Finite Group Databasebeta

← #28610125

Subgroups — #28610125

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×#16837740C211yesmaximal, characteristic
2#21426987C211yesmaximal, characteristic
3#21424831C211yesmaximal, characteristic, complemented
4#21424789C211yesmaximal, characteristic, complemented
5#21424727C211yesmaximal, characteristic, complemented
6#21424688C211yesmaximal, characteristic, complemented
7#21424678C211yesmaximal, characteristic, complemented
8#21424520C211yesmaximal, characteristic
9#21424350C211yesmaximal, characteristic, complemented
10#21424295C211yesmaximal, characteristic
11#21423994C211yesmaximal, characteristic, complemented
12#21423926C211yesmaximal, characteristic, complemented
13#21423874C211yesmaximal, characteristic
14#21423812C211yesmaximal, characteristic
15C59×#2362C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18#16837740C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837296—21no—
33#16837296—21no—
34#16837269—21no—
35#16837262C2211yescharacteristic
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837253—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C22⋊Q8×C59C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C2×#117×C59C2211yescharacteristic
52C59×#391C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#388—21no—
55C59×#388—21no—
56C59×#387—21no—
57C4×C4⋊C4×C59—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59C2211yescharacteristic
60C22×C59⋊C8—22no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273686—22no—
68#14273686C2×C414yes—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—22no—
81#14273660C2×C414yes—
82#14273658—41no—
83#14273658C2×C414yes—
84#14273651—22no—
85#14273651—41no—
86#14273651C2×C414yes—
87#14273651C2×C414yes—
88C22⋊(C59⋊C8)C2×C414yes—
89#14273638—22no—
90#14273638—41no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×Q8×C59C2311yescharacteristic
97C23×C4×C59C2311yescharacteristic
98C22⋊Q8×C59—22no—
99C22⋊Q8×C59C2×C414yes—
100C4×Q8×C59—22no—
101C4×Q8×C59—41no—
102#117×C59—22no—
103#117×C59—41no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59D411yescharacteristic
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C42×C59—21no—
115C2×C42×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59C22×C412yes—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59#4112yes—
147C22×C4×C59C2411yescharacteristic
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C4⋊C4×C59—24no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—42no—
158C4⋊C4×C59—42no—
159C4⋊C4×C59C22×C412yes—
160C4⋊C4×C59#4112yes—
161C22⋊C4×C59—24no—
162C22⋊C4×C59—41no—
163C22⋊C4×C59—41no—
164C22⋊C4×C59—42no—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—42no—
170C42×C59—42no—
171C42×C59—42no—
172C22⋊Dic59—24no—
173C22⋊Dic59—81no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59C22×C412yes—
181C4⋊Dic59C22×C412yes—
182Dic59⋊C4—42no—
183Dic59⋊C4—42no—
184C4×Dic59—24no—
185C4×Dic59—81no—
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⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—44no—
197C2×C59⋊C8C22×C412yes—
198C2×C59⋊C8C22×C412yes—
199C2×C59⋊C8#4112yes—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59C4×D412yes—
207C23×C59C22×D411yescharacteristic
208Q8×C59—22no—
209Q8×C59—22no—
210Q8×C59—41no—
211Q8×C59—41no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—22no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59C4×D412yes—
243C2×C4×C59C23×C411yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59H(5,𝔽2)11yescharacteristic
248C2×C4×C592−1+411yescharacteristic
249C2×Dic59—22no—
250C2×Dic59—22no—
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×Dic59—41no—
261C59⋊C8—22no—
262C59⋊C8—22no—
263C59⋊C8—22no—
264C59⋊C8—41no—
265#22937—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—42no—
284C4×C59—42no—
285C4×C59—41no—
286C4×C59—42no—
287C4×C59—41no—
288C4×C59—81no—
289C4×C59C4×#4111yescharacteristic
290C4×C59#52011yescharacteristic
291C4×C59#55111yescharacteristic
292C4×C59#55211yescharacteristic
293Dic59—42no—
294Dic59—42no—
295Dic59—81no—
296Dic59—81no—
297C2×#485—591no—
298#2362D5911yescharacteristic
299#2347—591no—
300#1943—591no—
301#1935—591no—
302#1927—591no—
303#1921—591no—
304#1920—591no—
305#1892—591no—
306#1863—591no—
307#1854—591no—
308#1776—591no—
309#1763—591no—
310#1752—591no—
311#1735—591no—
312C2×C59—42no—
313C2×C59#235911yescharacteristic
314C2×C59#294311yescharacteristic
315C2×C59#294711yescharacteristic
316C2×C59#330711yescharacteristic
317C2×C59#331311yescharacteristic
318C2×C59#334911yescharacteristic
319C2×C59#335011yescharacteristic
320C23×C8—591no—
321C2×C22⋊Q8C2×D5911yescharacteristic
322C2×C4×Q8C2×D5911yescharacteristic
323C2×#117—591no—
324C2×#117C2×D5911yescharacteristic
