Finite Group Databasebeta

← #28599264

Subgroups — #28599264

Order 15104.

786 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837456C211yesmaximal, characteristic
2#21426499C211yesmaximal, characteristic
3#21426491C211yesmaximal, characteristic
4#21424867C211yesmaximal, characteristic, complemented
5#21424865C211yesmaximal, characteristic, complemented
6#21424863C211yesmaximal, characteristic, complemented
7#21424861C211yesmaximal, characteristic, complemented
8#21424297C211yesmaximal, characteristic
9#21424295C211yesmaximal, characteristic
10#21424293C211yesmaximal, characteristic
11#21424263C211yesmaximal, characteristic, complemented
12#21424261C211yesmaximal, characteristic, complemented
13#21424228C211yesmaximal, characteristic, complemented
14#21424228C211yesmaximal, characteristic, complemented
15C59×#1854C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C22×C4×Dic59C2211yescharacteristic
18#16837456C418yescomplemented
19C2×#14273661C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×Dic59⋊C8C2211yescharacteristic
26C2×Dic59⋊C8C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30#16837271C2211yescharacteristic
31#16837271C2211yescharacteristic
32#16837269—21no—
33#16837267C2211yescharacteristic
34#16837267C2211yescharacteristic
35#16837267C2211yescharacteristic
36#16837267C2211yescharacteristic
37#16837263—21no—
38#16837262C2211yescharacteristic
39#16837260—21no—
40#16837260—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4×Dic59⋊C4—21no—
44C4×Dic59⋊C4—21no—
45C22⋊C4×Dic59—21no—
46#16837247—21no—
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#117×C59C2211yescharacteristic
54C2×#107×C59C2211yescharacteristic
55C2×#106×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C23×Dic59D411yescharacteristic
61C23×Dic59D411yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273686—41no—
65#14273686—41no—
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C22⋊Dic59—21no—
75C2×C22⋊Dic59—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59C2×C414yes—
80C2×C4×Dic59D411yescharacteristic
81C2×C4×Dic59D411yescharacteristic
82C2×C4×Dic59Q814yes—
83C2×C4×Dic59C2311yescharacteristic
84C2×C4×Dic59C2311yescharacteristic
85#14273661—41no—
86#14273660—41no—
87#14273660—41no—
88#14273659—41no—
89#14273658—41no—
90#14273658—41no—
91Dic59⋊C8C2×C414yes—
92Dic59⋊C8C2×C414yes—
93C22⋊(C59⋊C8)C2×C414yes—
94#14273637C2×C414yes—
95#14273637C2×C414yes—
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⋊#95—21no—
106C59⋊#95—21no—
107C23×C4×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110#117×C59—41no—
111#117×C59—41no—
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C42×C59—21no—
119#107×C59—41no—
120#106×C59—41no—
121C22⋊C8×C59C2×C414yes—
122#95×C59—21no—
123C24×C59C2×D411yescharacteristic
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59C22×C412yes—
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×Q812yes—
135C22×C4×C59C2411yescharacteristic
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—41no—
141C22×Dic59—41no—
142C22×Dic59—41no—
143C22×Dic59—41no—
144C22×Dic59C4⋊C414yes—
145C22×Dic59C4⋊C414yes—
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59C22×C412yes—
155C2×C8×C59C22×C412yes—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C22⋊C4×C59—42no—
165C22⋊C4×C59—42no—
166C42×C59—42no—
167C22⋊Dic59—42no—
168C22⋊Dic59—42no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177Dic59⋊C4—42no—
178Dic59⋊C4—42no—
179Dic59⋊C4—42no—
180Dic59⋊C4—42no—
181C4×Dic59—42no—
182C4×Dic59—42no—
183C4×Dic59—42no—
184C4×Dic59C4⋊C414yes—
