Finite Group Databasebeta

← #28745193

Subgroups — #28745193

Order 15104.

692 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837837C211yesmaximal, characteristic, complemented
2C2×#16837832C211yesmaximal, characteristic, complemented
3#21435783C211yesmaximal, characteristic, complemented
4#21426365C211yesmaximal, characteristic, complemented
5#21426202C211yesmaximal, characteristic, complemented
6#21426194C211yesmaximal, characteristic, complemented
7#21426151C211yesmaximal, characteristic, complemented
8#21426138C211yesmaximal, characteristic, complemented
9#21426036C211yesmaximal, characteristic, complemented
10#21426034C211yesmaximal, characteristic, complemented
11#21426024C211yesmaximal, characteristic, complemented
12#21426019C211yesmaximal, characteristic, complemented
13#21425970C211yesmaximal, characteristic, complemented
14#21425922C211yesmaximal, characteristic, complemented
15C59×#3283C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×D472C2211yescharacteristic
18#16838048—21no—
19#16837837—24no—
20#16837832—24no—
21C2×#14273673C2211yescharacteristic, complemented
22C2×#14273673C2211yescharacteristic, complemented
23C2×#14273671C2211yescharacteristic, complemented
24C2×#14273670C2211yescharacteristic
25C2×#14273670C2211yescharacteristic
26C2×#14273670C2211yescharacteristic
27C2×#14273670C2211yescharacteristic
28C2×#14273668C2211yescharacteristic
29C2×#14273668C2211yescharacteristic
30C2×#14273667C2211yescharacteristic
31C22×C59⋊C16C2211yescharacteristic, complemented
32#16837786C2211yescharacteristic
33#16837643C2211yescharacteristic
34#16837638C2211yescharacteristic
35#16837618—21no—
36#16837555—21no—
37#16837549—21no—
38C4×D472C2211yescharacteristic
39#16837229C2211yescharacteristic, complemented
40C4⋊(C59⋊C16)C2211yescharacteristic, complemented
41#16837206C2211yescharacteristic, complemented
42#16837206C2211yescharacteristic, complemented
43#16837205C2211yescharacteristic, complemented
44#16837203C2211yescharacteristic, complemented
45#16837202C2211yescharacteristic
46#16837202C2211yescharacteristic
47C22⋊(C59⋊C16)C2211yescharacteristic
48C22⋊(C59⋊C16)C2211yescharacteristic, complemented
49#16837178C2211yescharacteristic, complemented
50C2×#135×C59C2211yescharacteristic
51C2×#135×C59C2211yescharacteristic, complemented
52C59×#547—21no—
53C59×#494C2211yescharacteristic, complemented
54C59×#467C2211yescharacteristic
55C59×#467C2211yescharacteristic, complemented
56C59×#461—21no—
57C59×#452—21no—
58C59×#452—21no—
59C4×D8×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—41no—
83#14273671—22no—
84#14273670—22no—
85#14273670—22no—
86#14273670—22no—
87#14273670—41no—
88#14273668—22no—
89#14273668D414yescomplemented
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—
100C2×#41×C59—21no—
101C2×#41×C59—21no—
102#135×C59—22no—
103#135×C59—22no—
104#135×C59—41no—
105#135×C59—41no—
106C2×Q16×C59—21no—
107C2×Q16×C59C2311yescharacteristic
108C2×Q16×C59C2311yescharacteristic
109C2×SD16×C59—21no—
110C2×SD16×C59—21no—
111C2×D8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C4⋊Q8×C59—21no—
115#123×C59—41no—
116#123×C59—41no—
117C22⋊Q8×C59—21no—
118C22⋊Q8×C59—21no—
119#121×C59—41no—
120C4×D4×C59—21no—
121C4×D4×C59D411yescharacteristic
122C2×C4⋊C4×C59—41no—
