Finite Group Databasebeta

← #28560732

Subgroups — #28560732

Order 15104.

598 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434664C211yesmaximal, characteristic
2#21433257C211yesmaximal, characteristic, complemented
3#21433251C211yesmaximal, characteristic, complemented
4#21422998C211yesmaximal, characteristic, complemented
5#21422995C211yesmaximal, characteristic, complemented
6#21422948C211yesmaximal, characteristic, complemented
7#21422945C211yesmaximal, characteristic, complemented
8#21422926C211yesmaximal, characteristic, complemented
9#21422908C211yesmaximal, characteristic, complemented
10#21422905C211yesmaximal, characteristic, complemented
11#335×D59C211yesmaximal, characteristic, complemented
12#21422506C211yesmaximal, characteristic, complemented
13#21422506C211yesmaximal, characteristic, complemented
14#21422384C211yesmaximal, characteristic, complemented
15C59×#1581C211yesmaximal, characteristic, complemented
16#16837934C2211yescharacteristic
17#16837934C2211yescharacteristic
18#16837803C411yescharacteristic
19#16837785C411yescharacteristic
20#16837755C411yescharacteristic
21#16837755C411yescharacteristic
22#16837644C411yescharacteristic
23#16837635C411yescharacteristic
24#16837630C411yescharacteristic
25#16837630C411yescharacteristic
26#16837532C2211yescharacteristic
27#16837529C2211yescharacteristic
28C8×Dic118C2211yescharacteristic
29#16837526C2211yescharacteristic
30C4×Dic236C2211yescharacteristic
31C4×#11745670C2211yescharacteristic
32C4×#11745670C2211yescharacteristic
33#16837480C2211yescharacteristic
34#16837476—21no—
35#16837471—21no—
36D4×C59⋊C8C2211yescharacteristic
37C8×C22⋊D59—21no—
38(C22⋊D59)⋊C8—21no—
39#16837460—21no—
40#16837460—21no—
41#16837453C2211yescharacteristic
42#16837430—21no—
43#16837428—21no—
44C4⋊C8×D59C2211yescharacteristic
45C4⋊C8×D59C2211yescharacteristic
46C4×#11745659C2211yescharacteristic
47C4×C8×D59C2211yescharacteristic
48#16837155C2211yescharacteristic
49#16837154C2211yescharacteristic
50#16837154C2211yescharacteristic
51#16837134C2211yescharacteristic
52#16837134C2211yescharacteristic
53#16837132C2211yescharacteristic
54#16837132C2211yescharacteristic
55D4⋊(C59⋊C8)C2211yescharacteristic
56D4⋊(C59⋊C8)C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837124C2211yescharacteristic
59#16837124C2211yescharacteristic
60C4×D8×C59C2211yescharacteristic
61C59×#436C2211yescharacteristic
62C8×D4×C59C2211yescharacteristic
63C59×#335C2211yescharacteristic
64C59×D4⋊C8C2211yescharacteristic
65C59×D4⋊C8C2211yescharacteristic
66C59×#322C2211yescharacteristic
67C2×Dic236C2×C411yescharacteristic
68C2×#11745670C2×C411yescharacteristic
69C2×#11745670C2×C411yescharacteristic
70C2×#11745664—21no—
71C2×#11745659—21no—
72C22×C59⋊C8—21no—
73C2×C8×D59—21no—
74C2×C8×D59—21no—
75C2×C8×D59C2×C411yescharacteristic
76C2×C8×D59C2×C411yescharacteristic
77#14273720—21no—
78#14273720—21no—
79#14273718C2×C411yescharacteristic
80#14273718C2×C411yescharacteristic
81#14273714C2×C411yescharacteristic
82#14273714C2×C411yescharacteristic
83#14273709—21no—
84#14273709—21no—
85#14273707C2×C411yescharacteristic
86#14273707C2×C411yescharacteristic
87#14273698C2×C411yescharacteristic
88#14273698C2×C411yescharacteristic
89C4×Dic118C2311yescharacteristic
90C4×Dic118C2311yescharacteristic
91C4×C22⋊D59—21no—
92C4×C22⋊D59—21no—
93#14273686—21no—
94#14273686—21no—
95C42×D59C2311yescharacteristic
96#14273661C2×C411yescharacteristic
97#14273658C2×C411yescharacteristic
