Finite Group Databasebeta

← #28610163

Subgroups — #28610163

Order 15104.

802 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×#16837744C211yesmaximal, characteristic
2#21427121C211yesmaximal, characteristic
3#21426979C211yesmaximal, characteristic
4#21424839C211yesmaximal, characteristic, complemented
5#21424786C211yesmaximal, characteristic, complemented
6#21424738C211yesmaximal, characteristic, complemented
7#21424689C211yesmaximal, characteristic, complemented
8#21424675C211yesmaximal, characteristic, complemented
9#21424519C211yesmaximal, characteristic
10#21424367C211yesmaximal, characteristic, complemented
11#21424297C211yesmaximal, characteristic
12#21423993C211yesmaximal, characteristic, complemented
13#21423927C211yesmaximal, characteristic, complemented
14#21423813C211yesmaximal, characteristic
15C59×#1752C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837744C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273661C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28C2×#14273652C2211yescharacteristic
29#16837308C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837296—21no—
34#16837290—21no—
35#16837290—21no—
36#16837290—21no—
37#16837284—21no—
38#16837266—21no—
39#16837262C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43#16837253—21no—
44#16837247—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C23×C8×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C4×C22⋊C4×C59C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61#14273706—22no—
62#14273706C2×C414yes—
63C4×Dic118—22no—
64C4×Dic118—41no—
65#14273686—22no—
66#14273686—22no—
67#14273686—41no—
68#14273686C2×C414yes—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91#14273661—41no—
92#14273661C2×C414yes—
93#14273659—22no—
94#14273659C2×C414yes—
95#14273652—22no—
96#14273652—41no—
97#14273652C2×C414yes—
98#14273652C2×C414yes—
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⋊#95—21no—
108C23×C4×C59C2311yescharacteristic
109C22×C8×C59—22no—
110C22×C8×C59—22no—
111C22×C8×C59—41no—
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C22⋊C8×C59C2×C414yes—
121#97×C59—22no—
122#97×C59—41no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59C22×C412yes—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59#4112yes—
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C2×Dic118—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118C22×C412yes—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—44no—
162C2×C8×C59C22×C412yes—
163C2×C8×C59C22×C412yes—
164C2×C8×C59#4112yes—
165C22⋊C4×C59—41no—
166C22⋊C4×C59—41no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59—21no—
171C22⋊Dic59—24no—
172C22⋊Dic59—24no—
173C22⋊Dic59—42no—
174C22⋊Dic59—81no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59C22×C412yes—
186C4⋊Dic59C22×C412yes—
187C4⋊Dic59C22×C412yes—
188C4⋊Dic59#4112yes—
189Dic59⋊C4—24no—
190Dic59⋊C4—42no—
191Dic59⋊C4—42no—
192Dic59⋊C4—81no—
193Dic59⋊C4—81no—
194C4×Dic59—24no—
195C4×Dic59—42no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C4×Dic59—42no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59C4×D412yes—
207C23×C59C22×D411yescharacteristic
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
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×C59C4×D412yes—
227C2×C4×C59C23×C411yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59H(5,𝔽2)11yescharacteristic
232C2×C4×C592−1+411yescharacteristic
233C2×Dic59—22no—
234C2×Dic59—22no—
235C2×Dic59—22no—
236C2×Dic59—22no—
237C2×Dic59—22no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
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—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257Dic118—22no—
258Dic118—22no—
259Dic118—41no—
260Dic118—41no—
261C8×C59—22no—
262C8×C59—22no—
263C8×C59—22no—
264C8×C59—41no—
265#22939—591nomaximal, Sylow
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—22no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59—41no—
273C22×C59C2×C4×D411yescharacteristic
274C22×C59#54611yescharacteristic
275C22×C59#54711yescharacteristic
276C22×C59#57511yescharacteristic
277C22×C59#57511yescharacteristic
278C22×C59#57711yescharacteristic
279C22×C59#57811yescharacteristic
280C4×C59—22no—
281C4×C59—41no—
282C4×C59—41no—
283C4×C59—41no—
284C4×C59—41no—
285C4×C59C4×#4111yescharacteristic
286C4×C59#52011yescharacteristic
287C4×C59#55111yescharacteristic
288C4×C59#55211yescharacteristic
289Dic59—42no—
290Dic59—42no—
291Dic59—42no—
292Dic59—42no—
293Dic59—42no—
294Dic59—81no—
295Dic59—81no—
296Dic59—81no—
297C2×#485—591no—
298#2362—591no—
299#2346—591no—
300#1944—591no—
301#1935—591no—
302#1928—591no—
303#1921—591no—
304#1919—591no—
305#1892—591no—
306#1865—591no—
307#1854—591no—
308#1776—591no—
309#1763—591no—
310#1752D5911yescharacteristic
311#1735—591no—
312C2×C59—42no—
313C2×C59#235911yescharacteristic
314C2×C59#294311yescharacteristic
315C2×C59#294711yescharacteristic
316C2×C59#331511yescharacteristic
317C2×C59#332111yescharacteristic
318C2×C59#334811yescharacteristic
319C2×C59#334911yescharacteristic
320C23×C8C2×D5911yescharacteristic
321C2×C22⋊Q8—591no—
322C2×C4×Q8—591no—
323C2×#117—591no—
