Finite Group Databasebeta

← #28596052

Subgroups — #28596052

Order 15104.

844 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426780C211yesmaximal, characteristic
2#21426452C211yesmaximal, characteristic
3#21424732C211yesmaximal, characteristic, complemented
4#21424732C211yesmaximal, characteristic, complemented
5#21424690C211yesmaximal, characteristic, complemented
6#21424676C211yesmaximal, characteristic, complemented
7#21424404C211yesmaximal, characteristic
8#21424291C211yesmaximal, characteristic
9#21424191C211yesmaximal, characteristic
10#21423995C211yesmaximal, characteristic, complemented
11#106×Dic59C211yesmaximal, characteristic, complemented
12#21423928C211yesmaximal, characteristic, complemented
13#103×Dic59C211yesmaximal, characteristic, complemented
14#21423864C211yesmaximal, characteristic
15C2×C59×#485C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19C2×#14273659C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273637C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26#16837309C2211yescharacteristic
27#16837301—21no—
28#16837301C2211yescharacteristic
29#16837295—21no—
30#16837295—21no—
31#16837288—21no—
32#16837262C2211yescharacteristic
33C4⋊C4×Dic59C2211yescharacteristic
34C4⋊C4×Dic59C2211yescharacteristic
35C4⋊C4×Dic59C2211yescharacteristic
36C4⋊C4×Dic59C2211yescharacteristic
37C4⋊C4×Dic59C2211yescharacteristic
38C4×C4⋊Dic59—21no—
39C4×C4⋊Dic59—21no—
40C22⋊C4×Dic59—21no—
41C22⋊C4×Dic59—21no—
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×C22⋊Q8×C59C2211yescharacteristic
53C2×#117×C59C2211yescharacteristic
54C59×#485C418yescomplemented
55C2×#107×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C2×C22⋊C8×C59C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63Q8×Dic59—22no—
64Q8×Dic59—22no—
65#14273686—22no—
66#14273686—22no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—22no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59D411yescharacteristic
73C2×C4⋊Dic59D411yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59D411yescharacteristic
80C2×C22⋊Dic59D411yescharacteristic
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59C2311yescharacteristic
88C2×C4×Dic59C2311yescharacteristic
89#14273659—22no—
90#14273659—22no—
91#14273651—22no—
92#14273651—22no—
93C22⋊(C59⋊C8)—22no—
94#14273637—22no—
95C8×Dic59—22no—
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—
105C22×Q8×C59C2311yescharacteristic
106C23×C4×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22⋊Q8×C59—22no—
110C22⋊Q8×C59C2×C414yes—
111#117×C59—22no—
112#117×C59C2×C414yes—
113C2×C4⋊C4×C59—21no—
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×C59C2×C414yescomplemented
121#106×C59C2×C414yescomplemented
122#103×C59C2×C414yescomplemented
123#103×C59C2×C414yescomplemented
124C22⋊C8×C59C2×C414yes—
125C24×C59C2×D411yescharacteristic
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59C22×C412yes—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59#4112yes—
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—24no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—81no—
150C22×Dic59—81no—
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59C22×C412yes—
160C2×C8×C59C22×C412yes—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59C22×C412yes—
166C4⋊C4×C59C22×C412yes—
167C4⋊C4×C59C22×C412yes—
168C22⋊C4×C59—24no—
169C22⋊C4×C59—24no—
170C42×C59—24no—
171C22⋊Dic59—22no—
172C22⋊Dic59—22no—
173C22⋊Dic59—81no—
174C4⋊Dic59—24no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—22no—
178C4⋊Dic59—22no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—42no—
