Finite Group Databasebeta

← #28554863

Subgroups — #28554863

Order 15104.

848 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838011C211yesmaximal, characteristic
2#21433180C211yesmaximal, characteristic, complemented
3#21433179C211yesmaximal, characteristic, complemented
4#21422466C211yesmaximal, characteristic
5#21422464C212yesmaximal
6#21422463C211yesmaximal, characteristic
7#21422452C211yesmaximal, characteristic, complemented
8#21422450C212yesmaximal, complemented
9#21422438C212yesmaximal, complemented
10#21422429C212yesmaximal
11C59×#1470C211yesmaximal, characteristic, complemented
12#16838011—22no—
13#16838011—22no—
14#16838011—22no—
15#16838011—22no—
16C2×#14273726C411yescharacteristic
17C2×#14273722C411yescharacteristic
18C2×#14273722C411yescharacteristic
19C2×#14273722C2211yescharacteristic
20C2×#14273721C2211yescharacteristic
21C2×#14273718C411yescharacteristic
22C2×#14273717C411yescharacteristic
23C2×#14273717C411yescharacteristic
24C2×#14273717C2211yescharacteristic
25C2×#14273705C411yescharacteristic
26C2×#14273705C411yescharacteristic
27C2×C4×Dic118C2211yescharacteristic
28C2×#14273692C2211yescharacteristic
29C2×#14273685C2211yescharacteristic
30#16837532—22no—
31#16837531—22no—
32#16837530—22no—
33#16837529—22no—
34#16837459C2211yescharacteristic
35#16837456C2212yes—
36#16837456C2212yes—
37#16837456C2212yes—
38#16837455C2211yescharacteristic
39#16837452C2212yes—
40#16837128C2212yes—
41#16837128C2212yes—
42#16837127C2212yes—
43#16837127C2212yes—
44#16837125C2212yes—
45#16837125C2212yes—
46C2×#125×C59C2211yescharacteristic
47C59×#432C2211yescharacteristic
48C59×#431C2211yescharacteristic
49C59×#324C2212yes—
50C59×#324C2212yes—
51C22×Dic118C2×C411yescharacteristic
52C22×Dic118C2×C411yescharacteristic
53C22×C4×D59C2×C411yescharacteristic
54#14273726—22no—
55#14273726—22no—
56#14273722—22no—
57#14273722—22no—
58#14273722—22no—
59#14273722—22no—
60#14273722—22no—
61#14273722—22no—
62#14273721—21no—
63#14273721—21no—
64#14273721—21no—
65#14273721—21no—
66#14273718—22no—
67#14273718—22no—
68#14273717—22no—
69#14273717—22no—
70#14273717—22no—
71#14273717—22no—
72#14273717—22no—
73#14273717—22no—
74#14273705—22no—
75#14273705—22no—
76#14273705—22no—
77#14273705—22no—
78C4×Dic118D412yes—
79C4×Dic118D412yes—
80C4×Dic118D412yes—
81C4×Dic118D412yes—
82#14273692—22no—
83#14273692—22no—
84#14273685—22no—
85#14273685—22no—
86C2×C4⋊Dic59C2×C411yescharacteristic
87C2×C4⋊Dic59C2×C411yescharacteristic
88C2×C4⋊Dic59C2×C411yescharacteristic
89C2×C4⋊Dic59C2×C411yescharacteristic
90C2×Dic59⋊C4C2×C411yescharacteristic
91C2×Dic59⋊C4C2×C411yescharacteristic
92C2×Dic59⋊C4C2×C411yescharacteristic
93C2×Dic59⋊C4C2×C411yescharacteristic
94C2×Dic59⋊C4C2×C411yescharacteristic
95C2×Dic59⋊C4C2×C411yescharacteristic
96C2×Dic59⋊C4C2×C411yescharacteristic
97C2×Dic59⋊C4C2×C411yescharacteristic
98C2×Dic59⋊C4C2×C411yescharacteristic
99C2×Dic59⋊C4C2311yescharacteristic
100C2×(C2×D59)⋊C4C2×C411yescharacteristic
101C2×(C2×D59)⋊C4C2×C411yescharacteristic
102C2×(C2×D59)⋊C4C2×C411yescharacteristic
103C2×(C2×D59)⋊C4C2×C411yescharacteristic
104C2×(C2×D59)⋊C4C2×C411yescharacteristic
105C2×(C2×D59)⋊C4C2311yescharacteristic
