Finite Group Databasebeta

← #28608420

Subgroups — #28608420

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×#16837489C211yesmaximal, characteristic
2#21427065C211yesmaximal, characteristic
3#21427057C211yesmaximal, characteristic, complemented
4#21424950C211yesmaximal, characteristic
5#21424948C211yesmaximal, characteristic
6#21424737C211yesmaximal, characteristic, complemented
7#21424735C211yesmaximal, characteristic, complemented
8#21424716C211yesmaximal, characteristic
9#21424714C211yesmaximal, characteristic
10#21424257C211yesmaximal, characteristic, complemented
11#21424110C211yesmaximal, characteristic, complemented
12#21424105C211yesmaximal, characteristic
13#106×Dic59C211yesmaximal, characteristic, complemented
14C8×Dic59⋊C4C211yesmaximal, characteristic, complemented
15C4×#106×C59C211yesmaximal, characteristic, complemented
16C22×#11745660C2211yescharacteristic
17C2×#14273692C2211yescharacteristic
18C2×#14273689C2211yescharacteristic
19#16837489C418yescomplemented
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273649C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31C2×C8×Dic59C2211yescharacteristic
32#16837308C2211yescharacteristic
33#16837307C2211yescharacteristic
34#16837304C2211yescharacteristic
35#16837303C2211yescharacteristic
36#16837293—21no—
37#16837291—21no—
38#16837286—21no—
39#16837284—21no—
40#16837269—21no—
41#16837267C2211yescharacteristic
42#16837267C2211yescharacteristic
43#16837264—21no—
44C4⋊C4×Dic59C2211yescharacteristic
45C4⋊C4×Dic59C2211yescharacteristic
46C4×Dic59⋊C4C2211yescharacteristic
47C4×(C2×D59)⋊C4—21no—
48#16837248—21no—
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C2×#11745660—22no—
64C2×#11745660—22no—
65C2×#11745660C2×C414yes—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273692—22no—
69#14273692C2×C414yes—
70#14273689—22no—
71#14273689C2×C414yes—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4D411yescharacteristic
76C2×Dic59⋊C4D411yescharacteristic
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59C2311yescharacteristic
86C2×C4×Dic59C2311yescharacteristic
87#14273660—41no—
88#14273660—41no—
89#14273659—22no—
90Dic59⋊C8—22no—
91#14273652—41no—
92#14273650—41no—
93#14273650C2×C414yes—
94#14273649—41no—
95#14273647—41no—
96#14273647C2×C414yes—
97C8×Dic59—22no—
98C8×Dic59C2×C414yes—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95D411yescharacteristic
110C59⋊#95D411yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C42×C59—21no—
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120#106×C59—22no—
121#106×C59C2×C414yes—
122C4×C8×C59—22no—
123#95×C59—21no—
124#95×C59—21no—
125C23×D59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2411yescharacteristic
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59C2×D411yescharacteristic
137C22×Dic59C2×D411yescharacteristic
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C2×D236—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236C22×C412yes—
146C2×C4×D59—24no—
147C2×C4×D59—81no—
148C2×C4×D59—81no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118C22×C412yes—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59C22×C412yes—
159C2×C8×C59#4112yes—
160C2×C8×C59#4112yes—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59C22×C412yes—
168C4⋊C4×C59C22×C412yes—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—42no—
172C42×C59—81no—
173(C2×D59)⋊C4—24no—
174(C2×D59)⋊C4—42no—
175(C2×D59)⋊C4—42no—
176(C2×D59)⋊C4—81no—
177(C2×D59)⋊C4—81no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182Dic59⋊C4—22no—
