Finite Group Databasebeta

← #28617616

Subgroups — #28617616

Order 15104.

892 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837420C211yesmaximal, characteristic
2#21432616C211yesmaximal, characteristic
3C2×#16837384C211yesmaximal, characteristic
4#21425547C218yesmaximal, complemented
5C2×#16837174C211yesmaximal, characteristic, complemented
6#21423823C211yesmaximal, characteristic, complemented
7C2×#16837158C211yesmaximal, characteristic, complemented
8C59×#2905C211yesmaximal, characteristic, complemented
9C22×#11745685C2211yescharacteristic
10C22×#11745664C2211yescharacteristic
11C22×#11745664C2211yescharacteristic
12C2×#14273686C2211yescharacteristic
13#16837420—22no—
14#16837420C2214yes—
15C2×#14273655—22no—
16#16837412—21no—
17#16837408—21no—
18#16837404—22no—
19#16837399C2214yes—
20#16837384—22no—
21#16837384C2214yes—
22C2×#14273645—21no—
23C2×#14273643—21no—
24#16837174C2214yescomplemented
25#16837167C2214yescomplemented
26#16837158C2214yescomplemented
27C22×M16×C59C2211yescharacteristic
28C22×M16×C59C2211yescharacteristic
29C2×#117×C59C2211yescharacteristic
30C59×#424—22no—
31C59×#424C2214yescomplemented
32C59×#423—22no—
33C2×#11745685—21no—
34C2×#11745685—21no—
35C2×#11745685—21no—
36C2×#11745685—21no—
37C2×#11745685—21no—
38C2×#11745685—21no—
39C2×#11745685—21no—
40C2×#11745685—21no—
41C2×#11745685—21no—
42C2×#11745685—21no—
43C2×#11745685D412yes—
44C2×#11745685C2312yes—
45C22×Dic118D411yescharacteristic
46C22×D236D411yescharacteristic
47C22×C22⋊D59—21no—
48C22×C4×D59—21no—
49C2×#11745664—21no—
50C2×#11745664—21no—
51C2×#11745664—21no—
52C2×#11745664—21no—
53C2×#11745664—21no—
54C2×#11745664—21no—
55C2×#11745664—21no—
56C2×#11745664—21no—
57C2×#11745664—21no—
58C2×#11745664—21no—
59C2×#11745664C2312yes—
60C2×#11745664C2312yes—
61C22×C59⋊C8—21no—
62C22×C59⋊C8—21no—
63#14273686—21no—
64#14273686—21no—
65#14273686—21no—
66#14273686C2312yes—
67C2×C4⋊Dic59—21no—
68C2×C22⋊Dic59—21no—
69C2×C4×Dic59—21no—
70#14273655—28no—
71#14273652—24no—
72#14273649—24no—
73#14273645—24no—
74#14273643—24no—
75C23×C4×C59C2311yescharacteristic
76C2×M16×C59—21no—
77C2×M16×C59—21no—
78C2×M16×C59—21no—
79C2×M16×C59—21no—
80C2×M16×C59—21no—
81C2×M16×C59—21no—
82C2×M16×C59—21no—
83C2×M16×C59—21no—
84C2×M16×C59—21no—
85C2×M16×C59—21no—
86C2×M16×C59—21no—
87C2×M16×C59C2312yes—
88C22×C8×C59—21no—
89C22×C8×C59C2312yes—
90#117×C59—21no—
91#117×C59—21no—
92#117×C59D412yes—
93#117×C59C2312yes—
94C2×C4⋊C4×C59—21no—
95C2×C22⋊C4×C59—21no—
96C2×C42×C59D412yes—
97C4⋊C8×C59—28no—
98C24×C59#4111yescharacteristic
99C23×D59—21no—
100C22×C4×C59—22no—
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59C2×D411yescharacteristic
104C22×C4×C59C2×D411yescharacteristic
105C22×C4×C59#4111yescharacteristic
106C22×C4×C59#4111yescharacteristic
107C22×C4×C59#4111yescharacteristic
108C22×C4×C59#4111yescharacteristic
109C22×C4×C59#4111yescharacteristic
110C22×C4×C59#4111yescharacteristic
111C22×C4×C59#4111yescharacteristic
112C22×C4×C59#4111yescharacteristic
113C22×C4×C59#4111yescharacteristic
114C22×C4×C59C2411yescharacteristic
115C2×C22⋊D59—22no—
116C2×C22⋊D59—22no—
117C2×C22⋊D59—41no—
118C2×C22⋊D59—41no—
119C2×C22⋊D59—41no—
120C2×C22⋊D59—41no—
121C22×Dic59—21no—
122C22×Dic59—41no—
123#11745685—22no—
124#11745685—22no—
125#11745685—22no—
126#11745685—22no—
127#11745685—22no—
128#11745685—22no—
129#11745685—22no—
130#11745685—22no—
131#11745685—41no—
132#11745685—41no—
133#11745685—41no—
134#11745685—41no—
135#11745685—41no—
136#11745685—41no—
137#11745685—41no—
