Finite Group Databasebeta

← #28593072

Subgroups — #28593072

Order 15104.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837499C211yesmaximal, characteristic
2#21426743C211yesmaximal, characteristic, complemented
3#21426733C211yesmaximal, characteristic
4#21424976C211yesmaximal, characteristic
5#21424951C211yesmaximal, characteristic
6#21424831C211yesmaximal, characteristic, complemented
7#21424828C211yesmaximal, characteristic, complemented
8#21424795C211yesmaximal, characteristic
9#21424784C211yesmaximal, characteristic, complemented
10#21424772C211yesmaximal, characteristic, complemented
11#21423992C211yesmaximal, characteristic, complemented
12#21423915C211yesmaximal, characteristic, complemented
13#102×Dic59C211yesmaximal, characteristic
14#21423782C211yesmaximal, characteristic, complemented
15C59×#1934C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×#14273692C2211yescharacteristic
18C2×D4×Dic59C2211yescharacteristic
19#16837499C418yescomplemented
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273652C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31C2×C8×Dic59C2211yescharacteristic
32#16837308C2211yescharacteristic
33#16837307C2211yescharacteristic
34#16837305C2211yescharacteristic
35#16837301C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837298C2211yescharacteristic
38#16837296—21no—
39#16837290—21no—
40#16837289—21no—
41#16837283—21no—
42#16837269—21no—
43#16837267—21no—
44#16837263—21no—
45#16837260—21no—
46C4⋊C4×Dic59C2211yescharacteristic
47C4⋊C4×Dic59C2211yescharacteristic
48#16837248C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#102×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C59×#390C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C23×Dic59—21no—
62C2×#11745670—22no—
63C2×#11745670—22no—
64C2×#11745670C2×C414yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273692—22no—
68#14273692C2×C414yes—
69D4×Dic59—22no—
70D4×Dic59C2×C414yes—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59C2311yescharacteristic
85C2×C4×Dic59C2311yescharacteristic
86#14273660—22no—
87#14273660C2×C414yes—
88#14273658—22no—
89#14273652—22no—
90#14273652C2×C414yes—
91#14273650—22no—
92#14273648—22no—
93#14273638—22no—
94#14273637—22no—
95#14273637C2×C414yes—
96C8×Dic59—22no—
97C8×Dic59—22no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C59⋊#95D411yescharacteristic
110C59⋊#95D411yescharacteristic
111C22×D4×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#102×C59—22no—
122#102×C59C2×C414yes—
123C4×C8×C59—22no—
124#95×C59—21no—
125C24×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59C22×C412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—24no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—81no—
144C22×Dic59—81no—
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118C22×C412yes—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59C22×C412yes—
160C2×C8×C59#4112yes—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—81no—
167C4⋊C4×C59C22×C412yes—
168C22⋊C4×C59—81no—
169C22⋊C4×C59—81no—
170C42×C59—21no—
171C42×C59—21no—
172C22⋊Dic59—24no—
173C22⋊Dic59—81no—
174C22⋊Dic59—81no—
175#11745670—28no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—81no—
182C4⋊Dic59C22×C412yes—
183Dic59⋊C4—24no—
184Dic59⋊C4—81no—
185Dic59⋊C4—81no—
186C4×Dic59—21no—
187C4×Dic59—21no—
188C4×Dic59—24no—
189C4×Dic59—81no—
190C4×Dic59—81no—
191C4×Dic59C22×C412yes—
192C4×Dic59#4112yes—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8C22×C412yes—
196C2×C59⋊C8#4112yes—
197C2×C59⋊C8#4112yes—
198C2×C59⋊C8#4112yes—
199C23×C59—22no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59C22×D411yescharacteristic
207D4×C59—22no—
208D4×C59—22no—
209D4×C59—41no—
210D4×C59—41no—
211C2×C4×C59—22no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59C23×C411yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—22no—
237C2×Dic59—22no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—84no—
255C2×Dic59C4×D412yes—
256C2×Dic59C4×D412yes—
257Dic118—22no—
258Dic118—22no—
259Dic118—41no—
260Dic118—41no—
261C8×C59—22no—
262C8×C59—22no—
263C59⋊C8—22no—
264C59⋊C8#11714yes—
265#21291—591nomaximal, Sylow
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—81no—
270C22×C59—81no—
271C22×C59—81no—
272C22×C59—81no—
273C22×C59C2×C4×D411yescharacteristic
274C22×C59#53211yescharacteristic
275C22×C59#53211yescharacteristic
276C22×C59#57611yescharacteristic
277C22×C59#57711yescharacteristic
278C22×C59#57711yescharacteristic
279C22×C59#57811yescharacteristic
280C4×C59—41no—
281C4×C59—41no—
282C4×C59—42no—
283C4×C59—81no—
284C4×C59—81no—
285C4×C59C2×#11711yescharacteristic
286C4×C59C4×#4111yescharacteristic
287C4×C59#52911yescharacteristic
288C4×C59#52911yescharacteristic
289Dic59—22no—
290Dic59—22no—
291Dic59—41no—
292Dic59—41no—
293Dic59—42no—
294Dic59—42no—
295Dic59—81no—
296Dic59—81no—
297C2×#440—591no—
298#2320—591no—
299#2319—591no—
300#1973—591no—
301#1968—591no—
302#1943—591no—
303#1942—591no—
304#1936—591no—
305#1935—591no—
306#1934D5911yescharacteristic
307#1933—591no—
308#1776—591no—
309C4×#103—591no—
310C4×#102—591no—
311#1726—591no—
312C2×C59—82no—
313C2×C59#229311yescharacteristic
314C2×C59#294611yescharacteristic
315C2×C59#294711yescharacteristic
316C2×C59#317411yescharacteristic
317C2×C59#317511yescharacteristic
