Finite Group Databasebeta

← #28559143

Subgroups — #28559143

Order 15104.

598 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
1#21434125C211yesmaximal, characteristic
2#21433178C211yesmaximal, characteristic, complemented
3#21433173C211yesmaximal, characteristic
4#21422998C211yesmaximal, characteristic, complemented
5#21422996C211yesmaximal, characteristic, complemented
6#21422988C211yesmaximal, characteristic, complemented
7#21422956C211yesmaximal, characteristic, complemented
8#21422950C211yesmaximal, characteristic, complemented
9#21422933C211yesmaximal, characteristic, complemented
10#21422931C211yesmaximal, characteristic, complemented
11#21422873C211yesmaximal, characteristic, complemented
12#21422541C211yesmaximal, characteristic, complemented
13#21422511C211yesmaximal, characteristic
14#21422390C211yesmaximal, characteristic, complemented
15C59×#1590C211yesmaximal, characteristic, complemented
16#16837949C2211yescharacteristic
17C4×Q8×D59C2211yescharacteristic
18#16837722C411yescharacteristic
19#16837719C411yescharacteristic
20#16837686C411yescharacteristic
21#16837685C411yescharacteristic
22#16837601C411yescharacteristic
23#16837600C411yescharacteristic
24#16837542C411yescharacteristic
25#16837539C411yescharacteristic
26#16837533C2211yescharacteristic
27#16837532C2211yescharacteristic
28#16837531—21no—
29#16837530—21no—
30#16837529C2211yescharacteristic
31C4×#11745660C2211yescharacteristic
32C4×#11745671C2211yescharacteristic
33C4×#11745670C2211yescharacteristic
34#16837481C2211yescharacteristic
35#16837471—21no—
36#16837470—21no—
37D4×C59⋊C8C2211yescharacteristic
38#16837467C2211yescharacteristic
39#16837458—21no—
40#16837458C2211yescharacteristic
41#16837457—21no—
42#16837454C2211yescharacteristic
43#16837453C2211yescharacteristic
44#16837453C2211yescharacteristic
45#16837370—21no—
46#16837367—21no—
47C4×#11745659C2211yescharacteristic
48#16837154C2211yescharacteristic
49#16837154C2211yescharacteristic
50#16837153C2211yescharacteristic
51#16837134C2211yescharacteristic
52Q8⋊(C59⋊C8)C2211yescharacteristic
53#16837132C2211yescharacteristic
54#16837131C2211yescharacteristic
55D4⋊(C59⋊C8)C2211yescharacteristic
56#16837129C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837124C2211yescharacteristic
59#16837123C2211yescharacteristic
60C59×#446C2211yescharacteristic
61C4×SD16×C59C2211yescharacteristic
62C59×#436C2211yescharacteristic
63C59×#334C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×D4⋊C8C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×#11745685—21no—
68C2×Q8×D59C2×C411yescharacteristic
69C2×#11745660C2×C411yescharacteristic
70C2×#11745670C2×C411yescharacteristic
71C2×#11745671C2×C411yescharacteristic
72C2×#11745659—21no—
73C2×#11745659—21no—
74C22×C59⋊C8—21no—
75C2×C8×D59—21no—
76#14273726C2×C411yescharacteristic
77#14273723C2×C411yescharacteristic
78#14273713—21no—
79#14273702C2×C411yescharacteristic
80#14273700—21no—
81#14273698C2×C411yescharacteristic
82Q8×Dic59C2×C411yescharacteristic
83C4×Dic118—21no—
84C4×Dic118C2311yescharacteristic
85#14273692—21no—
86#14273692C2×C411yescharacteristic
87C4×D236C2311yescharacteristic
88C4×C22⋊D59—21no—
89C4⋊C4×D59—21no—
90C4⋊C4×D59C2×C411yescharacteristic
91C42×D59—21no—
92C42×D59C2311yescharacteristic
93#14273659C2×C411yescharacteristic
94#14273658C2×C411yescharacteristic
95#14273658C2×C411yescharacteristic
96C4⋊(C59⋊C8)—21no—
97C4⋊(C59⋊C8)C2311yescharacteristic
98C4⋊(C59⋊C8)C2311yescharacteristic
99Dic59⋊C8—21no—
100Dic59⋊C8—21no—