123#107×C59C2311yescharacteristic
124C4⋊C8×C59D411yescharacteristic
125#103×C59—21no—
126#103×C59—21no—
127#102×C59—21no—
128C22⋊C8×C59D411yescharacteristic
129C22⋊C8×C59D411yescharacteristic
130C4×C8×C59C2311yescharacteristic
131C23×D59—41no—
132#41×C59—42no—
133#41×C59—42no—
134#41×C59—81no—
135#41×C59—81no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—41no—
140C2×D4×C59—41no—
141C2×D4×C59D811yescharacteristic
142C22×C4×C59—41no—
143C22×C4×C59—41no—
144C22×C4×C59—41no—
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C2×C22⋊D59—41no—
148C2×C22⋊D59—41no—
149#11745685—42no—
150#11745685—81no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—42no—
154C2×D236—41no—
155C2×D236—81no—
156C2×C4×D59—41no—
157C2×C4×D59—41no—
158C2×Dic118—21no—
159Q16×C59—21no—
160Q16×C59—21no—
161Q16×C59—21no—
162Q16×C59—21no—
163Q16×C59—81no—
164SD16×C59—42no—
165SD16×C59—42no—
166D8×C59—21no—
167D8×C59C2×D412yes—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59—21no—
172C2×C8×C59C2×D411yescharacteristic
173C2×C8×C59C2×D411yescharacteristic
174C2×C8×C59C2×D411yescharacteristic
175C2×C8×C59C2411yescharacteristic
176C4⋊C4×C59—21no—
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×C59—41no—
185C22⋊C4×C59D811yescharacteristic
186C22⋊C4×C59D811yescharacteristic
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×C59—41no—
209C23×C59C2×D811yescharacteristic
210C23×C59C2×D811yescharacteristic
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—82no—
214C22×D59—82no—
215C22×D59—161no—
216Q8×C59—41no—
217Q8×C59—41no—
218Q8×C59—41no—
219Q8×C59—41no—
220D4×C59—22no—
221D4×C59—41no—
222D4×C59—41no—
223D4×C59—82no—
224D4×C59—82no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—82no—
233C2×C4×C59—82no—
234C2×C4×C59—82no—
235C2×C4×C59C2×D811yescharacteristic
236C2×C4×C59C2×D811yescharacteristic
237C2×C4×C59C2×D811yescharacteristic
238C2×C4×C59C2×D811yescharacteristic
239C2×C4×C59C22×D411yescharacteristic
240C22⋊D59—82no—
241C22⋊D59—161no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244D236—22no—
245D236—41no—
246D236—41no—
247D236—41no—
248D236—82no—
249C4×D59—82no—
250C4×D59—161no—
251Dic118—41no—
252Dic118—41no—
253C8×C59—21no—
254C8×C59—21no—
255C8×C59—21no—
256C8×C59C22×D411yescharacteristic
257C8×C59H(5,𝔽2)11yescharacteristic
258#33833—591nomaximal, Sylow
259C22×C59—21no—
260C22×C59—21no—
261C22×C59—21no—
262C22×C59—21no—
263C22×C59—21no—
264C22×C59—21no—
265C22×C59—42no—
266C22×C59—81no—
267C22×C59—81no—
268C22×C59C22×D811yescharacteristic
269C2×D59—42no—
270C2×D59—81no—
271C2×D59—81no—
272C2×D59—81no—
273C2×D59—81no—
274C2×D59—81no—
275C2×D59—164no—
276C4×C59—21no—
277C4×C59—21no—
278C4×C59—21no—
279C4×C59—21no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59D4211yescharacteristic
284C4×C59#57611yescharacteristic
285Dic59—81no—
286Dic59—81no—
287C2×#509—591no—
288C2×#508—591no—
289#3283D5911yescharacteristic, complemented
290#3280—591no—
291#2251—591no—
292#2219—591no—
293#2217—591no—
294#2210—591no—
295#2208—591no—
296#2189—591no—
297#2189—591no—
298#2187—591no—
299#2186—591no—
300#2177—591no—
301#2168—591no—
302C2×C59—41no—
303C2×C59—41no—
304C2×C59—82no—
305C2×C59D4×D811yescharacteristic
306C2×C59#341411yescharacteristic
307C2×C59#341611yescharacteristic
