Finite Group Databasebeta

← #28596692

Subgroups — #28596692

Order 15104.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837755C211yesmaximal, characteristic
2#21426836C211yesmaximal, characteristic, complemented
3#21426806C211yesmaximal, characteristic
4#21425063C211yesmaximal, characteristic, complemented
5#21424911C211yesmaximal, characteristic
6#21424796C211yesmaximal, characteristic
7#21424785C211yesmaximal, characteristic, complemented
8#21424772C211yesmaximal, characteristic, complemented
9#21424222C211yesmaximal, characteristic, complemented
10#21423925C211yesmaximal, characteristic, complemented
11#21423923C211yesmaximal, characteristic
12#21423915C211yesmaximal, characteristic, complemented
13#102×Dic59C211yesmaximal, characteristic
14#21423782C211yesmaximal, characteristic, complemented
15C59×#1934C211yesmaximal, characteristic
16C2×#14273718C2211yescharacteristic
17C2×#14273714C2211yescharacteristic
18C2×#14273692C2211yescharacteristic
19C2×D4×Dic59C2211yescharacteristic
20#16837755C418yescomplemented
21C2×#14273652C2211yescharacteristic
22C2×#14273652C2211yescharacteristic
23C2×#14273650C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31C2×C8×Dic59C2211yescharacteristic
32#16837308C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837298—21no—
35#16837297—21no—
36#16837296—21no—
37#16837296—21no—
38#16837287—21no—
39#16837284—21no—
40#16837274—21no—
41#16837274—21no—
42#16837271C2211yescharacteristic
43#16837271C2211yescharacteristic
44C4⋊C4×Dic59C2211yescharacteristic
45C4⋊C4×Dic59C2211yescharacteristic
46#16837248C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#102×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C59×#390C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C23×Dic59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273718—22no—
65#14273718C2×C414yescomplemented
66#14273714—22no—
67#14273714C2×C414yes—
68#14273692—22no—
69#14273692—22no—
70D4×Dic59—22no—
71D4×Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59—21no—
85C2×C22⋊Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59C2311yescharacteristic
90C2×C4×Dic59C2311yescharacteristic
91#14273652—22no—
92#14273652C2×C414yescomplemented
93#14273650—22no—
94#14273650C2×C414yescomplemented
95#14273648—22no—
96#14273648C2×C414yes—
97#14273638—22no—
98#14273637—22no—
99#14273637—22no—
100C8×Dic59—22no—
101C8×Dic59C2×C414yescomplemented
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C59⋊#95D411yescharacteristic
110C59⋊#95D411yescharacteristic
111C22×D4×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#102×C59—22no—
122#102×C59C2×C414yes—
123C4×C8×C59—22no—
124#95×C59—21no—
125C24×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59C22×C412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—42no—
143C22×Dic59—81no—
144C22×Dic59—81no—
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
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×C59—21no—
159C2×C8×C59C22×C412yes—
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—81no—
167C4⋊C4×C59C22×C412yes—
168C22⋊C4×C59—81no—
169C22⋊C4×C59—81no—
170C42×C59—21no—
171C42×C59—21no—
172C22⋊Dic59—24no—
173C22⋊Dic59—24no—
174C22⋊Dic59—42no—
175C4⋊Dic59—21no—
176C4⋊Dic59—24no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C412yes—
182Dic59⋊C4—24no—
183Dic59⋊C4—42no—
184Dic59⋊C4—81no—
185Dic59⋊C4—81no—
186Dic59⋊C4—81no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189C4×Dic59—42no—
190C4×Dic59—81no—
191C4×Dic59—81no—
192C4×Dic59C22×C412yes—
193C4×Dic59#4112yes—
194C4×Dic59#4112yes—
195C4×Dic59#4112yes—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8C22×C412yes—
201C2×C59⋊C8#4112yes—
202C23×C59—22no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59C22×D411yescharacteristic
210D4×C59—22no—
211D4×C59—22no—
212D4×C59—41no—
213D4×C59—41no—
214C2×C4×C59—22no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
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×C59C23×C411yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—41no—
246C2×Dic59—41no—
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—84no—
256C2×Dic59C4×D412yes—
257C2×Dic59C4×D412yes—
258C2×Dic59#12412yes—
259C2×Dic59#12412yes—
260Dic118—22no—
261Dic118—22no—
262Dic118—41no—
263Dic118—41no—
264C8×C59—22no—
265C8×C59—22no—
266C59⋊C8—22no—
267C59⋊C8—22no—
268#21643—591nomaximal, Sylow
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—81no—
273C22×C59—81no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59C2×C4×D411yescharacteristic
277C22×C59C2×#12411yescharacteristic
278C22×C59#53211yescharacteristic
279C22×C59#57611yescharacteristic
280C22×C59#57711yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57811yescharacteristic
283C4×C59—41no—
284C4×C59—41no—
285C4×C59—42no—
286C4×C59—81no—
287C4×C59—81no—
288C4×C59C4×#4111yescharacteristic
289C4×C59C4×#4111yescharacteristic
290C4×C59#52911yescharacteristic
291C4×C59#52911yescharacteristic
292Dic59—22no—
293Dic59—22no—
294Dic59—22no—
295Dic59—22no—
296Dic59—42no—
297Dic59—42no—
298Dic59—81no—
299Dic59—81no—
300C2×#488—591no—
301#2330—591no—
302#2326—591no—
303#1992—591no—
304#1959—591no—
305#1936—591no—
306#1935—591no—
307#1934D5911yescharacteristic
308#1933—591no—
309#1834—591no—
310#1763—591no—
311#1762—591no—
312C4×#103—591no—
313C4×#102—591no—
314#1726—591no—
315C2×C59—82no—
