Finite Group Databasebeta

← #28608424

Subgroups — #28608424

Order 15104.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic59C211yesmaximal, characteristic
2#21427068C211yesmaximal, characteristic
3#21427060C211yesmaximal, characteristic, complemented
4#21424952C211yesmaximal, characteristic
5#21424740C211yesmaximal, characteristic, complemented
6#21424734C211yesmaximal, characteristic, complemented
7#21424719C211yesmaximal, characteristic
8#21424713C211yesmaximal, characteristic
9#21424257C211yesmaximal, characteristic, complemented
10#21424109C211yesmaximal, characteristic, complemented
11#21424104C211yesmaximal, characteristic
12#106×Dic59C211yesmaximal, characteristic, complemented
13C4×#14273659C211yesmaximal, characteristic, complemented
14C8×Dic59⋊C4C211yesmaximal, characteristic, complemented
15C59×#1968C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18SD16×Dic59C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273659C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29C2×C8×Dic59C2211yescharacteristic
30#16837309C2211yescharacteristic
31#16837305C2211yescharacteristic
32#16837296—21no—
33#16837290—21no—
34#16837287—21no—
35#16837285—21no—
36#16837269—21no—
37#16837267C2211yescharacteristic
38#16837267C2211yescharacteristic
39#16837263—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×Dic59⋊C4C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45C22⋊C4×Dic59—21no—
46#16837247—21no—
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C22×SD16×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C2×C4×C8×C59C2211yescharacteristic
58C59×#391C2211yescharacteristic
59C59×#390C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63Q8×Dic59—22no—
64Q8×Dic59C2×C414yes—
65D4×Dic59—22no—
66D4×Dic59C2×C414yes—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4D411yescharacteristic
73C2×Dic59⋊C4D411yescharacteristic
74C2×C22⋊Dic59—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59C2311yescharacteristic
83C2×C4×Dic59C2311yescharacteristic
84#14273660—41no—
85#14273660—41no—
86#14273659—22no—
87#14273659C2×C414yes—
88Dic59⋊C8—22no—
89#14273651—41no—
90#14273651C2×C414yes—
91#14273648—41no—
92#14273648C2×C414yes—
93C8×Dic59—22no—
94C8×Dic59C2×C414yes—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C22×Q8×C59C2311yescharacteristic
108C22×D4×C59C2311yescharacteristic
109C2×SD16×C59—22no—
110C2×SD16×C59—22no—
111C2×SD16×C59C2×C414yes—
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C42×C59C2311yescharacteristic
119#106×C59—22no—
120#103×C59—41no—
121#102×C59—41no—
122C4×C8×C59—22no—
123#95×C59—21no—
124#95×C59—21no—
125C24×C59—21no—
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59C22×C412yes—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—24no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—81no—
150C22×Dic59—81no—
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155SD16×C59—28no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59C22×C412yes—
160C2×C8×C59#4112yes—
161C2×C8×C59#4112yes—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C22⋊C4×C59—42no—
167C22⋊C4×C59—81no—
168C42×C59—21no—
169C42×C59—21no—
170C22⋊Dic59—24no—
171C22⋊Dic59—42no—
172C22⋊Dic59—81no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—24no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59C22×C412yes—
181C4⋊Dic59C22×C412yes—
182Dic59⋊C4—22no—
183Dic59⋊C4—22no—
184C4×Dic59—21no—
185C4×Dic59—21no—
186C4×Dic59—24no—
187C4×Dic59—42no—
188C4×Dic59—42no—
189C4×Dic59—81no—
190C4×Dic59—81no—
191C4×Dic59C22×C412yes—
192C4×Dic59#4112yes—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C22×C412yes—
199C23×C59—22no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59C22×D411yescharacteristic
207Q8×C59—22no—
208Q8×C59—22no—
209Q8×C59—41no—
210Q8×C59—41no—
211D4×C59—22no—
212D4×C59—22no—
213D4×C59—41no—
214D4×C59—41no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—22no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59#13512yes—
230C2×C4×C59C23×C411yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59H(5,𝔽2)11yescharacteristic
235C2×C4×C592−1+411yescharacteristic
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—84no—
258C2×Dic59C4×D412yes—
259C2×Dic59C4×D412yes—
260C2×Dic59#13512yes—
261C8×C59—41no—
262C8×C59#11714yes—
263C59⋊C8—22no—
264C59⋊C8—41no—
265#22806—591nomaximal, Sylow
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—81no—
270C22×C59—81no—
271C22×C59—81no—
272C22×C59—81no—
273C22×C59C2×C4×D411yescharacteristic
274C22×C59#54711yescharacteristic
275C22×C59#55011yescharacteristic
276C22×C59C2×#13511yescharacteristic
277C22×C59C2×#13511yescharacteristic
278C22×C59#57711yescharacteristic
279C22×C59#57711yescharacteristic
280C4×C59—22no—
281C4×C59—41no—
282C4×C59—42no—
283C4×C59—81no—
284C4×C59—81no—
285C4×C59C2×#11711yescharacteristic
286C4×C59#52011yescharacteristic
287C4×C59#55511yescharacteristic
288C4×C59#55611yescharacteristic
289Dic59—22no—
290Dic59—22no—
291Dic59—22no—
292Dic59—41no—
293Dic59—42no—
294Dic59—42no—
295Dic59—81no—
296Dic59—81no—
297C2×C4×SD16—591no—
298#2357—591no—
299#2356—591no—
300#1968—591no—
301#1968D5911yescharacteristic
302#1928—591no—
303#1928—591no—
304#1925—591no—
305#1925—591no—
306#1846—591no—
307#1805—591no—
308#1804—591no—
309C4×#106—591no—
