Finite Group Databasebeta

← #28599222

Subgroups — #28599222

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837452C211yesmaximal, characteristic
2#21426499C211yesmaximal, characteristic
3#21426498C211yesmaximal, 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#21424254C211yesmaximal, characteristic, complemented
12#21424251C211yesmaximal, characteristic, complemented
13#21424222C211yesmaximal, characteristic, complemented
14#21424222C211yesmaximal, characteristic, complemented
15C59×#1854C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C2×#14273683—21no—
18C22×C4×Dic59C2211yescharacteristic
19#16837452C418yescomplemented
20C2×#14273661C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×Dic59⋊C8C2211yescharacteristic
27C2×Dic59⋊C8C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×C8×Dic59C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31#16837271C2211yescharacteristic
32#16837271C2211yescharacteristic
33#16837271C2211yescharacteristic
34#16837271C2211yescharacteristic
35#16837267C2211yescharacteristic
36#16837267C2211yescharacteristic
37#16837266—21no—
38#16837262C2211yescharacteristic
39#16837260—21no—
40#16837260—21no—
41#16837260—21no—
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4×Dic59⋊C4—21no—
45C4×Dic59⋊C4—21no—
46#16837253—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—
66#14273683—44no—
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59C2×C414yes—
85C2×C4×Dic59D411yescharacteristic
86C2×C4×Dic59D411yescharacteristic
87C2×C4×Dic59Q814yes—
88C2×C4×Dic59C2311yescharacteristic
89C2×C4×Dic59C2311yescharacteristic
90#14273661—41no—
91#14273660—41no—
92#14273660—41no—
93#14273659—41no—
94#14273658—41no—
95#14273658—41no—
96Dic59⋊C8C2×C414yes—
97Dic59⋊C8C2×C414yes—
98C22⋊(C59⋊C8)C2×C414yes—
99C8×Dic59C2×C414yes—
100C8×Dic59C2×C414yes—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C23×C4×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111#117×C59—41no—
112#117×C59—41no—
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C42×C59—21no—
120#107×C59—41no—
121#106×C59—41no—
122C22⋊C8×C59C2×C414yes—
123#95×C59—21no—
124C24×C59C2×D411yescharacteristic
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×C59—21no—
133C22×C4×C59C22×C412yes—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×Q812yes—
136C22×C4×C59C2411yescharacteristic
137C22×Dic59—22no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—22no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59C4⋊C414yes—
146C22×Dic59C4⋊C414yes—
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59C22×C412yes—
156C2×C8×C59C22×C412yes—
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—
164C4⋊C4×C59—21no—
165C22⋊C4×C59—42no—
166C22⋊C4×C59—42no—
167C42×C59—42no—
168C22⋊Dic59—42no—
169C22⋊Dic59—42no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178Dic59⋊C4—42no—
179Dic59⋊C4—42no—
180Dic59⋊C4—42no—
181Dic59⋊C4—42no—
182Dic59⋊C4—42no—
183Dic59⋊C4—42no—
184Dic59⋊C4—42no—
185Dic59⋊C4—42no—
186C4×Dic59—42no—
187C4×Dic59—42no—
188C4×Dic59—42no—
189C4×Dic59C4⋊C414yes—
190C4×Dic59C4⋊C414yes—
191C4×Dic59C22×C412yes—
192C4×Dic59C22×C412yes—
193C4×Dic59C2×Q812yes—
194C4×Dic59C2×Q812yes—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8C22×C412yes—
200C2×C59⋊C8C22×C412yes—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59C2×C4⋊C412yes—
208C23×C59C22×D411yescharacteristic
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