185C4×Dic59C4⋊C414yes—
186C4×Dic59C22×C412yes—
187C4×Dic59C22×C412yes—
188C4×Dic59C2×Q812yes—
189C4×Dic59C2×Q812yes—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8C22×C412yes—
195C2×C59⋊C8C22×C412yes—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59C2×C4⋊C412yes—
203C23×C59C22×D411yescharacteristic
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—22no—
207C2×C4×C59—22no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
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×C59C2×C4⋊C412yes—
223C2×C4×C59C23×C411yescharacteristic
224C2×C4×C59C22×Q811yescharacteristic
225C2×C4×C59H(5,𝔽2)11yescharacteristic
226C2×C4×C59H(5,𝔽2)11yescharacteristic
227C2×C4×C592−1+411yescharacteristic
228C2×C4×C592−1+411yescharacteristic
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—44no—
238C2×Dic59—44no—
239C2×Dic59—41no—
240C2×Dic59—41no—
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×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59C2×C4⋊C412yes—
252C2×Dic59C2×C4⋊C412yes—
253C2×Dic59C2×C4⋊C412yes—
254C2×Dic59C2×C4⋊C412yes—
255C8×C59—22no—
256C8×C59—22no—
257C59⋊C8—22no—
258C59⋊C8—22no—
259#21989—591nomaximal, Sylow
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59—41no—
266C22×C59—41no—
267C22×C59C22×C4⋊C411yescharacteristic
268C22×C59#53511yescharacteristic
269C22×C59#53611yescharacteristic
270C22×C59#57511yescharacteristic
271C22×C59#57511yescharacteristic
272C22×C59#57511yescharacteristic
273C22×C59#57511yescharacteristic
274C4×C59—22no—
275C4×C59—81no—
276C4×C59—81no—
277C4×C59—81no—
278C4×C59—81no—
279C4×C59#52011yescharacteristic
280C4×C59#52011yescharacteristic
281C4×C59#54411yescharacteristic
282C4×C59#54411yescharacteristic
283Dic59—22no—
284Dic59—22no—
285Dic59—22no—
286Dic59—22no—
287Dic59—81no—
288Dic59—81no—
289Dic59—81no—
290Dic59—81no—
291C2×#432—591no—
292#2285—591no—
293#2283—591no—
294#1949—591no—
295#1949—591no—
296#1949—591no—
297#1949—591no—
298#1854—591no—
299#1854—591no—
300#1854—591no—
301#1854D5911yescharacteristic
302#1847—591no—
303#1847—591no—
304#1836—591no—
305#1836—591no—
306C2×C59—42no—
307C2×C59#231711yescharacteristic
308C2×C59#291111yescharacteristic
309C2×C59#291111yescharacteristic
310C2×C59#321911yescharacteristic
311C2×C59#321911yescharacteristic
312C2×C59#324011yescharacteristic
313C2×C59#324011yescharacteristic
314C2×#117—591no—
315C2×#117C2×D5911yescharacteristic
316C22×C42—591no—
317#432—598no—
318C2×#107—591no—
319C2×#107—591no—
320C2×#107—591no—
321C2×#107C2×D5911yescharacteristic
322C2×#106—591no—
323C2×#106—591no—
324C2×#106—591no—
325C2×#106C2×D5911yescharacteristic
326C2×C4⋊C8—591no—
327C2×C4⋊C8—591no—
328C2×C22⋊C8—591no—
329C2×C22⋊C8C2×D5911yescharacteristic
330C2×#97—591no—
331C2×#97—591no—
332#384—591no—
333#384—591no—
334#383—1181no—
335#382—591no—
336#382—591no—
337#382—591no—
338#382—591no—
339#381—1181no—
340#380—591no—
341#380—1181no—
342#380—1181no—
343#380C2×D5911yescharacteristic
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C4⋊C4—1181no—
347C4×C4⋊C4—1181no—
348C4×C22⋊C4—1181no—
349#376—1181no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340C2×D5911yescharacteristic
357#340C2×D5911yescharacteristic
358C59#2198911yescharacteristic, complemented, Sylow
359C23×C4—591no—
360C23×C4—591no—
361C23×C4C22×D5911yescharacteristic
362C22×C8—591no—
363C22×C8—591no—
364C22×C8C22×D5911yescharacteristic
365C22×C8C22×D5911yescharacteristic
366#117—41no—
367#117—41no—
368#117—2361no—
369#117—2361no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C42—21no—
387C2×C42—594no—
