Finite Group Databasebeta

← #28559834

Subgroups — #28559834

Order 15104.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434168C211yesmaximal, characteristic
2#21433114C211yesmaximal, characteristic
3#21433043C211yesmaximal, characteristic, complemented
4#21423000C211yesmaximal, characteristic, complemented
5#21422988C211yesmaximal, characteristic, complemented
6#21422976C211yesmaximal, characteristic, complemented
7#21422969C211yesmaximal, characteristic, complemented
8#21422948C211yesmaximal, characteristic, complemented
9#21422941C211yesmaximal, characteristic, complemented
10#21422916C211yesmaximal, characteristic, complemented
11C8×Dic236C211yesmaximal, characteristic, complemented
12#21422881C211yesmaximal, characteristic, complemented
13#21422570C211yesmaximal, characteristic, complemented
14#21422484C211yesmaximal, characteristic
15C59×#1453C211yesmaximal, characteristic, complemented
16#16837959C2211yescharacteristic
17C4×#11745685C2211yescharacteristic
18#16837744C411yescharacteristic
19#16837744C411yescharacteristic
20#16837688C411yescharacteristic
21#16837683C411yescharacteristic
22#16837613C411yescharacteristic
23#16837608C411yescharacteristic
24#16837566C411yescharacteristic
25#16837566C411yescharacteristic
26#16837532C2211yescharacteristic
27#16837532C2211yescharacteristic
28#16837530—21no—
29#16837530—21no—
30C8×Dic118C2211yescharacteristic
31C8×Dic118C2211yescharacteristic
32C8×Dic118C2211yescharacteristic
33C8×Dic118C2211yescharacteristic
34C4×Dic236C2211yescharacteristic
35C4×Dic236C2211yescharacteristic
36C4×#11745660C2211yescharacteristic
37C4×#11745660C2211yescharacteristic
38#16837481C2211yescharacteristic
39#16837481C2211yescharacteristic
40#16837478—21no—
41#16837478—21no—
42#16837459C2211yescharacteristic
43#16837457—21no—
44#16837430—21no—
45C2×C4⋊(C59⋊C8)C2211yescharacteristic
46#16837360—21no—
47#16837354—21no—
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
61C59×#431C2211yescharacteristic
62C4×M16×C59C2211yescharacteristic
63C59×#322C2211yescharacteristic
64C59×#322C2211yescharacteristic
65C59×#322C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×#11745685C2×C411yescharacteristic
68C2×Dic236C2×C411yescharacteristic
69C2×Dic236C2×C411yescharacteristic
70C2×#11745660C2×C411yescharacteristic
71C2×#11745660C2×C411yescharacteristic
72C2×#11745659—21no—
73C2×#11745659—21no—
74C22×C59⋊C8—21no—
75#14273727C2×C411yescharacteristic
76#14273727C2×C411yescharacteristic
77#14273719—21no—
78#14273706—21no—
79#14273706C2×C411yescharacteristic
80#14273706C2×C411yescharacteristic
81C4×Dic118—21no—
82C4×Dic118C2×C411yescharacteristic
83C4×Dic118C2311yescharacteristic
84C4×Dic118C2311yescharacteristic
85C4×Dic118C2311yescharacteristic
86C4×D236C2×C411yescharacteristic
87C4×D236C2311yescharacteristic
88C4×C22⋊D59—21no—
89C4×C22⋊D59—21no—
90#14273686—21no—
91#14273686C2×C411yescharacteristic
92#14273684—21no—
93C42×D59—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—
123C8×Dic59—21no—
124C8×Dic59—21no—
125C8×Dic59—21no—
126C8×Dic59—21no—
127C2×M16×C59D411yescharacteristic
128C2×M16×C59D411yescharacteristic
129C2×C42×C59C2311yescharacteristic
