Finite Group Databasebeta

← #28560648

Subgroups — #28560648

Order 15104.

572 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434696C211yesmaximal, characteristic
2#21433258C211yesmaximal, characteristic
3#21423003C211yesmaximal, characteristic, complemented
4#21422986C211yesmaximal, characteristic, complemented
5#21422974C211yesmaximal, characteristic, complemented
6#21422965C211yesmaximal, characteristic, complemented
7#21422946C211yesmaximal, characteristic, complemented
8#21422930C211yesmaximal, characteristic, complemented
9#21422906C211yesmaximal, characteristic, complemented
10C8×#11745672C211yesmaximal, characteristic, complemented
11#21422849C211yesmaximal, characteristic, complemented
12#21422502C211yesmaximal, characteristic
13#21422499C211yesmaximal, characteristic, complemented
14#21422391C211yesmaximal, characteristic, complemented
15C59×#2986C211yesmaximal, characteristic, complemented
16#16837936C2211yescharacteristic
17#16837808C411yescharacteristic
18#16837794C411yescharacteristic
19#16837770C411yescharacteristic
20#16837764C411yescharacteristic
21#16837658C411yescharacteristic
22#16837658C411yescharacteristic
23#16837650C411yescharacteristic
24#16837650C411yescharacteristic
25#16837532C2211yescharacteristic
26C8×Dic118C2211yescharacteristic
27C4×#11745672C2211yescharacteristic
28C4×#11745672C2211yescharacteristic
29C4×#11745671C2211yescharacteristic
30C4×#11745671C2211yescharacteristic
31#16837481C2211yescharacteristic
32#16837470—21no—
33C8×D236C2211yescharacteristic
34(C22⋊D59)⋊C8—21no—
35#16837462—21no—
36#16837455C2211yescharacteristic
37#16837427—21no—
38C2×C4⋊(C59⋊C8)C2211yescharacteristic
39C4×#11745664C2211yescharacteristic
40#16837154C2211yescharacteristic
41#16837154C2211yescharacteristic
42#16837152C2211yescharacteristic
43#16837152C2211yescharacteristic
44Q8⋊(C59⋊C8)C2211yescharacteristic
45Q8⋊(C59⋊C8)C2211yescharacteristic
46Q8⋊(C59⋊C8)C2211yescharacteristic
47Q8⋊(C59⋊C8)C2211yescharacteristic
48#16837132C2211yescharacteristic
49#16837132C2211yescharacteristic
50#16837129C2211yescharacteristic
51#16837129C2211yescharacteristic
52#16837124C2211yescharacteristic
53#16837124C2211yescharacteristic
54#16837123C2211yescharacteristic
55C8×C59⋊C8C2211yescharacteristic
56C59×#533C2211yescharacteristic
57C59×#446—21no—
58C59×#446C2211yescharacteristic
59C59×#446C2211yescharacteristic
60C8×Q8×C59—21no—
61C8×Q8×C59C2211yescharacteristic
62C8×Q8×C59C2211yescharacteristic
63C59×#433—21no—
64C59×#433—21no—
65C59×#431C2211yescharacteristic
66C59×#431C2211yescharacteristic
67C2×#11745672C2×C411yescharacteristic
68C2×#11745672C2×C411yescharacteristic
69C2×#11745671C2×C411yescharacteristic
70C2×#11745671C2×C411yescharacteristic
71C2×#11745664C2×C411yescharacteristic
72C2×#11745664C2×C411yescharacteristic
73C2×#11745659—21no—
74C22×C59⋊C8—21no—
75C2×C8×D59—21no—
76#14273721—21no—
77#14273719C2×C411yescharacteristic
78#14273715C2×C411yescharacteristic
79#14273710—21no—
80#14273706C2×C411yescharacteristic
81#14273701C2×C411yescharacteristic
82C4×Dic118C2311yescharacteristic
83C4×D236C2311yescharacteristic
84C4×C22⋊D59—21no—
85#14273684—21no—
86#14273660C2×C411yescharacteristic
87#14273660C2×C411yescharacteristic
88#14273658C2×C411yescharacteristic
89#14273658C2×C411yescharacteristic
90C4⋊(C59⋊C8)—21no—
91C4⋊(C59⋊C8)—21no—
92C4⋊(C59⋊C8)C2311yescharacteristic
93C4⋊(C59⋊C8)C2311yescharacteristic
94Dic59⋊C8—21no—
