Finite Group Databasebeta

← #28559327

Subgroups — #28559327

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#21434698C211yesmaximal, characteristic
2#21433246C211yesmaximal, characteristic
3#21433187C211yesmaximal, characteristic, complemented
4#21423004C211yesmaximal, characteristic, complemented
5#21422999C211yesmaximal, characteristic, complemented
6#21422990C211yesmaximal, characteristic, complemented
7#21422989C211yesmaximal, characteristic, complemented
8#21422913C211yesmaximal, characteristic, complemented
9#21422908C211yesmaximal, characteristic, complemented
10C8×Dic236C211yesmaximal, characteristic, complemented
11#21422860C211yesmaximal, characteristic, complemented
12#21422553C211yesmaximal, characteristic
13#21422549C211yesmaximal, characteristic, complemented
14C8×#11745659C211yesmaximal, characteristic, complemented
15C8×SD16×C59C211yesmaximal, characteristic, complemented
16#16837956C2211yescharacteristic
17#16837934C2211yescharacteristic
18#16837809C411yescharacteristic
19#16837793C411yescharacteristic
20#16837769C411yescharacteristic
21#16837760C411yescharacteristic
22#16837661C411yescharacteristic
23#16837659C411yescharacteristic
24#16837647C411yescharacteristic
25#16837645C411yescharacteristic
26#16837533C2211yescharacteristic
27#16837532—21no—
28#16837531—21no—
29#16837529C2211yescharacteristic
30#16837529C2211yescharacteristic
31C8×Dic118C2211yescharacteristic
32C8×Dic118C2211yescharacteristic
33C4×Dic236C2211yescharacteristic
34C4×#11745672C2211yescharacteristic
35C4×#11745670C2211yescharacteristic
36#16837480C2211yescharacteristic
37#16837477—21no—
38#16837473—21no—
39#16837462—21no—
40#16837461—21no—
41#16837460—21no—
42#16837458C2211yescharacteristic
43#16837453C2211yescharacteristic
44#16837428—21no—
45#16837424C2211yescharacteristic
46C4×#11745659C2211yescharacteristic
47C4×C8×D59C2211yescharacteristic
48#16837155C2211yescharacteristic
49#16837154C2211yescharacteristic
50#16837152C2211yescharacteristic
51#16837134C2211yescharacteristic
52#16837134C2211yescharacteristic
53Q8⋊(C59⋊C8)C2211yescharacteristic
54#16837132C2211yescharacteristic
55#16837132C2211yescharacteristic
56D4⋊(C59⋊C8)C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837123C2211yescharacteristic
59C8×C59⋊C8C2211yescharacteristic
60C8×Q8×C59C2211yescharacteristic
61C4×SD16×C59C2211yescharacteristic
62C8×D4×C59C2211yescharacteristic
63C59×#334C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×D4⋊C8C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×Dic236C2×C411yescharacteristic
68C2×#11745672C2×C411yescharacteristic
69C2×#11745670C2×C411yescharacteristic
70C2×#11745664—21no—
71C2×#11745659—21no—
72C2×#11745659C2×C411yescharacteristic
73C2×#11745659C2×C411yescharacteristic
74C2×C8×D59—21no—
75#14273726C2×C411yescharacteristic
76#14273725C2×C411yescharacteristic
77#14273720—21no—
78#14273718C2×C411yescharacteristic
79#14273717—21no—
80#14273714C2×C411yescharacteristic
81#14273709—21no—
82#14273707C2×C411yescharacteristic
83#14273707C2×C411yescharacteristic
84#14273705—21no—
85#14273703C2×C411yescharacteristic
86#14273698C2×C411yescharacteristic
87C4×Dic118—21no—
88C4×Dic118C2311yescharacteristic
89C4×Dic118C2311yescharacteristic
90C4×C22⋊D59—21no—
