Finite Group Databasebeta

← #28593034

Subgroups — #28593034

Order 15104.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837502C211yesmaximal, characteristic
2#21426570C211yesmaximal, characteristic, complemented
3#21426561C211yesmaximal, characteristic
4#21424978C211yesmaximal, characteristic
5#21424951C211yesmaximal, characteristic
6#21424833C211yesmaximal, characteristic, complemented
7#21424828C211yesmaximal, characteristic, complemented
8#21424790C211yesmaximal, characteristic
9#21424787C211yesmaximal, characteristic, complemented
10#21424776C211yesmaximal, characteristic
11#21424773C211yesmaximal, characteristic, complemented
12#21423994C211yesmaximal, characteristic, complemented
13#21423927C211yesmaximal, characteristic, complemented
14#97×Dic59C211yesmaximal, characteristic, complemented
15C59×#1762C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273686—21no—
19#16837502C418yescomplemented
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#16837307C2211yescharacteristic
33#16837307C2211yescharacteristic
34#16837305C2211yescharacteristic
35#16837305C2211yescharacteristic
36#16837301C2211yescharacteristic
37#16837301C2211yescharacteristic
38#16837298C2211yescharacteristic
39#16837298C2211yescharacteristic
40#16837295—21no—
41#16837295—21no—
42#16837284—21no—
43#16837284—21no—
44C4×Dic59⋊C4—21no—
45C4×C4⋊Dic59C2211yescharacteristic
46C4×C22⋊Dic59—21no—
47#16837248—21no—
48C42×Dic59C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C2×#11745670—22no—
62C2×#11745670—22no—
63C2×#11745670C2×C414yes—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C4×Dic118—22no—
67C4×Dic118C2×C414yes—
68#14273686—44no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59D411yescharacteristic
84C2×C4×Dic59D411yescharacteristic
85C2×C4×Dic59D411yescharacteristic
86C2×C4×Dic59D411yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88C2×C4×Dic59C2311yescharacteristic
89#14273660—22no—
90#14273660C2×C414yes—
91#14273658—22no—
92#14273652—22no—
93#14273652—22no—
94#14273650C2×C414yes—
95#14273648C2×C414yes—
96#14273638C2×C414yes—
97#14273637—22no—
98#14273637—22no—
99C8×Dic59—22no—
100C8×Dic59—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C22×D4×C59C2311yescharacteristic
111C23×C4×C59—21no—
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C4×D4×C59—22no—
115C4×D4×C59C2×C414yes—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C42×C59—21no—
121C2×C42×C59C2311yescharacteristic
122#102×C59—22no—
123#102×C59—22no—
124#97×C59—22no—
125#95×C59—21no—
126C24×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59C22×C412yes—
133C22×C4×C59—21no—
134C22×C4×C59—24no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2411yescharacteristic
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118C22×C412yes—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59#4112yes—
161C2×C8×C59#4112yes—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59C22×C412yes—
168C22⋊C4×C59—24no—
169C42×C59—42no—
170C42×C59C22×C412yes—
171C42×C59#4112yes—
172C22⋊Dic59—42no—
173C22⋊Dic59—42no—
174C22⋊Dic59—42no—
175C22⋊Dic59—42no—
176#11745670—28no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—42no—
183C4⋊Dic59—42no—
184C4⋊Dic59C22×C412yes—
185Dic59⋊C4—24no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188C4×Dic59—24no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—22no—
194C4×Dic59—22no—
195C4×Dic59—41no—
196C4×Dic59—41no—
197C4×Dic59—42no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8C22×C412yes—
201C2×C59⋊C8C22×C412yes—
202C2×C59⋊C8#4112yes—
203C2×C59⋊C8#4112yes—
204C23×C59—22no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59C22×D411yescharacteristic
212D4×C59—22no—
213D4×C59—22no—
214D4×C59—41no—
215D4×C59—41no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—22no—
221C2×C4×C59—48no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59C4×D412yes—
231C2×C4×C59C4×D412yes—
232C2×C4×C59C23×C411yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59AGL(1,ℤ/8)12yes—
261C2×Dic59#13712yes—
262Dic118—22no—
263Dic118—22no—
264Dic118—41no—
265Dic118—41no—
266C8×C59—22no—
267C8×C59—22no—
268C59⋊C8—22no—
269C59⋊C8#11714yescomplemented
270#21288—591nomaximal, Sylow
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59—81no—
277C22×C59—81no—
278C22×C59C2×C4×D411yescharacteristic
279C22×C59#53211yescharacteristic
280C22×C59#53211yescharacteristic
281C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C59C2×#13711yescharacteristic
283C22×C59#57511yescharacteristic
284C22×C59#57511yescharacteristic
285C4×C59—22no—
286C4×C59—22no—
287C4×C59—42no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59C2×#11711yescharacteristic
291C4×C59C4×#4111yescharacteristic
292C4×C59#52911yescharacteristic
293C4×C59#52911yescharacteristic
294Dic59—22no—
295Dic59—22no—
296Dic59—41no—
297Dic59—41no—
298Dic59—42no—
299Dic59—42no—
300Dic59—81no—
301Dic59—81no—
302C2×#440—591no—
303#2295—591no—
304#2293—591no—
305#1973—591no—
306#1968—591no—
307#1943—591no—
308#1942—591no—
309#1936—591no—
310#1935—591no—
311#1934—591no—
312#1933—591no—
313#1776—591no—
314#1763—591no—
315#1762D5911yescharacteristic
316C4×#97—591no—
317C2×C59—82no—
318C2×C59#229311yescharacteristic
319C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
