q16×C59C2311yescharacteristic
111C2×Q16×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C4⋊Q8×C59—21no—
115C4⋊Q8×C59—41no—
116C22⋊Q8×C59—21no—
117C22⋊Q8×C59—21no—
118C22⋊Q8×C59—41no—
119C22⋊Q8×C59—41no—
120C4×Q8×C59—21no—
121C4×D4×C59—41no—
122C4×D4×C59D411yescharacteristic
123#107×C59C2311yescharacteristic
124C4⋊C8×C59D411yescharacteristic
125#103×C59—21no—
126#103×C59—21no—
127#103×C59—21no—
128C22⋊C8×C59D411yescharacteristic
129C22⋊C8×C59D411yescharacteristic
130C4×C8×C59C2311yescharacteristic
131C23×D59—41no—
132C2×Q8×C59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—42no—
136C2×Q8×C59—42no—
137C2×Q8×C59—41no—
138C2×Q8×C59—41no—
139C2×Q8×C59—81no—
140C2×Q8×C59—81no—
141C2×D4×C59D811yescharacteristic
142C22×C4×C59—41no—
143C22×C4×C59—41no—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C2×C22⋊D59—41no—
147C2×C22⋊D59—41no—
148#11745685—42no—
149#11745685—81no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236—42no—
153C2×D236—41no—
154C2×D236—81no—
155C2×C4×D59—41no—
156C2×C4×D59—41no—
157C2×Dic118—21no—
158Q16×C59—21no—
159Q16×C59—21no—
160Q16×C59—21no—
161Q16×C59—21no—
162Q16×C59—21no—
163Q16×C59—42no—
164Q16×C59—42no—
165Q16×C59—81no—
166Q16×C59C2×D412yes—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59C2×D411yescharacteristic
172C2×C8×C59C2×D411yescharacteristic
173C2×C8×C59C2×D411yescharacteristic
174C2×C8×C59C2411yescharacteristic
175C4⋊C4×C59—21no—
176C4⋊C4×C59—41no—
177C4⋊C4×C59—41no—
178C4⋊C4×C59—41no—
179C4⋊C4×C59—41no—
180C4⋊C4×C59—41no—
181C4⋊C4×C59D811yescharacteristic
182C22⋊C4×C59—41no—
183C22⋊C4×C59—41no—
184C22⋊C4×C59D811yescharacteristic
185C22⋊C4×C59D811yescharacteristic
186C42×C59—41no—
187C42×C59C2×D411yescharacteristic
188C22⋊Dic59—41no—
189(C2×D59)⋊C4—41no—
190(C2×D59)⋊C4—41no—
191(C2×D59)⋊C4—41no—
192(C2×D59)⋊C4—41no—
193C4⋊Dic59—21no—
194Dic59⋊C4—41no—
195Dic236—21no—
196Dic236—21no—
197D472—21no—
198D472—21no—
199D472—21no—
200D472—42no—
201D472C2×D412yes—
202#11745660—42no—
203#11745660—81no—
204C59⋊C16—21no—
205C59⋊C16—21no—
206C59⋊C16—21no—
207C59⋊C16C2×D412yes—
208C23×C59C2×D811yescharacteristic
209C23×C59C2×D811yescharacteristic
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—82no—
213C22×D59—82no—
214C22×D59—161no—
215Q8×C59—22no—
216Q8×C59—41no—
217Q8×C59—41no—
218Q8×C59—41no—
219Q8×C59—41no—
220Q8×C59—41no—
221Q8×C59—82no—
222Q8×C59—82no—
223D4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—82no—
233C2×C4×C59—82no—
234C2×C4×C59C2×D811yescharacteristic
235C2×C4×C59C2×D811yescharacteristic
236C2×C4×C59C2×D811yescharacteristic
237C2×C4×C59C2×D811yescharacteristic
238C2×C4×C59C22×D411yescharacteristic
239C22⋊D59—82no—
240C22⋊D59—161no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243D236—22no—
244D236—41no—
245D236—41no—
246D236—41no—
247D236—82no—
248C4×D59—82no—
249C4×D59—161no—
250Dic118—41no—
251Dic118—41no—
252C8×C59—21no—
253C8×C59—21no—
254C8×C59—21no—
255C8×C59C22×D411yescharacteristic
256C8×C59H(5,𝔽2)11yescharacteristic
257#33832—591nomaximal, Sylow
258C22×C59—21no—
259C22×C59—21no—
260C22×C59—21no—
261C22×C59—21no—
262C22×C59—21no—
263C22×C59#51012yescomplemented
264C22×C59C22×D811yescharacteristic
265C2×D59—42no—
266C2×D59—81no—
267C2×D59—81no—
268C2×D59—81no—
269C2×D59—81no—
270C2×D59—81no—
271C2×D59—164no—
272C4×C59—21no—
273C4×C59—21no—
274C4×C59—21no—
275C4×C59—21no—
276C4×C59—42no—
277C4×C59—81no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59D4211yescharacteristic
281C4×C59#57611yescharacteristic
