Finite Group Databasebeta

← #28559338

Subgroups — #28559338

Order 15104.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434702C211yesmaximal, characteristic
2#21433247C211yesmaximal, characteristic
3#21433190C211yesmaximal, characteristic, complemented
4#21423006C211yesmaximal, characteristic, complemented
5#21423000C211yesmaximal, characteristic, complemented
6#21422994C211yesmaximal, characteristic, complemented
7#21422988C211yesmaximal, characteristic, complemented
8#21422916C211yesmaximal, characteristic, complemented
9#21422910C211yesmaximal, characteristic, complemented
10C8×Dic236C211yesmaximal, characteristic, complemented
11C8×#11745660C211yesmaximal, characteristic, complemented
12#21422863C211yesmaximal, characteristic, complemented
13#21422557C211yesmaximal, characteristic
14#21422552C211yesmaximal, characteristic, complemented
15C8×M16×C59C211yesmaximal, characteristic, complemented
16#16837959C2211yescharacteristic
17#16837936C2211yescharacteristic
18#16837811C411yescharacteristic
19#16837796C411yescharacteristic
20#16837771C411yescharacteristic
21#16837762C411yescharacteristic
22#16837660C411yescharacteristic
23#16837660C411yescharacteristic
24#16837648C411yescharacteristic
25#16837648C411yescharacteristic
26#16837532C2211yescharacteristic
27#16837532C2211yescharacteristic
28#16837532C2211yescharacteristic
29#16837531—21no—
30#16837531—21no—
31C8×Dic118C2211yescharacteristic
32C8×Dic118C2211yescharacteristic
33C8×Dic118C2211yescharacteristic
34C4×Dic236C2211yescharacteristic
35C4×Dic236C2211yescharacteristic
36C4×#11745660C2211yescharacteristic
37C4×#11745660C2211yescharacteristic
38#16837481C2211yescharacteristic
39#16837477—21no—
40#16837476—21no—
41C8×D236C2211yescharacteristic
42#16837462—21no—
43#16837460—21no—
44#16837460—21no—
45#16837459C2211yescharacteristic
46#16837427—21no—
47#16837426C2211yescharacteristic
48#16837155C2211yescharacteristic
49#16837155C2211yescharacteristic
50#16837153C2211yescharacteristic
51#16837153C2211yescharacteristic
52#16837134C2211yescharacteristic
53#16837134C2211yescharacteristic
54#16837134C2211yescharacteristic
55#16837134C2211yescharacteristic
56#16837134C2211yescharacteristic
57#16837134C2211yescharacteristic
58#16837131C2211yescharacteristic
59#16837131C2211yescharacteristic
60C59×#431C2211yescharacteristic
61C4×M16×C59C2211yescharacteristic
62C2×C4×C8×C59C2211yescharacteristic
63C59×#322C2211yescharacteristic
64C59×#322C2211yescharacteristic
65C82×C59C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×Dic236C2×C411yescharacteristic
68C2×Dic236C2×C411yescharacteristic
69C2×#11745660C2×C411yescharacteristic
70C2×#11745660C2×C411yescharacteristic
71C2×#11745664—21no—
72C2×#11745659—21no—
73C2×C8×D59—21no—
74#14273727C2×C411yescharacteristic
75#14273727C2×C411yescharacteristic
76#14273721—21no—
77#14273719—21no—
78#14273719C2×C411yescharacteristic
79#14273715C2×C411yescharacteristic
80#14273710—21no—
81#14273706—21no—
82#14273706C2×C411yescharacteristic
83#14273706C2×C411yescharacteristic
84#14273706C2×C411yescharacteristic
85#14273701C2×C411yescharacteristic
86C4×Dic118—21no—
87C4×Dic118C2311yescharacteristic
88C4×Dic118C2311yescharacteristic
89C4×Dic118C2311yescharacteristic
90C4×D236C2311yescharacteristic
91C4×C22⋊D59—21no—
92#14273686—21no—
93#14273684—21no—
94#14273661C2×C411yescharacteristic
95#14273661C2×C411yescharacteristic
96#14273659C2×C411yescharacteristic
97#14273659C2×C411yescharacteristic
98C4⋊(C59⋊C8)—21no—
