Finite Group Databasebeta

← #28599295

Subgroups — #28599295

Order 15104.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837460C211yesmaximal, characteristic
2#21426716C211yesmaximal, characteristic
3#21426703C211yesmaximal, characteristic
4#21424867C211yesmaximal, characteristic, complemented
5#21424861C211yesmaximal, characteristic, complemented
6#21424856C211yesmaximal, characteristic, complemented
7#21424853C211yesmaximal, characteristic, complemented
8#21424295C211yesmaximal, characteristic
9#21424288C211yesmaximal, characteristic
10#21424286C211yesmaximal, characteristic
11#21424261C211yesmaximal, characteristic, complemented
12#21424259C211yesmaximal, characteristic, complemented
13#21424228C211yesmaximal, characteristic, complemented
14#21424222C211yesmaximal, characteristic, complemented
15C59×#1854C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18#16837460C418yescomplemented
19C2×#14273661C2211yescharacteristic
20C2×#14273661C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×Dic59⋊C8C2211yescharacteristic
26C2×Dic59⋊C8C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29C2×C8×Dic59C2211yescharacteristic
30#16837300—21no—
31#16837300—21no—
32#16837296—21no—
33#16837290—21no—
34#16837287—21no—
35#16837285—21no—
36#16837274—21no—
37#16837274—21no—
38#16837271C2211yescharacteristic
39#16837271C2211yescharacteristic
40#16837271C2211yescharacteristic
41#16837267C2211yescharacteristic
42#16837267C2211yescharacteristic
43#16837267C2211yescharacteristic
44#16837262C2211yescharacteristic
45C4⋊C4×Dic59C2211yescharacteristic
46C4⋊C4×Dic59C2211yescharacteristic
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×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63#14273686C2×C414yes—
64#14273686Q814yes—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—41no—
67C2×C4⋊Dic59—41no—
68C2×C4⋊Dic59D411yescharacteristic
69C2×C4⋊Dic59D411yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59D411yescharacteristic
82C2×C22⋊Dic59D411yescharacteristic
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59C2311yescharacteristic
86C2×C4×Dic59C2311yescharacteristic
87#14273661—41no—
88#14273661—41no—
89#14273660—41no—
90#14273658—41no—
91#14273658—41no—
92#14273658—41no—
93Dic59⋊C8C2×C414yescomplemented
94Dic59⋊C8C2×C414yescomplemented
95C22⋊(C59⋊C8)C2×C414yes—
96#14273637C2×C414yescomplemented
97C8×Dic59C2×C414yescomplemented
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—42no—
141C22×Dic59—81no—
142C22×Dic59—81no—
143C22×Dic59C2×D411yescharacteristic
144C22×Dic59C2×D411yescharacteristic
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59C22×C412yes—
152C2×C8×C59C22×C412yes—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C22⋊C4×C59—42no—
162C22⋊C4×C59—81no—
163C42×C59—42no—
164C22⋊Dic59—81no—
165C22⋊Dic59C4⋊C414yes—
166C22⋊Dic59C4⋊C414yes—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—42no—
176C4⋊Dic59C4⋊C414yes—
177C4⋊Dic59C4⋊C414yes—
178Dic59⋊C4—81no—
179Dic59⋊C4—81no—
180Dic59⋊C4—81no—
181Dic59⋊C4—81no—
182Dic59⋊C4—81no—
183Dic59⋊C4—81no—
184C4×Dic59—81no—
185C4×Dic59—81no—
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—41no—
238C2×Dic59—41no—
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—84no—
250C2×Dic59C2×C4⋊C412yes—
251C2×Dic59C2×C4⋊C412yes—
252C2×Dic59C2×C4⋊C412yes—
253C2×Dic59C2×C4⋊C412yes—
254C8×C59—22no—
255C8×C59—22no—
256C59⋊C8—22no—
257C59⋊C8—22no—
258#21992—591nomaximal, Sylow
259C22×C59—22no—
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—41no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59—41no—
266C22×C59C22×C4⋊C411yescharacteristic
267C22×C59#53511yescharacteristic
268C22×C59#53611yescharacteristic
269C22×C59#57611yescharacteristic
270C22×C59#57711yescharacteristic
271C22×C59#57711yescharacteristic
272C22×C59#57811yescharacteristic
273C4×C59—22no—
274C4×C59—81no—
275C4×C59—81no—
276C4×C59—81no—
277C4×C59—81no—
278C4×C59#52011yescharacteristic
279C4×C59#52011yescharacteristic
280C4×C59#54411yescharacteristic
281C4×C59#54411yescharacteristic
282Dic59—22no—
283Dic59—22no—
284Dic59—22no—
285Dic59—22no—
286Dic59—81no—
287Dic59—81no—
288Dic59—81no—
289Dic59—81no—
290C2×#433—591no—
291#2317—591no—
292#2314—591no—
293#1949—591no—
294#1949—591no—
295#1948—591no—
296#1947—591no—
297#1854—591no—
298#1854D5911yescharacteristic
299#1853—591no—
300#1852—591no—
301#1847—591no—
302#1846—591no—
303#1836—591no—
304#1834—591no—
305C2×C59—42no—
306C2×C59#231711yescharacteristic
307C2×C59#291211yescharacteristic
308C2×C59#291311yescharacteristic
309C2×C59#322411yescharacteristic
310C2×C59#322511yescharacteristic
311C2×C59#324111yescharacteristic
312C2×C59#324211yescharacteristic
313C2×#117—591no—
314C2×#117C2×D5911yescharacteristic
315C22×C4⋊C4—591no—
316#433—598no—
317C2×#107—591no—
318C2×#107—591no—
319C2×#107—591no—