91#14273686—21no—
92#14273684—21no—
93C42×D59C2311yescharacteristic
94#14273661C2×C411yescharacteristic
95#14273660C2×C411yescharacteristic
96#14273658C2×C411yescharacteristic
97C4⋊(C59⋊C8)—21no—
98C4⋊(C59⋊C8)C2311yescharacteristic
99C4⋊(C59⋊C8)C2311yescharacteristic
100Dic59⋊C8—21no—
101Dic59⋊C8—21no—
102Dic59⋊C8—21no—
103Dic59⋊C8—21no—
104Dic59⋊C8—21no—
105Dic59⋊C8D411yescharacteristic
106Dic59⋊C8D411yescharacteristic
107#14273652C2×C411yescharacteristic
108#14273652C2×C411yescharacteristic
109#14273651C2×C411yescharacteristic
110#14273650C2×C411yescharacteristic
111#14273650C2×C411yescharacteristic
112#14273648C2×C411yescharacteristic
113C22⋊(C59⋊C8)—21no—
114(C2×D59)⋊C8—21no—
115(C2×D59)⋊C8D411yescharacteristic
116(C2×D59)⋊C8D411yescharacteristic
117#14273638—21no—
118#14273637—21no—
119#14273637—21no—
120#14273637—21no—
121#14273637C2×C411yescharacteristic
122#14273637C2×C411yescharacteristic
123C4×C59⋊C8C2311yescharacteristic
124C4×C59⋊C8C2311yescharacteristic
125C4×C59⋊C8C2311yescharacteristic
126C8×Dic59—21no—
127C8×Dic59—21no—
128C8×Dic59—21no—
129C2×SD16×C59C2×C411yescharacteristic
130C22×C8×C59—21no—
131C4×Q8×C59C2311yescharacteristic
132C4×D4×C59C2311yescharacteristic
133#106×C59C2×C411yescharacteristic
134C4⋊C8×C59—21no—
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137#103×C59C2×C411yescharacteristic
138#102×C59C2×C411yescharacteristic
139C22⋊C8×C59—21no—
140C4×C8×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59C22×C411yescharacteristic
145C2×D4×C59C22×C411yescharacteristic
146C22×C4×C59—21no—
147C2×C22⋊D59—21no—
148C2×C4×D59C22×C411yescharacteristic
149C2×C4×D59C2×D411yescharacteristic
150C2×C4×D59#4111yescharacteristic
151C2×Dic118—21no—
152C2×Dic118C22×C411yescharacteristic
153C2×Dic118C22×C411yescharacteristic
154SD16×C59—22no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—81no—
159C2×C8×C59C22×C411yescharacteristic
160C2×C8×C59C22×C411yescharacteristic
161C2×C8×C59C2×D411yescharacteristic
162C2×C8×C59C2×D411yescharacteristic
163C4⋊C4×C59—21no—
164C4⋊C4×C59C22×C411yescharacteristic
165C4⋊C4×C59C22×C411yescharacteristic
166C22⋊C4×C59—21no—
167C42×C59—21no—
168C42×C59C2411yescharacteristic
169C22⋊Dic59—21no—
170C22⋊Dic59—21no—
171C22⋊Dic59—21no—
172#11745672—22no—
173#11745670—22no—
174(C2×D59)⋊C4—21no—
175(C2×D59)⋊C4—21no—
176(C2×D59)⋊C4—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59C22×C411yescharacteristic
181C4⋊Dic59C22×C411yescharacteristic
182Dic59⋊C4—21no—
183Dic59⋊C4—21no—
184Dic59⋊C4—21no—
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59C22×C411yescharacteristic
192C4×Dic59C2×D411yescharacteristic
193C4×Dic59#4111yescharacteristic
194#11745664—81no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C22×C411yescharacteristic
199C2×C59⋊C8C22×C411yescharacteristic
200C2×C59⋊C8C2×D411yescharacteristic
201C2×C59⋊C8C2×D411yescharacteristic
202Dic236—22no—
203#11745659—22no—
204#11745659—22no—
205#11745659—81no—
206C8×D59—42no—
207C23×C59—21no—
208C22×D59C4×D411yescharacteristic
209Q8×C59—21no—
210Q8×C59—21no—
211D4×C59—21no—
212D4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59C23×C411yescharacteristic
219C2×C4×C59C22×D411yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C22⋊D59—81no—