99C4⋊(C59⋊C8)C2311yescharacteristic
100C4⋊(C59⋊C8)C2311yescharacteristic
101C4⋊(C59⋊C8)C2311yescharacteristic
102C4⋊(C59⋊C8)C2311yescharacteristic
103Dic59⋊C8—21no—
104Dic59⋊C8—21no—
105Dic59⋊C8—21no—
106Dic59⋊C8—21no—
107Dic59⋊C8—21no—
108Dic59⋊C8—21no—
109#14273652C2×C411yescharacteristic
110#14273652C2×C411yescharacteristic
111#14273652C2×C411yescharacteristic
112#14273652C2×C411yescharacteristic
113#14273652C2×C411yescharacteristic
114#14273652C2×C411yescharacteristic
115#14273649C2×C411yescharacteristic
116#14273649C2×C411yescharacteristic
117C22⋊(C59⋊C8)—21no—
118(C2×D59)⋊C8—21no—
119(C2×D59)⋊C8—21no—
120#14273638—21no—
121#14273637—21no—
122#14273637—21no—
123#14273637—21no—
124C8×Dic59—21no—
125C8×Dic59—21no—
126C8×Dic59—21no—
127C2×M16×C59C2×C411yescharacteristic
128C2×M16×C59C2×C411yescharacteristic
129C22×C8×C59—21no—
130C2×C42×C59C2311yescharacteristic
131C4⋊C8×C59D411yescharacteristic
132C4⋊C8×C59D411yescharacteristic
133C22⋊C8×C59D411yescharacteristic
134C22⋊C8×C59D411yescharacteristic
135#97×C59C2×C411yescharacteristic
136#97×C59C2×C411yescharacteristic
137C4×C8×C59—21no—
138C4×C8×C59C2311yescharacteristic
139C4×C8×C59C2311yescharacteristic
140C4×C8×C59C2311yescharacteristic
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C22×C4×C59C22×C411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59#4111yescharacteristic
147C2×C22⋊D59—21no—
148C2×D236C22×C411yescharacteristic
149C2×C4×D59—21no—
150C2×Dic118—21no—
151C2×Dic118C22×C411yescharacteristic
152C2×Dic118C22×C411yescharacteristic
153C2×Dic118C22×C411yescharacteristic
154M16×C59—22no—
155M16×C59—22no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—42no—
159C2×C8×C59C22×C411yescharacteristic
160C2×C8×C59C22×C411yescharacteristic
161C2×C8×C59C22×C411yescharacteristic
162C2×C8×C59C22×C411yescharacteristic
163C2×C8×C59C2×D411yescharacteristic
164C2×C8×C59C2×D411yescharacteristic
165C2×C8×C59C2×D411yescharacteristic
166C2×C8×C59C2×D411yescharacteristic
167C42×C59C22×C411yescharacteristic
168C42×C59C2×D411yescharacteristic
169C42×C59#4111yescharacteristic
170C42×C59C2411yescharacteristic
171C22⋊Dic59—21no—
172C22⋊Dic59—21no—
173C22⋊Dic59—21no—
174(C2×D59)⋊C4—21no—
175(C2×D59)⋊C4—21no—
176(C2×D59)⋊C4—21no—
177(C2×D59)⋊C4—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C411yescharacteristic
182C4⋊Dic59C22×C411yescharacteristic
183C4⋊Dic59C22×C411yescharacteristic
184C4⋊Dic59C22×C411yescharacteristic
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189Dic59⋊C4—21no—
190Dic59⋊C4—21no—
191Dic59⋊C4—21no—
192Dic59⋊C4—21no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—21no—
196#11745664—81no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201Dic236—22no—
202Dic236—22no—
203#11745660—22no—
204#11745660—22no—
205#11745659—81no—
206C8×D59—81no—
207C23×C59C4×D411yescharacteristic
208C22×D59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—41no—
214C2×C4×C59C4×D411yescharacteristic
215C2×C4×C59C4×D411yescharacteristic
216C2×C4×C59C4×D411yescharacteristic
217C2×C4×C59C23×C411yescharacteristic
218C2×C4×C59C22×D411yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C22⋊D59—81no—
221C2×Dic59—21no—
222C2×Dic59—21no—
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—21no—
228D236—21no—
229D236—21no—
230C4×D59—41no—
231C4×D59—41no—
232Dic118—21no—
233Dic118—21no—
234Dic118—21no—
235Dic118—21no—
236Dic118—21no—
237Dic118—21no—
238Dic118—81no—
239C8×C59—22no—
240C8×C59—21no—
241C8×C59—21no—
242C8×C59—21no—
243C8×C59—21no—