138#11745685—41no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236C2×D412yes—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—41no—
148C2×C4×D59—41no—
149C2×C4×D59—41no—
150C2×C4×D59—41no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118C2×D412yes—
157M16×C59—22no—
158M16×C59—22no—
159M16×C59—41no—
160M16×C59—41no—
161M16×C59—41no—
162M16×C59—41no—
163M16×C59—41no—
164M16×C59—41no—
165C2×C8×C59—22no—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59—22no—
169C2×C8×C59—41no—
170C2×C8×C59—41no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—22no—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—22no—
175C42×C59—22no—
176C42×C59C2×D414yes—
177C22⋊Dic59—22no—
178C22⋊Dic59—41no—
179C4⋊Dic59—22no—
180C4⋊Dic59—41no—
181C4×Dic59—22no—
182C4×Dic59—41no—
183#11745664—22no—
184#11745664—22no—
185#11745664—22no—
186#11745664—22no—
187#11745664—41no—
188#11745664—41no—
189#11745664—41no—
190#11745664—41no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C2×C59⋊C8—41no—
194C2×C59⋊C8—41no—
195C2×C59⋊C8—41no—
196C2×C59⋊C8—41no—
197C23×C59—81no—
198C23×C59#12311yescharacteristic
199C23×C59#12311yescharacteristic
200C23×C59#12311yescharacteristic
201C23×C59#12311yescharacteristic
202C23×C59#12611yescharacteristic
203C23×C59#12611yescharacteristic
204C23×C59C2×#4111yescharacteristic
205C22×D59—22no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—82no—
209C2×C4×C59—24no—
210C2×C4×C59—82no—
211C2×C4×C59#12111yescharacteristic
212C2×C4×C59#12111yescharacteristic
213C2×C4×C59#12111yescharacteristic
214C2×C4×C59#12111yescharacteristic
215C2×C4×C59#12311yescharacteristic
216C2×C4×C59#12311yescharacteristic
217C2×C4×C59#12311yescharacteristic
218C2×C4×C59#12311yescharacteristic
219C2×C4×C59#12311yescharacteristic
220C2×C4×C59#12311yescharacteristic
221C2×C4×C59#12311yescharacteristic
222C2×C4×C59#12311yescharacteristic
223C2×C4×C59#12411yescharacteristic
224C2×C4×C59#12411yescharacteristic
225C2×C4×C59#12411yescharacteristic
226C2×C4×C59#12411yescharacteristic
227C2×C4×C59#12611yescharacteristic
228C2×C4×C59#12611yescharacteristic
229C2×C4×C59#12611yescharacteristic
230C2×C4×C59#12611yescharacteristic
231C2×C4×C59#12611yescharacteristic
232C2×C4×C59#12611yescharacteristic
233C2×C4×C59C22×D411yescharacteristic
234C2×C4×C59C22×D411yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C22⋊D59—42no—
240C22⋊D59—42no—
241C22⋊D59—42no—
242C22⋊D59—42no—
243C2×Dic59—22no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—42no—
247C2×Dic59—82no—
248D236—22no—
249D236—22no—
250D236—41no—
251D236—41no—
252C4×D59—42no—
253C4×D59—42no—
254C4×D59—42no—
255C4×D59—42no—
256Dic118—22no—
257Dic118—22no—
258Dic118—41no—
259Dic118—41no—
260C8×C59—42no—
261C8×C59—42no—
262C59⋊C8—42no—
263C59⋊C8—42no—
264#23649—591nomaximal, Sylow
265C22×C59—41no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59#39711yescharacteristic
273C22×C59#39711yescharacteristic
274C22×C59#39711yescharacteristic
275C22×C59#39711yescharacteristic
276C22×C59C2×#12311yescharacteristic
277C22×C59C2×#12311yescharacteristic
278C22×C59C2×#12611yescharacteristic
279C2×D59—42no—
280C2×D59—82no—
281C2×D59—82no—
282C2×D59—82no—
283C4×C59—44no—
284C4×C59#39711yescharacteristic
285C4×C59#39711yescharacteristic
286C4×C59#39711yescharacteristic
287C4×C59#39711yescharacteristic
288C4×C59C2×#12111yescharacteristic
289C4×C59C2×#12311yescharacteristic
290C4×C59C2×#12411yescharacteristic
291C4×C59C2×#12611yescharacteristic
292Dic59—42no—
293Dic59—82no—
294#2905D5911yescharacteristic, complemented
