Finite Group Databasebeta

← #28593067

Subgroups — #28593067

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×#16837502C211yesmaximal, characteristic
2#21426748C211yesmaximal, characteristic, complemented
3#21424976C211yesmaximal, characteristic
4#21424949C211yesmaximal, characteristic
5#21424831C211yesmaximal, characteristic, complemented
6#21424826C211yesmaximal, characteristic, complemented
7#21424796C211yesmaximal, characteristic
8#21424785C211yesmaximal, characteristic, complemented
9#21424782C211yesmaximal, characteristic
10#21424771C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12C4×#14273650C211yesmaximal, characteristic, complemented
13C4×#14273648C211yesmaximal, characteristic
14#21423783C211yesmaximal, characteristic, complemented
15C59×#2319C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18#16837502C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273650C2211yescharacteristic
23C2×#14273650C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32C2×C4×C59⋊C8C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837308C2211yescharacteristic
35#16837308C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837284—21no—
39#16837284—21no—
40#16837269—21no—
41#16837266—21no—
42C4×C4⋊Dic59C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C59×#390C2211yescharacteristic
51C59×#390C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#388—21no—
55C59×#388—21no—
56C59×#381—21no—
57C59×#380—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C59×#376C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C2×#11745670—22no—
62C2×#11745670—22no—
63C2×#11745670C2×C414yes—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C4×Dic118—22no—
69C4×Dic118C2×C414yes—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59—21no—
78#14273660—22no—
79#14273660C2×C414yes—
80#14273658—22no—
81#14273650—22no—
82#14273650—22no—
83#14273650C2×C414yes—
84#14273648—22no—
85#14273648—22no—
86#14273648C2×C414yes—
87#14273638—22no—
88#14273638—22no—
89#14273638C2×C414yes—
90C4×C59⋊C8—22no—
91C4×C59⋊C8—22no—
92C4×C59⋊C8—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C22×D4×C59C2311yescharacteristic
101C23×C4×C59—21no—
102C4×D4×C59—22no—
103C4×D4×C59C2×C414yes—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C42×C59—21no—
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121#95×C59D411yescharacteristic
122#95×C59D411yescharacteristic
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
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—24no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—81no—
137C22×C4×C59—81no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2411yescharacteristic
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118C22×C412yes—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—81no—
161C4⋊C4×C59—81no—
162C4⋊C4×C59C22×C412yes—
163C22⋊C4×C59—24no—
164C22⋊C4×C59—81no—
165C22⋊C4×C59—81no—
166C22⋊C4×C59—81no—
167C22⋊C4×C59—81no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—81no—
173C42×C59C22×C412yes—
174C42×C59#4112yes—
175#11745670—28no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C412yes—
182Dic59⋊C4—24no—
183Dic59⋊C4—81no—
184Dic59⋊C4—81no—
185C4×Dic59—24no—
186C4×Dic59—81no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8C22×C412yes—
194C2×C59⋊C8C22×C412yes—
195C2×C59⋊C8#4112yes—
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—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—22no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
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—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—84no—
237C2×C4×C59C4×D412yes—
238C2×C4×C59C4×D412yes—
239C2×C4×C59C23×C411yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×Dic59—22no—
246C2×Dic59—22no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257Dic118—22no—
258Dic118—22no—
259Dic118—41no—
260Dic118—41no—
261C59⋊C8—22no—
262C59⋊C8—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—22no—
281C4×C59—22no—
282C4×C59—41no—
283C4×C59—41no—
284C4×C59—41no—
285C4×C59—41no—
286C4×C59—42no—
287C4×C59—81no—
288C4×C59—81no—
289C4×C59C2×#11711yescharacteristic
290C4×C59C4×#4111yescharacteristic
291C4×C59#52911yescharacteristic
292C4×C59#52911yescharacteristic
293Dic59—42no—
294Dic59—42no—
295Dic59—81no—
296Dic59—81no—
297C2×#440—591no—
298#2320—591no—
299#2319D5911yescharacteristic
300#1973—591no—
301#1968—591no—
302#1943—591no—
303#1942—591no—
304#1936—591no—
305#1935—591no—
306#1934—591no—
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×D4C2×D5911yescharacteristic