244C8×C59—21no—
245C8×C59—21no—
246C8×C59—41no—
247C8×C59#13112yes—
248C8×C59#13112yes—
249C59⋊C8—41no—
250C59⋊C8—41no—
251C59⋊C8—41no—
252C59⋊C8—41no—
253#17969—591nomaximal, Sylow
254C22×C59—21no—
255C22×C59—21no—
256C22×C59—21no—
257C22×C59C2×C4×D411yescharacteristic
258C2×D59—41no—
259C2×D59—41no—
260C2×D59—41no—
261C4×C59—21no—
262C4×C59—21no—
263C4×C59—21no—
264C4×C59—21no—
265C4×C59C2×#13111yescharacteristic
266C4×C59C2×#13111yescharacteristic
267C4×C59#57711yescharacteristic
268C4×C59#57811yescharacteristic
269Dic59—41no—
270Dic59—41no—
271Dic59—41no—
272Dic59—41no—
273Dic59—41no—
274Dic59—41no—
275Dic59—41no—
276#3172—591no—
277#2993—591no—
278#2985—591no—
279#1593—591no—
280#1592—591no—
281#1591—591no—
282#1590—591no—
283#1579—591no—
284SD16⋊C8—591no—
285C8×Q16—591no—
286C8×SD16—591no—
287#1569—591no—
288#1494—591no—
289#1493—591no—
290C8×M16D5911yescharacteristic, complemented
291C2×C59—41no—
292C2×C59#293111yescharacteristic
293C2×C59#294711yescharacteristic
294C2×C59#295611yescharacteristic
295D59—81no—
296#533—591no—
297#529—591no—
298#501—591no—
299#497—591no—
300#491—591no—
301#489—591no—
302#470—591no—
303#470—591no—
304#468—591no—
305#468—591no—
306#446—591no—
307#446—591no—
308#446—591no—
309#446—1181no—
310#446—1181no—
311C8×Q8—591no—
312C8×Q8—591no—
313C8×Q8—591no—
314C4×Q16—591no—
315C4×Q16—591no—
316C4×SD16—591no—
317C4×SD16—591no—
318#436—591no—
319#436—1181no—
320#435—1181no—
321C8×D4—591no—
322#433—1181no—
323#433—1181no—
324#433—1181no—
325#432—591no—
326#431C2×D5911yescharacteristic
327#424—1181no—
328#423—591no—
329C4×M16C2×D5911yescharacteristic
330C2×C4×C8C2×D5911yescharacteristic
331#335—591no—
332#335—591no—
333#334—591no—
334#334—591no—
335Q8⋊C8—591no—
336Q8⋊C8—591no—
337Q8⋊C8—591no—
338Q8⋊C8—591no—
339Q8⋊C8—591no—
340Q8⋊C8—591no—
341D4⋊C8—591no—
342D4⋊C8—591no—
343#322C2×D5911yescharacteristic
344#322C2×D5911yescharacteristic
345C82C2×D5911yescharacteristic
346C82C2×D5911yescharacteristic
347C59#1796911yescharacteristic, complemented, Sylow
348C2×Q16—591no—
349C2×Q16—591no—
350C2×SD16—591no—
351C2×SD16—591no—
352C2×M16—1181no—
353C2×M16—1181no—
354C2×M16C4×D5911yescharacteristic
355C2×M16C4×D5911yescharacteristic
356C22×C8—21no—
357C22×C8—1181no—
358C4⋊Q8—591no—
359C4⋊Q8—591no—
360#126—1181no—
361#125—591no—
362#125—1181no—
363#124—591no—
364#123—1181no—
365C22⋊Q8—591no—
366C22⋊Q8—591no—
367C22⋊Q8—591no—
368C22⋊Q8—1181no—
369#121—591no—
370C4×Q8—591no—
371C4×Q8—591no—
372C4×Q8—591no—
373C4×Q8—1181no—
374C4×D4—591no—
375C4×D4—1181no—
376#117—1181no—
377#117—1181no—
378C2×C42C22×D5911yescharacteristic
379#107—591no—
380#107—591no—
381#106—591no—
382#106—591no—
383C4⋊C8—591no—
384C4⋊C8—591no—
385C4⋊C8—591no—
386C4⋊C8—591no—
387C4⋊C8—1181no—
388C4⋊C8—1181no—
389C4⋊C8—1181no—
390C4⋊C8—1181no—
391C4⋊C8—1181no—
392C4⋊C8—1181no—
393C4⋊C8—1181no—
394C4⋊C8D23611yescharacteristic
395C4⋊C8D23611yescharacteristic
396#103—591no—
397#103—591no—
398#103—591no—
399#103—591no—
400#103—591no—
401#103—591no—
402#102—591no—
403#102—591no—
404C22⋊C8—1181no—
405C22⋊C8—1181no—
406C22⋊C8—1181no—
407C22⋊C8D23611yescharacteristic
408C22⋊C8D23611yescharacteristic
409#97—1181no—
410#97—1181no—
411#97—1181no—
412#97—1181no—
413#97C4×D5911yescharacteristic
414#97C4×D5911yescharacteristic
415C4×C8—21no—
416C4×C8—1181no—
417C4×C8—1181no—
418C4×C8—1181no—
419C4×C8C22×D5911yescharacteristic
420C4×C8C22×D5911yescharacteristic
421C4×C8C22×D5911yescharacteristic
422C4×C8C22×D5911yescharacteristic