106C2×C4×Dic59C2×C411yescharacteristic
107C2×C4×Dic59C2×C411yescharacteristic
108C2×C4×Dic59C2×C411yescharacteristic
109C2×C4×Dic59C2311yescharacteristic
110C2×C4×Dic59C2311yescharacteristic
111C4⋊(C59⋊C8)—21no—
112C4⋊(C59⋊C8)—21no—
113Dic59⋊C8—22no—
114Dic59⋊C8—22no—
115Dic59⋊C8—22no—
116Dic59⋊C8—22no—
117C22⋊(C59⋊C8)C2312yes—
118C22⋊(C59⋊C8)C2312yes—
119#14273638—21no—
120#14273637—22no—
121#14273637—22no—
122#14273637—22no—
123C4×C59⋊C8—21no—
124C8×Dic59—22no—
125#125×C59—21no—
126#125×C59—21no—
127#125×C59—21no—
128#125×C59—21no—
129C2×C4⋊C4×C59C2×C411yescharacteristic
130C2×C4⋊C4×C59C2×C411yescharacteristic
131C2×C4⋊C4×C59C2×C411yescharacteristic
132C2×C4⋊C4×C59C2×C411yescharacteristic
133C2×C4⋊C4×C59C2311yescharacteristic
134C2×C4⋊C4×C59C2311yescharacteristic
135C2×C42×C59C2311yescharacteristic
136C4⋊C8×C59—21no—
137C4⋊C8×C59—21no—
138C22⋊C8×C59C2312yes—
139C22⋊C8×C59C2312yes—
140#97×C59—21no—
141C4×C8×C59—21no—
142C23×D59C22×C411yescharacteristic
143C22×C4×C59C22×C411yescharacteristic
144C22×C4×C59C22×C411yescharacteristic
145C22×C4×C59C22×C411yescharacteristic
146C22×C4×C59C22×C411yescharacteristic
147C22×C4×C59C22×C411yescharacteristic
148C22×C4×C59C22×C411yescharacteristic
149C22×C4×C59C2411yescharacteristic
150C22×Dic59C22×C411yescharacteristic
151C22×Dic59C22×C411yescharacteristic
152C22×Dic59C22×C411yescharacteristic
153C22×Dic59C22×C411yescharacteristic
154C22×Dic59C22×C411yescharacteristic
155C22×Dic59C22×C411yescharacteristic
156C22×Dic59C22×C411yescharacteristic
157C2×C4×D59—22no—
158C2×C4×D59—42no—
159C2×Dic118—22no—
160C2×Dic118—42no—
161C2×Dic118—42no—
162C2×Dic118C22⋊C412yes—
163C2×Dic118C22⋊C412yes—
164C2×C8×C59—22no—
165C2×C8×C59—22no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C42×C59C2×D411yescharacteristic
179C42×C59C2×D411yescharacteristic
180C42×C59C2×D411yescharacteristic
181C42×C59C2×D411yescharacteristic
182(C2×D59)⋊C4—21no—
183(C2×D59)⋊C4—21no—
184(C2×D59)⋊C4—21no—
185(C2×D59)⋊C4—21no—
186(C2×D59)⋊C4—21no—
187(C2×D59)⋊C4—21no—
188(C2×D59)⋊C4—22no—
189(C2×D59)⋊C4—22no—
190(C2×D59)⋊C4—22no—
191C4⋊Dic59—22no—
192C4⋊Dic59—22no—
193C4⋊Dic59—22no—
194C4⋊Dic59C22⋊C412yes—
195C4⋊Dic59C22⋊C412yes—
196Dic59⋊C4—21no—
197Dic59⋊C4—21no—
198Dic59⋊C4—21no—
199Dic59⋊C4—21no—
200Dic59⋊C4—21no—
201Dic59⋊C4—21no—
202Dic59⋊C4—22no—
203Dic59⋊C4—22no—
204Dic59⋊C4—22no—
205Dic59⋊C4—22no—
206Dic59⋊C4—22no—
207Dic59⋊C4C22⋊C412yes—
208Dic59⋊C4C22⋊C412yes—
209Dic59⋊C4C22⋊C412yes—
210Dic59⋊C4C22⋊C412yes—
211C4×Dic59—22no—
212C4×Dic59—22no—
213C4×Dic59—22no—
214C4×Dic59C2×D412yes—
215C4×Dic59C2×D412yes—
216C4×Dic59C2×D412yes—
217C4×Dic59C2×D412yes—
218C2×C59⋊C8—22no—
219C2×C59⋊C8—22no—
220C23×C59C23×C411yescharacteristic
221C22×D59—21no—
222C22×D59—22no—
223C22×D59—41no—
224C22×D59—41no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59C2×C22⋊C411yescharacteristic
237C2×C4×C59C2×C22⋊C411yescharacteristic
238C2×C4×C59C2×C22⋊C411yescharacteristic
239C2×C4×C59C2×C22⋊C411yescharacteristic
240C2×C4×C59C22×D411yescharacteristic
241C2×C4×C59C22×D411yescharacteristic
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—22no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59C2×C22⋊C412yes—
253C2×Dic59C2×C22⋊C412yes—
