Finite Group Databasebeta

← #28571470

Subgroups — #28571470

Order 15104.

760 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434556C211yesmaximal, characteristic, complemented
2C2×#16837682C211yesmaximal, characteristic, complemented
3D4×#11745664C211yesmaximal, characteristic, complemented
4#21423583C212yesmaximal, complemented
5#21423559C212yesmaximal, complemented
6#21423459C212yesmaximal, complemented
7#21423435C212yesmaximal, complemented
8#21422849C211yesmaximal, characteristic, complemented
9#21422773C211yesmaximal, characteristic, complemented
10#21422767C211yesmaximal, characteristic, complemented
11C59×#3100C211yesmaximal, characteristic
12C2×#14273757—21no—
13C2×#14273757—21no—
14C22×#11745664—21no—
15C2×#14273727C2211yescharacteristic
16#16837797—22no—
17#16837781C2212yescomplemented
18#16837774C2212yescomplemented
19#16837682—21no—
20#16837682—21no—
21#16837682C2212yes—
22#16837682C2212yes—
23#16837480C2212yes—
24#16837474—22no—
25D4×C59⋊C8C2212yes—
26C2×#14273660—21no—
27#16837426C2211yescharacteristic
28C2×C4⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273648—21no—
30#16837366—21no—
31C4×#11745664C2211yescharacteristic
32#16837154C2212yescomplemented
33#16837152C2212yescomplemented
34#16837148C2212yescomplemented
35#16837147C2212yescomplemented
36#16837147C2212yescomplemented
37#16837146C2212yescomplemented
38C59×#532C2211yescharacteristic
39C2×C4×D4×C59C2211yescharacteristic
40C59×#480—21no—
41C59×#474C2211yescharacteristic
42C59×#474C2211yescharacteristic
43C59×#459C2211yescharacteristic
44C59×#459C2211yescharacteristic
45C59×#447—21no—
46C4×D8×C59—21no—
47C4×D8×C59—21no—
48C59×#431C2211yescharacteristic
49C22×Dic118—21no—
50#14273757—44no—
51#14273757—44no—
52C2×#11745672—22no—
53C2×#11745672—22no—
54C2×#11745670—22no—
55C2×#11745670—22no—
56C2×#11745664—21no—
57C2×#11745664—22no—
58C2×#11745664—44no—
59C2×#11745664D411yescharacteristic
60C2×#11745664D411yescharacteristic
61C22×C59⋊C8—21no—
62C22×C59⋊C8—22no—
63#14273727—21no—
64#14273727—21no—
65#14273727—21no—
66#14273727C2312yes—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69#14273660—22no—
70#14273660—22no—
71C4⋊(C59⋊C8)—21no—
72C4⋊(C59⋊C8)—21no—
73C4⋊(C59⋊C8)C2312yes—
74C4⋊(C59⋊C8)C2312yes—
75#14273648—22no—
76#14273648—22no—
77C22⋊(C59⋊C8)—22no—
78C22⋊(C59⋊C8)—22no—
79#14273638D412yes—
80C4×C59⋊C8C2312yes—
81C2×#41×C59—21no—
82C22×D4×C59D411yescharacteristic
83C23×C4×C59—21no—
84C2×D8×C59—21no—
85C2×D8×C59—21no—
86C4⋊Q8×C59C2311yescharacteristic
87#127×C59C2311yescharacteristic
88#124×C59—21no—
89#121×C59—21no—
90#121×C59—21no—
91C4×D4×C59—21no—
92C4×D4×C59—21no—
93C4×D4×C59—21no—
94C4×D4×C59—21no—
95C4×D4×C59D411yescharacteristic
96C4×D4×C59D411yescharacteristic
97C4×D4×C59C2311yescharacteristic
98C4×D4×C59C2311yescharacteristic
99C2×C4⋊C4×C59D411yescharacteristic
100C2×C22⋊C4×C59—21no—
101C2×C42×C59C2311yescharacteristic
102#107×C59—21no—
103#107×C59—21no—
104C4⋊C8×C59C2311yescharacteristic