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—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59C2×C4⋊C412yes—
228C2×C4×C59C23×C411yescharacteristic
229C2×C4×C59C22×Q811yescharacteristic
230C2×C4×C59H(5,𝔽2)11yescharacteristic
231C2×C4×C59H(5,𝔽2)11yescharacteristic
232C2×C4×C592−1+411yescharacteristic
233C2×C4×C592−1+411yescharacteristic
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—28no—
239C2×Dic59—28no—
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×Dic59—41no—
252C2×Dic59C2×C4⋊C412yes—
253C2×Dic59C2×C4⋊C412yes—
254C2×Dic59C2×C4⋊C412yes—
255C2×Dic59C2×C4⋊C412yes—
256C2×Dic59#13512yes—
257C2×Dic59#13512yes—
258C2×Dic59AGL(1,ℤ/8)12yes—
259C2×Dic59#13712yes—
260C8×C59—22no—
261C8×C59—22no—
262C59⋊C8—22no—
263C59⋊C8—22no—
264#21984—591nomaximal, Sylow
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59C22×C4⋊C411yescharacteristic
273C22×C59#53511yescharacteristic
274C22×C59#53611yescharacteristic
275C22×C59C2×#13511yescharacteristic
276C22×C59C2×#13511yescharacteristic
277C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C59C2×#13711yescharacteristic
279C4×C59—22no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59—81no—
284C4×C59#52011yescharacteristic
285C4×C59#52011yescharacteristic
286C4×C59#54411yescharacteristic
287C4×C59#54411yescharacteristic
288Dic59—22no—
289Dic59—22no—
290Dic59—81no—
291Dic59—81no—
292Dic59—81no—
293Dic59—81no—
294Dic59#42714yes—
295Dic59#42814yes—
296C2×#431—591no—
297#2285—591no—
298#2285—591no—
299#1949—591no—
300#1949—591no—
301#1949—591no—
302#1949—591no—
303#1854—591no—
304#1854—591no—
305#1854—591no—
306#1854D5911yescharacteristic
307#1845—591no—
308#1844—591no—
309#1834—591no—
310#1834—591no—
311C2×C59—42no—
312C2×C59#231711yescharacteristic
313C2×C59C2×#42711yescharacteristic
314C2×C59C2×#42811yescharacteristic
315C2×C59#320711yescharacteristic
316C2×C59#321011yescharacteristic
317C2×C59#322811yescharacteristic
318C2×C59#323111yescharacteristic
319C2×#117—591no—
320C2×#117—1181no—
321C2×#117C2×D5911yescharacteristic
322C22×C42—591no—
323#431—598no—
324C2×#107—591no—
325C2×#107—591no—
326C2×#107—591no—
327C2×#107C2×D5911yescharacteristic
328C2×#106—591no—
329C2×#106—591no—
330C2×#106—591no—
331C2×#106C2×D5911yescharacteristic
332C2×C4⋊C8—591no—
333C2×C4⋊C8—591no—
334C2×C22⋊C8—591no—
335C2×C22⋊C8C2×D5911yescharacteristic
336C2×C4×C8—591no—
337C2×C4×C8—591no—
338#384—591no—
339#384—591no—
340#384—591no—
341#384—591no—
342#382—591no—
343#382—591no—
344#382—1181no—
345#380—591no—
346#380—1181no—
347#380—1181no—
348#380—1181no—
349#380C2×D5911yescharacteristic
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—1181no—
353C4×C4⋊C4—1181no—
354C4×C4⋊C4—1181no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340C2×D5911yescharacteristic
362#340C2×D5911yescharacteristic
363C59#2198411yescharacteristic, complemented, Sylow
364C23×C4—591no—
365C23×C4—591no—
366C23×C4C22×D5911yescharacteristic
367C22×C8—591no—
368C22×C8—591no—
369C22×C8C22×D5911yescharacteristic
370C22×C8C22×D5911yescharacteristic
371#117—41no—
372#117—41no—
373#117—2364no—
374#117—2361no—
375#117—2361no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C4⋊C4C22×D5911yescharacteristic
390C2×C4⋊C4C22×D5911yescharacteristic
391C2×C4⋊C4C22×D5911yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1181no—
395C2×C22⋊C4—1181no—
396C2×C42—21no—
397C2×C42—594no—
398C2×C42—591no—
399C2×C42—591no—
