Finite Group Databasebeta

← #28559154

Subgroups — #28559154

Order 15104.

606 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434102C211yesmaximal, characteristic
2#21433192C211yesmaximal, characteristic, complemented
3#21433134C211yesmaximal, characteristic
4#21422998C211yesmaximal, characteristic, complemented
5#21422996C211yesmaximal, characteristic, complemented
6#21422988C211yesmaximal, characteristic, complemented
7#21422977C211yesmaximal, characteristic, complemented
8#21422971C211yesmaximal, characteristic, complemented
9#21422940C211yesmaximal, characteristic, complemented
10#21422938C211yesmaximal, characteristic, complemented
11#21422873C211yesmaximal, characteristic, complemented
12#21422537C211yesmaximal, characteristic
13#21422515C211yesmaximal, characteristic, complemented
14#21422390C211yesmaximal, characteristic, complemented
15C59×#1590C211yesmaximal, characteristic, complemented
16#16837956C2211yescharacteristic
17C4×D4×D59C2211yescharacteristic
18#16837734C411yescharacteristic
19#16837731C411yescharacteristic
20#16837691C411yescharacteristic
21#16837690C411yescharacteristic
22#16837606C411yescharacteristic
23#16837605C411yescharacteristic
24#16837554C411yescharacteristic
25#16837551C411yescharacteristic
26#16837532C2211yescharacteristic, complemented
27#16837530—21no—
28#16837529—21no—
29Q8×C59⋊C8C2211yescharacteristic, complemented
30#16837526C2211yescharacteristic, complemented
31C4×#11745660C2211yescharacteristic
32C4×#11745671C2211yescharacteristic
33C4×#11745670C2211yescharacteristic
34#16837481C2211yescharacteristic
35#16837480C2211yescharacteristic
36#16837479C2211yescharacteristic
37#16837478—21no—
38#16837477—21no—
39#16837464—21no—
40#16837462—21no—
41#16837458C2211yescharacteristic
42#16837454C2211yescharacteristic
43#16837453C2211yescharacteristic
44#16837453C2211yescharacteristic
45#16837365—21no—
46#16837364—21no—
47C4×#11745659C2211yescharacteristic
48#16837154C2211yescharacteristic, complemented
49#16837154C2211yescharacteristic, complemented
50#16837153C2211yescharacteristic, complemented
51#16837134C2211yescharacteristic, complemented
52Q8⋊(C59⋊C8)C2211yescharacteristic, complemented
53#16837132C2211yescharacteristic, complemented
54#16837131C2211yescharacteristic
55D4⋊(C59⋊C8)C2211yescharacteristic
56#16837129C2211yescharacteristic
57#16837124C2211yescharacteristic, complemented
58#16837124C2211yescharacteristic, complemented
59#16837123C2211yescharacteristic, complemented
60C59×#446C2211yescharacteristic, complemented
61C4×SD16×C59C2211yescharacteristic
62C59×#436C2211yescharacteristic
63C59×#334C2211yescharacteristic, complemented
64C59×Q8⋊C8C2211yescharacteristic, complemented
65C59×D4⋊C8C2211yescharacteristic
66C82×C59C2211yescharacteristic, complemented
67C2×D4×D59C2×C411yescharacteristic
68C22×C4×D59—21no—
69C2×#11745660C2×C411yescharacteristic
70C2×#11745670C2×C411yescharacteristic
71C2×#11745671C2×C411yescharacteristic