222C2×Dic59—21no—
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59C4×D411yescharacteristic
227C2×Dic59C4×D411yescharacteristic
228C2×Dic59C4×D411yescharacteristic
229C4×D59—21no—
230C4×D59—21no—
231C4×D59—21no—
232C4×D59—21no—
233C4×D59—41no—
234Dic118—21no—
235Dic118—21no—
236Dic118—21no—
237Dic118—21no—
238Dic118—81no—
239C8×C59—22no—
240C8×C59—21no—
241C8×C59—21no—
242C8×C59—21no—
243C8×C59—41no—
244C8×C59—41no—
245C8×C59#13112yes—
246C59⋊C8—21no—
247C59⋊C8—21no—
248C59⋊C8—21no—
249C59⋊C8—41no—
250C59⋊C8—41no—
251C59⋊C8—41no—
252C59⋊C8#13112yes—
253#17969—591nomaximal, Sylow
254C22×C59—41no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59C2×C4×D411yescharacteristic
258C2×D59—21no—
259C2×D59—21no—
260C2×D59—21no—
261C4×C59—21no—
262C4×C59—41no—
263C4×C59—41no—
264C4×C59—41no—
265C4×C59C2×#13111yescharacteristic
266C4×C59C2×#13111yescharacteristic
267C4×C59#57711yescharacteristic
268C4×C59#57811yescharacteristic
269Dic59—21no—
270Dic59—21no—
271Dic59—21no—
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×SD16D5911yescharacteristic, complemented
287#1569—591no—
288#1494—591no—
289#1493—591no—
290C8×M16—591no—
291C2×C59—81no—
292C2×C59#293111yescharacteristic
293C2×C59#294711yescharacteristic
294C2×C59#295611yescharacteristic
295D59—41no—
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×Q8C2×D5911yescharacteristic
314C4×Q16—591no—
315C4×Q16—591no—
316C4×SD16—591no—
317C4×SD16C2×D5911yescharacteristic
318#436—591no—
319#436—1181no—
320#435—1181no—
321C8×D4C2×D5911yescharacteristic
322#433—1181no—
323#433—1181no—
324#433—1181no—
325#432—591no—
326#431—591no—
327#424—1181no—
328#423—591no—
329C4×M16—591no—
330C2×C4×C8—591no—
331#335—591no—
332#335—591no—
333#334—591no—
334#334C2×D5911yescharacteristic
335Q8⋊C8—591no—
336Q8⋊C8—591no—
337Q8⋊C8—591no—
338Q8⋊C8—591no—
339Q8⋊C8—591no—
340Q8⋊C8C2×D5911yescharacteristic
341D4⋊C8—591no—
342D4⋊C8C2×D5911yescharacteristic
343#322—591no—
344#322—591no—
345C82—591no—
346C82C2×D5911yescharacteristic
347C59#1796911yescharacteristic, complemented, Sylow
348C2×Q16—591no—
349C2×Q16—591no—
350C2×SD16—591no—
351C2×SD16C4×D5911yescharacteristic
352C2×M16—591no—
353C2×M16—591no—
354C2×M16—1181no—
355C2×M16—1181no—
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—1181no—
373C4×Q8C22×D5911yescharacteristic
374C4×D4—1181no—
375C4×D4C22×D5911yescharacteristic
376#117—1181no—
377#117—1181no—
378C2×C42—591no—
379#107—591no—
380#107—591no—
381#106—591no—
382#106C4×D5911yescharacteristic
383C4⋊C8—21no—
384C4⋊C8—591no—
385C4⋊C8—591no—
386C4⋊C8—591no—
387C4⋊C8—591no—
388C4⋊C8—1181no—
389C4⋊C8—1181no—
390C4⋊C8—1181no—
391C4⋊C8—1181no—
392C4⋊C8—1181no—
393C4⋊C8—1181no—
394C4⋊C8C22×D5911yescharacteristic
395C4⋊C8C22×D5911yescharacteristic
396#103—591no—
397#103—591no—
398#103—591no—
399#103—591no—
400#103—591no—
401#103C4×D5911yescharacteristic
402#102—591no—
403#102C4×D5911yescharacteristic
404C22⋊C8—21no—
405C22⋊C8—591no—
406C22⋊C8—591no—
407C22⋊C8—1181no—
408C22⋊C8—1181no—
409#97—591no—
410#97—591no—
411#97—1181no—
412#97—1181no—
413#97—1181no—
414#97—1181no—
415C4×C8—21no—
416C4×C8—591no—
417C4×C8—591no—
418C4×C8—591no—
419C4×C8—1181no—
420C4×C8—1181no—
421C4×C8—1181no—
422C4×C8C22×D5911yescharacteristic