130C4⋊C8×C59—21no—
131C4⋊C8×C59C2×C411yescharacteristic
132C4⋊C8×C59C2×C411yescharacteristic
133C22⋊C8×C59—21no—
134C22⋊C8×C59C2×C411yescharacteristic
135C22⋊C8×C59C2×C411yescharacteristic
136#97×C59D411yescharacteristic
137#97×C59D411yescharacteristic
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—
148#11745685—22no—
149#11745685—41no—
150C2×D236C22×C411yescharacteristic
151C2×C4×D59—21no—
152C2×C4×D59—21no—
153C2×C4×D59—21no—
154C2×Dic118—21no—
155C2×Dic118C22×C411yescharacteristic
156C2×Dic118C22×C411yescharacteristic
157C2×Dic118C22×C411yescharacteristic
158M16×C59—22no—
159M16×C59—41no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C22×C411yescharacteristic
163C2×C8×C59C22×C411yescharacteristic
164C2×C8×C59C22×C411yescharacteristic
165C2×C8×C59C22×C411yescharacteristic
166C2×C8×C59C2×D411yescharacteristic
167C2×C8×C59C2×D411yescharacteristic
168C2×C8×C59C2×D411yescharacteristic
169C2×C8×C59C2×D411yescharacteristic
170C42×C59C22×C411yescharacteristic
171C42×C59C2×D411yescharacteristic
172C42×C59#4111yescharacteristic
173C42×C59C2411yescharacteristic
174C22⋊Dic59—21no—
175C22⋊Dic59—21no—
176C22⋊Dic59—21no—
177(C2×D59)⋊C4—21no—
178(C2×D59)⋊C4—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59C22×C411yescharacteristic
183C4⋊Dic59C22×C411yescharacteristic
184C4⋊Dic59C22×C411yescharacteristic
185C4⋊Dic59C22×C411yescharacteristic
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189Dic59⋊C4—21no—
190Dic59⋊C4—21no—
191Dic59⋊C4—21no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—21no—
196C4×Dic59—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—42no—
202Dic236—22no—
203Dic236—22no—
204#11745660—41no—
205#11745660—41no—
206#11745659—42no—
207#11745659—42no—
208C23×C59C4×D411yescharacteristic
209C22×D59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—41no—
215C2×C4×C59C4×D411yescharacteristic
216C2×C4×C59C4×D411yescharacteristic
217C2×C4×C59C4×D411yescharacteristic
218C2×C4×C59C23×C411yescharacteristic
219C2×C4×C59C22×D411yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C22⋊D59—42no—
222C2×Dic59—21no—
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—21no—
229D236—21no—
230D236—21no—
231C4×D59—42no—
232C4×D59—41no—
233C4×D59—41no—
234C4×D59—81no—
235Dic118—21no—
236Dic118—21no—
237Dic118—21no—
238Dic118—21no—
239Dic118—21no—
240Dic118—81no—
241Dic118#13112yes—
242C8×C59—21no—
243C8×C59—21no—
244C8×C59—21no—
245C8×C59—21no—
246C8×C59—21no—
247C8×C59—21no—
248C8×C59—21no—
249C8×C59—41no—
250C8×C59—41no—
251C8×C59#13712yes—
252C59⋊C8—41no—
253C59⋊C8—41no—
254C59⋊C8—41no—
255C59⋊C8—41no—
256#18013—591nomaximal, Sylow
257C22×C59—21no—
258C22×C59—21no—
259C22×C59—21no—
260C22×C59C2×C4×D411yescharacteristic
261C2×D59—41no—
262C2×D59—41no—
263C2×D59—41no—
264C4×C59—21no—
265C4×C59—21no—
266C4×C59—21no—
267C4×C59—21no—
268C4×C59C2×#13111yescharacteristic
269C4×C59#56811yescharacteristic
270C4×C59C2×#13711yescharacteristic
271C4×C59#57511yescharacteristic
272Dic59—22no—
273Dic59—41no—
274Dic59—41no—
275Dic59—41no—
276Dic59—41no—
277Dic59—41no—
278Dic59—41no—
279#3107—591no—
280#2971—591no—
281#2962—591no—
282#1592—591no—