388C2×C42—591no—
389C2×C42—591no—
390C2×C42—594no—
391C2×C42—591no—
392C2×C42—591no—
393C2×C42—1181no—
394C2×C42—1181no—
395C2×C42—1181no—
396#107—41no—
397#107—2361no—
398#107—2361no—
399#107—2361no—
400#106—41no—
401#106—2361no—
402#106—2361no—
403#106—2361no—
404C4⋊C8—594no—
405C4⋊C8—594no—
406C22⋊C8—594no—
407C22⋊C8C4×D5914yes—
408#97—594no—
409#97—594no—
410#95—21no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422C24D4×D5911yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—591no—
432C22×C4—591no—
433C22×C4—591no—
434C22×C4—591no—
435C22×C4—594no—
436C22×C4—594no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—2361no—
442C22×C4—2361no—
443C22×C4—2361no—
444C22×C4—2361no—
445C22×C4C2×C4×D5912yes—
446C22×C4D4×D5911yescharacteristic
447C22×C4Q8×D5912yes—
448C22×C4C23×D5911yescharacteristic
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—592no—
454C2×C8—592no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8C2×C4×D5912yes—
460C2×C8C2×C4×D5912yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
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—2362no—
478C4⋊C4—2362no—
479C4⋊C4—2362no—
480C4⋊C4—2362no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—2362no—
484C22⋊C4—2362no—
485C42—42no—
486C42—594no—
487C42—594no—
488C42—592no—
489C42—592no—
490C42—592no—
491C42—592no—
492C42—2362no—
493C42—2362no—
494C42—2362no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23C4⋊C4×D5912yes—
502C23C2×D4×D5911yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—592no—
522C2×C4—592no—
523C2×C4—592no—
524C2×C4—592no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—2364no—
534C2×C4—2364no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4C4⋊C4×D5912yes—
548C2×C4C22×C4×D5911yescharacteristic
549C2×C4C2×Q8×D5911yescharacteristic
550C2×C4#1427377211yescharacteristic
551C2×C4#1427377211yescharacteristic
552C2×C4#1427377411yescharacteristic
553C2×C4#1427377411yescharacteristic
554C8—22no—
555C8—22no—
556C8—1182no—
557C8—1182no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22C2×C4⋊C4×D5911yescharacteristic
566C22#1683796711yescharacteristic
567C22#1683797411yescharacteristic
568C22#1683819811yescharacteristic
569C22#1683819911yescharacteristic
570C22#1683819911yescharacteristic
571C22#1683820111yescharacteristic
572C4—22no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—1182no—
578C4—1182no—
579C4—1182no—
580C4—1182no—
581C4—4721no—
582C4—4721no—
583C4—4721no—
584C4—4721no—
585C4#1683788111yescharacteristic
586C4#1683788111yescharacteristic
587C4#1683803111yescharacteristic
588C4#1683803111yescharacteristic
589C2—42no—
590C2#2142671511yescharacteristic
591C2#2143268311yescharacteristic
592C2#2143268411yescharacteristic
593C2#2143513111yescharacteristic
594C2#2143513711yescharacteristic
595C2#2143533411yescharacteristic
596C2#2143534011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1854C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1854D59
hypocenterC59#21989
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #21424867C2semidirect
5. #21424865C2semidirect
6. #21424863C2semidirect
7. #21424861C2semidirect
11. #21424263C2semidirect
12. #21424261C2semidirect
13. #21424228C2semidirect
14. #21424228C2semidirect
18. #16837456C4semidirect
358. C59#21989semidirect

Maximal subgroups

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

subgroupindex
C2×#168374562
#214264992
#214264912
#214248672
#214248652
#214248632
#214248612
#214242972
#214242952
#214242932
#214242632
#214242612
#214242282
C59×#18542
#2198959