101Dic59⋊C8C2×C411yescharacteristic
102Dic59⋊C8C2×C411yescharacteristic
103Dic59⋊C8D411yescharacteristic
104Dic59⋊C8D411yescharacteristic
105#14273652C2×C411yescharacteristic
106#14273651C2×C411yescharacteristic
107#14273650C2×C411yescharacteristic
108#14273649C2×C411yescharacteristic
109#14273648C2×C411yescharacteristic
110#14273647C2×C411yescharacteristic
111C22⋊(C59⋊C8)—21no—
112(C2×D59)⋊C8—21no—
113(C2×D59)⋊C8—21no—
114(C2×D59)⋊C8—21no—
115(C2×D59)⋊C8—21no—
116(C2×D59)⋊C8C2×C411yescharacteristic
117(C2×D59)⋊C8C2×C411yescharacteristic
118(C2×D59)⋊C8D411yescharacteristic
119(C2×D59)⋊C8D411yescharacteristic
120#14273638—21no—
121#14273637—21no—
122#14273637—21no—
123#14273637—21no—
124#14273637—21no—
125C4×C59⋊C8C2311yescharacteristic
126C4×C59⋊C8C2311yescharacteristic
127C4×C59⋊C8C2311yescharacteristic
128C8×Dic59—21no—
129C2×SD16×C59C2×C411yescharacteristic
130C2×M16×C59—21no—
131C4×Q8×C59C2311yescharacteristic
132C4×D4×C59C2311yescharacteristic
133#106×C59C2×C411yescharacteristic
134C4⋊C8×C59—21no—
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137#103×C59C2×C411yescharacteristic
138#102×C59C2×C411yescharacteristic
139C22⋊C8×C59—21no—
140#97×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59C22×C411yescharacteristic
145C2×D4×C59C22×C411yescharacteristic
146C22×C4×C59—21no—
147C2×C22⋊D59—21no—
148C2×C22⋊D59—21no—
149C2×C22⋊D59—21no—
150Q8×D59—41no—
151Q8×D59—41no—
152#11745685—81no—
153#11745685—81no—
154C2×D236—21no—
155C2×D236C22×C411yescharacteristic
156C2×C4×D59—21no—
157C2×C4×D59—21no—
158C2×C4×D59—21no—
159C2×C4×D59C22×C411yescharacteristic
160C2×C4×D59C2×D411yescharacteristic
161C2×C4×D59#4111yescharacteristic
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118C22×C411yescharacteristic
165SD16×C59—41no—
166M16×C59—81no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59C22×C411yescharacteristic
171C2×C8×C59C22×C411yescharacteristic
172C2×C8×C59C2×D411yescharacteristic
173C2×C8×C59C2×D411yescharacteristic
174C4⋊C4×C59—21no—
175C4⋊C4×C59C22×C411yescharacteristic
176C4⋊C4×C59C22×C411yescharacteristic
177C22⋊C4×C59—21no—
178C42×C59—21no—
179C42×C59C2411yescharacteristic
180C22⋊Dic59—21no—
181#11745671—41no—
182#11745670—41no—
183(C2×D59)⋊C4—21no—
184(C2×D59)⋊C4—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59C22×C411yescharacteristic
187C4⋊Dic59C22×C411yescharacteristic
188Dic59⋊C4—21no—
189Dic59⋊C4—21no—
190Dic59⋊C4—21no—
191Dic59⋊C4—21no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59C22×C411yescharacteristic
196C4×Dic59C2×D411yescharacteristic
197C4×Dic59#4111yescharacteristic
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—81no—
202C2×C59⋊C8C22×C411yescharacteristic
203C2×C59⋊C8C22×C411yescharacteristic
204C2×C59⋊C8C2×D411yescharacteristic
205C2×C59⋊C8C2×D411yescharacteristic
206#11745660—41no—
207#11745659—42no—
208#11745659—81no—
209C8×D59—81no—
210C23×C59—21no—
211C22×D59—21no—
212C22×D59C4×D411yescharacteristic
213Q8×C59—21no—
214Q8×C59—21no—
215D4×C59—21no—
216D4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59C23×C411yescharacteristic
223C2×C4×C59C22×D411yescharacteristic
224C2×C4×C59C2×#4111yescharacteristic
225C22⋊D59—81no—
226C22⋊D59—81no—
227C22⋊D59—81no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59C4×D411yescharacteristic
232C2×Dic59C4×D411yescharacteristic
233C2×Dic59C4×D411yescharacteristic
234D236—21no—
235D236—21no—
236D236—81no—
237C4×D59—21no—
238C4×D59—21no—
239C4×D59—21no—
240C4×D59—21no—
241C4×D59—41no—
242C4×D59—42no—
243C4×D59—41no—
244C4×D59—41no—
245C4×D59—81no—