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⋊C8—591no—
335C2×#97—591no—
336C2×#97—591no—
337#391C2×D5911yescharacteristic
338#389—591no—
339#389—591no—
340#389C2×D5911yescharacteristic
341#388—21no—
342#388—21no—
343#388—1181no—
344#388—1181no—
345#387—21no—
346#383—1181no—
347#380—591no—
348C4×C4⋊C4—21no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—1181no—
352C4×C4⋊C4C2×D5911yescharacteristic
353C4×C22⋊C4C2×D5911yescharacteristic
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#336—591no—
363#336—591no—
364C59#2293711yescharacteristic, complemented, Sylow
365C22×Q8C22×D5911yescharacteristic
366C23×C4C22×D5911yescharacteristic
367C22×C8—591no—
368C22×C8—591no—
369C22×C8—591no—
370C22×C8—591no—
371C22×C8—1182no—
372C22×C8—1182no—
373C22×C8—2361no—
374C22⋊Q8—22no—
375C22⋊Q8C4×D5914yes—
376C4×Q8—22no—
377C4×Q8—41no—
378#117—22no—
379#117—41no—
380#117—594no—
381#117—1182no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
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⋊C4C22×D5911yescharacteristic
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1181no—
397C2×C22⋊C4—1181no—
398C2×C22⋊C4C22⋊D5911yescharacteristic
399C2×C22⋊C4C22⋊D5911yescharacteristic
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—1181no—
404C2×C42—1181no—
405C2×C42—1181no—
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408#107—594no—
409#107—1182no—
410#106—594no—
411#106—2361no—
412#103—594no—
413#103—594no—
414#103—1182no—
415#103—2361no—
416C22⋊C8—594no—
417#97—1182no—
418#97—2361no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95C22⋊D5911yescharacteristic
429#95C22⋊D5911yescharacteristic
430C24C2×C22⋊D5911yescharacteristic
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×C4×D5912yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4C2×C4×D5912yes—
450C22×C4#1174568512yes—
451C22×C4C2×C22⋊D5911yescharacteristic
452C22×C4C2×C22⋊D5911yescharacteristic
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C2×C22⋊D5911yescharacteristic
456C22×C4C23×D5911yescharacteristic
457C2×C8—592no—
458C2×C8—592no—
459C2×C8—592no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—2364no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—592no—
478C4⋊C4—592no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—2362no—
486C4⋊C4—2362no—
487C4⋊C4C2×C4×D5912yes—
488C4⋊C4#1174568712yes—
489C22⋊C4—24no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—42no—
493C22⋊C4—1184no—
494C22⋊C4—4721no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—1184no—
503C42—4721no—
504C42—4721no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C4×C22⋊D5912yes—
512C23C22×C22⋊D5911yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
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—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—1182no—
548C2×C4—1182no—
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×C22⋊D5912yes—
560C2×C4C22×C4×D5911yescharacteristic
561C2×C4#41×D5911yescharacteristic
562C2×C4C2×#1174568711yescharacteristic
563C2×C4C2×#1174568511yescharacteristic
564C2×C4#1427377211yescharacteristic
565C2×C4#1427377411yescharacteristic
566C8—1182no—
567C8—1182no—
568C8—1182no—
569C8—2361no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×C4×C22⋊D5911yescharacteristic
578C22#1683804611yescharacteristic
579C22#1683805111yescharacteristic
580C22#1683820011yescharacteristic
581C22#1683820011yescharacteristic
582C22#1683821011yescharacteristic
583C22#1683821711yescharacteristic
584C4—22no—
585C4—41no—
586C4—42no—
587C4—42no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—41no—
592C4—81no—
593C4—4721no—
594C4—4721no—
595C4—2362no—
596C4—2362no—
597C4C4×#1174568711yescharacteristic
598C4#1683788111yescharacteristic
599C4#1683808111yescharacteristic
600C4#1683809211yescharacteristic
601C2—42no—
602C2#2142708211yescharacteristic
603C2#2143289411yescharacteristic
604C2#2143293911yescharacteristic
605C2#2143607211yescharacteristic
606C2#2143616211yescharacteristic
607C2#2143662611yescharacteristic
608C2#2143664111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2362C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#2362D59
hypocenterC59#22937
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
3. #21424831C2semidirect
4. #21424789C2semidirect
5. #21424727C2semidirect
6. #21424688C2semidirect
7. #21424678C2semidirect
9. #21424350C2semidirect
11. #21423994C2semidirect
12. #21423926C2semidirect
18. #16837740C4semidirect
364. C59#22937semidirect

Maximal subgroups

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

subgroupindex
C2×#168377402
#214269872
#214248312
#214247892
#214247272
#214246882
#214246782
#214245202
#214243502
#214242952
#214239942
#214239262
#214238742
#214238122
C59×#23622
#2293759