72C2×#11745664—21no—
73C2×#11745659—21no—
74C2×#11745659—21no—
75C2×#11745659—21no—
76#14273726C2×C411yescharacteristic
77#14273725C2×C411yescharacteristic
78#14273717—21no—
79#14273707C2×C411yescharacteristic
80#14273705—21no—
81#14273703C2×C411yescharacteristic
82C4×Dic118—21no—
83C4×Dic118C2311yescharacteristic
84C4×D236C2311yescharacteristic
85D4×Dic59C2×C411yescharacteristic
86#14273689C2×C411yescharacteristic
87C4×C22⋊D59—21no—
88(C22⋊D59)⋊C4—21no—
89#14273684—21no—
90C4⋊C4×D59C2×C411yescharacteristic
91C22⋊C4×D59—21no—
92C42×D59C2311yescharacteristic
93#14273659C2×C411yescharacteristic
94#14273658C2×C411yescharacteristic
95#14273658C2×C411yescharacteristic
96C4⋊(C59⋊C8)—21no—
97C4⋊(C59⋊C8)C2311yescharacteristic
98C4⋊(C59⋊C8)C2311yescharacteristic
99Dic59⋊C8—21no—
100Dic59⋊C8—21no—
101Dic59⋊C8C2×C411yescharacteristic
102Dic59⋊C8C2×C411yescharacteristic
103Dic59⋊C8D411yescharacteristic
104Dic59⋊C8D411yescharacteristic
105#14273652C2×C411yescharacteristic
106#14273651C2×C411yescharacteristic
107#14273650C2×C411yescharacteristic
108#14273649C2×C411yescharacteristic
109#14273648C2×C411yescharacteristic
110#14273647C2×C411yescharacteristic
111C22⋊(C59⋊C8)—21no—
112(C2×D59)⋊C8—21no—
113(C2×D59)⋊C8—21no—
114(C2×D59)⋊C8—21no—
115(C2×D59)⋊C8—21no—
116(C2×D59)⋊C8C2×C411yescharacteristic
117(C2×D59)⋊C8C2×C411yescharacteristic
118(C2×D59)⋊C8D411yescharacteristic
119(C2×D59)⋊C8D411yescharacteristic
120#14273637—21no—
121#14273637—21no—
122#14273637—21no—
123C4×C59⋊C8—21no—
124C4×C59⋊C8C2311yescharacteristic
125C4×C59⋊C8C2311yescharacteristic
126C4×C59⋊C8C2311yescharacteristic
127C8×Dic59—21no—
128C8×Dic59—21no—
129C2×SD16×C59C2×C411yescharacteristic
130C2×M16×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—
140#97×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C23×D59—21no—
145C2×Q8×C59C22×C411yescharacteristic
146C2×D4×C59C22×C411yescharacteristic
147C22×C4×C59—21no—
148C2×C22⋊D59—21no—
149C22×Dic59—21no—
150D4×D59—41no—
151D4×D59—41no—
152C2×D236C22×C411yescharacteristic
153C2×C4×D59—21no—
154C2×C4×D59—21no—
155C2×C4×D59—21no—
156C2×C4×D59—81no—
157C2×C4×D59—81no—
158C2×C4×D59C22×C411yescharacteristic
159C2×C4×D59C2×D411yescharacteristic
160C2×C4×D59#4111yescharacteristic
161C2×Dic118—21no—
162C2×Dic118C22×C411yescharacteristic
163SD16×C59—41no—
164M16×C59—81no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59C22×C411yescharacteristic
169C2×C8×C59C22×C411yescharacteristic
170C2×C8×C59C2×D411yescharacteristic
171C2×C8×C59C2×D411yescharacteristic
172C4⋊C4×C59—21no—
173C4⋊C4×C59C22×C411yescharacteristic
174C4⋊C4×C59C22×C411yescharacteristic
175C22⋊C4×C59—21no—
176C42×C59—21no—
177C42×C59C2411yescharacteristic
178C22⋊Dic59—21no—
179#11745671—41no—
180#11745670—41no—
181(C2×D59)⋊C4—21no—
182(C2×D59)⋊C4—21no—
183(C2×D59)⋊C4—21no—
184(C2×D59)⋊C4—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59C22×C411yescharacteristic
187C4⋊Dic59C22×C411yescharacteristic
