Finite Group Databasebeta

← #28589463

Subgroups — #28589463

Order 15104.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837442C211yesmaximal, characteristic, complemented
2C2×#16837438C211yesmaximal, characteristic, complemented
3C2×#16837369C211yesmaximal, characteristic
4#21426688C211yesmaximal, characteristic
5#21424295C212yesmaximal, complemented
6#21424295C212yesmaximal, complemented
7#21424289C212yesmaximal, complemented
8#21424284C212yesmaximal, complemented
9#21423817C211yesmaximal, characteristic, complemented
10C2×#16837164C211yesmaximal, characteristic, complemented
11C2×C59×#520C211yesmaximal, characteristic
12C22×#11745664C2211yescharacteristic
13C23×C59⋊C8C2211yescharacteristic
14C2×#14273686C2211yescharacteristic
15C22×C4⋊Dic59C2211yescharacteristic
16#16837442—24no—
17#16837438—24no—
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2212yes—
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2212yes—
24#16837369C418yescomplemented
25C2×C22⋊(C59⋊C8)C2212yes—
26C2×C22⋊(C59⋊C8)C2212yes—
27#16837309—21no—
28#16837305—21no—
29#16837282—22no—
30#16837262C2212yes—
31#16837262C2212yes—
32#16837164C2212yes—
33#16837164C2212yes—
34#16837164C2212yes—
35#16837164C2212yes—
36C22×#41×C59C2211yescharacteristic
37C59×#520—24no—
38C59×#520—24no—
39C2×C4×Q8×C59—21no—
40C2×C4×D4×C59—22no—
41C2×C4×D4×C59—21no—
42C2×#117×C59C2212yes—
43C2×#117×C59C2211yescharacteristic
44C22×C4⋊C4×C59C2212yes—
45C22×C4⋊C4×C59C2211yescharacteristic
46C23×Dic59—21no—
47C2×#11745664—22no—
48C2×#11745664—22no—
49C2×#11745664C2×C414yescomplemented
50C22×C59⋊C8—22no—
51C22×C59⋊C8—22no—
52C22×C59⋊C8C2×C414yescomplemented
53C22×C59⋊C8C2311yescharacteristic
54C22×C59⋊C8C2311yescharacteristic
55C22×C59⋊C8C2312yes—
56#14273686—22no—
57#14273686—22no—
58C2×C4⋊Dic59—22no—
59C2×C4⋊Dic59—22no—
60C2×C4⋊Dic59—21no—
61C2×C4⋊Dic59C2312yes—
62C2×C4⋊Dic59C2311yescharacteristic
63C2×C4⋊Dic59C2311yescharacteristic
64C2×C22⋊Dic59—21no—
65C2×C22⋊Dic59—22no—
66C2×C22⋊Dic59—22no—
67C2×C4×Dic59—21no—
68#14273660—22no—
69#14273660—22no—
70#14273660—24no—
71#14273658—22no—
72#14273658—22no—
73#14273658—24no—
74C22⋊(C59⋊C8)C2×C418yescomplemented
75C22⋊(C59⋊C8)C2×C418yescomplemented
76C59⋊#95—22no—
77C59⋊#95—22no—
78C2×#41×C59—22no—
79C2×#41×C59—22no—
80C2×#41×C59—22no—
81C2×#41×C59—22no—
82C2×#41×C59C2×C414yes—
83C2×#41×C59Q814yes—
84C22×Q8×C59D411yescharacteristic
85C22×D4×C59D412yes—
86C22×D4×C59D411yescharacteristic
87C23×C4×C59—22no—
88C23×C4×C59—21no—
89C23×C4×C59C2312yes—
90C23×C4×C59C2311yescharacteristic
91C4×Q8×C59—28no—
92C4×D4×C59—216no—
93C4×D4×C59—28no—
94#117×C59—22no—
95#117×C59—24no—
96#117×C59—24no—
97#117×C59D414yes—
98C2×C4⋊C4×C59—22no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—22no—
101C2×C4⋊C4×C59—24no—
102C2×C4⋊C4×C59—24no—
103C2×C4⋊C4×C59D414yes—
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C4⋊C4×C59C2312yes—