320C2×C59C4×#13711yescharacteristic
321C2×C59#318011yescharacteristic
322C2×C59#318111yescharacteristic
323C2×C59#318211yescharacteristic
324C2×C59#318311yescharacteristic
325C22×SD16—591no—
326C2×C4×Q8—591no—
327C2×C4×D4C2×D5911yescharacteristic
328C2×#117—1181no—
329#440—598no—
330C2×#107—591no—
331C2×#107—591no—
332C2×#106—591no—
333C2×#103—591no—
334C2×#103—591no—
335C2×#103—591no—
336C2×#102—591no—
337C2×#102C2×D5911yescharacteristic
338C2×#102C2×D5911yescharacteristic
339C2×#97—591no—
340C2×#97—591no—
341C2×#97—591no—
342C2×#97C2×D5911yescharacteristic
343C2×C4×C8—591no—
344C2×C4×C8—591no—
345#391—591no—
346#391—591no—
347#390—591no—
348#390—591no—
349#389—591no—
350#389—591no—
351#389—591no—
352#389—591no—
353#388—1181no—
354#388—1181no—
355#387—1181no—
356#387—1181no—
357C4×C4⋊C4—591no—
358C4×C4⋊C4—1181no—
359C4×C4⋊C4C2×D5911yescharacteristic
360C4×C22⋊C4—1181no—
361#376—1181no—
362C43—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340C2×D5911yescharacteristic
368#340C2×D5911yescharacteristic
369C59#2128811yescharacteristic, complemented, Sylow
370C22×Q8—591no—
371C22×D4C22×D5911yescharacteristic
372C23×C4—21no—
373C2×SD16—594no—
374C2×SD16—1182no—
375C2×SD16—1182no—
376C22×C8—591no—
377C22×C8—591no—
378C22×C8C22×D5911yescharacteristic
379C22×C8C22×D5911yescharacteristic
380C4×Q8—594no—
381C4×Q8—1182no—
382C4×D4—22no—
383C4×D4C4×D5914yes—
384#117—2364no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C42—21no—
402C2×C42—591no—
403C2×C42—591no—
404C2×C42—591no—
405C2×C42—591no—
406C2×C42—591no—
407C2×C42—591no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42C22×D5911yescharacteristic
411#107—594no—
412#107—1182no—
413#106—1182no—
414#103—594no—
415#103—1182no—
416#103—1182no—
417#102—22no—
418#102—22no—
419#102—594no—
420#97—22no—
421#97—594no—
422#97—1182no—
423#97—1182no—
424C4×C8—1182no—
425C4×C8—1182no—
426#95—21no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436C24—21no—
437C2×Q8—592no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C2×Q8—1181no—
442C2×Q8—1181no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D5912yes—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—591no—
458C22×C4—591no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4D4×D5911yescharacteristic
464C22×C4D4×D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466SD16—1188no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—592no—
472C2×C8—592no—
473C2×C8—592no—
474C2×C8—592no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8#1174568512yes—
478C2×C8#1174568512yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—592no—
485C4⋊C4—1184no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—2362no—
492C4⋊C4—2362no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4C2×C4×D5912yes—
496C22⋊C4—24no—
497C22⋊C4—2362no—
498C22⋊C4—2362no—
499C22⋊C4—2362no—
500C22⋊C4—2362no—
501C42—42no—
502C42—1182no—
503C42—1182no—
504C42—1184no—
505C42—1181no—
506C42—1181no—
507C42—1181no—
508C42—1181no—
509C42—2361no—
510C42—2361no—
511C42—2362no—
512C42C2×C4×D5912yes—
513C42#1174568712yes—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D5911yescharacteristic
522Q8—1182no—
523Q8—1182no—
524Q8—2361no—
525Q8—2361no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—592no—
545C2×C4—592no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1182no—
557C2×C4—1182no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4(C22⋊D59)⋊C412yes—
569C2×C4(C22⋊D59)⋊C412yes—
570C2×C4C22×C4×D5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C2×C4#41×D5911yescharacteristic
573C2×C4C2×#1174568711yescharacteristic
574C2×C4C2×#1174568511yescharacteristic
575C2×C4C2×#1174568511yescharacteristic
576C8—22no—
577C8—22no—
578C8—594no—
579C8—1182no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×(C22⋊D59)⋊C411yescharacteristic
588C22#1683794911yescharacteristic
589C22#1683794911yescharacteristic
590C22AGL(1,ℤ/8)×D5911yescharacteristic
591C22#137×D5911yescharacteristic
592C22#1683819811yescharacteristic
593C22#1683819811yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—1182no—
600C4—1182no—
601C4—4721no—
602C4—4721no—
603C4—2362no—
604C4—2362no—
605C4—2361no—
606C4—2361no—
607C4#117×D5911yescharacteristic
608C4C4×#1174568511yescharacteristic
609C4#1683793211yescharacteristic
610C4#1683793211yescharacteristic
611C2—82no—
612C2#2142655911yescharacteristic
613C2#2143290011yescharacteristic
614C2#2143292011yescharacteristic
615C2#2143479211yescharacteristic
616C2#2143480111yescharacteristic
617C2#2143481111yescharacteristic
618C2#2143482311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1762C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1762D59
hypocenterC59#21288
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #21426570C2semidirect
6. #21424833C2semidirect
7. #21424828C2semidirect
9. #21424787C2semidirect
11. #21424773C2semidirect
12. #21423994C2semidirect
13. #21423927C2semidirect
14. #97×Dic59C2semidirect
19. #16837502C4semidirect
269. C59⋊C8#117semidirect
369. C59#21288semidirect

Maximal subgroups

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

subgroupindex
C2×#168375022
#214265702
#214265612
#214249782
#214249512
#214248332
#214248282
#214247902
#214247872
#214247762
#214247732
#214239942
#214239272
#97×Dic592
C59×#17622
#2128859