282Dic59—81no—
283Dic59—81no—
284C2×#510—591no—
285C22×SD32—591no—
286D4×Q16D5911yescharacteristic, complemented
287#3280—591no—
288#2251—591no—
289#2220—591no—
290#2216—591no—
291#2211—591no—
292#2208—591no—
293#2189—591no—
294#2188—591no—
295#2187—591no—
296#2185—591no—
297#2175—591no—
298#2168—591no—
299C2×C59—22no—
300C2×C59—41no—
301C2×C59D4×D811yescharacteristic
302C2×C59C2×#51011yescharacteristic
303C2×C59#341611yescharacteristic
304D59—82no—
305D59—161no—
306C2×#135—591no—
307C22×Q16C2×D5911yescharacteristic
308C22×Q16C2×D5911yescharacteristic, complemented
309C22×D8—591no—
310D4×Q8—21no—
311#546—1181no—
312#510—1184no—
313C2×Q32—591no—
314C2×Q32—591no—
315C2×Q32—591no—
316C2×SD32—591no—
317C2×SD32—591no—
318C2×SD32—591no—
319C2×SD32—591no—
320C2×SD32—591no—
321C2×SD32—591no—
322C2×SD32—1184no—
323C2×M32—591no—
324C22×C16—591no—
325#495—591no—
326#494C2×D5911yescharacteristic, complemented
327#467—591no—
328#467C2×D5911yescharacteristic
329#467C2×D5911yescharacteristic, complemented
330#466—591no—
331#463—1181no—
332#462—21no—
333#451—21no—
334#451—21no—
335#450—1181no—
336#449—1181no—
337C4×Q16C2×D5911yescharacteristic, complemented
338C4×D8—591no—
339C8×D4C2×D5911yescharacteristic
340#366—591no—
341C4⋊C16—591no—
342#358—591no—
343#358—591no—
344#358—591no—
345#358—591no—
346#357—591no—
347#357—591no—
348C22⋊C16—591no—
349C22⋊C16—591no—
350#346—591no—
351C59#3383211yescharacteristic, complemented, Sylow
352C2×#41—1181no—
353C22×Q8—21no—
354C22×Q8—21no—
355C22×D4—1181no—
356#135—1182no—
357#135—2361no—
358C2×Q16—22no—
359C2×Q16—22no—
360C2×Q16—21no—
361C2×Q16—21no—
362C2×Q16—21no—
363C2×Q16—41no—
364C2×Q16—41no—
365C2×Q16—591no—
366C2×Q16C22×D5911yescharacteristic
367C2×Q16C22×D5911yescharacteristic
368C2×Q16C22×D5911yescharacteristic
369C2×SD16—1181no—
370C2×SD16—1181no—
371C2×D8—591no—
372C2×D8—591no—
373C2×D8—1182no—
374C2×D8—1181no—
375C2×D8—2361no—
376C22×C8C22×D5911yescharacteristic
377C22×C8C22×D5911yescharacteristic
378C4⋊Q8—21no—
379C4⋊Q8—41no—
380#124—1181no—
381#123—2361no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—41no—
385C22⋊Q8—41no—
386C22⋊Q8—1181no—
387#121—1181no—
388#121—2361no—
389#120—2361no—
390C4×Q8—21no—
391C4×D4—41no—
392C4×D4—1181no—
393C4×D4C22⋊D5911yescharacteristic
394C2×C22⋊C4—2361no—
395Q32—1182no—
396Q32—1182no—
397Q32—2361no—
398SD32—594no—
399SD32—1182no—
400SD32—1182no—
401SD32—1182no—
402SD32—1182no—
403SD32—2361no—
404M32—1182no—
405C2×C16—591no—
406C2×C16—591no—
407C2×C16—591no—
408C2×C16—591no—
409C2×C16—1182no—
410#107—591no—
411#107C22×D5911yescharacteristic
412C4⋊C8C22⋊D5911yescharacteristic
413#103—21no—
414#103—21no—
415#103—21no—
416#103—1181no—
417#102—1181no—
418#102—1181no—
419C22⋊C8C22⋊D5911yescharacteristic
420C22⋊C8C22⋊D5911yescharacteristic
421C4×C8C22×D5911yescharacteristic
422C24—2361no—
423#41—4721no—
424#41—2362no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—81no—
433C2×Q8—81no—
434C2×Q8—1181no—
435C2×D4—1181no—
436C2×D4—1181no—
437C2×D4—4721no—
438C2×D4—2362no—
439C2×D4—2361no—
440C2×D4—2361no—
441C2×D4—2361no—
442C2×D4#1174566911yescharacteristic
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—2361no—
446C22×C4—2361no—
447C22×C4C2×C22⋊D5911yescharacteristic
448C22×C4C2×C22⋊D5911yescharacteristic
449Q16—21no—
450Q16—21no—
451Q16—21no—
452Q16—21no—
453Q16—21no—
454Q16—42no—
455Q16—42no—
456Q16—81no—
457Q16—1181no—
458Q16—1181no—
459Q16D4×D5912yes—
460SD16—4721no—
461SD16—2362no—
462D8—592no—
463D8—1181no—