246Dic118—21no—
247Dic118—21no—
248Dic118—42no—
249Dic118—81no—
250C8×C59—22no—
251C8×C59—21no—
252C8×C59—21no—
253C8×C59—21no—
254C8×C59—21no—
255C8×C59—41no—
256C8×C59—41no—
257C59⋊C8—21no—
258C59⋊C8—21no—
259C59⋊C8—21no—
260C59⋊C8—21no—
261C59⋊C8—41no—
262C59⋊C8—41no—
263C59⋊C8—41no—
264#17951—591nomaximal, Sylow
265C22×C59—41no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59C2×C4×D411yescharacteristic
269C2×D59—21no—
270C2×D59—21no—
271C2×D59—21no—
272C2×D59—41no—
273C2×D59—41no—
274C2×D59—41no—
275C4×C59—21no—
276C4×C59—41no—
277C4×C59—41no—
278C4×C59—41no—
279C4×C59#56811yescharacteristic
280C4×C59#56811yescharacteristic
281C4×C59#57711yescharacteristic
282C4×C59#57711yescharacteristic
283Dic59—21no—
284Dic59—21no—
285Dic59—21no—
286Dic59—41no—
287Dic59—41no—
288Dic59—41no—
289#3103—591no—
290#2983—591no—
291#2981—591no—
292#1592—591no—
293#1592—591no—
294#1590—591no—
295#1590D5911yescharacteristic, complemented
296#1585—591no—
297#1585—591no—
298#1582—591no—
299#1582—591no—
300#1572—591no—
301#1491—591no—
302#1484—591no—
303#1453—591no—
304C2×C59—81no—
305C2×C59#293011yescharacteristic
306C2×C59#294911yescharacteristic
307C2×C59#295611yescharacteristic
308D59—41no—
309D59—81no—
310#532—591no—
311C2×C4×Q8—591no—
312#481—591no—
313#481—591no—
314#475—591no—
315#475—591no—
316#460—591no—
317#460—591no—
318#448—591no—
319#448—591no—
320#446—591no—
321#446—591no—
322#446—591no—
323#446—1181no—
324#446—1181no—
325#446C2×D5911yescharacteristic
326C4×SD16—591no—
327C4×SD16—591no—
328C4×SD16—591no—
329C4×SD16C2×D5911yescharacteristic
330#436—591no—
331#436C2×D5911yescharacteristic
332#435—1181no—
333#435—1181no—
334C8×D4—591no—
335C8×D4—591no—
336#432—591no—
337#432—1181no—
338#432—1181no—
339#431—591no—
340#431—591no—
341#431—591no—
342#408—1181no—
343#408—1181no—
344C4×M16—591no—
345#335—591no—
346#335—591no—
347#334—591no—
348#334C2×D5911yescharacteristic
349Q8⋊C8—591no—
350Q8⋊C8—591no—
351Q8⋊C8—591no—
352Q8⋊C8C2×D5911yescharacteristic
353D4⋊C8—591no—
354D4⋊C8—591no—
355D4⋊C8—591no—
356D4⋊C8C2×D5911yescharacteristic
357#322—591no—
358#322—591no—
359#322—591no—
360C82C2×D5911yescharacteristic
361C59#1795111yescharacteristic, complemented, Sylow
362C2×#41—1181no—
363C22×Q8—591no—
364C2×SD16—591no—
365C2×SD16—591no—
366C2×SD16—591no—
367C2×SD16C4×D5911yescharacteristic
368C2×M16—21no—
369C2×M16—1181no—
370C2×M16—1181no—
371C22×C8—1181no—
372C22×C8—1181no—
373C4⋊Q8—591no—
374#127—591no—
375#124—1181no—
376#121—591no—
377#121—591no—
378#121—1181no—
379C4×Q8—591no—
380C4×Q8—591no—
381C4×Q8—591no—
382C4×Q8—1181no—
383C4×Q8—1181no—
384C4×Q8C22×D5911yescharacteristic
385C4×D4—591no—
386C4×D4—1181no—
387C4×D4C22×D5911yescharacteristic
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C42—591no—
391C2×C42—1181no—
392#106—591no—
393#106—591no—
394#106—591no—
395#106C4×D5911yescharacteristic
396C4⋊C8—21no—
397C4⋊C8—591no—
398C4⋊C8—591no—
399C4⋊C8—591no—
400C4⋊C8—591no—
401C4⋊C8—591no—
402C4⋊C8—591no—
403C4⋊C8—1181no—
404C4⋊C8—1181no—
405C4⋊C8—1181no—
406C4⋊C8C22×D5911yescharacteristic
407C4⋊C8C22×D5911yescharacteristic
408#103—591no—
409#103—591no—
410#103—591no—
411#103C4×D5911yescharacteristic
412#102—591no—
413#102—591no—
414#102—591no—
415#102C4×D5911yescharacteristic
416C22⋊C8—21no—
417C22⋊C8—591no—
418C22⋊C8—591no—
419C22⋊C8—591no—
420C22⋊C8—591no—
421C22⋊C8—1181no—
422C22⋊C8—1181no—
423C22⋊C8—1181no—
424C22⋊C8—1181no—
425C22⋊C8—1181no—
426#97—21no—
427#97—1181no—
428#97—1181no—
429#97—1181no—
430#97—1181no—
