Finite Group Databasebeta

← #28599299

Subgroups — #28599299

Order 15104.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837462C211yesmaximal, characteristic
2#21426721C211yesmaximal, characteristic
3#21426701C211yesmaximal, characteristic
4#21424862C211yesmaximal, characteristic, complemented
5#21424859C211yesmaximal, characteristic, complemented
6#21424855C211yesmaximal, characteristic, complemented
7#21424298C211yesmaximal, characteristic
8#21424292C211yesmaximal, characteristic
9#21424287C211yesmaximal, characteristic
10#21424285C211yesmaximal, characteristic
11#21424261C211yesmaximal, characteristic, complemented
12#21424259C211yesmaximal, characteristic, complemented
13#21424228C211yesmaximal, characteristic, complemented
14#21424222C211yesmaximal, characteristic, complemented
15C59×#1949C211yesmaximal, characteristic, complemented
16C2×#14273685C2211yescharacteristic
17C2×#14273684C2211yescharacteristic
18C2×C4⋊C4×D59C2211yescharacteristic
19#16837462C418yescomplemented
20C2×#14273661C2211yescharacteristic
21C2×#14273661C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×(C2×D59)⋊C8C2211yescharacteristic
28C2×(C2×D59)⋊C8C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31#16837299—21no—
32#16837299—21no—
33#16837293—21no—
34#16837291—21no—
35#16837286—21no—
36#16837284—21no—
37#16837276—21no—
38#16837276—21no—
39#16837272C2211yescharacteristic
40#16837271C2211yescharacteristic
41#16837271C2211yescharacteristic
42#16837267C2211yescharacteristic
43#16837267C2211yescharacteristic
44#16837259C2211yescharacteristic
45#16837259C2211yescharacteristic
46C4×C4⋊Dic59C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×C4⋊C8×C59C2211yescharacteristic
56C59×#382C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×C4×D59—21no—
61C22×C4×D59C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273685—41no—
65#14273685—41no—
66#14273684C2×C414yes—
67#14273684Q814yes—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4D411yescharacteristic
80C2×Dic59⋊C4D411yescharacteristic
81C4⋊C4×D59—41no—
82C4⋊C4×D59—41no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4D411yescharacteristic
87C2×(C2×D59)⋊C4D411yescharacteristic
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59C2311yescharacteristic
91#14273661—41no—
92#14273661—41no—
93#14273660—41no—
94#14273658—41no—
95#14273658—41no—
96#14273658—41no—
97C4⋊(C59⋊C8)C2×C414yescomplemented
98(C2×D59)⋊C8C2×C414yes—
99(C2×D59)⋊C8C2×C414yes—
100#14273637C2×C414yescomplemented
101C8×Dic59C2×C414yescomplemented
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117#107×C59—41no—
118#106×C59—41no—
119C4⋊C8×C59C2×C414yescomplemented
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123C23×D59C2×D411yescharacteristic
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2411yescharacteristic
131C22×Dic59—21no—
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59C2×D411yescharacteristic
136C22×Dic59C2×D411yescharacteristic
137C22×Dic59C2×D411yescharacteristic
138C2×C4×D59—21no—
139C2×C4×D59—21no—
140C2×C4×D59—21no—
141C2×C4×D59—21no—
142C2×C4×D59—42no—
143C2×C4×D59—81no—
144C2×C4×D59—81no—
145C2×C4×D59C22×C412yes—
146C2×C4×D59C2×Q812yes—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59C22×C412yes—
152C2×C8×C59C22×C412yes—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C42×C59—81no—
162C42×C59C22×C412yes—
163C42×C59C2×Q812yes—
164(C2×D59)⋊C4—42no—
165(C2×D59)⋊C4—81no—
166(C2×D59)⋊C4—81no—
167(C2×D59)⋊C4C4⋊C414yes—
168(C2×D59)⋊C4C4⋊C414yes—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—81no—
178C4⋊Dic59—81no—
179Dic59⋊C4—42no—
180Dic59⋊C4—81no—
181Dic59⋊C4—81no—
182Dic59⋊C4—81no—
183Dic59⋊C4—81no—
184Dic59⋊C4C4⋊C414yes—