308D59—82no—
309D59—161no—
310C2×#135—591no—
311C2×#135C2×D5911yescharacteristic
312C2×#135C2×D5911yescharacteristic, complemented
313C22×D8—591no—
314#547—21no—
315#546—1181no—
316#509—1184no—
317#508—1184no—
318C2×Q32—591no—
319C2×Q32—591no—
320C2×SD32—591no—
321C2×SD32—591no—
322C2×SD32—591no—
323C2×SD32—591no—
324C2×SD32—591no—
325C2×D16—591no—
326C2×D16—591no—
327C2×M32—591no—
328C22×C16—591no—
329#495—591no—
330#494C2×D5911yescharacteristic, complemented
331#467—591no—
332#467C2×D5911yescharacteristic
333#467C2×D5911yescharacteristic, complemented
334#466—591no—
335#463—1181no—
336#461—21no—
337#452—21no—
338#452—21no—
339#450—1181no—
340#449—1181no—
341C4×D8—591no—
342C4×D8C2×D5911yescharacteristic, complemented
343C8×D4C2×D5911yescharacteristic
344#367—591no—
345C4⋊C16—591no—
346#358—591no—
347#358—591no—
348#358—591no—
349#357—591no—
350#357—591no—
351#357—591no—
352C22⋊C16—591no—
353C22⋊C16—591no—
354#346—591no—
355C59#3383311yescharacteristic, complemented, Sylow
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—1181no—
359C22×D4—1181no—
360#135—22no—
361#135—22no—
362#135—41no—
363#135—41no—
364#135—1182no—
365#135—2361no—
366C2×Q16—21no—
367C2×Q16—591no—
368C2×Q16C22×D5911yescharacteristic
369C2×Q16C22×D5911yescharacteristic
370C2×SD16—21no—
371C2×SD16—21no—
372C2×SD16—1181no—
373C2×SD16—1181no—
374C2×D8—591no—
375C2×D8—591no—
376C2×D8—1182no—
377C2×D8—1181no—
378C2×D8—2361no—
379C2×D8C22×D5911yescharacteristic
380C22×C8C22×D5911yescharacteristic
381C22×C8C22×D5911yescharacteristic
382C4⋊Q8—21no—
383#124—1181no—
384#123—41no—
385#123—41no—
386#123—2361no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—1181no—
390#121—41no—
391#121—1181no—
392#121—2361no—
393#120—2361no—
394C4×D4—21no—
395C4×D4—1181no—
396C4×D4C22⋊D5911yescharacteristic
397C2×C4⋊C4—41no—
398C2×C22⋊C4—2361no—
399Q32—1182no—
400Q32—2361no—
401SD32—1182no—
402SD32—1182no—
403SD32—1182no—
404SD32—1182no—
405SD32—2361no—
406D16—594no—
407D16—1182no—
408M32—1182no—
409C2×C16—591no—
410C2×C16—591no—
411C2×C16—591no—
412C2×C16—591no—
413C2×C16—1182no—
414#107—591no—
415#107C22×D5911yescharacteristic
416C4⋊C8C22⋊D5911yescharacteristic
417#103—21no—
418#103—21no—
419#103—1181no—
420#102—21no—
421#102—1181no—
422#102—1181no—
423C22⋊C8C22⋊D5911yescharacteristic
424C22⋊C8C22⋊D5911yescharacteristic
425C4×C8C22×D5911yescharacteristic
426C24—2361no—
427#41—42no—
428#41—42no—
429#41—81no—
430#41—81no—
431#41—4721no—
432#41—2362no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—1181no—
436C2×D4—21no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—1181no—
440C2×D4—1181no—
441C2×D4—4721no—
442C2×D4—2362no—
443C2×D4—2361no—
444C2×D4—2361no—
445C2×D4—2361no—
446C2×D4#1174566911yescharacteristic
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—2361no—
451C22×C4—2361no—
452C22×C4C2×C22⋊D5911yescharacteristic
453C22×C4C2×C22⋊D5911yescharacteristic
454Q16—21no—
455Q16—21no—
456Q16—21no—
457Q16—21no—
458Q16—81no—
459Q16—1181no—
460Q16—1181no—
461SD16—42no—
462SD16—42no—
463SD16—4721no—
464SD16—2362no—
465D8—21no—
466D8—592no—
467D8—1181no—
468D8—1181no—
469D8—1181no—
470D8—2362no—
471D8D4×D5912yes—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8C2×C22⋊D5911yescharacteristic