283#1590—591no—
284#1588—591no—
285#1587—591no—
286#1584—591no—
287#1583—591no—
288#1579—591no—
289C8×Q16—591no—
290#1573—591no—
291#1497—591no—
292#1477—591no—
293#1453D5911yescharacteristic, complemented
294C2×C59—41no—
295C2×C59#293311yescharacteristic
296C2×C59C4×#13711yescharacteristic
297C2×C59#295711yescharacteristic
298D59—81no—
299#533—591no—
300C4×#41—591no—
301#485—591no—
302#485—591no—
303#475—591no—
304#474—591no—
305#462—591no—
306#461—591no—
307#452—591no—
308#452—591no—
309#446—591no—
310#446—591no—
311#446—1181no—
312#446—1181no—
313C8×Q8—591no—
314C8×Q8—591no—
315C8×Q8—591no—
316C8×Q8—591no—
317C4×Q16—591no—
318C4×Q16—591no—
319C4×SD16—591no—
320C4×SD16—591no—
321#436—591no—
322#436—591no—
323#436—1181no—
324#436—1181no—
325#432—591no—
326#432—1181no—
327#431C2×D5911yescharacteristic
328#431C2×D5911yescharacteristic
329#424—1181no—
330C2×C4⋊C8—591no—
331#405—1181no—
332C4×M16—1181no—
333C4×M16C2×D5911yescharacteristic
334#335—591no—
335#335—591no—
336#334—591no—
337#334—591no—
338Q8⋊C8—591no—
339Q8⋊C8—591no—
340Q8⋊C8—591no—
341Q8⋊C8—591no—
342Q8⋊C8—591no—
343Q8⋊C8—591no—
344D4⋊C8—591no—
345D4⋊C8—591no—
346#322C2×D5911yescharacteristic
347#322C2×D5911yescharacteristic
348#322C2×D5911yescharacteristic
349C82C2×D5911yescharacteristic
350C59#1801311yescharacteristic, complemented, Sylow
351C2×#41—591no—
352C2×Q16—591no—
353C2×Q16—591no—
354C2×SD16—591no—
355C2×SD16—591no—
356C2×M16—1181no—
357C2×M16—1181no—
358C2×M16D23611yescharacteristic
359C2×M16D23611yescharacteristic
360C22×C8—1181no—
361C4⋊Q8—591no—
362C4⋊Q8—591no—
363#125—1181no—
364C22⋊Q8—591no—
365C22⋊Q8—591no—
366C22⋊Q8—1181no—
367C4×Q8—591no—
368C4×Q8—591no—
369C4×Q8—591no—
370C4×Q8—591no—
371C4×Q8—1181no—
372C4×D4—591no—
373C4×D4—591no—
374C4×D4—1181no—
375C4×D4—1181no—
376#117—591no—
377#117—1181no—
378#117—1181no—
379C2×C42—1181no—
380C2×C42C22×D5911yescharacteristic
381#107—591no—
382#107—591no—
383#106—591no—
384#106—591no—
385C4⋊C8—21no—
386C4⋊C8—591no—
387C4⋊C8—591no—
388C4⋊C8—591no—
389C4⋊C8—591no—
390C4⋊C8—1181no—
391C4⋊C8—1181no—
392C4⋊C8—1181no—
393C4⋊C8—1181no—
394C4⋊C8—1181no—
395C4⋊C8—1181no—
396C4⋊C8—1181no—
397C4⋊C8C4×D5911yescharacteristic
398C4⋊C8C4×D5911yescharacteristic
399#103—591no—
400#103—591no—
401#103—591no—
402#103—591no—
403#103—591no—
404#103—591no—
405#102—591no—
406#102—591no—
407C22⋊C8—21no—
408C22⋊C8—1181no—
409C22⋊C8—1181no—
410C22⋊C8—1181no—
411C22⋊C8C4×D5911yescharacteristic
412C22⋊C8C4×D5911yescharacteristic
413#97—1181no—
414#97—1181no—
415#97—1181no—
416#97D23611yescharacteristic
417#97D23611yescharacteristic
418C4×C8—1181no—
419C4×C8—1181no—
420C4×C8—1181no—
421C4×C8—1181no—
422C4×C8C22×D5911yescharacteristic
423C4×C8C22×D5911yescharacteristic
424C4×C8C22×D5911yescharacteristic
425C4×C8C22×D5911yescharacteristic
426C4×C8C22×D5911yescharacteristic
427C4×C8C22×D5911yescharacteristic
428#41—1182no—
429#41—2361no—
430C2×Q8—591no—
431C2×Q8—591no—
432C2×Q8—591no—
433C2×Q8—1181no—
434C2×D4—591no—
435C2×D4—1181no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4C2×C4×D5911yescharacteristic
440C22×C4C2×D23611yescharacteristic
441C22×C4#1174568511yescharacteristic