320C2×#107C2×D5911yescharacteristic
321C2×#106—591no—
322C2×#106—591no—
323C2×#106—591no—
324C2×#106C2×D5911yescharacteristic
325C2×C4⋊C8—591no—
326C2×C4⋊C8—591no—
327C2×C22⋊C8—591no—
328C2×C22⋊C8C2×D5911yescharacteristic
329C2×#97—591no—
330C2×C4×C8—591no—
331#389—1181no—
332#389—1181no—
333#388—1181no—
334#388—1181no—
335#387—1181no—
336#387—1181no—
337#385—1181no—
338#385—1181no—
339#384—591no—
340#384—591no—
341#384—591no—
342#382—591no—
343#382—591no—
344#382—591no—
345#380—591no—
346#380C2×D5911yescharacteristic
347C4×C4⋊C4—591no—
348C4×C4⋊C4—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340C2×D5911yescharacteristic
356#340C2×D5911yescharacteristic
357C59#2199211yescharacteristic, complemented, Sylow
358C23×C4—1181no—
359C23×C4C22×D5911yescharacteristic
360C22×C8—591no—
361C22×C8—591no—
362C22×C8C22×D5911yescharacteristic
363C22×C8C22×D5911yescharacteristic
364#117—41no—
365#117—41no—
366#117—594no—
367#117—594no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
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⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—2361no—
382C2×C4⋊C4—2361no—
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—591no—
390C2×C22⋊C4—591no—
391C2×C22⋊C4—1181no—
392C2×C42—21no—
393C2×C42—591no—
394C2×C42—591no—
395C2×C42—1181no—
396C2×C42—1181no—
397#107—41no—
398#107—2361no—
399#107—2361no—
400#107—2361no—
401#106—41no—
402#106—2361no—
403#106—2361no—
404#106—2361no—
405C4⋊C8—594no—
406C4⋊C8—594no—
407C22⋊C8—594no—
408C22⋊C8C4×D5914yes—
409#97—594no—
410C4×C8—594no—
411#95—21no—
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—
421C24D4×D5911yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—591no—
431C22×C4—591no—
432C22×C4—591no—
433C22×C4—591no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4—4721no—
439C22×C4—4721no—
440C22×C4—2362no—
441C22×C4C2×C4×D5912yes—
442C22×C4D4×D5911yescharacteristic
443C22×C4Q8×D5912yes—
444C22×C4C23×D5911yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—592no—
450C2×C8—592no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8C2×C4×D5912yes—
456C2×C8C2×C4×D5912yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—594no—
466C4⋊C4—594no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—4721no—
476C4⋊C4—4721no—
477C4⋊C4—4721no—
478C4⋊C4—4721no—
479C4⋊C4—4721no—
480C4⋊C4—4721no—
481C4⋊C4—2362no—
482C22⋊C4—42no—
483C22⋊C4—81no—
484C22⋊C4—594no—
485C22⋊C4—594no—
486C22⋊C4—4721no—
487C42—42no—
488C42—592no—
489C42—592no—
490C42—592no—
491C42—592no—
492C42—4721no—
493C42—4721no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C4⋊C4×D5912yes—
501C23C2×D4×D5911yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
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—592no—
521C2×C4—592no—
522C2×C4—592no—
523C2×C4—592no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—4724no—
533C2×C4—2361no—
534C2×C4—2361no—
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×C4C4⋊C4×D5912yes—
546C2×C4C22×C4×D5911yescharacteristic
547C2×C4C2×Q8×D5911yescharacteristic
548C2×C4#1427377211yescharacteristic
549C2×C4#1427377211yescharacteristic
550C2×C4#1427377411yescharacteristic
551C2×C4#1427377411yescharacteristic
552C8—22no—
553C8—22no—
554C8—1182no—
555C8—1182no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C2×C4⋊C4×D5911yescharacteristic
564C22#1683796711yescharacteristic
565C22#1683797411yescharacteristic
566C22#1683820411yescharacteristic
567C22#1683821111yescharacteristic
568C22#1683821211yescharacteristic
569C22#1683821611yescharacteristic
570C4—22no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—1182no—
576C4—1182no—
577C4—1182no—
578C4—1182no—
579C4—4721no—
580C4—4721no—
581C4—4721no—
582C4—4721no—
583C4#1683788111yescharacteristic
584C4#1683788111yescharacteristic
585C4#1683803111yescharacteristic
586C4#1683803111yescharacteristic
587C2—42no—
588C2#2142671511yescharacteristic
589C2#2143269011yescharacteristic
590C2#2143269311yescharacteristic
591C2#2143519611yescharacteristic
592C2#2143520111yescharacteristic
593C2#2143535511yescharacteristic
594C2#2143536211yescharacteristic
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#21992
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. #21424861C2semidirect
6. #21424856C2semidirect
7. #21424853C2semidirect
11. #21424261C2semidirect
12. #21424259C2semidirect
13. #21424228C2semidirect
14. #21424222C2semidirect
18. #16837460C4semidirect
93. Dic59⋊C8C2×C4semidirect
94. Dic59⋊C8C2×C4semidirect
96. #14273637C2×C4semidirect
97. C8×Dic59C2×C4semidirect
357. C59#21992semidirect

Maximal subgroups

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

subgroupindex
C2×#168374602
#214267162
#214267032
#214248672
#214248612
#214248562
#214248532
#214242952
#214242882
#214242862
#214242612
#214242592
#214242282
#214242222
C59×#18542
#2199259