95Dic59⋊C8—21no—
96#14273651C2×C411yescharacteristic
97#14273651C2×C411yescharacteristic
98#14273651C2×C411yescharacteristic
99#14273651C2×C411yescharacteristic
100#14273650C2×C411yescharacteristic
101#14273650C2×C411yescharacteristic
102#14273647C2×C411yescharacteristic
103#14273647C2×C411yescharacteristic
104C22⋊(C59⋊C8)—21no—
105(C2×D59)⋊C8—21no—
106(C2×D59)⋊C8—21no—
107#14273638C2×C411yescharacteristic
108#14273638C2×C411yescharacteristic
109#14273637—21no—
110C4×C59⋊C8C2311yescharacteristic
111C4×C59⋊C8C2311yescharacteristic
112C4×C59⋊C8C2311yescharacteristic
113C4×C59⋊C8C2311yescharacteristic
114C8×Dic59—21no—
115C4⋊Q8×C59C2×C411yescharacteristic
116C4⋊Q8×C59C2×C411yescharacteristic
117#125×C59—21no—
118C22⋊Q8×C59—21no—
119C22⋊Q8×C59C2×C411yescharacteristic
120C22⋊Q8×C59C2×C411yescharacteristic
121C4×Q8×C59—21no—
122C4×Q8×C59C2311yescharacteristic
123C4×Q8×C59C2311yescharacteristic
124#117×C59—21no—
125C2×C42×C59C2311yescharacteristic
126C4⋊C8×C59—21no—
127C4⋊C8×C59—21no—
128C4⋊C8×C59—21no—
129C4⋊C8×C59—21no—
130C4⋊C8×C59D411yescharacteristic
131C4⋊C8×C59D411yescharacteristic
132C4⋊C8×C59C2311yescharacteristic
133C4⋊C8×C59C2311yescharacteristic
134C22⋊C8×C59—21no—
135C22⋊C8×C59D411yescharacteristic
136C22⋊C8×C59D411yescharacteristic
137#97×C59—21no—
138#97×C59—21no—
139C4×C8×C59—21no—
140C4×C8×C59—21no—
141C4×C8×C59—21no—
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59—21no—
145C2×Q8×C59C22×C411yescharacteristic
146C2×Q8×C59C22×C411yescharacteristic
147C22×C4×C59C22×C411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59#4111yescharacteristic
150C2×C22⋊D59—21no—
151C2×D236C22×C411yescharacteristic
152C2×C4×D59—21no—
153C2×Dic118C22×C411yescharacteristic
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59C2×D411yescharacteristic
157C2×C8×C59C2×D411yescharacteristic
158C2×C8×C59C2×D411yescharacteristic
159C2×C8×C59C2×D411yescharacteristic
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59C22×C411yescharacteristic
168C4⋊C4×C59C22×C411yescharacteristic
169C22⋊C4×C59—21no—
170C22⋊C4×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59C22×C411yescharacteristic
174C42×C59C2×D411yescharacteristic
175C42×C59#4111yescharacteristic
176C42×C59C2411yescharacteristic
177C22⋊Dic59—21no—
178#11745672—22no—
179#11745672—41no—
180#11745671—22no—
181#11745671—41no—
182(C2×D59)⋊C4—21no—
183(C2×D59)⋊C4—21no—
184(C2×D59)⋊C4—21no—
185(C2×D59)⋊C4—21no—
186C4⋊Dic59C22×C411yescharacteristic
187C4⋊Dic59C22×C411yescharacteristic
188Dic59⋊C4—21no—
189Dic59⋊C4—21no—
190Dic59⋊C4—21no—
191Dic59⋊C4—21no—
192C4×Dic59—21no—
193#11745664—22no—
194#11745664—41no—
195C2×C59⋊C8—21no—
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⋊C8C22×C411yescharacteristic
203C2×C59⋊C8C22×C411yescharacteristic
204#11745659—81no—
205C8×D59—81no—
206C23×C59C4×D411yescharacteristic
207C22×D59—21no—
208Q8×C59—21no—
209Q8×C59—21no—
210Q8×C59—21no—
211Q8×C59—21no—
212Q8×C59—42no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—41no—
222C2×C4×C59C4×D411yescharacteristic
223C2×C4×C59C4×D411yescharacteristic
224C2×C4×C59C4×D411yescharacteristic
225C2×C4×C59C23×C411yescharacteristic
226C2×C4×C59C22×D411yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C22⋊D59—42no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232D236—21no—
233D236—21no—
234C4×D59—41no—
235C4×D59—41no—
236Dic118—21no—
237Dic118—21no—
238C8×C59—21no—
239C8×C59—21no—
240C8×C59—21no—