184C4⋊Dic59—81no—
185C4⋊Dic59#4112yes—
186C4⋊Dic59#4112yes—
187C4×Dic59—21no—
188C4×Dic59—21no—
189C4×Dic59—42no—
190C4×Dic59—81no—
191C4×Dic59—81no—
192C4×Dic59—81no—
193C4×Dic59—81no—
194C4×Dic59—81no—
195C4×Dic59#4112yes—
196C4×Dic59#4112yes—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59C4×D412yes—
210C23×C59C22×D411yescharacteristic
211Q8×C59—22no—
212Q8×C59—22no—
213Q8×C59—41no—
214Q8×C59—41no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59C4×D412yes—
234C2×C4×C59C23×C411yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—22no—
247C2×Dic59—22no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—84no—
263C2×Dic59#12312yes—
264C2×Dic59#12312yes—
265C8×C59—22no—
266C8×C59—22no—
267C59⋊C8—41no—
268C59⋊C8—41no—
269#21591—591nomaximal, Sylow
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—22no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59C2×C4×D411yescharacteristic
278C22×C59C2×#12311yescharacteristic
279C22×C59#52511yescharacteristic
280C22×C59#57711yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57711yescharacteristic
283C22×C59#57811yescharacteristic
284C4×C59—22no—
285C4×C59—42no—
286C4×C59—42no—
287C4×C59—42no—
288C4×C59—42no—
289C4×C59C4×#4111yescharacteristic
290C4×C59C4×#4111yescharacteristic
291C4×C59#52911yescharacteristic
292C4×C59#52911yescharacteristic
293Dic59—22no—
294Dic59—22no—
295Dic59—41no—
296Dic59—41no—
297Dic59—42no—
298Dic59—42no—
299Dic59—81no—
300Dic59—81no—
301C2×#485D5911yescharacteristic
302#2323—591no—
303#2274—591no—
304#1927—591no—
305#1927—591no—
306#1921—591no—
307#1919—591no—
308#1872—591no—
309#1853—591no—
310#1825—591no—
311#1776—591no—
312C4×#106—591no—
313#1763—591no—
314C4×#103—591no—
315#1750—591no—
316C2×C59—42no—
317C2×C59C4×#12311yescharacteristic
318C2×C59#294711yescharacteristic
319C2×C59#295211yescharacteristic
320C2×C59#309711yescharacteristic
321C2×C59#309911yescharacteristic
322C2×C59#312111yescharacteristic
323C2×C59#312311yescharacteristic
324C2×C22⋊Q8C2×D5911yescharacteristic
325C2×C4×Q8—591no—
326C2×#117—591no—
327C2×#117C2×D5911yescharacteristic
328C22×C4⋊C4—591no—
329#485Dic5918yescomplemented
330C2×#107C2×D5911yescharacteristic
331C2×#106—591no—
332C2×#106—591no—
333C2×#106C2×D5911yescharacteristic
334C2×#103—591no—
335C2×#103—591no—
336C2×#103C2×D5911yescharacteristic
337C2×#103C2×D5911yescharacteristic
338C2×C22⋊C8—591no—
339C2×C22⋊C8C2×D5911yescharacteristic
340C2×#97—591no—
341C2×C4×C8—591no—
342#391—591no—
343#389—591no—
344#389—1181no—
345#388—1181no—
346#388—1181no—
347#387—1181no—
348#380—591no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—1181no—
355C4×C4⋊C4—1181no—
356C4×C22⋊C4—1181no—
357C4×C22⋊C4—1181no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#336—591no—
367#336—591no—
368C59#2159111yescharacteristic, complemented, Sylow
369C22×Q8C22×D5911yescharacteristic
370C23×C4—1181no—
371C23×C4C22×D5911yescharacteristic
372C22×C8—591no—
373C22×C8—591no—
374C22×C8C22×D5911yescharacteristic
375C22×C8C22×D5911yescharacteristic
376C22⋊Q8—22no—
377C22⋊Q8C2×Dic5914yes—
378C4×Q8—1182no—
379C4×Q8—1182no—
380#117—22no—
381#117—1182no—
382#117—1182no—
383#117C2×Dic5914yes—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1182no—
392C2×C4⋊C4—1182no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—591no—
402C2×C22⋊C4—591no—
403C2×C22⋊C4—1181no—
404C2×C42—21no—
405C2×C42—591no—
406C2×C42—591no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42—1181no—
413#107C2×Dic5914yescomplemented
414#106—1182no—
415#106—1182no—
416#106C2×Dic5914yescomplemented
417#103—1182no—
418#103—1182no—
