Finite Group Databasebeta

← #28599302

Subgroups — #28599302

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×#16837463C211yesmaximal, characteristic
2#21426705C211yesmaximal, characteristic
3#21424862C211yesmaximal, characteristic, complemented
4#21424862C211yesmaximal, characteristic, complemented
5#21424857C211yesmaximal, characteristic, complemented
6#21424852C211yesmaximal, characteristic, complemented
7#21424295C211yesmaximal, characteristic
8#21424295C211yesmaximal, characteristic
9#21424289C211yesmaximal, characteristic
10#21424284C211yesmaximal, characteristic
11#21424262C211yesmaximal, characteristic, complemented
12#21424258C211yesmaximal, characteristic, complemented
13#21424229C211yesmaximal, characteristic, complemented
14#21424223C211yesmaximal, characteristic, complemented
15C59×#2317C211yesmaximal, characteristic
16#16837463C418yescomplemented
17C2×#14273660C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837295—21no—
32#16837295—21no—
33#16837288—21no—
34#16837288—21no—
35#16837272C2211yescharacteristic
36#16837268C2211yescharacteristic
37#16837262C2211yescharacteristic
38#16837262C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49C2×#117×C59C2211yescharacteristic
50C2×#117×C59C2211yescharacteristic
51C22×C4⋊C4×C59C2211yescharacteristic
52C59×#389—21no—
53C59×#389—21no—
54C59×#385—21no—
55C59×#385—21no—
56C59×#384C2211yescharacteristic
57C59×#384C2211yescharacteristic
58C59×#382C2211yescharacteristic
59C59×#382C2211yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C22⋊Dic59—21no—
71C2×C22⋊Dic59—21no—
72C2×C4×Dic59—21no—
73C2×C4×Dic59—21no—
74#14273660—41no—
75#14273660—41no—
76#14273660—41no—
77#14273660—41no—
78#14273658—41no—
79#14273658—41no—
80#14273658—41no—
81#14273658—41no—
82C4⋊(C59⋊C8)C2×C414yescomplemented
83C4⋊(C59⋊C8)C2×C414yescomplemented
84C22⋊(C59⋊C8)C2×C414yes—
85C22⋊(C59⋊C8)C2×C414yes—
86#14273638C2×C414yescomplemented
87C4×C59⋊C8C2×C414yescomplemented
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C23×C4×C59—21no—
95C23×C4×C59C2311yescharacteristic
96#117×C59—41no—
97#117×C59—41no—
98#117×C59C2×C414yes—
99#117×C59Q814yes—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—41no—
106C2×C4⋊C4×C59—41no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C22⋊C4×C59D411yescharacteristic
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—
123C24×C59C2×D411yescharacteristic
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—42no—
133C22×C4×C59—81no—
134C22×C4×C59—81no—
135C22×C4×C59C22×C412yes—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×Q812yes—
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—42no—
156C4⋊C4×C59—81no—
157C4⋊C4×C59—81no—
158C4⋊C4×C59—81no—
159C4⋊C4×C59—81no—
160C4⋊C4×C59C4⋊C414yes—
161C4⋊C4×C59C4⋊C414yes—
162C22⋊C4×C59—42no—
163C22⋊C4×C59C4⋊C414yes—
164C22⋊C4×C59C4⋊C414yes—
165C42×C59—42no—
166C42×C59C22×C412yes—
167C42×C59C22×C412yes—
168C42×C59C2×Q812yes—
169C42×C59C2×Q812yes—
170C22⋊Dic59—81no—
171C22⋊Dic59—81no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—81no—
181C4⋊Dic59—81no—
182C4×Dic59—81no—
183C4×Dic59—81no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8C22×C412yes—
193C2×C59⋊C8C22×C412yes—
194C2×C59⋊C8C22×C412yes—
195C2×C59⋊C8C22×C412yes—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59C2×C4⋊C412yes—