188Dic59⋊C4—21no—
189Dic59⋊C4—21no—
190Dic59⋊C4—21no—
191Dic59⋊C4—21no—
192Dic59⋊C4—21no—
193Dic59⋊C4—21no—
194C4×Dic59—21no—
195C4×Dic59C22×C411yescharacteristic
196C4×Dic59C2×D411yescharacteristic
197C4×Dic59#4111yescharacteristic
198#11745664—81no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8C22×C411yescharacteristic
203C2×C59⋊C8C22×C411yescharacteristic
204C2×C59⋊C8C2×D411yescharacteristic
205C2×C59⋊C8C2×D411yescharacteristic
206#11745660—41no—
207#11745659—42no—
208#11745659—81no—
209#11745659—81no—
210C23×C59—21no—
211C22×D59—21no—
212C22×D59—42no—
213C22×D59—81no—
214C22×D59—81no—
215C22×D59C4×D411yescharacteristic
216Q8×C59—21no—
217Q8×C59—21no—
218D4×C59—21no—
219D4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59C23×C411yescharacteristic
226C2×C4×C59C22×D411yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C22⋊D59—42no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—81no—
233C2×Dic59C4×D411yescharacteristic
234C2×Dic59C4×D411yescharacteristic
235C2×Dic59C4×D411yescharacteristic
236D236—21no—
237D236—21no—
238C4×D59—21no—
239C4×D59—21no—
240C4×D59—21no—
241C4×D59—21no—
242C4×D59—41no—
243C4×D59—41no—
244C4×D59—41no—
245C4×D59—81no—
246C4×D59—81no—
247Dic118—21no—
248Dic118—21no—
249Dic118—81no—
250C8×C59—22no—
251C8×C59—21no—
252C8×C59—21no—
253C8×C59—21no—
254C8×C59—21no—
255C8×C59—41no—
256C8×C59—41no—
257C59⋊C8—21no—
258C59⋊C8—21no—
259C59⋊C8—21no—
260C59⋊C8—21no—
261C59⋊C8—41no—
262C59⋊C8—41no—
263C59⋊C8—41no—
264#17952—591nomaximal, Sylow
265C22×C59—41no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59C2×C4×D411yescharacteristic
269C2×D59—21no—
270C2×D59—21no—
271C2×D59—21no—
272C2×D59—41no—
273C2×D59—41no—
274C2×D59—41no—
275C2×D59—84no—
276C4×C59—21no—
277C4×C59—41no—
278C4×C59—41no—
279C4×C59—41no—
280C4×C59#56811yescharacteristic
281C4×C59#56811yescharacteristic
282C4×C59#57711yescharacteristic
283C4×C59#57711yescharacteristic
284Dic59—21no—
285Dic59—21no—
286Dic59—21no—
287Dic59—41no—
288Dic59—41no—
289Dic59—41no—
290#3101—591no—
291#2985—591no—
292#2975—591no—
293#1592—591no—
294#1592—591no—
295#1590—591no—
296#1590D5911yescharacteristic, complemented
297#1588—591no—
298#1588—591no—
299#1583—591no—
300#1583—591no—
301#1572—591no—
302#1490—591no—
303#1485—591no—
304#1453—591no—
305C2×C59—81no—
306C2×C59#293011yescharacteristic
307C2×C59#294911yescharacteristic
308C2×C59#295611yescharacteristic
309D59—41no—
310D59—81no—
311#533—591no—
312C2×C4×D4—591no—
313#483—591no—
314#483—591no—
315#476—591no—
316#476—591no—
317#461—591no—
318#461—591no—
319#450—591no—
320#450—591no—
321#446—591no—
322#446—1181no—
323#446—1181no—
324#446C2×D5911yescharacteristic, complemented
325C8×Q8—591no—
326C8×Q8—591no—
327C4×SD16—591no—
328C4×SD16—591no—
329C4×SD16—591no—
330C4×SD16C2×D5911yescharacteristic
331#436—591no—
332#436—591no—
333#436—591no—
334#436—1181no—
335#436—1181no—
336#436C2×D5911yescharacteristic
337#433—1181no—