400C2×C42—594no—
401C2×C42—591no—
402C2×C42—591no—
403C2×C42—1181no—
404C2×C42—1181no—
405C2×C42—1181no—
406#107—41no—
407#107—2361no—
408#107—2361no—
409#107—2361no—
410#106—41no—
411#106—2361no—
412#106—2361no—
413#106—2361no—
414C4⋊C8—594no—
415C4⋊C8—594no—
416C22⋊C8—594no—
417C22⋊C8C4×D5914yes—
418C4×C8—594no—
419C4×C8—594no—
420#95—21no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428C24D4×D5911yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—591no—
438C22×C4—591no—
439C22×C4—594no—
440C22×C4—594no—
441C22×C4—591no—
442C22×C4—591no—
443C22×C4—1182no—
444C22×C4—1182no—
445C22×C4—1182no—
446C22×C4—1182no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4C2×C4×D5912yes—
452C22×C4D4×D5911yescharacteristic
453C22×C4Q8×D5912yes—
454C22×C4C23×D5911yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—592no—
460C2×C8—592no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8C2×C4×D5912yes—
466C2×C8C2×C4×D5912yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—2362no—
484C4⋊C4—2362no—
485C4⋊C4—2362no—
486C4⋊C4—2362no—
487C4⋊C4—2362no—
488C4⋊C4—2362no—
489C4⋊C4—2362no—
490C4⋊C4—2362no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—2362no—
494C22⋊C4—2362no—
495C42—42no—
496C42—594no—
497C42—594no—
498C42—592no—
499C42—592no—
500C42—592no—
501C42—592no—
502C42—2362no—
503C42—2362no—
504C42—2362no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C4⋊C4×D5912yes—
512C23C2×D4×D5911yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—592no—
532C2×C4—592no—
533C2×C4—592no—
534C2×C4—592no—
535C2×C4—592no—
536C2×C4—592no—
537C2×C4—592no—
538C2×C4—592no—
539C2×C4—1188no—
540C2×C4—1188no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
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×C4C4⋊C4×D5912yes—
558C2×C4C22×C4×D5911yescharacteristic
559C2×C4C2×Q8×D5911yescharacteristic
560C2×C4#1427377211yescharacteristic
561C2×C4#1427377211yescharacteristic
562C2×C4#1427377411yescharacteristic
563C2×C4#1427377411yescharacteristic
564C8—22no—
565C8—22no—
566C8—1182no—
567C8—1182no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×C4⋊C4×D5911yescharacteristic
576C22#1683796711yescharacteristic
577C22#1683797411yescharacteristic
578C22#135×D5911yescharacteristic
579C22#135×D5911yescharacteristic
580C22AGL(1,ℤ/8)×D5911yescharacteristic
581C22#137×D5911yescharacteristic
582C4—22no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—594no—
588C4—594no—
589C4—1182no—
590C4—1182no—
591C4—4721no—
592C4—4721no—
593C4—4721no—
594C4—4721no—
595C4#1683788111yescharacteristic
596C4#1683788111yescharacteristic
597C4#1683803111yescharacteristic
598C4#1683803111yescharacteristic
599C2—42no—
600C2#2142671511yescharacteristic
601C2#427×D5911yescharacteristic
602C2#428×D5911yescharacteristic
603C2#2143502711yescharacteristic
604C2#2143505011yescharacteristic
605C2#2143523011yescharacteristic
606C2#2143525311yescharacteristic
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#21984
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. #21424254C2semidirect
12. #21424251C2semidirect
13. #21424222C2semidirect
14. #21424222C2semidirect
19. #16837452C4semidirect
363. C59#21984semidirect

Maximal subgroups

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

subgroupindex
C2×#168374522
#214264992
#214264982
#214248672
#214248652
#214248632
#214248612
#214242972
#214242952
#214242932
#214242542
#214242512
#214242222
C59×#18542
#2198459