203C23×C59C22×D411yescharacteristic
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
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—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
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×C59—84no—
231C2×C4×C59C2×C4⋊C412yes—
232C2×C4×C59C2×C4⋊C412yes—
233C2×C4×C59C2×C4⋊C412yes—
234C2×C4×C59C2×C4⋊C412yes—
235C2×C4×C59C2×C4⋊C412yes—
236C2×C4×C59C23×C411yescharacteristic
237C2×C4×C59C22×Q811yescharacteristic
238C2×C4×C59H(5,𝔽2)11yescharacteristic
239C2×C4×C59H(5,𝔽2)11yescharacteristic
240C2×C4×C592−1+411yescharacteristic
241C2×C4×C592−1+411yescharacteristic
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—
254C59⋊C8—22no—
255C59⋊C8—22no—
256C59⋊C8—22no—
257C59⋊C8—22no—
258#21992—591nomaximal, Sylow
259C22×C59—22no—
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—41no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59—41no—
266C22×C59C22×C4⋊C411yescharacteristic
267C22×C59#53511yescharacteristic
268C22×C59#53611yescharacteristic
269C22×C59#57611yescharacteristic
270C22×C59#57711yescharacteristic
271C22×C59#57711yescharacteristic
272C22×C59#57811yescharacteristic
273C4×C59—22no—
274C4×C59—22no—
275C4×C59—22no—
276C4×C59—22no—
277C4×C59—22no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59#52011yescharacteristic
283C4×C59#52011yescharacteristic
284C4×C59#54411yescharacteristic
285C4×C59#54411yescharacteristic
286Dic59—81no—
287Dic59—81no—
288Dic59—81no—
289Dic59—81no—
290C2×#433—591no—
291#2317D5911yescharacteristic
292#2314—591no—
293#1949—591no—
294#1949—591no—
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—42no—
306C2×C59#231711yescharacteristic
307C2×C59#291211yescharacteristic
308C2×C59#291311yescharacteristic
309C2×C59#322411yescharacteristic
310C2×C59#322511yescharacteristic
311C2×C59#324111yescharacteristic
312C2×C59#324211yescharacteristic
313C2×#117C2×D5911yescharacteristic
314C2×#117C2×D5911yescharacteristic
315C22×C4⋊C4C2×D5911yescharacteristic
316#433—598no—
317C2×#107—591no—
318C2×#107—591no—
319C2×#107—591no—
320C2×#107—591no—
321C2×#106—591no—
322C2×#106—591no—
323C2×#106—591no—
324C2×#106—591no—
325C2×C4⋊C8—591no—
326C2×C4⋊C8—591no—
327C2×C22⋊C8—591no—
328C2×C22⋊C8—591no—
329C2×#97—591no—
330C2×C4×C8—591no—
331#389—21no—
332#389—21no—
333#388—1181no—
334#388—1181no—
335#387—1181no—
336#387—1181no—
337#385—21no—
338#385—21no—
339#384—591no—
340#384C2×D5911yescharacteristic
341#384C2×D5911yescharacteristic
342#382—591no—
343#382C2×D5911yescharacteristic
344#382C2×D5911yescharacteristic
345#380—591no—
346#380—591no—
347C4×C4⋊C4—591no—
348C4×C4⋊C4—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357C59#2199211yescharacteristic, complemented, Sylow
358C23×C4—21no—
359C23×C4C22×D5911yescharacteristic
360C22×C8—591no—
361C22×C8—591no—
362C22×C8—591no—
363C22×C8—591no—
364#117—41no—
365#117—41no—
366#117Dic11814yes—
367#117C4×D5914yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4—41no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4C22⋊D5911yescharacteristic
382C2×C4⋊C4C22⋊D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—1181no—
390C2×C22⋊C4C22⋊D5911yescharacteristic
391C2×C22⋊C4C22⋊D5911yescharacteristic
392C2×C42—21no—
393C2×C42—1181no—
394C2×C42—1181no—
395C2×C42C22×D5911yescharacteristic
396C2×C42C22×D5911yescharacteristic