338#433—1181no—
339#432—591no—
340#431—591no—
341#431—591no—
342#431—591no—
343#407—1181no—
344#407—1181no—
345C4×M16—591no—
346#335—591no—
347#335—591no—
348#334—591no—
349#334C2×D5911yescharacteristic, complemented
350Q8⋊C8—591no—
351Q8⋊C8—591no—
352Q8⋊C8—591no—
353Q8⋊C8C2×D5911yescharacteristic, complemented
354D4⋊C8—591no—
355D4⋊C8—591no—
356D4⋊C8—591no—
357D4⋊C8C2×D5911yescharacteristic
358#322—591no—
359#322—591no—
360#322—591no—
361C82C2×D5911yescharacteristic, complemented
362C59#1795211yescharacteristic, complemented, Sylow
363C22×D4—591no—
364C23×C4—1181no—
365C2×SD16—591no—
366C2×SD16—591no—
367C2×SD16—591no—
368C2×SD16C4×D5911yescharacteristic
369C2×M16—21no—
370C2×M16—1181no—
371C2×M16—1181no—
372C2×M16—1181no—
373C2×M16—1181no—
374C4⋊Q8—591no—
375C4⋊Q8—591no—
376#125—1181no—
377C22⋊Q8—591no—
378C22⋊Q8—591no—
379C22⋊Q8—1181no—
380C4×Q8—591no—
381C4×Q8—1181no—
382C4×Q8C22×D5911yescharacteristic
383C4×D4—591no—
384C4×D4—591no—
385C4×D4—591no—
386C4×D4—1181no—
387C4×D4—1181no—
388C4×D4C22×D5911yescharacteristic
389#117—1181no—
390C2×C4⋊C4—591no—
391C2×C22⋊C4—1181no—
392C2×C42—591no—
393#106—591no—
394#106—591no—
395#106—591no—
396#106C4×D5911yescharacteristic
397C4⋊C8—21no—
398C4⋊C8—591no—
399C4⋊C8—591no—
400C4⋊C8—591no—
401C4⋊C8—591no—
402C4⋊C8—591no—
403C4⋊C8—591no—
404C4⋊C8—1181no—
405C4⋊C8—1181no—
406C4⋊C8—1181no—
407C4⋊C8C22×D5911yescharacteristic
408C4⋊C8C22×D5911yescharacteristic
409#103—591no—
410#103—591no—
411#103—591no—
412#103C4×D5911yescharacteristic
413#102—591no—
414#102—591no—
415#102—591no—
416#102C4×D5911yescharacteristic
417C22⋊C8—21no—
418C22⋊C8—591no—
419C22⋊C8—591no—
420C22⋊C8—591no—
421C22⋊C8—591no—
422C22⋊C8—1181no—
423C22⋊C8—1181no—
424C22⋊C8—1181no—
425C22⋊C8—1181no—
426C22⋊C8—1181no—
427#97—21no—
428#97—1181no—
429#97—1181no—
430#97—1181no—
431C4×C8—591no—
432C4×C8—591no—
433C4×C8—591no—
434C4×C8—1181no—
435C4×C8—1181no—
436C4×C8—1181no—
437C4×C8C22×D5911yescharacteristic
438C4×C8C22×D5911yescharacteristic
439C4×C8C22×D5911yescharacteristic
440C24—1181no—
441C2×Q8—591no—
442C2×Q8—1181no—
443C2×Q8C2×C4×D5911yescharacteristic
444C2×D4—591no—
445C2×D4—1181no—
446C2×D4—2361no—
447C2×D4—2361no—
448C2×D4C2×C4×D5911yescharacteristic
449C22×C4—21no—
450C22×C4—591no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—4721no—
458C22×C4—4721no—
459SD16—41no—
460SD16—2361no—
461SD16—2361no—
462SD16—2361no—
463M16—81no—
464M16—4721no—
465M16—4721no—
466M16—4721no—
467M16—2362no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—591no—
472C2×C8—591no—
473C2×C8—591no—
474C2×C8—591no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8C2×C4×D5911yescharacteristic
479C2×C8C2×C4×D5911yescharacteristic
480C2×C8D4×D5911yescharacteristic
481C2×C8D4×D5911yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—591no—
484C4⋊C4—591no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4C2×C4×D5911yescharacteristic