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C22⋊C4×C59—22no—
108C2×C22⋊C4×C59—21no—
109C2×C42×C59—22no—
110C2×C42×C59—21no—
111C24×C59C2×D412yes—
112C24×C59C2×D411yescharacteristic
113#41×C59—28no—
114#41×C59—28no—
115#41×C59—28no—
116#41×C59—28no—
117C2×Q8×C59—22no—
118C2×Q8×C59—22no—
119C2×Q8×C59C4⋊C414yes—
120C2×D4×C59—24no—
121C2×D4×C59—24no—
122C2×D4×C59—22no—
123C2×D4×C59—22no—
124C2×D4×C59C4⋊C418yes—
125C2×D4×C59C4⋊C414yes—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—21no—
131C22×C4×C59—22no—
132C22×C4×C59—21no—
133C22×C4×C59—28no—
134C22×C4×C59—24no—
135C22×C4×C59—81no—
136C22×C4×C59—82no—
137C22×C4×C59—81no—
138C22×C4×C59—82no—
139C22×C4×C59C22×C414yes—
140C22×C4×C59C22×C412yes—
141C22×C4×C59C2×D412yes—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D412yes—
144C22×C4×C59C2×D412yes—
145C22×C4×C59C2×Q814yes—
146C22×C4×C59C2×Q812yes—
147C22×C4×C59#4112yes—
148C22×C4×C59#4112yes—
149C22×C4×C59C2411yescharacteristic
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—22no—
153C22×Dic59—42no—
154C22×Dic59—81no—
155C22×Dic59—81no—
156C4⋊C4×C59—28no—
157C4⋊C4×C59—24no—
158C4⋊C4×C59—22no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59C2×D412yes—
162C4⋊C4×C59C2×D414yes—
163C4⋊C4×C59C2×D412yes—
164C22⋊C4×C59—28no—
165C22⋊C4×C59—24no—
166C42×C59—28no—
167C42×C59—24no—
168C22⋊Dic59—42no—
169C22⋊Dic59—82no—
170C22⋊Dic59—82no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—22no—
176C4⋊Dic59—22no—
177C4⋊Dic59—42no—
178C4×Dic59—42no—
179#11745664—28no—
180C2×C59⋊C8—28no—
181C2×C59⋊C8—21no—
182C2×C59⋊C8—22no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—22no—
187C2×C59⋊C8C22×C412yes—
188C2×C59⋊C8C22×C412yes—
189C2×C59⋊C8C22×C414yes—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—22no—
195C23×C59—22no—
196C23×C59—22no—
197C23×C59—22no—
198C23×C59—22no—
199C23×C59—22no—
200C23×C59C2×C4⋊C414yes—
201C23×C59C2×C4⋊C412yes—
202C23×C59#12012yes—
203C23×C59#12312yes—
204C23×C59C22×D411yescharacteristic
205Q8×C59—28no—
206D4×C59—216no—
207D4×C59—28no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—24no—
211C2×C4×C59—24no—
212C2×C4×C59—24no—
213C2×C4×C59—24no—
214C2×C4×C59—22no—
215C2×C4×C59—24no—
216C2×C4×C59—22no—
217C2×C4×C59—42no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—42no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—84no—
224C2×C4×C59—88no—
225C2×C4×C59C2×C4⋊C414yes—
226C2×C4×C59C2×C4⋊C412yes—
227C2×C4×C59C4×D414yes—
228C2×C4×C59C4×D414yes—
229C2×C4×C59#12012yes—
230C2×C4×C59C22⋊Q814yes—
231C2×C4×C59C22⋊Q814yes—
232C2×C4×C59#12312yes—
233C2×C4×C59C23×C411yescharacteristic
234C2×C4×C59C22×D411yescharacteristic
235C2×C4×C59C22×D411yescharacteristic
236C2×C4×C59C22×Q811yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—42no—
242C2×Dic59—41no—
243C2×Dic59—42no—
244C2×Dic59—41no—