464D8—1181no—
465D8—1181no—
466D8—2362no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8C2×C22⋊D5911yescharacteristic
472C2×C8C2×C22⋊D5911yescharacteristic
473C2×C8C2×C22⋊D5911yescharacteristic
474C2×C8C23×D5911yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—1181no—
482C4⋊C4—2361no—
483C4⋊C4#1174566911yescharacteristic
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—2361no—
487C22⋊C4—2361no—
488C22⋊C4—2361no—
489C22⋊C4—2361no—
490C22⋊C4—2361no—
491C22⋊C4#1174566911yescharacteristic
492C22⋊C4#1174566911yescharacteristic
493C42—41no—
494C42C2×C22⋊D5911yescharacteristic
495C16—592no—
496C16—1181no—
497C16—1181no—
498C16—1181no—
499C23—9441no—
500C23—4722no—
501C23—4722no—
502C23—2361no—
503C23—2361no—
504C23C2×#1174566911yescharacteristic
505C23C2×#1174566911yescharacteristic
506Q8—22no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—82no—
513Q8—82no—
514Q8—2361no—
515Q8—2361no—
516D4—22no—
517D4—1182no—
518D4—9441no—
519D4—4722no—
520D4—4722no—
521D4—2361no—
522D4—2361no—
523D4—2361no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—9441no—
535C2×C4—4722no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4C2×#1174566911yescharacteristic
539C2×C4C2×#1174566911yescharacteristic
540C2×C4C2×#1174566911yescharacteristic
541C2×C4C2×#1174566911yescharacteristic
542C2×C4C22×C22⋊D5911yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8C2×D4×D5911yescharacteristic
547C8#1427377211yescharacteristic
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—9444no—
554C22—4721no—
555C22—4721no—
556C22—4721no—
557C22—4721no—
558C22—4721no—
559C22—2362no—
560C22#1683784312yescomplemented
561C22C22×#1174566911yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—4721no—
571C4—4721no—
572C4D4×C22⋊D5911yescharacteristic
573C4#1683820511yescharacteristic
574C2—22no—
575C2—41no—
576C2—9441no—
577C2—4722no—
578C2D4×#1174566911yescharacteristic
579C2C2×#1683784311yescharacteristic
580C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingD4×Q16×C59C2
socleC22×C59C22×D8
prime radical#21435783C2
hypercenterD4×Q16D59
hypocenterC59#33832
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#16837843C2semidirect
2. C22×#14273670C2semidirect
4. #21426365C2semidirect
5. #21426211C2semidirect
6. #21426185C2semidirect
7. #21426158C2semidirect
8. #21426138C2semidirect
9. #21426034C2semidirect
10. #21426030C2semidirect
11. #21426024C2semidirect
12. #21426013C2semidirect
13. #21425958C2semidirect
14. #21425922C2semidirect
15. D4×Q16×C59C2semidirect
20. C2×#14273673C22semidirect
21. C2×#14273673C22semidirect
22. C2×#14273673C22semidirect
30. C2×#14273667C22semidirect
39. #16837227C22semidirect
40. C4⋊(C59⋊C16)C22semidirect
41. #16837206C22semidirect
42. #16837206C22semidirect
43. #16837206C22semidirect
44. #16837205C22semidirect
48. C22⋊(C59⋊C16)C22semidirect
49. #16837178C22semidirect
51. C22×Q16×C59C22semidirect
53. C59×#494C22semidirect
55. C59×#467C22semidirect
59. C4×Q16×C59C22semidirect
89. #14273670D4semidirect
263. C22×C59#510semidirect
286. D4×Q16D59semidirect
308. C22×Q16C2×D59semidirect
326. #494C2×D59semidirect
329. #467C2×D59semidirect
337. C4×Q16C2×D59semidirect
351. C59#33832semidirect
560. C22#16837843semidirect

Maximal subgroups

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

subgroupindex
C2×#168378432
C22×#142736702
#214357832
#214263652
#214262112
#214261852
#214261582
#214261382
#214260342
#214260302
#214260242
#214260132
#214259582
#214259222
D4×Q16×C592
#3383259