493C4⋊C4C2×C4×D5911yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—1181no—
496C22⋊C4—1181no—
497C22⋊C4—1181no—
498C22⋊C4—1181no—
499C22⋊C4—1181no—
500C42—21no—
501C42—591no—
502C42—591no—
503C42—591no—
504C42—1181no—
505C42C23×D5911yescharacteristic
506C23—21no—
507C23—591no—
508C23—1181no—
509C23—4721no—
510C23—4721no—
511C23—2362no—
512Q8—21no—
513Q8—21no—
514Q8—1181no—
515Q8—1181no—
516Q8—4721no—
517D4—21no—
518D4—21no—
519D4—1181no—
520D4—1181no—
521D4—2362no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—591no—
528C2×C4—591no—
529C2×C4—591no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—4721no—
538C2×C4—4721no—
539C2×C4—4721no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4C22×C4×D5911yescharacteristic
544C2×C4C2×D4×D5911yescharacteristic
545C2×C4#41×D5911yescharacteristic
546C8—22no—
547C8—21no—
548C8—21no—
549C8—21no—
550C8—21no—
551C8—41no—
552C8—41no—
553C8—1181no—
554C8—1181no—
555C8—1181no—
556C8—1181no—
557C8—2361no—
558C8—2361no—
559C8—2361no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—1181no—
564C22—1181no—
565C22—1181no—
566C22—4724no—
567C22—2361no—
568C22—2361no—
569C22—2361no—
570C22C4×D4×D5911yescharacteristic
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—1181no—
576C4—1181no—
577C4—1181no—
578C4—2361no—
579C4—2361no—
580C4—2361no—
581C4#1683816711yescharacteristic
582C4#1683816811yescharacteristic
583C4#1683820811yescharacteristic
584C4#1683820911yescharacteristic
585C2—81no—
586C2—4721no—
587C2—2361no—
588C2#2143279611yescharacteristic
589C2#2143295011yescharacteristic
590C2#2143300111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1590C2
socleC22×C59C2×C4×D4
prime radicalC2×D4×D59C2×C4
cosocleC42×C59C24
hypercenter#1590D59
hypocenterC59#17952
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #21433192C2semidirect
4. #21422998C2semidirect
5. #21422996C2semidirect
6. #21422988C2semidirect
7. #21422977C2semidirect
8. #21422971C2semidirect
9. #21422940C2semidirect
10. #21422938C2semidirect
11. #21422873C2semidirect
13. #21422515C2semidirect
14. #21422390C2semidirect
15. C59×#1590C2semidirect
26. #16837532C22semidirect
29. Q8×C59⋊C8C22semidirect
30. #16837526C22semidirect
48. #16837154C22semidirect
49. #16837154C22semidirect
50. #16837153C22semidirect
51. #16837134C22semidirect
52. Q8⋊(C59⋊C8)C22semidirect
53. #16837132C22semidirect
57. #16837124C22semidirect
58. #16837124C22semidirect
59. #16837123C22semidirect
60. C59×#446C22semidirect
63. C59×#334C22semidirect
64. C59×Q8⋊C8C22semidirect
66. C82×C59C22semidirect
296. #1590D59semidirect
324. #446C2×D59semidirect
349. #334C2×D59semidirect
353. Q8⋊C8C2×D59semidirect
361. C82C2×D59semidirect
362. C59#17952semidirect

Maximal subgroups

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

subgroupindex
#214341022
#214331922
#214331342
#214229982
#214229962
#214229882
#214229772
#214229712
#214229402
#214229382
#214228732
#214225372
#214225152
#214223902
C59×#15902
#1795259