105C4⋊C8×C59C2311yescharacteristic
106#102×C59—21no—
107#102×C59—21no—
108#102×C59—21no—
109#102×C59—21no—
110C22⋊C8×C59—21no—
111C4×C8×C59—21no—
112C24×C59—21no—
113#41×C59—42no—
114#41×C59—42no—
115C2×Q8×C59—21no—
116C2×D4×C59—22no—
117C2×D4×C59—21no—
118C2×D4×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—41no—
122C2×D4×C59—41no—
123C2×D4×C59C2×D411yescharacteristic
124C2×D4×C59C2×D411yescharacteristic
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—42no—
131C22×C4×C59—42no—
132C22×C4×C59—41no—
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59#4111yescharacteristic
136C22×Dic59—41no—
137C2×Dic118—22no—
138C2×Dic118—42no—
139C2×Dic118—42no—
140C2×Dic118—41no—
141D8×C59—42no—
142D8×C59—42no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59C2×D411yescharacteristic
149C4⋊C4×C59C2×D411yescharacteristic
150C22⋊C4×C59—21no—
151C22⋊C4×C59—21no—
152C22⋊C4×C59—21no—
153C22⋊C4×C59—21no—
154C22⋊C4×C59—41no—
155C22⋊C4×C59—41no—
156C42×C59C2×D411yescharacteristic
157C42×C59C2×D411yescharacteristic
158C42×C59#4111yescharacteristic
159C42×C59C2411yescharacteristic
160#11745672—44no—
161#11745672—44no—
162#11745670—44no—
163#11745670—44no—
164C4⋊Dic59—22no—
165C4⋊Dic59—22no—
166C4⋊Dic59—41no—
167C4⋊Dic59—41no—
168#11745664—22no—
169#11745664—42no—
170#11745664—44no—
171#11745664D814yescomplemented
172C2×C59⋊C8—22no—
173C2×C59⋊C8—22no—
174C2×C59⋊C8—42no—
175C2×C59⋊C8—44no—
176C2×C59⋊C8C2×D412yes—
177C2×C59⋊C8C2×D412yes—
178C23×C59—21no—
179C23×C59—21no—
180C23×C59—42no—
181C23×C59—42no—
182C23×C59—41no—
183C23×C59—81no—
184C23×C59#12111yescharacteristic
185Q8×C59—41no—
186Q8×C59—41no—
187D4×C59—21no—
188D4×C59—21no—
189D4×C59—21no—
190D4×C59—42no—
191D4×C59—41no—
192D4×C59—41no—
193D4×C59—41no—
194D4×C59—41no—
195D4×C59—81no—
196D4×C59—82no—
197D4×C59#13712yescomplemented
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—41no—
203C2×C4×C59—42no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—42no—
209C2×C4×C59—42no—
210C2×C4×C59—41no—
211C2×C4×C59—82no—
212C2×C4×C59#12111yescharacteristic
213C2×C4×C59#12111yescharacteristic
214C2×C4×C59#12111yescharacteristic
215C2×C4×C59C2×D812yes—
216C2×C4×C59AGL(1,ℤ/8)12yes—
217C2×C4×C59C22×D411yescharacteristic
218C2×C4×C59C22×D411yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C2×Dic59—42no—
221C2×Dic59—82no—
222Dic118—42no—
223Dic118—42no—
224Dic118—82no—
225C8×C59—41no—
226C8×C59—41no—
227C59⋊C8—22no—
228C59⋊C8—42no—
229C59⋊C8—42no—
230C59⋊C8C2×D814yes—
231#19163—591nomaximal, Sylow
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—22no—
235C22×C59—41no—
236C22×C59—41no—
237C22×C59—41no—
238C22×C59—41no—
239C22×C59—41no—
240C22×C59—84no—
241C22×C59—81no—
242C22×C59—81no—
243C22×C59—81no—
244C22×C59#45912yes—
245C22×C59C2×#12111yescharacteristic
246C4×C59—21no—
247C4×C59—21no—
248C4×C59—21no—
249C4×C59—41no—
250C4×C59—41no—
251C4×C59—81no—
252C4×C59#45912yes—
253C4×C59C22×D811yescharacteristic