245C2×Dic59—42no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—84no—
249C59⋊C8—22no—
250C59⋊C8—22no—
251C59⋊C8—24no—
252#20849—591nomaximal, Sylow
253C22×C59—22no—
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—22no—
259C22×C59—24no—
260C22×C59—24no—
261C22×C59—24no—
262C22×C59—42no—
263C22×C59—42no—
264C22×C59—42no—
265C22×C59—42no—
266C22×C59#38514yes—
267C22×C59C22×C4⋊C411yescharacteristic
268C22×C59C2×#12011yescharacteristic
269C22×C59C2×#12311yescharacteristic
270C22×C59#57611yescharacteristic
271C22×C59#57711yescharacteristic
272C22×C59#57711yescharacteristic
273C22×C59#57811yescharacteristic
274C4×C59—24no—
275C4×C59—44no—
276C4×C59—42no—
277C4×C59—42no—
278C4×C59#38514yes—
279C4×C59C2×C4×D411yescharacteristic
280C4×C59C2×C4×D411yescharacteristic
281C4×C59C2×C22⋊Q811yescharacteristic
282C4×C59C2×C22⋊Q811yescharacteristic
283Dic59—82no—
284Dic59—81no—
285Dic59—81no—
286C2×#520D5911yescharacteristic
287C2×#428—591no—
288C2×#427—591no—
289C2×#408—591no—
290#2311—591no—
291#1854—592no—
292#1854—592no—
293#1853—592no—
294#1852—592no—
295#1736—591no—
296C2×#340—591no—
297C2×C59—24no—
298C2×C59—44no—
299C2×C59C2×#38511yescharacteristic
300C2×C59#291211yescharacteristic
301C2×C59#291311yescharacteristic
302C2×C59#301211yescharacteristic
303C2×C59#301311yescharacteristic
304C2×C59#309611yescharacteristic
305C2×C59#309711yescharacteristic
306C22×#41C2×D5911yescharacteristic
307C22×M16—591no—
308C23×C8—591no—
309#520—24no—
310#520—24no—
311C2×C4×Q8—21no—
312C2×C4×D4—22no—
313C2×C4×D4—21no—
314C2×#117—591no—
315C2×#117C2×D5912yes—
316C2×#117C2×D5911yescharacteristic
317C22×C4⋊C4—591no—
318C22×C4⋊C4C2×D5912yes—
319C22×C4⋊C4C2×D5911yescharacteristic
320#428—1184no—
321#427—1184no—
322C2×#107—591no—
323C2×#107—592no—
324C2×#107—591no—
325C2×#106—591no—
326C2×#106—592no—
327C2×#106—591no—
328#408—598no—
329C2×C22⋊C8—592no—
330C2×C22⋊C8—592no—
331#391—1181no—
332#390—1181no—
333#386—1182no—
334#380—592no—
335#380—592no—
336#340—592no—
337#340—592no—
338#340—592no—
339#340—592no—
340C59#2084911yescharacteristic, complemented, Sylow
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41Dic11814yes—
346C2×#41C4×D5914yes—
347C22×Q8C22⋊D5911yescharacteristic
348C22×D4C22⋊D5912yes—
349C22×D4C22⋊D5911yescharacteristic
350C23×C4—22no—
351C23×C4—21no—
352C23×C4—1181no—
353C23×C4C22×D5912yes—
354C23×C4C22×D5911yescharacteristic
355C2×M16—594no—
356C2×M16—1182no—
357C2×M16—1182no—
358C22×C8—594no—
359C22×C8—591no—
360C22×C8—591no—
361C22×C8—592no—
362C22×C8—1182no—
363C22×C8—1182no—
364C4×Q8—28no—
365C4×D4—216no—
366C4×D4—28no—
367#117—22no—
368#117—24no—
369#117—24no—
370#117—1182no—
371#117—1182no—
372#117C22⋊D5914yes—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—592no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—1182no—
382C2×C4⋊C4—1182no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4C22⋊D5914yes—