241C8×C59—21no—
242C8×C59—41no—
243C8×C59—41no—
244C59⋊C8—21no—
245C59⋊C8—21no—
246C59⋊C8—21no—
247C59⋊C8—41no—
248C59⋊C8—41no—
249C59⋊C8—41no—
250C59⋊C8—41no—
251C59⋊C8#13112yes—
252#18083—591nomaximal, Sylow
253C22×C59—21no—
254C22×C59—21no—
255C22×C59—21no—
256C22×C59C2×C4×D411yescharacteristic
257C2×D59—41no—
258C2×D59—41no—
259C2×D59—41no—
260C4×C59—21no—
261C4×C59—21no—
262C4×C59—21no—
263C4×C59—21no—
264C4×C59—41no—
265C4×C59—41no—
266C4×C59—41no—
267C4×C59—41no—
268C4×C59C2×#13111yescharacteristic
269C4×C59#56811yescharacteristic
270C4×C59#57711yescharacteristic
271C4×C59#57811yescharacteristic
272Dic59—41no—
273Dic59—41no—
274Dic59—41no—
275#3172—591no—
276#2994—591no—
277#2986D5911yescharacteristic, complemented
278#1593—591no—
279#1590—591no—
280#1588—591no—
281#1587—591no—
282#1584—591no—
283#1582—591no—
284SD16⋊C8—591no—
285C8×Q16—591no—
286#1565—591no—
287#1482—591no—
288#1481—591no—
289#1453—591no—
290C2×C59—41no—
291C2×C59#293511yescharacteristic
292C2×C59#294711yescharacteristic
293C2×C59#295511yescharacteristic
294D59—81no—
295#533C2×D5911yescharacteristic
296#529—591no—
297#501—591no—
298#497—591no—
299#491—591no—
300#489—591no—
301#470—591no—
302#470—591no—
303#468—591no—
304#468—591no—
305#446—21no—
306#446—591no—
307#446C2×D5911yescharacteristic
308#446C2×D5911yescharacteristic
309C8×Q8—21no—
310C8×Q8—591no—
311C8×Q8C2×D5911yescharacteristic
312C8×Q8C2×D5911yescharacteristic
313C4×Q16—591no—
314C4×Q16—591no—
315C4×SD16—591no—
316C4×SD16—591no—
317#436—591no—
318#435—1181no—
319C8×D4—591no—
320C8×D4—1181no—
321#433—21no—
322#433—21no—
323#433—1181no—
324#431—591no—
325#431C2×D5911yescharacteristic
326#431C2×D5911yescharacteristic
327#424—1181no—
328C2×C4⋊C8—591no—
329C4×M16—591no—
330#335—591no—
331#335—591no—
332#334—591no—
333#334—591no—
334Q8⋊C8—591no—
335Q8⋊C8—591no—
336Q8⋊C8—591no—
337Q8⋊C8—591no—
338Q8⋊C8—591no—
339Q8⋊C8—591no—
340D4⋊C8—591no—
341D4⋊C8—591no—
342#322—591no—
343#322—591no—
344#322—591no—
345C82—591no—
346C59#1808311yescharacteristic, complemented, Sylow
347C2×Q16—591no—
348C2×Q16—591no—
349C2×SD16—591no—
350C2×SD16—591no—
351C2×M16—591no—
352C2×M16—591no—
353C2×M16—1181no—
354C22×C8—1181no—
355C22×C8—1181no—
356C4⋊Q8C4×D5911yescharacteristic
357C4⋊Q8C4×D5911yescharacteristic
358#126—1181no—
359#125—21no—
360#125—591no—
361#124—591no—
362#123—1181no—
363C22⋊Q8—21no—
364C22⋊Q8—591no—
365C22⋊Q8C4×D5911yescharacteristic
366C22⋊Q8C4×D5911yescharacteristic
367#121—591no—
368C4×Q8—21no—
369C4×Q8—591no—
370C4×Q8C22×D5911yescharacteristic
371C4×Q8C22×D5911yescharacteristic
372C4×D4—591no—
373C4×D4—1181no—
374#117—21no—
375#117—1181no—
376C2×C42C22×D5911yescharacteristic
377#107—591no—
378#107—591no—
379#106—591no—
380#106—591no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—591no—
386C4⋊C8—591no—
387C4⋊C8—1181no—
388C4⋊C8—1181no—
389C4⋊C8—1181no—
390C4⋊C8—1181no—
391C4⋊C8C22⋊D5911yescharacteristic
392C4⋊C8C22⋊D5911yescharacteristic
393C4⋊C8C22×D5911yescharacteristic
394C4⋊C8C22×D5911yescharacteristic
395#103—591no—
396#103—591no—
397#103—591no—
398#103—591no—
399#103—591no—
400#103—591no—
401#102—591no—
402#102—591no—
403C22⋊C8—21no—
404C22⋊C8—1181no—
405C22⋊C8—1181no—
406C22⋊C8—1181no—
407C22⋊C8C22⋊D5911yescharacteristic
408C22⋊C8C22⋊D5911yescharacteristic
409#97—21no—
410#97—21no—
411#97—591no—
412#97—591no—
413#97—1181no—
414C4×C8—21no—
415C4×C8—21no—
416C4×C8—21no—
417C4×C8—591no—
418C4×C8—591no—