98#14273658C2×C411yescharacteristic
99C4⋊(C59⋊C8)C2311yescharacteristic
100C4⋊(C59⋊C8)C2311yescharacteristic
101Dic59⋊C8—21no—
102Dic59⋊C8—21no—
103Dic59⋊C8—21no—
104Dic59⋊C8—21no—
105Dic59⋊C8—21no—
106Dic59⋊C8—21no—
107Dic59⋊C8D411yescharacteristic
108Dic59⋊C8D411yescharacteristic
109#14273652C2×C411yescharacteristic
110#14273652C2×C411yescharacteristic
111#14273650C2×C411yescharacteristic
112#14273650C2×C411yescharacteristic
113#14273648C2×C411yescharacteristic
114#14273648C2×C411yescharacteristic
115C22⋊(C59⋊C8)—21no—
116C22⋊(C59⋊C8)—21no—
117(C2×D59)⋊C8D411yescharacteristic
118(C2×D59)⋊C8D411yescharacteristic
119#14273637—21no—
120#14273637—21no—
121#14273637—21no—
122C4×C59⋊C8C2311yescharacteristic
123C4×C59⋊C8C2311yescharacteristic
124C4×C59⋊C8C2311yescharacteristic
125C8×Dic59—21no—
126C8×Dic59—21no—
127C8×Dic59C2×C411yescharacteristic
128C8×Dic59C2×C411yescharacteristic
129C2×D8×C59C2×C411yescharacteristic
130C2×M16×C59—21no—
131C22×C8×C59—21no—
132C4×D4×C59C2311yescharacteristic
133C4×D4×C59C2311yescharacteristic
134#107×C59C2×C411yescharacteristic
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137#102×C59C2×C411yescharacteristic
138#102×C59C2×C411yescharacteristic
139C22⋊C8×C59—21no—
140C22⋊C8×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×D4×C59C22×C411yescharacteristic
145C2×D4×C59C22×C411yescharacteristic
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C2×C22⋊D59—21no—
149C2×C22⋊D59—21no—
150C2×C4×D59C22×C411yescharacteristic
151C2×C4×D59C2×D411yescharacteristic
152C2×C4×D59#4111yescharacteristic
153C2×Dic118C22×C411yescharacteristic
154C2×Dic118C22×C411yescharacteristic
155D8×C59—41no—
156M16×C59—81no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—81no—
161C2×C8×C59C22×C411yescharacteristic
162C2×C8×C59C22×C411yescharacteristic
163C2×C8×C59C2×D411yescharacteristic
164C2×C8×C59C2×D411yescharacteristic
165C4⋊C4×C59C22×C411yescharacteristic
166C4⋊C4×C59C22×C411yescharacteristic
167C22⋊C4×C59—21no—
168C22⋊C4×C59—21no—
169C42×C59C2411yescharacteristic
170C22⋊Dic59—21no—
171C22⋊Dic59—21no—
172C22⋊Dic59—21no—
173C22⋊Dic59—21no—
174C22⋊Dic59—21no—
175C22⋊Dic59—21no—
176#11745670—22no—
177#11745670—22no—
178(C2×D59)⋊C4—21no—
179(C2×D59)⋊C4—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59C22×C411yescharacteristic
185C4⋊Dic59C22×C411yescharacteristic
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189Dic59⋊C4—21no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59C22×C411yescharacteristic
193C4×Dic59C2×D411yescharacteristic
194C4×Dic59#4111yescharacteristic
195#11745664—81no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—81no—
200C2×C59⋊C8C22×C411yescharacteristic
201C2×C59⋊C8C22×C411yescharacteristic
202C2×C59⋊C8C2×D411yescharacteristic
203C2×C59⋊C8C2×D411yescharacteristic
204Dic236—41no—
205#11745659—42no—
206C8×D59—22no—
207C8×D59—41no—
208C8×D59—81no—
209C8×D59—81no—
210C23×C59—21no—
211C23×C59—21no—
212C22×D59C4×D411yescharacteristic
213D4×C59—21no—
214D4×C59—21no—
215D4×C59—21no—
216D4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59C23×C411yescharacteristic
224C2×C4×C59C22×D411yescharacteristic
225C2×C4×C59C2×#4111yescharacteristic
226C22⋊D59—42no—
227C22⋊D59—42no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59C4×D411yescharacteristic
233C2×Dic59C4×D411yescharacteristic
234C2×Dic59C4×D411yescharacteristic
235C4×D59—22no—
236C4×D59—21no—
237C4×D59—21no—
238C4×D59—21no—
239C4×D59—21no—