295C2×#421—591no—
296#2896—591no—
297C2×#413—591no—
298#2080—598no—
299C2×#344—591no—
300#1738—591no—
301C2×H(3,ℤ/4)—591no—
302C2×C59—21no—
303C2×C59—21no—
304C2×C59—21no—
305C2×C59—21no—
306C2×C59—21no—
307C2×C59—21no—
308C2×C59—21no—
309C2×C59C2×#39711yescharacteristic
310D59—82no—
311C22×#41—591no—
312C22×M16—591no—
313C22×M16—591no—
314C22×M16C2×D5911yescharacteristic
315C22×M16C2×D5911yescharacteristic
316C2×#117—591no—
317C2×#117C2×D5911yescharacteristic
318#424—22no—
319#424C2×D5914yescomplemented
320#423—22no—
321#421—594no—
322#421—1182no—
323C2×#104—1182no—
324#419—1181no—
325#418—1181no—
326#417—1182no—
327#416—594no—
328#413—594no—
329#413—1182no—
330C2×#101—1181no—
331C2×#100—1181no—
332#344—594no—
333#341—594no—
334H(3,ℤ/4)—594no—
335C59#2364911yescharacteristic, complemented, Sylow
336C2×#41—592no—
337C2×#41—592no—
338C2×#41—1181no—
339C2×#41—1181no—
340C2×#41—1181no—
341C2×#41—1181no—
342C2×#41—1181no—
343C2×#41—1181no—
344C2×#41—1181no—
345C2×#41—1181no—
346C2×#41—1181no—
347C2×#41—1181no—
348C22×Q8—591no—
349C22×D4—591no—
350C22×D4—1181no—
351C23×C4—1181no—
352C23×C4C22×D5911yescharacteristic
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—592no—
365C2×M16—592no—
366C2×M16—1181no—
367C2×M16—1181no—
368C2×M16—1181no—
369C2×M16—1181no—
370C2×M16—1181no—
371C2×M16—1181no—
372C2×M16—1181no—
373C2×M16—1181no—
374C2×M16—1181no—
375C2×M16—1181no—
376C2×M16C22×D5912yes—
377C22×C8—21no—
378C22×C8—1181no—
379C22×C8—1181no—
380C22×C8C22×D5912yes—
381#117—21no—
382#117—21no—
383#117—592no—
384#117—1181no—
385#117—1181no—
386#117—1181no—
387#117C22⋊D5912yes—
388#117C22×D5912yes—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—1181no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1181no—
393C2×C42—1181no—
394C2×C42C22⋊D5912yes—
395C4⋊C8—28no—
396#104—1188no—
397#103—1184no—
398#102—1184no—
399#101—1184no—
400#100—1184no—
401C24—1181no—
402C24#1174568511yescharacteristic
403#41—1182no—
404#41—1182no—
405#41—1182no—
406#41—1182no—
407#41—1182no—
408#41—1182no—
409#41—1182no—
410#41—1182no—
411#41—2361no—
412#41—2361no—
413#41—2361no—
414#41—2361no—
415#41—2361no—
416#41—2361no—
417#41—2361no—
418#41—2361no—
419C2×Q8—592no—
420C2×Q8—1181no—
421C2×Q8—1181no—
422C2×Q8—1181no—
423C2×Q8—1181no—
424C2×Q8—1181no—
425C2×D4—592no—
426C2×D4—1182no—
427C2×D4—1182no—
428C2×D4—1181no—
429C2×D4—1181no—
430C2×D4—1181no—
431C2×D4—1181no—
432C2×D4—1181no—
433C2×D4—2361no—
434C2×D4—2361no—
435C2×D4—2361no—
436C2×D4—2361no—
437C22×C4—22no—
438C22×C4—1181no—
439C22×C4—1182no—
440C22×C4—1182no—
441C22×C4—2361no—
442C22×C4—2361no—
443C22×C4—2361no—
444C22×C4—2361no—
445C22×C4—2361no—
446C22×C4#1174568511yescharacteristic
447C22×C4#1174568511yescharacteristic
448C22×C4#1174568511yescharacteristic
449C22×C4D4×D5911yescharacteristic
450C22×C4D4×D5911yescharacteristic
451C22×C4#1174568711yescharacteristic
452C22×C4#1174568711yescharacteristic
453C22×C4#1174568911yescharacteristic
454C22×C4#1174568911yescharacteristic
455C22×C4#1174568911yescharacteristic
456C22×C4#1174568911yescharacteristic
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460M16—22no—
461M16—22no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—1182no—
469M16—1182no—
470M16—1182no—
471M16—1182no—
472M16—2361no—
473M16—2361no—
474M16—2361no—
475M16—2361no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—1182no—
483C2×C8—1182no—
484C2×C8—2361no—
485C2×C8—2361no—
486C2×C8—2361no—
487C2×C8—2361no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—1182no—