254C2×Dic59C2×C22⋊C412yes—
255C2×Dic59C2×C22⋊C412yes—
256C4×D59—44no—
257Dic118—42no—
258Dic118—42no—
259Dic118—44no—
260C8×C59—42no—
261C8×C59—42no—
262C59⋊C8—42no—
263C59⋊C8—42no—
264#17410—591nomaximal, Sylow
265C22×C59—21no—
266C22×C59—21no—
267C22×C59C22×C22⋊C411yescharacteristic
268C22×C59#56812yes—
269C2×D59—41no—
270C2×D59—41no—
271C2×D59—41no—
272C2×D59—82no—
273C4×C59—21no—
274C4×C59—21no—
275C4×C59—21no—
276C4×C59—21no—
277C4×C59—21no—
278C4×C59—21no—
279C4×C59—41no—
280C4×C59—41no—
281C4×C59—41no—
282C4×C59—41no—
283Dic59—22no—
284Dic59—22no—
285Dic59—22no—
286Dic59—22no—
287Dic59—41no—
288Dic59—41no—
289Dic59—41no—
290C2×#541—591no—
291#2983—591no—
292#2983—591no—
293#1473—591no—
294#1473—591no—
295#1473—592no—
296#1470—592no—
297#1470—591no—
298#1470D5911yescharacteristic, complemented
299#1467—592no—
300#1464—592no—
301C2×C59—22no—
302C2×C59#287911yescharacteristic
303C2×C59#288411yescharacteristic
304C2×C59#288911yescharacteristic
305D59—81no—
306#541—1182no—
307#541—1182no—
308#541—1182no—
309#541—1182no—
310C2×C4⋊Q8—591no—
311C2×#126—591no—
312C2×#126—591no—
313C2×#126—591no—
314C2×#126—591no—
315C2×#125—591no—
316C2×#125—591no—
317C2×#125—591no—
318C2×#125—591no—
319C2×#125C2×D5911yescharacteristic
320C2×C22⋊Q8—591no—
321C2×C22⋊Q8—591no—
322C2×C4×Q8—591no—
323C2×C4×Q8—591no—
324C2×#117—591no—
325#446—1182no—
326#446—1182no—
327#446—1182no—
328#446—1182no—
329#432—591no—
330#432—592no—
331#432—592no—
332#432—592no—
333#432C2×D5911yescharacteristic
334#431—591no—
335#431—592no—
336#431C2×D5911yescharacteristic
337#324—592no—
338#324—592no—
339#324—592no—
340#324—592no—
341#324C2×D5912yes—
342#324C2×D5912yes—
343#323—592no—
344#323—592no—
345C59#1741011yescharacteristic, complemented, Sylow
346C22×Q8—591no—
347C22×Q8—591no—
348C23×C4—591no—
349C4⋊Q8—1182no—
350C4⋊Q8—1182no—
351#126—1182no—
352#126—1182no—
353#126—1182no—
354#126—1182no—
355#126—1182no—
356#126—1182no—
357#126—1181no—
358#126—1181no—
359#126—1181no—
360#126—1181no—
361#125—21no—
362#125—21no—
363#125—21no—
364#125—21no—
365#125—1182no—
366#125—1182no—
367#125—1182no—
368#125—1182no—
369#125—1182no—
370#125—1182no—
371#125—1182no—
372#125—1182no—
373C22⋊Q8—1182no—
374C22⋊Q8—1182no—
375C22⋊Q8—1182no—
376C22⋊Q8—1182no—
377C4×Q8—592no—
378C4×Q8—592no—
379C4×Q8—592no—
380C4×Q8—592no—
381C4×Q8—1182no—
382C4×Q8—1182no—
383#117—1182no—
384#117—1182no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4C4×D5911yescharacteristic
400C2×C4⋊C4C4×D5911yescharacteristic
401C2×C4⋊C4C4×D5911yescharacteristic
402C2×C4⋊C4C4×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C22⋊C4—591no—
406C2×C22⋊C4—591no—
407C2×C22⋊C4—591no—
408C2×C22⋊C4—591no—
409C2×C22⋊C4—591no—
410C2×C22⋊C4—591no—
411C2×C42—591no—
412C2×C42—591no—
413C2×C42—591no—
414C2×C42—591no—
415C2×C42—591no—
416C2×C42C22×D5911yescharacteristic
417C4⋊C8—21no—
418C4⋊C8—21no—
419C4⋊C8—1182no—
420C4⋊C8—1182no—
421C4⋊C8—1182no—
422C4⋊C8—1182no—
423C4⋊C8—1181no—
424C4⋊C8—1181no—
425C22⋊C8—592no—
426C22⋊C8—592no—
427C22⋊C8C22×D5912yes—
428C22⋊C8C22×D5912yes—
429#97—21no—
430#97—1182no—
431#97—1182no—
432#97—1182no—
433#97—1181no—
434C4×C8—21no—
435C4×C8—1182no—
436C4×C8—1181no—
437C24—591no—