419#103C2×Dic5914yescomplemented
420#103C2×Dic5914yescomplemented
421C22⋊C8—1182no—
422C22⋊C8C2×Dic5914yes—
423#97—1182no—
424C4×C8—1182no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434C24D4×D5911yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C22×Dic5912yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—591no—
450C22×C4—591no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—1184no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—4721no—
459C22×C4—4721no—
460C22×C4D4×D5911yescharacteristic
461C22×C4#1174568712yes—
462C22×C4C22×Dic5912yes—
463C22×C4C23×D5911yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8C22×Dic5912yes—
475C2×C8C22×Dic5912yes—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—592no—
481C4⋊C4—592no—
482C4⋊C4—1182no—
483C4⋊C4—1182no—
484C4⋊C4—1184no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—4721no—
492C4⋊C4—2362no—
493C4⋊C4C22×Dic5912yes—
494C4⋊C4C22×Dic5912yes—
495C4⋊C4C22×Dic5912yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—1182no—
499C22⋊C4—1182no—
500C22⋊C4—4721no—
501C42—24no—
502C42—592no—
503C42—592no—
504C42—1181no—
505C42—1181no—
506C42—4721no—
507C42—4721no—
508C42—4721no—
509C42—4721no—
510C42—4721no—
511C42—2362no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23D4×Dic5912yes—
519C23C2×D4×D5911yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—592no—
543C2×C4—592no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1182no—
551C2×C4—1182no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—4724no—
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×C4—2361no—
567C2×C4D4×Dic5912yes—
568C2×C4C23×Dic5911yescharacteristic
569C2×C4#41×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C2×C4#41×D5911yescharacteristic
573C2×C4C2×#1174568711yescharacteristic
574C8—22no—
575C8—22no—
576C8—2361no—
577C8—2361no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×D4×Dic5911yescharacteristic
586C22#123×D5911yescharacteristic
587C22#1683791011yescharacteristic
588C22#1683820911yescharacteristic
589C22#1683820911yescharacteristic
590C22#1683821111yescharacteristic
591C22#1683821811yescharacteristic
592C4—22no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—1182no—
598C4—1182no—
599C4—4721no—
600C4—4721no—
601C4—2361no—
602C4—2361no—
603C4—2362no—
604C4—2362no—
605C4#41×Dic5911yescharacteristic
606C4#41×Dic5911yescharacteristic
607C4#1683792911yescharacteristic
608C4#1683792911yescharacteristic
609C2—42no—
610C2#123×Dic5911yescharacteristic
611C2#2143294011yescharacteristic
612C2#2143298011yescharacteristic
613C2#2143406211yescharacteristic
614C2#2143408211yescharacteristic
615C2#2143432811yescharacteristic
616C2#2143434411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#485C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterC2×#485D59
hypocenterC59#21591
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
3. #21424732C2semidirect
4. #21424732C2semidirect
5. #21424690C2semidirect
6. #21424676C2semidirect
10. #21423995C2semidirect
11. #106×Dic59C2semidirect
12. #21423928C2semidirect
13. #103×Dic59C2semidirect
54. C59×#485C4semidirect
120. #107×C59C2×C4semidirect
121. #106×C59C2×C4semidirect
122. #103×C59C2×C4semidirect
123. #103×C59C2×C4semidirect
329. #485Dic59semidirect
368. C59#21591semidirect
413. #107C2×Dic59semidirect
416. #106C2×Dic59semidirect
419. #103C2×Dic59semidirect
420. #103C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214267802
#214264522
#214247322
#214246902
#214246762
#214244042
#214242912
#214241912
#214239952
#106×Dic592
#214239282
#103×Dic592
#214238642
C2×C59×#4852
#2159159