185Dic59⋊C4C4⋊C414yes—
186C4×Dic59—42no—
187C4×Dic59—81no—
188C4×Dic59C22×C412yes—
189C4×Dic59C2×Q812yes—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8C22×C412yes—
195C2×C59⋊C8C22×C412yes—
196C23×C59C22×D411yescharacteristic
197C22×D59—21no—
198C22×D59—21no—
199C22×D59—21no—
200C22×D59—21no—
201C22×D59—21no—
202C22×D59—21no—
203C22×D59C2×C4⋊C412yes—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59C2×C4⋊C412yes—
221C2×C4×C59C2×C4⋊C412yes—
222C2×C4×C59C23×C411yescharacteristic
223C2×C4×C59C22×Q811yescharacteristic
224C2×C4×C59H(5,𝔽2)11yescharacteristic
225C2×C4×C59H(5,𝔽2)11yescharacteristic
226C2×C4×C592−1+411yescharacteristic
227C2×C4×C592−1+411yescharacteristic
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
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×Dic59C2×C4⋊C412yes—
247C2×Dic59C2×C4⋊C412yes—
248C2×Dic59C2×C4⋊C412yes—
249C4×D59—22no—
250C4×D59—22no—
251C4×D59—22no—
252C4×D59—22no—
253C4×D59—84no—
254C8×C59—22no—
255C8×C59—22no—
256C59⋊C8—22no—
257C59⋊C8—22no—
258#21992—591nomaximal, Sylow
259C22×C59C22×C4⋊C411yescharacteristic
260C22×C59#53511yescharacteristic
261C22×C59#53611yescharacteristic
262C22×C59#57611yescharacteristic
263C22×C59#57711yescharacteristic
264C22×C59#57711yescharacteristic
265C22×C59#57811yescharacteristic
266C2×D59—22no—
267C2×D59—22no—
268C2×D59—22no—
269C2×D59—41no—
270C2×D59—41no—
271C2×D59—41no—
272C2×D59—41no—
273C4×C59—22no—
274C4×C59—22no—
275C4×C59—81no—
276C4×C59—81no—
277C4×C59—81no—
278C4×C59—81no—
279C4×C59#52011yescharacteristic
280C4×C59#52011yescharacteristic
281C4×C59#54411yescharacteristic
282C4×C59#54411yescharacteristic
283Dic59—22no—
284Dic59—22no—
285Dic59—22no—
286Dic59—81no—
287Dic59—81no—
288Dic59—81no—
289Dic59—81no—
290C2×#433—591no—
291#2317—591no—
292#2314—591no—
293#1949—591no—
294#1949D5911yescharacteristic, complemented
295#1948—591no—
296#1947—591no—
297#1854—591no—
298#1854—591no—
299#1853—591no—
300#1852—591no—
301#1847—591no—
302#1846—591no—
303#1836—591no—
304#1834—591no—
305C2×C59#231711yescharacteristic
306C2×C59#291211yescharacteristic
307C2×C59#291311yescharacteristic
308C2×C59#322411yescharacteristic
309C2×C59#322511yescharacteristic
310C2×C59#324111yescharacteristic
311C2×C59#324211yescharacteristic
312D59—42no—
313C2×#117—591no—
314C2×#117—591no—
315C22×C4⋊C4—591no—
316#433—598no—
317C2×#107—591no—
318C2×#107—591no—
319C2×#107—591no—
320C2×#107C2×D5911yescharacteristic
321C2×#106—591no—
322C2×#106—591no—
323C2×#106—591no—
324C2×#106C2×D5911yescharacteristic
325C2×C4⋊C8—591no—
326C2×C4⋊C8C2×D5911yescharacteristic
327C2×C22⋊C8—591no—
328C2×C22⋊C8—591no—
329C2×#97—591no—
330C2×C4×C8—591no—
331#389—1181no—
332#389—1181no—
333#388—1181no—
334#388—1181no—
335#387—1181no—
336#387—1181no—
337#385—1181no—
338#385—1181no—
339#384—591no—
340#384—591no—
341#384—591no—
342#382—591no—
343#382—591no—
344#382C2×D5911yescharacteristic
345#380—591no—
346#380—591no—
347C4×C4⋊C4—591no—
348C4×C4⋊C4C2×D5911yescharacteristic
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340C2×D5911yescharacteristic
356#340C2×D5911yescharacteristic
357C59#2199211yescharacteristic, complemented, Sylow
358C23×C4—591no—
359C23×C4—1181no—
360C22×C8—591no—
361C22×C8—591no—
362C22×C8C22×D5911yescharacteristic
363C22×C8C22×D5911yescharacteristic
364#117—594no—
365#117—594no—
366#117—2361no—
367#117—2361no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—2361no—
382C2×C4⋊C4—2361no—
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C22⋊C4—591no—
388C2×C22⋊C4—591no—
389C2×C22⋊C4—1181no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C42—21no—
393C2×C42—591no—
394C2×C42—1181no—
395C2×C42—1181no—
396C2×C42C22×D5911yescharacteristic