419C4×C8—591no—
420C4×C8—591no—
421C4×C8—1181no—
422C4×C8C22×D5911yescharacteristic
423C4×C8C22×D5911yescharacteristic
424C2×Q8—21no—
425C2×Q8—591no—
426C2×Q8C2×C4×D5911yescharacteristic
427C2×Q8C2×C4×D5911yescharacteristic
428C2×D4—591no—
429C2×D4—1181no—
430C22×C4—1181no—
431C22×C4C2×C4×D5911yescharacteristic
432C22×C4#1174568511yescharacteristic
433C22×C4C2×C22⋊D5911yescharacteristic
434Q16—1182no—
435Q16—2361no—
436SD16—1182no—
437SD16—2361no—
438M16—1182no—
439M16—4721no—
440M16—2361no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—591no—
444C2×C8—591no—
445C2×C8—591no—
446C2×C8—591no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—4721no—
452C2×C8—4721no—
453C2×C8C2×C22⋊D5911yescharacteristic
454C2×C8C2×C22⋊D5911yescharacteristic
455C2×C8C2×C22⋊D5911yescharacteristic
456C2×C8C2×C22⋊D5911yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—591no—
465C4⋊C4—591no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4C2×C4×D5911yescharacteristic
471C4⋊C4C2×C4×D5911yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—1181no—
475C22⋊C4—1181no—
476C22⋊C4—1181no—
477C22⋊C4—1181no—
478C22⋊C4—1181no—
479C42—21no—
480C42—21no—
481C42—1181no—
482C42C2×C4×D5911yescharacteristic
483C42#1174568511yescharacteristic
484C42C2×C22⋊D5911yescharacteristic
485C42C23×D5911yescharacteristic
486C23—1181no—
487C23C4×C22⋊D5911yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—42no—
493Q8—1181no—
494Q8—1181no—
495D4—1181no—
496D4—1181no—
497D4—2362no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—1181no—
508C2×C4—1181no—
509C2×C4—1181no—
510C2×C4—2361no—
511C2×C4—2361no—
512C2×C4C4×C22⋊D5911yescharacteristic
513C2×C4C4×C22⋊D5911yescharacteristic
514C2×C4C4×C22⋊D5911yescharacteristic
515C2×C4C22×C4×D5911yescharacteristic
516C2×C4C22×C22⋊D5911yescharacteristic
517C2×C4C2×#1174568511yescharacteristic
518C8—21no—
519C8—21no—
520C8—21no—
521C8—21no—
522C8—41no—
523C8—41no—
524C8—592no—
525C8—1181no—
526C8—1181no—
527C8—1181no—
528C8—2361no—
529C8—2361no—
530C8—2361no—
531C8—2361no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—2361no—
536C22—2361no—
537C22—2361no—
538C22C2×C4×C22⋊D5911yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—2361no—
548C4—2361no—
549C4—2361no—
550C4#131×D5911yescharacteristic
551C4#1683816711yescharacteristic
552C4#1683821011yescharacteristic
553C4#1683821711yescharacteristic
554C2—41no—
555C2—4721no—
556C2#2143283811yescharacteristic
557C2#2143293911yescharacteristic
558C2#2143299711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#2986C2
socleC22×C59C2×C4×D4
prime radical#14273701C2×C4
cosocleC42×C59C24
hypercenter#2986D59
hypocenterC59#18083
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2986

Complements

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

NH ≅ G/Ntype
3. #21423003C2semidirect
4. #21422986C2semidirect
5. #21422974C2semidirect
6. #21422965C2semidirect
7. #21422946C2semidirect
8. #21422930C2semidirect
9. #21422906C2semidirect
10. C8×#11745672C2semidirect
11. #21422849C2semidirect
13. #21422499C2semidirect
14. #21422391C2semidirect
15. C59×#2986C2semidirect
277. #2986D59semidirect
346. C59#18083semidirect

Maximal subgroups

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

subgroupindex
#214346962
#214332582
#214230032
#214229862
#214229742
#214229652
#214229462
#214229302
#214229062
C8×#117456722
#214228492
#214225022
#214224992
#214223912
C59×#29862
#1808359