240Dic118—21no—
241Dic118—21no—
242Dic118—21no—
243Dic118—21no—
244C8×C59—21no—
245C8×C59—21no—
246C8×C59—21no—
247C8×C59—21no—
248C8×C59—41no—
249C8×C59—41no—
250C8×C59—41no—
251C59⋊C8—21no—
252C59⋊C8—21no—
253C59⋊C8—21no—
254C59⋊C8—41no—
255C59⋊C8—41no—
256C59⋊C8—41no—
257C59⋊C8#13112yes—
258#18092—591nomaximal, Sylow
259C22×C59—41no—
260C22×C59—41no—
261C22×C59—41no—
262C22×C59—41no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59C2×C4×D411yescharacteristic
266C2×D59—21no—
267C2×D59—21no—
268C2×D59#44412yes—
269C4×C59—21no—
270C4×C59—41no—
271C4×C59—41no—
272C4×C59C2×#13111yescharacteristic
273C4×C59#56811yescharacteristic
274C4×C59#57611yescharacteristic
275C4×C59#57811yescharacteristic
276Dic59—21no—
277Dic59—21no—
278Dic59—21no—
279Dic59—41no—
280Dic59—41no—
281Dic59—41no—
282Dic59—41no—
283#3168—591no—
284#2994—591no—
285#2994—591no—
286#1592—591no—
287#1592—591no—
288#1584—591no—
289#1584—591no—
290#1581—591no—
291#1581D5911yescharacteristic, complemented
292SD16⋊C8—591no—
293SD16⋊C8—591no—
294C2×#335—591no—
295#1483—591no—
296#1483—591no—
297#1451—591no—
298C2×C59—81no—
299C2×C59—81no—
300C2×C59#293511yescharacteristic
301C2×C59#295111yescharacteristic
302C2×C59C2×#44411yescharacteristic
303D59—22no—
304#529—591no—
305#529—591no—
306#499—591no—
307#495—591no—
308#488—591no—
309#488—591no—
310#467—591no—
311#466—591no—
312#465—591no—
313#465—591no—
314#446—591no—
315#446—591no—
316C8×Q8—591no—
317C8×Q8—591no—
318C4×Q16—591no—
319C4×SD16—591no—
320C4×SD16—591no—
321C4×D8C2×D5911yescharacteristic
322#436—591no—
323#436C2×D5911yescharacteristic
324#435—1181no—
325#435—1181no—
326C8×D4—591no—
327C8×D4—1181no—
328C8×D4—1181no—
329C8×D4C2×D5911yescharacteristic
330#433—1181no—
331#433—1181no—
332#431—591no—
333#424—1181no—
334#424—1181no—
335C2×C4⋊C8—591no—
336C2×C4⋊C8—591no—
337C4×M16—591no—
338C2×C4×C8—591no—
339#335—591no—
340#335—591no—
341#335—591no—
342#335C2×D5911yescharacteristic
343Q8⋊C8—591no—
344Q8⋊C8—591no—
345Q8⋊C8—591no—
346Q8⋊C8—591no—
347D4⋊C8—591no—
348D4⋊C8—591no—
349D4⋊C8C2×D5911yescharacteristic
350D4⋊C8C2×D5911yescharacteristic
351#322—591no—
352#322—591no—
353#322—591no—
354#322C2×D5911yescharacteristic
355C59#1809211yescharacteristic, complemented, Sylow
356C2×Q16—591no—
357C2×SD16—591no—
358C2×SD16—591no—
359C2×D8C4×D5911yescharacteristic
360C2×M16—21no—
361C2×M16—1181no—
362C2×M16—1181no—
363C22×C8—21no—
364C22×C8—591no—
365C22×C8—591no—
366C22×C8—1181no—
367C22×C8—1181no—
368C22×C8—1181no—
369#126—1181no—
370#126—1181no—
371#125—591no—
372#125—591no—
373#124—591no—
374#124—591no—
375#123—1181no—
376#123—1181no—
377C22⋊Q8—591no—
378C22⋊Q8—591no—
379#121—591no—
380#121—591no—
381C4×Q8—591no—
382C4×Q8—591no—
383C4×D4—1181no—
384C4×D4—1181no—
385C4×D4C22×D5911yescharacteristic
386C4×D4C22×D5911yescharacteristic
387#117—1181no—
388#117—1181no—
389C2×C42—591no—
390#107—591no—
391#107C4×D5911yescharacteristic
392#106—591no—
393#106—591no—
394C4⋊C8—591no—
395C4⋊C8—591no—
396C4⋊C8—591no—
397C4⋊C8—591no—
398C4⋊C8—1181no—
399C4⋊C8—1181no—
400C4⋊C8—1181no—
401C4⋊C8—1181no—
402C4⋊C8—1181no—
403C4⋊C8—1181no—
404C4⋊C8C22×D5911yescharacteristic
405C4⋊C8C22×D5911yescharacteristic
406#103—591no—
407#103—591no—
408#103—591no—
409#103—591no—
410#102—591no—
411#102—591no—
412#102C4×D5911yescharacteristic
413#102C4×D5911yescharacteristic
414C22⋊C8—21no—
415C22⋊C8—21no—
416C22⋊C8—591no—
417C22⋊C8—591no—
418C22⋊C8—1181no—
419C22⋊C8—1181no—