310C4×#106—591no—
311C8×C4⋊C4—591no—
312C2×C59—82no—
313C2×C59#235411yescharacteristic
314C2×C59C4×#13511yescharacteristic
315C2×C59#294911yescharacteristic
316C2×C59#333711yescharacteristic
317C2×C59#333711yescharacteristic
318C2×C59#339411yescharacteristic
319C2×C59#339411yescharacteristic
320C22×SD16C2×D5911yescharacteristic
321C2×C4×Q8—591no—
322C2×C4×D4—591no—
323C4×SD16—598no—
324C2×#107—591no—
325C2×#107—591no—
326C2×#106—591no—
327C2×#106—591no—
328C2×#106C2×D5911yescharacteristic
329C2×C4⋊C8—591no—
330C2×#103—591no—
331C2×#103—591no—
332C2×#103C2×D5911yescharacteristic
333C2×#102—591no—
334C2×#102—591no—
335C2×#102C2×D5911yescharacteristic
336C2×C4×C8—591no—
337C2×C4×C8—591no—
338C2×C4×C8C2×D5911yescharacteristic
339#391—591no—
340#391C2×D5911yescharacteristic
341#390—591no—
342#390C2×D5911yescharacteristic
343#388—1181no—
344#388—1181no—
345#387—1181no—
346#387—1181no—
347#383—1181no—
348#382—591no—
349#382—591no—
350#381—1181no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C22⋊C4—1181no—
357#376—1181no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#336—591no—
363#336—591no—
364C59#2280611yescharacteristic, complemented, Sylow
365C22×Q8C22×D5911yescharacteristic
366C22×D4C22×D5911yescharacteristic
367C23×C4—1181no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×SD16C4×D5914yes—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375C4×Q8—594no—
376C4×Q8—1182no—
377C4×D4—594no—
378C4×D4—1182no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C4⋊C4C22×D5911yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C22⋊C4—1181no—
393C2×C42—591no—
394C2×C42—591no—
395C2×C42—1181no—
396C2×C42—1181no—
397C2×C42—1181no—
398C2×C42—1181no—
399C2×C42—1181no—
400C2×C42C22×D5911yescharacteristic
401#107—2361no—
402#107—2361no—
403#106—22no—
404#106—594no—
405#106—1182no—
406C4⋊C8—1182no—
407#103—41no—
408#103—594no—
409#103—2361no—
410#102—41no—
411#102—594no—
412#102—2361no—
413C4×C8—22no—
414C4×C8—594no—
415C4×C8—1182no—
416#95—21no—
417#95—21no—
418#95—591no—
419#95—591no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430C24—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×C4×D5912yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D5912yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—591no—
447C22×C4—591no—
448C22×C4—591no—
449C22×C4—591no—
450C22×C4—1184no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—4721no—
456C22×C4—4721no—
457C22×C4D4×D5911yescharacteristic
458C22×C4D4×D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460SD16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—592no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8C2×C4×D5912yes—
471C2×C8#1174568712yes—
472C2×C8#1174568712yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—592no—
478C4⋊C4—592no—
479C4⋊C4—1182no—
480C4⋊C4—1184no—
481C4⋊C4—1182no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C22⋊C4—42no—
489C22⋊C4—81no—
490C22⋊C4—1184no—
491C22⋊C4—4721no—
492C22⋊C4—2362no—
493C42—21no—
494C42—21no—
495C42—592no—
496C42—592no—
497C42—1184no—
498C42—1181no—
499C42—1181no—
500C42—4721no—
501C42—4721no—
502C42—2362no—
503C42—2362no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D5911yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—592no—
535C2×C4—592no—
536C2×C4—592no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1182no—
547C2×C4—1182no—
548C2×C4—4724no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4#1427374812yes—
560C2×C4C22×C4×D5911yescharacteristic
561C2×C4#41×D5911yescharacteristic
562C2×C4C2×#1174568711yescharacteristic
563C2×C4C2×#1174568711yescharacteristic
564C2×C4#1427377211yescharacteristic
565C2×C4#1427377411yescharacteristic
566C8—41no—
567C8—1182no—
568C8—2361no—
569C8#1427368314yes—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×D4×D5911yescharacteristic
578C22#1683805211yescharacteristic
579C22#1683807211yescharacteristic
580C22#135×D5911yescharacteristic
581C22C2×#1427374811yescharacteristic
582C22#1683820811yescharacteristic
583C22#1683820911yescharacteristic
584C4—22no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—1182no—
590C4—1182no—
591C4—1182no—
592C4—4721no—
593C4—4721no—
594C4—2362no—
595C4—2361no—
596C4—2362no—
597C4C2×#1427368311yescharacteristic
598C4#1683788111yescharacteristic
599C4#1683811611yescharacteristic
600C4#1683812111yescharacteristic
601C2—82no—
602C2#2142704411yescharacteristic
603C2C4×#1427374811yescharacteristic
604C2#2143295011yescharacteristic
605C2#2143647011yescharacteristic
606C2#2143648111yescharacteristic
607C2#2143708911yescharacteristic
608C2#2143709711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1968C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1968D59
hypocenterC59#22806
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
3. #21427060C2semidirect
5. #21424740C2semidirect
6. #21424734C2semidirect
9. #21424257C2semidirect
10. #21424109C2semidirect
12. #106×Dic59C2semidirect
13. C4×#14273659C2semidirect
14. C8×Dic59⋊C4C2semidirect
18. SD16×Dic59C4semidirect
364. C59#22806semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic592
#214270682
#214270602
#214249522
#214247402
#214247342
#214247192
#214247132
#214242572
#214241092
#214241042
#106×Dic592
C4×#142736592
C8×Dic59⋊C42
C59×#19682
#2280659