183Dic59⋊C4—24no—
184Dic59⋊C4—22no—
185C4×Dic59—21no—
186C4×Dic59—21no—
187C4×Dic59—24no—
188C4×Dic59—42no—
189C4×Dic59—81no—
190C4×Dic59C22×C412yes—
191C4×Dic59#4112yes—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8C22×C412yes—
198#11745660—28no—
199C23×C59C22×D411yescharacteristic
200C22×D59—22no—
201C22×D59—41no—
202C22×D59—41no—
203C22×D59—41no—
204C22×D59—41no—
205C22×D59—41no—
206C22×D59—41no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59#13512yes—
225C2×C4×C59C23×C411yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59H(5,𝔽2)11yescharacteristic
230C2×C4×C592−1+411yescharacteristic
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—22no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59C4×D412yes—
250C2×Dic59C4×D412yes—
251C2×Dic59#13512yes—
252D236—22no—
253D236—22no—
254D236—41no—
255D236—41no—
256C4×D59—84no—
257Dic118—22no—
258Dic118—22no—
259Dic118—41no—
260Dic118—41no—
261C8×C59—41no—
262C8×C59#11714yes—
263C59⋊C8—22no—
264C59⋊C8—41no—
265#22806—591nomaximal, Sylow
266C22×C59C2×C4×D411yescharacteristic
267C22×C59#54711yescharacteristic
268C22×C59#55011yescharacteristic
269C22×C59C2×#13511yescharacteristic
270C22×C59C2×#13511yescharacteristic
271C22×C59#57711yescharacteristic
272C22×C59#57711yescharacteristic
273C2×D59—42no—
274C2×D59—42no—
275C2×D59—42no—
276C2×D59—81no—
277C2×D59—81no—
278C2×D59—81no—
279C2×D59—81no—
280C4×C59—22no—
281C4×C59—41no—
282C4×C59—42no—
283C4×C59—42no—
284C4×C59—81no—
285C4×C59—81no—
286C4×C59C2×#11711yescharacteristic
287C4×C59#52011yescharacteristic
288C4×C59#55511yescharacteristic
289C4×C59#55611yescharacteristic
290Dic59—22no—
291Dic59—22no—
292Dic59—22no—
293Dic59—41no—
294Dic59—42no—
295Dic59—81no—
296Dic59—81no—
297C2×C4×SD16—591no—
298#2357—591no—
299#2356—591no—
300#1968—591no—
301#1968—591no—
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×#106D5911yescharacteristic, complemented
311C8×C4⋊C4—591no—
312C2×C59#235411yescharacteristic
313C2×C59C4×#13511yescharacteristic
314C2×C59#294911yescharacteristic
315C2×C59#333711yescharacteristic
316C2×C59#333711yescharacteristic
317C2×C59#339411yescharacteristic
318C2×C59#339411yescharacteristic
319D59—82no—
320C22×SD16—591no—
321C2×C4×Q8—591no—
322C2×C4×D4—591no—
323C4×SD16—598no—
324C2×#107—591no—
325C2×#107—591no—
326C2×#106—591no—
327C2×#106C2×D5911yescharacteristic
328C2×#106C2×D5911yescharacteristic
329C2×C4⋊C8—591no—
330C2×#103—591no—
331C2×#103—591no—
332C2×#103—591no—
333C2×#102—591no—
334C2×#102—591no—
335C2×#102—591no—
336C2×C4×C8—591no—
337C2×C4×C8—591no—
338C2×C4×C8C2×D5911yescharacteristic
339#391—591no—
340#391—591no—
341#390—591no—
342#390—591no—
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⋊C4C2×D5911yescharacteristic
355C4×C4⋊C4C2×D5911yescharacteristic
356C4×C22⋊C4—1181no—
357#376—1181no—
358#340—591no—
359#340—591no—
360#340C2×D5911yescharacteristic
361#340C2×D5911yescharacteristic
362#336—591no—
363#336—591no—
364C59#2280611yescharacteristic, complemented, Sylow
365C22×Q8—591no—
366C22×D4—591no—
367C23×C4—1181no—
368C2×SD16—594no—
369C2×SD16—1182no—
370C2×SD16—1182no—
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—1181no—
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C4⋊C4C22×D5911yescharacteristic
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—1181no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C22⋊C4—1181no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—591no—
396C2×C42—591no—
397C2×C42—1181no—
398C2×C42—1181no—
399C2×C42—1181no—
400C2×C42C22×D5911yescharacteristic