438C2×Q8—592no—
439C2×Q8—592no—
440C2×Q8—1182no—
441C2×Q8—2362no—
442C2×Q8—2362no—
443C22×C4—591no—
444C22×C4—591no—
445C22×C4—591no—
446C22×C4—591no—
447C22×C4—591no—
448C22×C4—591no—
449C22×C4—591no—
450C22×C4—1182no—
451C22×C4—2362no—
452C22×C4C2×C4×D5911yescharacteristic
453C22×C4C2×C4×D5911yescharacteristic
454C22×C4C2×C4×D5911yescharacteristic
455C22×C4C2×C4×D5911yescharacteristic
456C22×C4C2×C4×D5911yescharacteristic
457C22×C4C2×C4×D5911yescharacteristic
458C22×C4C23×D5911yescharacteristic
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—1182no—
462C2×C8—1182no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—592no—
476C4⋊C4—592no—
477C4⋊C4—592no—
478C4⋊C4—592no—
479C4⋊C4—592no—
480C4⋊C4—592no—
481C4⋊C4—1182no—
482C4⋊C4—1182no—
483C4⋊C4—1182no—
484C4⋊C4—1182no—
485C4⋊C4—1182no—
486C4⋊C4—1182no—
487C4⋊C4—1182no—
488C4⋊C4—1182no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C22⋊C4—1182no—
496C22⋊C4—1182no—
497C22⋊C4—1182no—
498C22⋊C4—1181no—
499C22⋊C4—1181no—
500C22⋊C4—1181no—
501C22⋊C4—1181no—
502C22⋊C4—1181no—
503C22⋊C4—1181no—
504C42—592no—
505C42—592no—
506C42—592no—
507C42—592no—
508C42—1182no—
509C42—1182no—
510C42—1182no—
511C42D4×D5911yescharacteristic
512C42D4×D5911yescharacteristic
513C42D4×D5911yescharacteristic
514C42D4×D5911yescharacteristic
515C23—1182no—
516C23—1181no—
517C23—2361no—
518C23—2361no—
519C23C22×C4×D5911yescharacteristic
520Q8—2362no—
521Q8—2362no—
522Q8—2364no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—592no—
535C2×C4—592no—
536C2×C4—592no—
537C2×C4—592no—
538C2×C4—1182no—
539C2×C4—1182no—
540C2×C4—1182no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—2364no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4C22⋊C4×D5911yescharacteristic
550C2×C4C22⋊C4×D5911yescharacteristic
551C2×C4C22⋊C4×D5911yescharacteristic
552C2×C4C22⋊C4×D5911yescharacteristic
553C2×C4C2×D4×D5911yescharacteristic
554C2×C4C2×D4×D5911yescharacteristic
555C8—42no—
556C8—42no—
557C8—2362no—
558C8—2362no—
559C22—21no—
560C22—21no—
561C22—4722no—
562C22—2361no—
563C22—2361no—
564C22—2361no—
565C22C2×C22⋊C4×D5911yescharacteristic
566C22#1683816712yes—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—1182no—
578C4—1182no—
579C4—1182no—
580C4—1182no—
581C4—2361no—
582C4—2361no—
583C4—2361no—
584C2—22no—
585C2—4721no—
586C2#2143253811yescharacteristic
587C2#2143255811yescharacteristic
588C2#2143257611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D59
commutatorC23×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1470C2
socleC22×C59C22×C22⋊C4
prime radicalC23×D59C22×C4
cosocleC22×C4×C59C24
hypercenter#1470D59
hypocenterC59#17410
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1470

Complements

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

NH ≅ G/Ntype
2. #21433180C2semidirect
3. #21433179C2semidirect
7. #21422452C2semidirect
8. #21422450C2semidirect
9. #21422438C2semidirect
11. C59×#1470C2semidirect
298. #1470D59semidirect
345. C59#17410semidirect

Maximal subgroups

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

subgroupindex
C2×#168380112
#214331802
#214331792
#214224662
#214224642
#214224632
#214224522
#214224502
#214224382
#214224292
C59×#14702
#1741059