397#107—2361no—
398#107—2361no—
399#107—2361no—
400#107—2361no—
401#106—2361no—
402#106—2361no—
403#106—2361no—
404#106—2361no—
405C4⋊C8—594no—
406C4⋊C8—594no—
407C22⋊C8—594no—
408C22⋊C8—594no—
409#97—594no—
410C4×C8—594no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421C24C2×C22⋊D5911yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4C2×Dic11812yes—
438C22×C4C2×C4×D5912yes—
439C22×C4C2×C22⋊D5911yescharacteristic
440C22×C4C2×C22⋊D5911yescharacteristic
441C22×C4C2×C22⋊D5911yescharacteristic
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C23×D5911yescharacteristic
445C2×C8—592no—
446C2×C8—592no—
447C2×C8—592no—
448C2×C8—592no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—1181no—
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—42no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—4721no—
479C4⋊C4—4721no—
480C4⋊C4Dic59⋊C414yes—
481C4⋊C4Dic59⋊C414yes—
482C22⋊C4—42no—
483C22⋊C4—4721no—
484C22⋊C4—4721no—
485C22⋊C4Dic59⋊C414yes—
486C22⋊C4Dic59⋊C414yes—
487C42—42no—
488C42—4721no—
489C42—4721no—
490C42C2×Dic11812yes—
491C42C2×Dic11812yes—
492C42C2×C4×D5912yes—
493C42C2×C4×D5912yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×Dic59⋊C412yes—
501C23C22×C22⋊D5911yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
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×C4C2×Dic59⋊C412yes—
542C2×C4C2×Dic59⋊C412yes—
543C2×C4C2×Dic59⋊C412yes—
544C2×C4C2×Dic59⋊C412yes—
545C2×C4C2×Dic59⋊C412yes—
546C2×C4C22×C4×D5911yescharacteristic
547C2×C4C22×Dic11811yescharacteristic
548C2×C4#1427377211yescharacteristic
549C2×C4#1427377211yescharacteristic
550C2×C4#1427377511yescharacteristic
551C2×C4#1427377511yescharacteristic
552C8—1182no—
553C8—1182no—
554C8—1182no—
555C8—1182no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×Dic59⋊C411yescharacteristic
564C22#1683796911yescharacteristic
565C22#1683797611yescharacteristic
566C22#1683820511yescharacteristic
567C22#1683821011yescharacteristic
568C22#1683821311yescharacteristic
569C22#1683821711yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—4721no—
580C4—4721no—
581C4—4721no—
582C4—4721no—
583C4#1683788311yescharacteristic
584C4#1683788311yescharacteristic
585C4#1683803311yescharacteristic
586C4#1683803311yescharacteristic
587C2—42no—
588C2#2142672011yescharacteristic
589C2#2143268911yescharacteristic
590C2#2143269411yescharacteristic
591C2#2143519011yescharacteristic
592C2#2143520511yescharacteristic
593C2#2143534911yescharacteristic
594C2#2143536411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2317C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#2317D59
hypocenterC59#21992
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. #21424862C2semidirect
4. #21424862C2semidirect
5. #21424857C2semidirect
6. #21424852C2semidirect
11. #21424262C2semidirect
12. #21424258C2semidirect
13. #21424229C2semidirect
14. #21424223C2semidirect
16. #16837463C4semidirect
82. C4⋊(C59⋊C8)C2×C4semidirect
83. C4⋊(C59⋊C8)C2×C4semidirect
86. #14273638C2×C4semidirect
87. C4×C59⋊C8C2×C4semidirect
357. C59#21992semidirect

Maximal subgroups

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

subgroupindex
C2×#168374632
#214267052
#214248622
#214248572
#214248522
#214242952
#214242892
#214242842
#214242622
#214242582
#214242292
#214242232
C59×#23172
#2199259