442Q16—1182no—
443Q16—1182no—
444SD16—2361no—
445SD16—2361no—
446M16—22no—
447M16—41no—
448M16—2362no—
449M16—2362no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—2362no—
457C2×C8C2×C4×D5911yescharacteristic
458C2×C8C2×C4×D5911yescharacteristic
459C2×C8C2×C4×D5911yescharacteristic
460C2×C8C2×C4×D5911yescharacteristic
461C2×C8C2×D23611yescharacteristic
462C2×C8C2×D23611yescharacteristic
463C2×C8C2×D23611yescharacteristic
464C2×C8C2×D23611yescharacteristic
465C4⋊C4—591no—
466C4⋊C4—591no—
467C4⋊C4—591no—
468C4⋊C4—591no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C22⋊C4—1181no—
479C22⋊C4—1181no—
480C22⋊C4—1181no—
481C22⋊C4—1181no—
482C22⋊C4—1181no—
483C42—1181no—
484C42—1181no—
485C42—1181no—
486C42—1181no—
487C42—1181no—
488C42C2×C4×D5911yescharacteristic
489C42C2×D23611yescharacteristic
490C42#1174568511yescharacteristic
491C42C23×D5911yescharacteristic
492C23—1181no—
493C23C4×D23611yescharacteristic
494Q8—592no—
495Q8—1181no—
496Q8—1181no—
497Q8—1181no—
498Q8—1181no—
499Q8—1181no—
500Q8—4721no—
501D4—1181no—
502D4—1181no—
503D4—2362no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—1181no—
510C2×C4—1181no—
511C2×C4—1181no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4—4721no—
517C2×C4—2362no—
518C2×C4—2361no—
519C2×C4—2361no—
520C2×C4C4×D23611yescharacteristic
521C2×C4C4×D23611yescharacteristic
522C2×C4C4×D23611yescharacteristic
523C2×C4C22×C4×D5911yescharacteristic
524C2×C4C22×D23611yescharacteristic
525C2×C4C2×#1174568511yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—21no—
533C8—41no—
534C8—41no—
535C8—2361no—
536C8—2361no—
537C8—2361no—
538C8—2361no—
539C8#1427376012yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—2361no—
544C22—2361no—
545C22—2361no—
546C22C2×C4×D23611yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—1182no—
552C4—2361no—
553C4—2361no—
554C4—2361no—
555C4—2361no—
556C4—2361no—
557C4—2361no—
558C4#131×D5911yescharacteristic
559C4#1683816711yescharacteristic
560C4C2×#1427376011yescharacteristic
561C4#1683820211yescharacteristic
562C2—41no—
563C2—4721no—
564C2#2143282411yescharacteristic
565C2C4×#1427376011yescharacteristic
566C2#2143301411yescharacteristic
subgroupG/H
centerC2×C4C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1453C2
socleC22×C59C2×C4×D4
prime radicalC2×#11745685C2×C4
cosocleC42×C59C24
hypercenter#1453D59
hypocenterC59#18013
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1453

Complements

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

NH ≅ G/Ntype
3. #21433043C2semidirect
4. #21423000C2semidirect
5. #21422988C2semidirect
6. #21422976C2semidirect
7. #21422969C2semidirect
8. #21422948C2semidirect
9. #21422941C2semidirect
10. #21422916C2semidirect
11. C8×Dic236C2semidirect
12. #21422881C2semidirect
13. #21422570C2semidirect
15. C59×#1453C2semidirect
293. #1453D59semidirect
350. C59#18013semidirect

Maximal subgroups

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

subgroupindex
#214341682
#214331142
#214330432
#214230002
#214229882
#214229762
#214229692
#214229482
#214229412
#214229162
C8×Dic2362
#214228812
#214225702
#214224842
C59×#14532
#1801359