423C4×C8C22×D5911yescharacteristic
424C4×C8C22×D5911yescharacteristic
425C2×Q8—591no—
426C2×Q8—591no—
427C2×Q8—591no—
428C2×Q8—1181no—
429C2×D4—591no—
430C2×D4—1181no—
431C22×C4—1181no—
432C22×C4C2×C4×D5911yescharacteristic
433C22×C4C2×D23611yescharacteristic
434C22×C4#1174568511yescharacteristic
435Q16—1182no—
436Q16—1182no—
437SD16—1182no—
438SD16—1182no—
439M16—22no—
440M16—22no—
441M16—4721no—
442M16—4721no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—42no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—4721no—
451C2×C8C2×C4×D5911yescharacteristic
452C2×C8C2×C4×D5911yescharacteristic
453C2×C8C2×C4×D5911yescharacteristic
454C2×C8C2×C4×D5911yescharacteristic
455C2×C8C2×D23611yescharacteristic
456C2×C8C2×D23611yescharacteristic
457C2×C8C2×D23611yescharacteristic
458C2×C8C2×D23611yescharacteristic
459C4⋊C4—591no—
460C4⋊C4—591no—
461C4⋊C4—591no—
462C4⋊C4—591no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C22⋊C4—1181no—
475C22⋊C4—1181no—
476C22⋊C4—1181no—
477C22⋊C4—1181no—
478C22⋊C4—1181no—
479C22⋊C4—1181no—
480C22⋊C4—1181no—
481C42—1181no—
482C42—1181no—
483C42—1181no—
484C42C2×C4×D5911yescharacteristic
485C42C2×D23611yescharacteristic
486C42#1174568511yescharacteristic
487C42C23×D5911yescharacteristic
488C23—1181no—
489C23C4×D23611yescharacteristic
490Q8—1181no—
491Q8—1181no—
492Q8—1181no—
493Q8—1181no—
494Q8—1181no—
495Q8—1181no—
496Q8—4721no—
497D4—1181no—
498D4—1181no—
499D4—4721no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—1181no—
506C2×C4—1181no—
507C2×C4—1181no—
508C2×C4—1181no—
509C2×C4—1181no—
510C2×C4—1181no—
511C2×C4—1181no—
512C2×C4—2361no—
513C2×C4—2361no—
514C2×C4C4×D23611yescharacteristic
515C2×C4C4×D23611yescharacteristic
516C2×C4C4×D23611yescharacteristic
517C2×C4C22×C4×D5911yescharacteristic
518C2×C4C22×D23611yescharacteristic
519C2×C4C2×#1174568511yescharacteristic
520C8—22no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—41no—
528C8—2361no—
529C8—2361no—
530C8—2361no—
531C8—2361no—
532C8#1427373312yes—
533C8#1427373312yes—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—2361no—
538C22—2361no—
539C22—2361no—
540C22C2×C4×D23611yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—2361no—
546C4—2361no—
547C4—2361no—
548C4—2361no—
549C4—2361no—
550C4—2361no—
551C4—2361no—
552C4C2×#1427373311yescharacteristic
553C4C2×#1427373311yescharacteristic
554C4#1683821411yescharacteristic
555C4#1683821911yescharacteristic
556C2—41no—
557C2—4721no—
558C2#2143280911yescharacteristic
559C2#2143294111yescharacteristic
560C2#2143300711yescharacteristic
subgroupG/H
centerC2×C4C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC8×M16×C59C2
socleC22×C59C2×C4×D4
prime radical#14273701C2×C4
cosocleC42×C59C24
hypercenterC8×M16D59
hypocenterC59#17969
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ #431 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
3. #21433190C2semidirect
4. #21423006C2semidirect
5. #21423000C2semidirect
6. #21422994C2semidirect
7. #21422988C2semidirect
8. #21422916C2semidirect
9. #21422910C2semidirect
10. C8×Dic236C2semidirect
11. C8×#11745660C2semidirect
12. #21422863C2semidirect
14. #21422552C2semidirect
15. C8×M16×C59C2semidirect
290. C8×M16D59semidirect
347. C59#17969semidirect

Maximal subgroups

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

subgroupindex
#214347022
#214332472
#214331902
#214230062
#214230002
#214229942
#214229882
#214229162
#214229102
C8×Dic2362
C8×#117456602
#214228632
#214225572
#214225522
C8×M16×C592
#1796959