491C4⋊C4—2361no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—1182no—
495C22⋊C4—2361no—
496C42—22no—
497C42—1182no—
498C42—2361no—
499C42C2×C22⋊D5914yes—
500C23—81no—
501C23—1182no—
502C23—4722no—
503C23—2361no—
504C23—2361no—
505C23#1427371011yescharacteristic
506C23#1427371011yescharacteristic
507C23#1427371211yescharacteristic
508C23#1427371211yescharacteristic
509C23#1427372211yescharacteristic
510C23#1427372211yescharacteristic
511C23C2×#1174568511yescharacteristic
512Q8—1182no—
513Q8—1182no—
514Q8—2361no—
515Q8—2361no—
516D4—1182no—
517D4—1182no—
518D4—2362no—
519D4—2362no—
520D4—2362no—
521D4—2362no—
522D4—2361no—
523D4—2361no—
524C2×C4—24no—
525C2×C4—82no—
526C2×C4—1182no—
527C2×C4—4722no—
528C2×C4—2362no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2362no—
532C2×C4—2362no—
533C2×C4—2362no—
534C2×C4—2362no—
535C2×C4#1427369811yescharacteristic
536C2×C4#1427369811yescharacteristic
537C2×C4#1427369811yescharacteristic
538C2×C4#1427369811yescharacteristic
539C2×C4#1427371011yescharacteristic
540C2×C4#1427371011yescharacteristic
541C2×C4#1427371211yescharacteristic
542C2×C4#1427371211yescharacteristic
543C2×C4#1427371211yescharacteristic
544C2×C4#1427371211yescharacteristic
545C2×C4#1427371211yescharacteristic
546C2×C4#1427371211yescharacteristic
547C2×C4#1427371611yescharacteristic
548C2×C4#1427371611yescharacteristic
549C2×C4#1427371611yescharacteristic
550C2×C4#1427371611yescharacteristic
551C2×C4#1427372211yescharacteristic
552C2×C4#1427372211yescharacteristic
553C2×C4#1427372211yescharacteristic
554C2×C4#1427372211yescharacteristic
555C2×C4#1427372211yescharacteristic
556C2×C4#1427372211yescharacteristic
557C2×C4C2×D4×D5911yescharacteristic
558C2×C4C22×C22⋊D5911yescharacteristic
559C2×C4C2×#1174568711yescharacteristic
560C2×C4C2×#1174568511yescharacteristic
561C2×C4C2×#1174568911yescharacteristic
562C2×C4C2×#1174568911yescharacteristic
563C8—42no—
564C8—42no—
565C8—2362no—
566C8—2362no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—4722no—
575C22—4722no—
576C22—4722no—
577C22—2362no—
578C22#1683732811yescharacteristic
579C22#1683732811yescharacteristic
580C22#1683732811yescharacteristic
581C22#1683732811yescharacteristic
582C22C2×#1427371011yescharacteristic
583C22C2×#1427371211yescharacteristic
584C22C2×#1427372211yescharacteristic
585C4—44no—
586C4—4722no—
587C4—2362no—
588C4#1683732811yescharacteristic
589C4#1683732811yescharacteristic
590C4#1683732811yescharacteristic
591C4#1683732811yescharacteristic
592C4C2×#1427369811yescharacteristic
593C4C2×#1427371211yescharacteristic
594C4C2×#1427371611yescharacteristic
595C4C2×#1427372211yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—4722no—
604C2C2×#1683732811yescharacteristic
subgroupG/H
centerC4C2×#14273716
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2905C2
socleC2×C59C2×#397
prime radicalC22×#11745685C22
hypercenter#2905D59
hypocenterC59#23649
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C23×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
4. #21425547C2semidirect
5. C2×#16837174C2semidirect
6. #21423823C2semidirect
7. C2×#16837158C2semidirect
8. C59×#2905C2semidirect
24. #16837174C22semidirect
25. #16837167C22semidirect
26. #16837158C22semidirect
31. C59×#424C22semidirect
294. #2905D59semidirect
319. #424C2×D59semidirect
335. C59#23649semidirect

Maximal subgroups

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

subgroupindex
C2×#168374202
#214326162
C2×#168373842
#214255472
C2×#168371742
#214238232
C2×#168371582
C59×#29052
#2364959