401#107—2361no—
402#107—2361no—
403#106—22no—
404#106—1182no—
405#106C4×D5914yes—
406C4⋊C8—1182no—
407#103—594no—
408#103—2361no—
409#103—2361no—
410#102—594no—
411#102—2361no—
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—1181no—
431C2×Q8—592no—
432C2×Q8—1181no—
433C2×Q8—1181no—
434C2×Q8—1181no—
435C2×Q8—1181no—
436C2×Q8—1181no—
437C2×D4—592no—
438C2×D4—1181no—
439C2×D4—1181no—
440C2×D4—1181no—
441C2×D4—1181no—
442C2×D4—1181no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—591no—
448C22×C4—591no—
449C22×C4—591no—
450C22×C4—591no—
451C22×C4—1184no—
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—1188no—
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#1174568512yes—
472C2×C8#1174568512yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—1182no—
480C4⋊C4—1184no—
481C4⋊C4—1182no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4C2×C4×D5912yes—
487C4⋊C4C2×C4×D5912yes—
488C22⋊C4—1184no—
489C22⋊C4—4721no—
490C22⋊C4—4721no—
491C22⋊C4—2362no—
492C22⋊C4—2362no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—81no—
497C42—592no—
498C42—592no—
499C42—1184no—
500C42—1181no—
501C42—1181no—
502C42—4721no—
503C42—2362no—
504C23—1182no—
505C23—2361no—
506C23—2361no—
507C23—2361no—
508C23—2361no—
509C23—2361no—
510C23—2361no—
511C23C2×D4×D5911yescharacteristic
512Q8—1182no—
513Q8—1182no—
514Q8—2361no—
515Q8—2361no—
516D4—1182no—
517D4—1182no—
518D4—2361no—
519D4—2361no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—592no—
538C2×C4—592no—
539C2×C4—592no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1182no—
550C2×C4—4724no—
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×#1174568511yescharacteristic
563C2×C4C2×#1174568511yescharacteristic
564C2×C4#1427377311yescharacteristic
565C2×C4#1427377511yescharacteristic
566C8—41no—
567C8—1182no—
568C8—2361no—
569C8#1427368414yes—
570C22—4721no—
571C22—4721no—
572C22—4721no—
573C22—4721no—
574C22—2362no—
575C22—2362no—
576C22—2362no—
577C22C4×D4×D5911yescharacteristic
578C22#1683805311yescharacteristic
579C22#1683807311yescharacteristic
580C22#135×D5911yescharacteristic
581C22C2×#1427374811yescharacteristic
582C22#1683820811yescharacteristic
583C22#1683820911yescharacteristic
584C4—22no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—81no—
589C4—81no—
590C4—1182no—
591C4—1182no—
592C4—1182no—
593C4—4721no—
594C4—4721no—
595C4—2361no—
596C4—2362no—
597C4C2×#1427368411yescharacteristic
598C4#1683788411yescharacteristic
599C4#1683811511yescharacteristic
600C4#1683812411yescharacteristic
601C2—4722no—
602C2#2142704111yescharacteristic
603C2C4×#1427374811yescharacteristic
604C2#2143295011yescharacteristic
605C2#2143647111yescharacteristic
606C2#2143647611yescharacteristic
607C2#2143709011yescharacteristic
608C2#2143709311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#106×C59C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenterC4×#106D59
hypocenterC59#22806
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
3. #21427057C2semidirect
6. #21424737C2semidirect
7. #21424735C2semidirect
10. #21424257C2semidirect
11. #21424110C2semidirect
13. #106×Dic59C2semidirect
14. C8×Dic59⋊C4C2semidirect
15. C4×#106×C59C2semidirect
19. #16837489C4semidirect
310. C4×#106D59semidirect
364. C59#22806semidirect

Maximal subgroups

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

subgroupindex
C2×#168374892
#214270652
#214270572
#214249502
#214249482
#214247372
#214247352
#214247162
#214247142
#214242572
#214241102
#214241052
#106×Dic592
C8×Dic59⋊C42
C4×#106×C592
#2280659