420#97—1181no—
421#97—1181no—
422#97—1181no—
423C4×C8—591no—
424C4×C8—591no—
425C4×C8—591no—
426C4×C8—591no—
427C4×C8—591no—
428C4×C8—1181no—
429C4×C8—1181no—
430C4×C8C22×D5911yescharacteristic
431C4×C8C22×D5911yescharacteristic
432C4×C8C22×D5911yescharacteristic
433C2×Q8—591no—
434C2×Q8—591no—
435C2×D4—1181no—
436C2×D4—1181no—
437C2×D4C2×C4×D5911yescharacteristic
438C2×D4C2×C4×D5911yescharacteristic
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—591no—
442C22×C4—591no—
443C22×C4—591no—
444Q16—2361no—
445SD16—1182no—
446SD16—1182no—
447D8—41no—
448M16—81no—
449M16—4721no—
450M16—2362no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—81no—
455C2×C8—591no—
456C2×C8—591no—
457C2×C8—591no—
458C2×C8—591no—
459C2×C8—1182no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—4721no—
464C2×C8—4721no—
465C2×C8—4721no—
466C2×C8—2361no—
467C2×C8C2×C4×D5911yescharacteristic
468C2×C8C2×C4×D5911yescharacteristic
469C2×C8D4×D5911yescharacteristic
470C2×C8D4×D5911yescharacteristic
471C4⋊C4—591no—
472C4⋊C4—591no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4C2×C4×D5911yescharacteristic
482C4⋊C4C2×C4×D5911yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—1181no—
486C22⋊C4—1181no—
487C22⋊C4—1181no—
488C22⋊C4—1181no—
489C22⋊C4—1181no—
490C22⋊C4—1181no—
491C22⋊C4—1181no—
492C22⋊C4—1181no—
493C42—591no—
494C42—591no—
495C42—591no—
496C42—1181no—
497C42—1181no—
498C42C23×D5911yescharacteristic
499C23—21no—
500C23—21no—
501C23—591no—
502Q8—1181no—
503Q8—1181no—
504Q8—1181no—
505Q8—1181no—
506D4—21no—
507D4—21no—
508D4—21no—
509D4—21no—
510D4—2362no—
511D4—2362no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—591no—
519C2×C4—591no—
520C2×C4—591no—
521C2×C4—1182no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4C22×C4×D5911yescharacteristic
531C2×C4C2×D4×D5911yescharacteristic
532C2×C4#41×D5911yescharacteristic
533C8—21no—
534C8—21no—
535C8—21no—
536C8—21no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—592no—
541C8—1181no—
542C8—1181no—
543C8—1181no—
544C8—2361no—
545C8—2361no—
546C8—2361no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—592no—
554C22—1181no—
555C22—1181no—
556C22C4×D4×D5911yescharacteristic
557C4—21no—
558C4—41no—
559C4—41no—
560C4—1181no—
561C4—1181no—
562C4—1181no—
563C4—2361no—
564C4—2361no—
565C4—2361no—
566C4—2361no—
567C4#131×D5911yescharacteristic
568C4#1683816811yescharacteristic
569C4#1683820311yescharacteristic
570C4#1683821611yescharacteristic
571C2—81no—
572C2—81no—
573C2—1182no—
574C2#2143283711yescharacteristic
575C2#2143296611yescharacteristic
576C2#444×D5911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1581C2
socleC22×C59C2×C4×D4
prime radical#16837635C4
cosocleC42×C59C24
hypercenter#1581D59
hypocenterC59#18092
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #21433257C2semidirect
3. #21433251C2semidirect
4. #21422998C2semidirect
5. #21422995C2semidirect
6. #21422948C2semidirect
7. #21422945C2semidirect
8. #21422926C2semidirect
9. #21422908C2semidirect
10. #21422905C2semidirect
11. #335×D59C2semidirect
12. #21422506C2semidirect
13. #21422506C2semidirect
14. #21422384C2semidirect
15. C59×#1581C2semidirect
291. #1581D59semidirect
355. C59#18092semidirect

Maximal subgroups

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

subgroupindex
#214346642
#214332572
#214332512
#214229982
#214229952
#214229482
#214229452
#214229262
#214229082
#214229052
#335×D592
#214225062
#214223842
C59×#15812
#1809259