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×#102—591no—
332C2×#102—591no—
333C2×#97—591no—
334C2×#97—591no—
335C2×#97—591no—
336C2×C4×C8—591no—
337C2×C4×C8—591no—
338C2×C4×C8—591no—
339#391—591no—
340#391—591no—
341#390C2×D5911yescharacteristic
342#390C2×D5911yescharacteristic
343#389—591no—
344#389—591no—
345#389C2×D5911yescharacteristic
346#389C2×D5911yescharacteristic
347#388—21no—
348#388—21no—
349#387—1181no—
350#387—1181no—
351#383—1181no—
352#382—1181no—
353#381—21no—
354#380—21no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4C2×D5911yescharacteristic
357#376C2×D5911yescharacteristic
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364C59#2129111yescharacteristic, complemented, Sylow
365C22×Q8—591no—
366C22×D4C22×D5911yescharacteristic
367C23×C4—21no—
368C2×SD16—594no—
369C2×SD16—1182no—
370C2×SD16—1182no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8—591no—
374C22×C8—591no—
375C4×Q8—594no—
376C4×Q8—1182no—
377C4×D4—22no—
378C4×D4C4×D5914yes—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
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—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C42—21no—
396C2×C42—1181no—
397C2×C42—1181no—
398C2×C42C22×D5911yescharacteristic
399C2×C42C22×D5911yescharacteristic
400C2×C42C22×D5911yescharacteristic
401#107—594no—
402#107—1182no—
403#106—1182no—
404#103—594no—
405#103—1182no—
406#103—1182no—
407#102—594no—
408#102—1182no—
409#102—1182no—
410#97—594no—
411#97—1182no—
412#97—1182no—
413C4×C8—1182no—
414C4×C8—1182no—
415C4×C8—1182no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95C22⋊D5911yescharacteristic
427#95C22⋊D5911yescharacteristic
428#95C22⋊D5911yescharacteristic
429#95C22⋊D5911yescharacteristic
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—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C2×C22⋊D5911yescharacteristic
456C22×C4C2×C22⋊D5911yescharacteristic
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460SD16—1188no—
461C2×C8—592no—
462C2×C8—592no—
463C2×C8—592no—
464C2×C8—592no—
465C2×C8—592no—
466C2×C8—592no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—592no—
481C4⋊C4—1184no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—4721no—
488C4⋊C4—4721no—
489C4⋊C4C2×C4×D5912yes—
490C22⋊C4—24no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—81no—
500C42—1184no—
501C42—4721no—
502C42C2×C4×D5912yes—
503C42#1174568512yes—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C22×C22⋊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—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—1182no—
547C2×C4—1182no—
548C2×C4—2361no—
549C2×C4—2361no—
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×C4C4×C22⋊D5912yes—
559C2×C4C4×C22⋊D5912yes—
560C2×C4C22×C4×D5911yescharacteristic
561C2×C4#41×D5911yescharacteristic
562C2×C4#41×D5911yescharacteristic
563C2×C4#41×D5911yescharacteristic
564C2×C4#41×D5911yescharacteristic
565C2×C4C2×#1174568511yescharacteristic
566C8—594no—
567C8—1182no—
568C8—1182no—
569C8—1182no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C2×C4×C22⋊D5911yescharacteristic
578C22#1683794811yescharacteristic
579C22#1683794811yescharacteristic
580C22#1683820511yescharacteristic
581C22#1683821311yescharacteristic
582C22#1683821311yescharacteristic
583C22#1683821711yescharacteristic
584C4—22no—
585C4—22no—
586C4—41no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—81no—
592C4—81no—
593C4—4721no—
594C4—4721no—
595C4—2362no—
596C4—2362no—
597C4#117×D5911yescharacteristic
598C4#1683787011yescharacteristic
599C4#1683793011yescharacteristic
600C4#1683793011yescharacteristic
601C2—82no—
602C2#2142655511yescharacteristic
603C2#2143292811yescharacteristic
604C2#2143293511yescharacteristic
605C2#2143471811yescharacteristic
606C2#2143472911yescharacteristic
607C2#2143475811yescharacteristic
608C2#2143478111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2319C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#2319D59
hypocenterC59#21291
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #21426748C2semidirect
5. #21424831C2semidirect
6. #21424826C2semidirect
8. #21424785C2semidirect
10. #21424771C2semidirect
11. #21423994C2semidirect
12. C4×#14273650C2semidirect
14. #21423783C2semidirect
18. #16837502C4semidirect
364. C59#21291semidirect

Maximal subgroups

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

subgroupindex
C2×#168375022
#214267482
#214249762
#214249492
#214248312
#214248262
#214247962
#214247852
#214247822
#214247712
#214239942
C4×#142736502
C4×#142736482
#214237832
C59×#23192
#2129159