431#97—1181no—
432C4×C8—591no—
433C4×C8—591no—
434C4×C8—591no—
435C4×C8—1181no—
436C4×C8C22×D5911yescharacteristic
437C4×C8C22×D5911yescharacteristic
438C4×C8C22×D5911yescharacteristic
439#41—4721no—
440#41—4721no—
441C2×Q8—591no—
442C2×Q8—1181no—
443C2×Q8—1181no—
444C2×Q8—2361no—
445C2×Q8—2361no—
446C2×Q8C2×C4×D5911yescharacteristic
447C2×D4—591no—
448C2×D4—1181no—
449C2×D4—1181no—
450C2×D4—1181no—
451C2×D4—1181no—
452C2×D4C2×C4×D5911yescharacteristic
453C22×C4—21no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460SD16—41no—
461SD16—2361no—
462SD16—2361no—
463SD16—2361no—
464M16—81no—
465M16—4721no—
466M16—2362no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—591no—
471C2×C8—591no—
472C2×C8—591no—
473C2×C8—591no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—4721no—
478C2×C8—4721no—
479C2×C8C2×C4×D5911yescharacteristic
480C2×C8C2×C4×D5911yescharacteristic
481C2×C8D4×D5911yescharacteristic
482C2×C8D4×D5911yescharacteristic
483C4⋊C4—21no—
484C4⋊C4—591no—
485C4⋊C4—591no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4C2×C4×D5911yescharacteristic
492C4⋊C4C2×C4×D5911yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—1181no—
495C22⋊C4—1181no—
496C22⋊C4—1181no—
497C42—21no—
498C42—591no—
499C42—591no—
500C42—591no—
501C42—1181no—
502C42—1181no—
503C42—1181no—
504C42C23×D5911yescharacteristic
505C23—21no—
506C23—591no—
507C23—1181no—
508Q8—21no—
509Q8—21no—
510Q8—1181no—
511Q8—1181no—
512Q8—4721no—
513Q8—2362no—
514D4—21no—
515D4—21no—
516D4—1181no—
517D4—1181no—
518D4—4721no—
519D4—4721no—
520D4—4721no—
521D4—4721no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—591no—
528C2×C4—591no—
529C2×C4—591no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—4721no—
538C2×C4—2361no—
539C2×C4—2362no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4C22×C4×D5911yescharacteristic
543C2×C4C2×D4×D5911yescharacteristic
544C2×C4#41×D5911yescharacteristic
545C8—22no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—21no—
550C8—41no—
551C8—41no—
552C8—1181no—
553C8—1181no—
554C8—1181no—
555C8—1181no—
556C8—2361no—
557C8—2361no—
558C8—2361no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—1181no—
563C22—1181no—
564C22—1181no—
565C22—2361no—
566C22—2361no—
567C22—2361no—
568C22C4×D4×D5911yescharacteristic
569C4—21no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—1181no—
574C4—1181no—
575C4—1181no—
576C4—2361no—
577C4—2361no—
578C4—2361no—
579C4#1683816711yescharacteristic
580C4#1683816811yescharacteristic
581C4#1683820811yescharacteristic
582C4#1683820911yescharacteristic
583C2—81no—
584C2—4721no—
585C2—2361no—
586C2#2143279611yescharacteristic
587C2#2143295011yescharacteristic
588C2#2143300111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1590C2
socleC22×C59C2×C4×D4
prime radical#16837949C22
cosocleC42×C59C24
hypercenter#1590D59
hypocenterC59#17951
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #21433178C2semidirect
4. #21422998C2semidirect
5. #21422996C2semidirect
6. #21422988C2semidirect
7. #21422956C2semidirect
8. #21422950C2semidirect
9. #21422933C2semidirect
10. #21422931C2semidirect
11. #21422873C2semidirect
12. #21422541C2semidirect
14. #21422390C2semidirect
15. C59×#1590C2semidirect
295. #1590D59semidirect
361. C59#17951semidirect

Maximal subgroups

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

subgroupindex
#214341252
#214331782
#214331732
#214229982
#214229962
#214229882
#214229562
#214229502
#214229332
#214229312
#214228732
#214225412
#214225112
#214223902
C59×#15902
#1795159