316C2×C59C4×#12411yescharacteristic
317C2×C59#295111yescharacteristic
318C2×C59#295211yescharacteristic
319C2×C59#313611yescharacteristic
320C2×C59#313711yescharacteristic
321C2×C59#317811yescharacteristic
322C2×C59#317911yescharacteristic
323C2×#125—591no—
324C2×#124—591no—
325C2×C4×Q8—591no—
326C2×C4×D4—591no—
327#488—598no—
328C2×#103—591no—
329C2×#103—591no—
330C2×#103—591no—
331C2×#103—591no—
332C2×#102—591no—
333C2×#102—591no—
334C2×#102C2×D5911yescharacteristic
335C2×#102C2×D5911yescharacteristic
336C2×#97—591no—
337C2×#97—591no—
338C2×#97—591no—
339C2×C4×C8—591no—
340C2×C4×C8—591no—
341C2×C4×C8C2×D5911yescharacteristic
342#391—591no—
343#390C2×D5911yescharacteristic
344#389—591no—
345#389—1181no—
346#389—1181no—
347#389C2×D5911yescharacteristic
348#388—1181no—
349#388—1181no—
350#387—1181no—
351#387—1181no—
352#385—1181no—
353#385—1181no—
354#384—591no—
355#384—591no—
356C4×C4⋊C4—591no—
357C4×C4⋊C4—591no—
358#376—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340C2×D5911yescharacteristic
366#340C2×D5911yescharacteristic
367C59#2164311yescharacteristic, complemented, Sylow
368C22×Q8—591no—
369C22×D4C22×D5911yescharacteristic
370C23×C4—1181no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375#125—594no—
376#125—1182no—
377#124—594no—
378#124—1182no—
379C4×Q8—1182no—
380C4×Q8—1182no—
381C4×D4—1182no—
382C4×D4—1182no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C42—591no—
404C2×C42—591no—
405C2×C42—1181no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42C22×D5911yescharacteristic
409#103—594no—
410#103—594no—
411#103—1182no—
412#103—1182no—
413#102—22no—
414#102—594no—
415#102—1182no—
416#102C4×D5914yes—
417#97—1182no—
418#97—1182no—
419#97—1182no—
420C4×C8—22no—
421C4×C8—594no—
422C4×C8—1182no—
423#95—21no—
424#95—591no—
425#95—591no—
426#95—591no—
427#95—591no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433C24—21no—
434C2×Q8—592no—
435C2×Q8—1181no—
436C2×Q8—1181no—
437C2×Q8—1181no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4C2×C4×D5912yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—591no—
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—
459C22×C4—2362no—
460C22×C4D4×D5911yescharacteristic
461C22×C4D4×D5911yescharacteristic
462C22×C4C23×D5911yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—592no—
468C2×C8—592no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8C2×C4×D5912yes—
474C2×C8#1174568712yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—81no—
481C4⋊C4—592no—
482C4⋊C4—1184no—
483C4⋊C4—1184no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—4721no—
490C4⋊C4—4721no—
491C4⋊C4—4721no—
492C4⋊C4—4721no—
493C4⋊C4—4721no—
494C4⋊C4—2362no—
495C4⋊C4C2×C4×D5912yes—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—1184no—
499C22⋊C4—1184no—
500C22⋊C4—2362no—
501C42—21no—
502C42—21no—
503C42—592no—
504C42—592no—
505C42—592no—
506C42—592no—
507C42—4721no—
508C42—4721no—
509C42—2362no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D5911yescharacteristic
518Q8—1182no—
519Q8—1182no—
520Q8—2361no—
521Q8—2361no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—592no—
542C2×C4—592no—
543C2×C4—592no—
544C2×C4—592no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1182no—
554C2×C4—1182no—
555C2×C4—4724no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4C22×C4×D5911yescharacteristic
567C2×C4#41×D5911yescharacteristic
568C2×C4#41×D5911yescharacteristic
569C2×C4#41×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4C2×#1174568711yescharacteristic
572C8—22no—
573C8—22no—
574C8—1182no—
575C8—1182no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C4×D4×D5911yescharacteristic
584C22#124×D5911yescharacteristic
585C22#1683794711yescharacteristic
586C22#1683820311yescharacteristic
587C22#1683820811yescharacteristic
588C22#1683820911yescharacteristic
589C22#1683821611yescharacteristic
590C4—42no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—1182no—
596C4—1182no—
597C4—1182no—
598C4—1182no—
599C4—4721no—
600C4—4721no—
601C4—2362no—
602C4—2362no—
603C4C4×#1174568711yescharacteristic
604C4#1683787011yescharacteristic
605C4#1683792911yescharacteristic
606C4#1683793511yescharacteristic
607C2—82no—
608C2#2142663711yescharacteristic
609C2#2143296611yescharacteristic
610C2#2143297411yescharacteristic
611C2#2143444511yescharacteristic
612C2#2143446011yescharacteristic
613C2#2143475911yescharacteristic
614C2#2143478211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1934C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1934D59
hypocenterC59#21643
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1934

Complements

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

NH ≅ G/Ntype
2. #21426836C2semidirect
4. #21425063C2semidirect
7. #21424785C2semidirect
8. #21424772C2semidirect
9. #21424222C2semidirect
10. #21423925C2semidirect
12. #21423915C2semidirect
14. #21423782C2semidirect
20. #16837755C4semidirect
65. #14273718C2×C4semidirect
92. #14273652C2×C4semidirect
94. #14273650C2×C4semidirect
101. C8×Dic59C2×C4semidirect
367. C59#21643semidirect

Maximal subgroups

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

subgroupindex
C2×#168377552
#214268362
#214268062
#214250632
#214249112
#214247962
#214247852
#214247722
#214242222
#214239252
#214239232
#214239152
#102×Dic592
#214237822
C59×#19342
#2164359