254C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C59C2×#13711yescharacteristic
256C4×C59#57511yescharacteristic
257Dic59—82no—
258#3152—591no—
259#3100D5911yescharacteristic
260C2×#474—591no—
261D4×M16—591no—
262#1689—592no—
263#1685—592no—
264#1669—592no—
265#1665—592no—
266#1565—591no—
267#1547—591no—
268#1546—591no—
269C2×C59—22no—
270C2×C59—42no—
271C2×C59—81no—
272C2×C59—161no—
273C2×C59C2×#45911yescharacteristic
274C2×C59#305611yescharacteristic
275C2×C59#306911yescharacteristic
276C2×#137—1181no—
277C2×#137—1181no—
278C22×M16—1181no—
279#532C2×D5911yescharacteristic
280C2×C4⋊Q8—591no—
281C2×C4×D4C2×D5911yescharacteristic
282#497—1182no—
283#494—592no—
284#492—592no—
285#480—21no—
286#474—592no—
287#474—592no—
288#474—1181no—
289#474—1181no—
290#474C2×D5911yescharacteristic
291#474C2×D5911yescharacteristic
292#459C2×D5911yescharacteristic
293#459C2×D5911yescharacteristic
294#447—21no—
295C4×D8—21no—
296C4×D8—21no—
297#436—592no—
298#435—1182no—
299C8×D4—592no—
300#431C2×D5911yescharacteristic
301C2×#107—1181no—
302#423—591no—
303C2×C4⋊C8—591no—
304C2×#102—1181no—
305#407—1181no—
306C4×M16—591no—
307#335—592no—
308#334—592no—
309#332—592no—
310#332—592no—
311#332—592no—
312#331—592no—
313C59#1916311yescharacteristic, complemented, Sylow
314C2×#41—21no—
315C22×Q8—1181no—
316C22×D4C22⋊D5911yescharacteristic
317C23×C4—21no—
318#137—2364no—
319#137—2364no—
320C2×Q16—1182no—
321C2×Q16—1182no—
322C2×SD16—1182no—
323C2×SD16—1182no—
324C2×D8—21no—
325C2×D8—21no—
326C2×M16—591no—
327C2×M16—591no—
328C2×M16—1182no—
329C2×M16—1181no—
330C2×M16—2364no—
331C22×C8—1182no—
332C22×C8—1181no—
333C4⋊Q8—592no—
334C4⋊Q8—1181no—
335C4⋊Q8—1181no—
336C4⋊Q8—1181no—
337C4⋊Q8C22×D5911yescharacteristic
338#127C22×D5911yescharacteristic
339#124—21no—
340#121—21no—
341#121—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4C22⋊D5911yescharacteristic
347C4×D4C22⋊D5911yescharacteristic
348C4×D4C22×D5911yescharacteristic
349C4×D4C22×D5911yescharacteristic
350C2×C4⋊C4—1181no—
351C2×C4⋊C4—1181no—
352C2×C4⋊C4C22⋊D5911yescharacteristic
353C2×C22⋊C4—21no—
354C2×C42C22×D5911yescharacteristic
355#107—21no—
356#107—21no—
357#107—1182no—
358#107—1182no—
359C4⋊C8—592no—
360C4⋊C8—592no—
361C4⋊C8—1181no—
362C4⋊C8—1181no—
363C4⋊C8C22×D5911yescharacteristic
364C4⋊C8C22×D5911yescharacteristic
365#102—21no—
366#102—21no—
367#102—21no—
368#102—21no—
369#102—1182no—
370#102—1182no—
371C22⋊C8—21no—
372C22⋊C8—1182no—
373C22⋊C8—1182no—
374#97—592no—
375C4×C8—21no—
376C4×C8—592no—
377C24—21no—
378#41—42no—
379#41—42no—
380C2×Q8—21no—
381C2×Q8—1182no—
382C2×Q8—2362no—
383C2×Q8—2362no—
384C2×Q8—2361no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4C2×C22⋊D5911yescharacteristic
393C2×D4C2×C22⋊D5911yescharacteristic
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—2361no—
403C22×C4D4×D5911yescharacteristic
404C22×C4#1174568711yescharacteristic
405C22×C4C2×C22⋊D5911yescharacteristic
406Q16—2364no—
407Q16—2364no—
408SD16—2364no—
409SD16—2364no—