318C2×C59#317811yescharacteristic
319C2×C59#317911yescharacteristic
320C22×SD16—591no—
321C2×C4×Q8—591no—
322C2×C4×D4—591no—
323#440—598no—
324C2×#107—591no—
325C2×#107—591no—
326C2×#106—591no—
327C2×#103—591no—
328C2×#103—591no—
329C2×#103—591no—
330C2×#102—591no—
331C2×#102C2×D5911yescharacteristic
332C2×#102C2×D5911yescharacteristic
333C2×#97—591no—
334C2×#97—591no—
335C2×#97—591no—
336C2×C4×C8—591no—
337C2×C4×C8—591no—
338C2×C4×C8C2×D5911yescharacteristic
339#391—591no—
340#391—591no—
341#390—591no—
342#390C2×D5911yescharacteristic
343#389—591no—
344#389—591no—
345#389—591no—
346#389C2×D5911yescharacteristic
347#388—1181no—
348#388—1181no—
349#387—1181no—
350#387—1181no—
351#383—1181no—
352#382—1181no—
353#381—1181no—
354#380—1181no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357#376—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340C2×D5911yescharacteristic
363#340C2×D5911yescharacteristic
364C59#2129111yescharacteristic, complemented, Sylow
365C22×Q8—591no—
366C22×D4C22×D5911yescharacteristic
367C23×C4—1181no—
368C2×SD16—594no—
369C2×SD16—1182no—
370C2×SD16—1182no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375C4×Q8—594no—
376C4×Q8—1182no—
377C4×D4—594no—
378C4×D4—1182no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4C22×D5911yescharacteristic
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C4⋊C4C22×D5911yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1181no—
393C2×C22⋊C4—1181no—
394C2×C22⋊C4—1181no—
395C2×C42—591no—
396C2×C42—591no—
397C2×C42—1181no—
398C2×C42—1181no—
399C2×C42—1181no—
400C2×C42C22×D5911yescharacteristic
401#107—594no—
402#107—1182no—
403#106—1182no—
404#103—594no—
405#103—1182no—
406#103—1182no—
407#102—22no—
408#102—1182no—
409#102C4×D5914yes—
410#97—594no—
411#97—1182no—
412#97—1182no—
413C4×C8—22no—
414C4×C8—1182no—
415C4×C8—1182no—
416#95—21no—
417#95—591no—
418#95—591no—
419#95—591no—
420#95—591no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430C24—21no—
431C2×Q8—592no—
432C2×Q8—1181no—
433C2×Q8—1181no—
434C2×Q8—1181no—
435C2×Q8—1181no—
436C2×Q8—1181no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D5912yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—591no—
447C22×C4—591no—
448C22×C4—591no—
449C22×C4—591no—
450C22×C4—1184no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—4721no—
456C22×C4—4721no—
457C22×C4D4×D5911yescharacteristic
458C22×C4D4×D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460SD16—1188no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—592no—
466C2×C8—592no—
467C2×C8—592no—
468C2×C8—592no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8C2×C4×D5912yes—
472C2×C8#1174568712yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—81no—
479C4⋊C4—592no—
480C4⋊C4—1184no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—4721no—
487C4⋊C4—4721no—
488C4⋊C4—4721no—
489C4⋊C4C2×C4×D5912yes—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—1184no—
493C22⋊C4—4721no—
494C22⋊C4—4721no—
495C42—21no—
496C42—21no—
497C42—592no—
498C42—592no—
499C42—1184no—
500C42—1181no—
501C42—1181no—
502C42—4721no—
503C42—4721no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D5911yescharacteristic
512Q8—1182no—
513Q8—1182no—
514Q8—2361no—
515Q8—2361no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—592no—
536C2×C4—592no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1182no—
548C2×C4—1182no—
549C2×C4—4724no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4C22×C4×D5911yescharacteristic
561C2×C4#41×D5911yescharacteristic
562C2×C4#41×D5911yescharacteristic
563C2×C4#41×D5911yescharacteristic
564C2×C4#41×D5911yescharacteristic
565C2×C4C2×#1174568711yescharacteristic
566C8—22no—
567C8—22no—
568C8—594no—
569C8—1182no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×D4×D5911yescharacteristic
578C22#1683794711yescharacteristic
579C22#1683795111yescharacteristic
580C22#1683820311yescharacteristic
581C22#1683820811yescharacteristic
582C22#1683820911yescharacteristic
583C22#1683821611yescharacteristic
584C4—42no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—1182no—
590C4—1182no—
591C4—4721no—
592C4—4721no—
593C4—2362no—
594C4—2362no—
595C4—2361no—
596C4—2361no—
597C4#117×D5911yescharacteristic
598C4C4×#1174568711yescharacteristic
599C4#1683792911yescharacteristic
600C4#1683793511yescharacteristic
601C2—82no—
602C2#2142655611yescharacteristic
603C2#2143292111yescharacteristic
604C2#2143293211yescharacteristic
605C2#2143471711yescharacteristic
606C2#2143472811yescharacteristic
607C2#2143476311yescharacteristic
608C2#2143478611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1934C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1934D59
hypocenterC59#21291
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. #21426743C2semidirect
6. #21424831C2semidirect
7. #21424828C2semidirect
9. #21424784C2semidirect
10. #21424772C2semidirect
11. #21423992C2semidirect
12. #21423915C2semidirect
14. #21423782C2semidirect
19. #16837499C4semidirect
364. C59#21291semidirect

Maximal subgroups

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

subgroupindex
C2×#168374992
#214267432
#214267332
#214249762
#214249512
#214248312
#214248282
#214247952
#214247842
#214247722
#214239922
#214239152
#102×Dic592
#214237822
C59×#19342
#2129159