477C2×C8C2×C22⋊D5911yescharacteristic
478C2×C8C2×C22⋊D5911yescharacteristic
479C2×C8C23×D5911yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—1181no—
486C4⋊C4—2361no—
487C4⋊C4#1174566911yescharacteristic
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—2361no—
492C22⋊C4—2361no—
493C22⋊C4—2361no—
494C22⋊C4—2361no—
495C22⋊C4—2361no—
496C22⋊C4#1174566911yescharacteristic
497C22⋊C4#1174566911yescharacteristic
498C42C2×C22⋊D5911yescharacteristic
499C16—592no—
500C16—1181no—
501C16—1181no—
502C16—1181no—
503C23—41no—
504C23—9441no—
505C23—4722no—
506C23—4722no—
507C23—2361no—
508C23—2361no—
509C23C2×#1174566911yescharacteristic
510C23C2×#1174566911yescharacteristic
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—2361no—
516Q8—2361no—
517D4—22no—
518D4—41no—
519D4—41no—
520D4—82no—
521D4—82no—
522D4—1182no—
523D4—9441no—
524D4—4722no—
525D4—4722no—
526D4—2361no—
527D4—2361no—
528D4—2361no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—9441no—
540C2×C4—4722no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4C2×#1174566911yescharacteristic
544C2×C4C2×#1174566911yescharacteristic
545C2×C4C2×#1174566911yescharacteristic
546C2×C4C2×#1174566911yescharacteristic
547C2×C4C22×C22⋊D5911yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8C2×D4×D5911yescharacteristic
552C8#1427377211yescharacteristic
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—9444no—
563C22—4721no—
564C22—4721no—
565C22—4721no—
566C22—4721no—
567C22—4721no—
568C22—2362no—
569C22C22×#1174566911yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—4721no—
578C4—4721no—
579C4D4×C22⋊D5911yescharacteristic
580C4#1683820511yescharacteristic
581C2—41no—
582C2—41no—
583C2—82no—
584C2—9441no—
585C2—4722no—
586C2D4×#1174566911yescharacteristic
587C2#2143723011yescharacteristic
588C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3283C2
socleC22×C59C22×D8
hypercenter#3283D59
hypocenterC59#33833
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. C2×#16837837C2semidirect
2. C2×#16837832C2semidirect
3. #21435783C2semidirect
4. #21426365C2semidirect
5. #21426202C2semidirect
6. #21426194C2semidirect
7. #21426151C2semidirect
8. #21426138C2semidirect
9. #21426036C2semidirect
10. #21426034C2semidirect
11. #21426024C2semidirect
12. #21426019C2semidirect
13. #21425970C2semidirect
14. #21425922C2semidirect
15. C59×#3283C2semidirect
21. C2×#14273673C22semidirect
22. C2×#14273673C22semidirect
23. C2×#14273671C22semidirect
31. C22×C59⋊C16C22semidirect
39. #16837229C22semidirect
40. C4⋊(C59⋊C16)C22semidirect
41. #16837206C22semidirect
42. #16837206C22semidirect
43. #16837205C22semidirect
44. #16837203C22semidirect
48. C22⋊(C59⋊C16)C22semidirect
49. #16837178C22semidirect
51. C2×#135×C59C22semidirect
53. C59×#494C22semidirect
55. C59×#467C22semidirect
59. C4×D8×C59C22semidirect
89. #14273668D4semidirect
289. #3283D59semidirect
312. C2×#135C2×D59semidirect
330. #494C2×D59semidirect
333. #467C2×D59semidirect
342. C4×D8C2×D59semidirect
355. C59#33833semidirect

Maximal subgroups

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

subgroupindex
C2×#168378372
C2×#168378322
#214357832
#214263652
#214262022
#214261942
#214261512
#214261382
#214260362
#214260342
#214260242
#214260192
#214259702
#214259222
C59×#32832
#3383359