397#107—41no—
398#107—2361no—
399#107—2361no—
400#107—2361no—
401#106—41no—
402#106—2361no—
403#106—2361no—
404#106—2361no—
405C4⋊C8—594no—
406C4⋊C8C4×D5914yescomplemented
407C22⋊C8—594no—
408C22⋊C8—594no—
409#97—594no—
410C4×C8—594no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421C24—591no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—591no—
427C22×C4—591no—
428C22×C4—591no—
429C22×C4—592no—
430C22×C4—592no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4—4721no—
440C22×C4—4721no—
441C22×C4—2362no—
442C22×C4D4×D5911yescharacteristic
443C22×C4D4×D5911yescharacteristic
444C22×C4C23×D5911yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—592no—
450C2×C8—592no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8C2×C4×D5912yes—
456C2×C8C2×C4×D5912yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—594no—
466C4⋊C4—594no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—4721no—
476C4⋊C4—4721no—
477C4⋊C4—4721no—
478C4⋊C4—4721no—
479C4⋊C4—4721no—
480C4⋊C4—4721no—
481C4⋊C4—2362no—
482C22⋊C4—594no—
483C22⋊C4—594no—
484C22⋊C4—4721no—
485C22⋊C4—4721no—
486C22⋊C4—2362no—
487C42—81no—
488C42—592no—
489C42—592no—
490C42—4721no—
491C42—2362no—
492C42C2×C4×D5912yes—
493C42Q8×D5912yes—
494C23—592no—
495C23—1181no—
496C23—1181no—
497C23—1181no—
498C23—1181no—
499C23—1181no—
500C23—1181no—
501C23C2×D4×D5911yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—592no—
519C2×C4—592no—
520C2×C4—592no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1182no—
528C2×C4—1182no—
529C2×C4—1182no—
530C2×C4—1182no—
531C2×C4—4724no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4C4⋊C4×D5912yes—
545C2×C4C4⋊C4×D5912yes—
546C2×C4C22×C4×D5911yescharacteristic
547C2×C4C2×Q8×D5911yescharacteristic
548C2×C4#1427377211yescharacteristic
549C2×C4#1427377311yescharacteristic
550C2×C4#1427377411yescharacteristic
551C2×C4#1427377511yescharacteristic
552C8—22no—
553C8—22no—
554C8—1182no—
555C8—1182no—
556C22—1182no—
557C22—1182no—
558C22—1182no—
559C22—2361no—
560C22—2361no—
561C22—2361no—
562C22—2361no—
563C22C2×C4⋊C4×D5911yescharacteristic
564C22#1683796811yescharacteristic
565C22#1683797511yescharacteristic
566C22#1683820411yescharacteristic
567C22#1683821111yescharacteristic
568C22#1683821211yescharacteristic
569C22#1683821611yescharacteristic
570C4—22no—
571C4—22no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—1182no—
577C4—1182no—
578C4—1182no—
579C4—4721no—
580C4—4721no—
581C4—4721no—
582C4—4721no—
583C4#1683788111yescharacteristic
584C4#1683788411yescharacteristic
585C4#1683802911yescharacteristic
586C4#1683803211yescharacteristic
587C2—2362no—
588C2#2142671811yescharacteristic
589C2#2143269011yescharacteristic
590C2#2143269311yescharacteristic
591C2#2143519711yescharacteristic
592C2#2143520211yescharacteristic
593C2#2143534611yescharacteristic
594C2#2143536911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1949C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1949D59
hypocenterC59#21992
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #21424862C2semidirect
5. #21424859C2semidirect
6. #21424855C2semidirect
11. #21424261C2semidirect
12. #21424259C2semidirect
13. #21424228C2semidirect
14. #21424222C2semidirect
15. C59×#1949C2semidirect
19. #16837462C4semidirect
97. C4⋊(C59⋊C8)C2×C4semidirect
100. #14273637C2×C4semidirect
101. C8×Dic59C2×C4semidirect
119. C4⋊C8×C59C2×C4semidirect
294. #1949D59semidirect
357. C59#21992semidirect
406. C4⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168374622
#214267212
#214267012
#214248622
#214248592
#214248552
#214242982
#214242922
#214242872
#214242852
#214242612
#214242592
#214242282
#214242222
C59×#19492
#2199259