324C2×#117—591no—
325#485—598no—
326C2×#107—591no—
327C2×#107—591no—
328C2×#106—591no—
329C2×#106—591no—
330C2×#103—591no—
331C2×#103—591no—
332C2×#103—591no—
333C2×#103—591no—
334C2×C22⋊C8C2×D5911yescharacteristic
335C2×#97C2×D5911yescharacteristic
336C2×#97C2×D5911yescharacteristic
337#391—591no—
338#389—591no—
339#389—591no—
340#389—591no—
341#388—1181no—
342#388—1181no—
343#388—1181no—
344#388—1181no—
345#387—1181no—
346#382—1181no—
347#380—591no—
348C4×C4⋊C4—591no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—1181no—
352C4×C22⋊C4C2×D5911yescharacteristic
353#376—1181no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#336C2×D5911yescharacteristic
363#336C2×D5911yescharacteristic
364C59#2293911yescharacteristic, complemented, Sylow
365C22×Q8—591no—
366C23×C4C22×D5911yescharacteristic
367C22×C8—22no—
368C22×C8—22no—
369C22×C8—41no—
370C22×C8C22×D5911yescharacteristic
371C22×C8C22×D5911yescharacteristic
372C22×C8C22×D5911yescharacteristic
373C22×C8C22×D5911yescharacteristic
374C22⋊Q8—594no—
375C22⋊Q8—1182no—
376C4×Q8—1182no—
377C4×Q8—2361no—
378#117—594no—
379#117—1182no—
380#117—1182no—
381#117—2361no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C22⋊C4—1181no—
395C2×C22⋊C4—1181no—
396C2×C22⋊C4—1181no—
397C2×C22⋊C4—1181no—
398C2×C22⋊C4D23611yescharacteristic
399C2×C22⋊C4D23611yescharacteristic
400C2×C42—1181no—
401C2×C42—1181no—
402C2×C42—1181no—
403C2×C42—1181no—
404C2×C42—1181no—
405C2×C42—1181no—
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408#107—594no—
409#107—2361no—
410#106—594no—
411#106—1182no—
412#103—594no—
413#103—594no—
414#103—1182no—
415#103—2361no—
416C22⋊C8C4×D5914yes—
417#97—22no—
418#97—41no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95D23611yescharacteristic
429#95D23611yescharacteristic
430C24C2×D23611yescharacteristic
431C2×Q8—592no—
432C2×Q8—1181no—
433C2×Q8—1181no—
434C2×Q8—1181no—
435C2×Q8—1181no—
436C2×Q8—1181no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—1181no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4C2×C4×D5912yes—
450C22×C4C2×D23611yescharacteristic
451C22×C4C2×D23611yescharacteristic
452C22×C4C2×D23611yescharacteristic
453C22×C4C2×D23611yescharacteristic
454C22×C4C2×D23611yescharacteristic
455C22×C4#1174568512yes—
456C22×C4C23×D5911yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—44no—
467C2×C8C2×C4×D5912yes—
468C2×C8C2×C4×D5912yes—
469C2×C8#1174568712yes—
470C4⋊C4—592no—
471C4⋊C4—592no—
472C4⋊C4—592no—
473C4⋊C4—592no—
474C4⋊C4—1184no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—4721no—
486C4⋊C4—4721no—
487C4⋊C4—2362no—
488C4⋊C4—2362no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—1184no—
492C22⋊C4—1184no—
493C22⋊C4—4721no—
494C22⋊C4—2362no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—1184no—
500C42—2362no—
501C42—2362no—
502C42—2362no—
503C42—2362no—
504C42—2362no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C4×D23612yes—
512C23C22×D23611yescharacteristic
513Q8—1182no—
514Q8—1182no—
515Q8—2361no—
516Q8—2361no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
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—1182no—
536C2×C4—1182no—
537C2×C4—1182no—
538C2×C4—1182no—
539C2×C4—1182no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
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×C4—2361no—
559C2×C4C4×D23612yes—
560C2×C4C22×C4×D5911yescharacteristic
561C2×C4#41×D5911yescharacteristic
562C2×C4C2×#1174568711yescharacteristic
563C2×C4C2×#1174568511yescharacteristic
564C2×C4#1427377211yescharacteristic
565C2×C4#1427377411yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8—41no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×C4×D23611yescharacteristic
578C22#1683804711yescharacteristic
579C22#1683805611yescharacteristic
580C22#1683820211yescharacteristic
581C22#1683820211yescharacteristic
582C22#1683821411yescharacteristic
583C22#1683821911yescharacteristic
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—4721no—
590C4—4721no—
591C4—4721no—
592C4—2362no—
593C4—2362no—
594C4—2362no—
595C4—2362no—
596C4—2362no—
597C4C4×#1174568711yescharacteristic
598C4#1683788111yescharacteristic
599C4#1683807811yescharacteristic
600C4#1683808511yescharacteristic
601C2—42no—
602C2#2142709011yescharacteristic
603C2#2143289811yescharacteristic
604C2#2143294111yescharacteristic
605C2#2143619411yescharacteristic
606C2#2143628411yescharacteristic
607C2#2143661311yescharacteristic
608C2#2143662811yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1752C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1752D59
hypocenterC59#22939
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
4. #21424839C2semidirect
5. #21424786C2semidirect
6. #21424738C2semidirect
7. #21424689C2semidirect
8. #21424675C2semidirect
10. #21424367C2semidirect
12. #21423993C2semidirect
13. #21423927C2semidirect
20. #16837744C4semidirect
364. C59#22939semidirect

Maximal subgroups

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

subgroupindex
C2×#168377442
#214271212
#214269792
#214248392
#214247862
#214247382
#214246892
#214246752
#214245192
#214243672
#214242972
#214239932
#214239272
#214238132
C59×#17522
#2293959