410D8—42no—
411D8—42no—
412M16—594no—
413M16—1182no—
414M16—2364no—
415M16—2362no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—592no—
419C2×C8—592no—
420C2×C8—1182no—
421C2×C8—1182no—
422C2×C8—2364no—
423C2×C8—2362no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—1182no—
428C4⋊C4—1182no—
429C4⋊C4—2361no—
430C4⋊C4—2361no—
431C4⋊C4C2×C22⋊D5911yescharacteristic
432C4⋊C4C2×C22⋊D5911yescharacteristic
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C42D4×D5911yescharacteristic
440C42#1174568711yescharacteristic
441C42C2×C22⋊D5911yescharacteristic
442C42C23×D5911yescharacteristic
443C23—21no—
444C23—21no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—81no—
449C23#1427369911yescharacteristic
450Q8—41no—
451Q8—41no—
452Q8—4722no—
453Q8—2362no—
454Q8—2362no—
455D4—21no—
456D4—21no—
457D4—21no—
458D4—42no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—81no—
464D4—82no—
465D4#1427375712yescomplemented
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—82no—
480C2×C4—4722no—
481C2×C4—2362no—
482C2×C4#1427369911yescharacteristic
483C2×C4#1427369911yescharacteristic
484C2×C4#1427369911yescharacteristic
485C2×C4D8×D5912yes—
486C2×C4#1427375112yes—
487C2×C4C2×D4×D5911yescharacteristic
488C2×C4C22×C22⋊D5911yescharacteristic
489C2×C4C2×#1174568711yescharacteristic
490C8—41no—
491C8—41no—
492C8—594no—
493C8—1182no—
494C8—2362no—
495C8—2362no—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—84no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22#1683759512yes—
509C22C2×#1427369911yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—4722no—
517C4#1683759512yes—
518C4C2×D8×D5911yescharacteristic
519C4C2×#1427375111yescharacteristic
520C4C2×#1427375711yescharacteristic
521C4#1683820011yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—161no—
526C2C2×#1683759511yescharacteristic
527C2#2143372811yescharacteristic
528C2#2143383511yescharacteristic
subgroupG/H
centerC22C2×#14273699
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
Fitting, prime radicalC59×#3100C2
socleC22×C59C2×#121
hypercenter#3100D59
hypocenterC59#19163
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #21434556C2semidirect
2. C2×#16837682C2semidirect
3. D4×#11745664C2semidirect
4. #21423583C2semidirect
5. #21423559C2semidirect
6. #21423459C2semidirect
7. #21423435C2semidirect
8. #21422849C2semidirect
9. #21422773C2semidirect
10. #21422767C2semidirect
17. #16837781C22semidirect
18. #16837774C22semidirect
32. #16837154C22semidirect
33. #16837152C22semidirect
34. #16837148C22semidirect
35. #16837147C22semidirect
36. #16837147C22semidirect
37. #16837146C22semidirect
171. #11745664D8semidirect
197. D4×C59#137semidirect
313. C59#19163semidirect
465. D4#14273757semidirect

Maximal subgroups

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

subgroupindex
#214345562
C2×#168376822
D4×#117456642
#214235832
#214235592
#214234592
#214234352
#214228492
#214227732
#214227672
C59×#31002
#1916359