423C4×C8C22×D5911yescharacteristic
424C4×C8C22×D5911yescharacteristic
425C2×Q8—591no—
426C2×Q8—591no—
427C2×Q8—1181no—
428C2×Q8C2×C4×D5911yescharacteristic
429C2×D4—1181no—
430C2×D4C2×C4×D5911yescharacteristic
431C22×C4—21no—
432C22×C4—591no—
433C22×C4—591no—
434C22×C4—591no—
435Q16—1182no—
436Q16—1182no—
437SD16—22no—
438SD16—1182no—
439M16—1182no—
440M16—1182no—
441M16—4721no—
442M16—4721no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—81no—
447C2×C8—591no—
448C2×C8—591no—
449C2×C8—591no—
450C2×C8—591no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—2362no—
455C2×C8C2×C4×D5911yescharacteristic
456C2×C8C2×C4×D5911yescharacteristic
457C2×C8D4×D5911yescharacteristic
458C2×C8D4×D5911yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—591no—
461C4⋊C4—591no—
462C4⋊C4—1181no—
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⋊C4C2×C4×D5911yescharacteristic
473C4⋊C4C2×C4×D5911yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—1181no—
476C22⋊C4—1181no—
477C22⋊C4—1181no—
478C22⋊C4—1181no—
479C22⋊C4—1181no—
480C22⋊C4—1181no—
481C42—21no—
482C42—591no—
483C42—591no—
484C42—591no—
485C42—1181no—
486C42—1181no—
487C42C23×D5911yescharacteristic
488C23—21no—
489C23—591no—
490Q8—21no—
491Q8—21no—
492Q8—1181no—
493Q8—1181no—
494Q8—1181no—
495Q8—1181no—
496Q8—4721no—
497D4—21no—
498D4—21no—
499D4—4721no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—591no—
506C2×C4—591no—
507C2×C4—591no—
508C2×C4—1181no—
509C2×C4—1181no—
510C2×C4—1181no—
511C2×C4—1181no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4—2361no—
517C2×C4C22×C4×D5911yescharacteristic
518C2×C4C2×D4×D5911yescharacteristic
519C2×C4#41×D5911yescharacteristic
520C8—22no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—592no—
527C8—1181no—
528C8—1181no—
529C8—1181no—
530C8—2361no—
531C8—2361no—
532C8—2361no—
533C8#1427373312yes—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—1181no—
538C22—1181no—
539C22—1181no—
540C22C4×D4×D5911yescharacteristic
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—1181no—
546C4—1181no—
547C4—1181no—
548C4—2361no—
549C4—2361no—
550C4—2361no—
551C4—2361no—
552C4#131×D5911yescharacteristic
553C4C2×#1427373311yescharacteristic
554C4#1683820811yescharacteristic
555C4#1683821611yescharacteristic
556C2—81no—
557C2—2361no—
558C2#2143280311yescharacteristic
559C2#2143293211yescharacteristic
560C2#2143300111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC8×SD16×C59C2
socleC22×C59C2×C4×D4
prime radical#14273698C2×C4
cosocleC42×C59C24
hypercenterC8×SD16D59
hypocenterC59#17969
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
3. #21433187C2semidirect
4. #21423004C2semidirect
5. #21422999C2semidirect
6. #21422990C2semidirect
7. #21422989C2semidirect
8. #21422913C2semidirect
9. #21422908C2semidirect
10. C8×Dic236C2semidirect
11. #21422860C2semidirect
13. #21422549C2semidirect
14. C8×#11745659C2semidirect
15. C8×SD16×C59C2semidirect
286. C8×SD16D59semidirect
347. C59#17969semidirect

Maximal subgroups

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

subgroupindex
#214346982
#214332462
#214331872
#214230042
#214229992
#214229902
#214229892
#214229132
#214229082
C8×Dic2362
#214228602
#214225532
#214225492
C8×#117456592
C8×SD16×C592
#1796959