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5912yes—
387C2×C4⋊C4C22×D5911yescharacteristic
388C2×C22⋊C4—22no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1182no—
391C2×C22⋊C4—1182no—
392C2×C22⋊C4—1181no—
393C2×C42—22no—
394C2×C42—21no—
395C2×C42—1181no—
396#107—1182no—
397#107—1182no—
398#107—1184no—
399#106—1182no—
400#106—1182no—
401#106—1184no—
402C22⋊C8—598no—
403C22⋊C8—598no—
404#95—1182no—
405#95—1182no—
406C24C2×C22⋊D5912yes—
407C24C2×C22⋊D5911yescharacteristic
408#41—28no—
409#41—28no—
410#41—28no—
411#41—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8Dic59⋊C414yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4Dic59⋊C418yes—
420C2×D4Dic59⋊C414yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—81no—
431C22×C4—82no—
432C22×C4—81no—
433C22×C4—82no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1182no—
437C22×C4—4721no—
438C22×C4—4721no—
439C22×C4—2362no—
440C22×C4C2×Dic11814yes—
441C22×C4C2×Dic11812yes—
442C22×C4C2×C4×D5914yes—
443C22×C4C2×C4×D5912yes—
444C22×C4#1174568512yes—
445C22×C4#1174568512yes—
446C22×C4C2×C22⋊D5912yes—
447C22×C4C2×C22⋊D5911yescharacteristic
448C22×C4C2×C22⋊D5912yes—
449C22×C4C2×C22⋊D5912yes—
450C22×C4C23×D5911yescharacteristic
451M16—1188no—
452C2×C8—592no—
453C2×C8—592no—
454C2×C8—594no—
455C2×C8—1188no—
456C2×C8—1181no—
457C2×C8—1182no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1182no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1182no—
472C4⋊C4—1182no—
473C4⋊C4—2362no—
474C4⋊C4C2×C22⋊D5912yes—
475C4⋊C4C2×C22⋊D5914yes—
476C4⋊C4C2×C22⋊D5912yes—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—4722no—
480C22⋊C4—4722no—
481C22⋊C4—2362no—
482C42—28no—
483C42—24no—
484C42—2362no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23C2×Dic59⋊C414yes—
496C23C2×Dic59⋊C412yes—
497C23#1427369612yes—
498C23#1427371012yes—
499C23C22×C22⋊D5911yescharacteristic
500Q8—28no—
501D4—216no—
502D4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—84no—
519C2×C4—88no—
520C2×C4—4724no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2362no—
524C2×C4—2361no—
525C2×C4—2362no—
526C2×C4—2361no—
527C2×C4—2362no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4C2×Dic59⋊C414yes—
531C2×C4C2×Dic59⋊C412yes—
532C2×C4C4×C22⋊D5914yes—
533C2×C4C4×C22⋊D5914yes—
534C2×C4#1427369612yes—
535C2×C4#1427370614yes—
536C2×C4#1427370614yes—
537C2×C4#1427371012yes—
538C2×C4C22×C4×D5911yescharacteristic
539C2×C4C22×C22⋊D5911yescharacteristic
540C2×C4C22×C22⋊D5911yescharacteristic
541C2×C4C22×Dic11811yescharacteristic
542C2×C4C2×#1174568511yescharacteristic
543C2×C4C2×#1174568511yescharacteristic
544C8—1182no—
545C8—1182no—
546C8—1184no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22#1683727514yes—
561C22C22×Dic59⋊C411yescharacteristic
562C22C2×#1427369611yescharacteristic
563C22C2×#1427371011yescharacteristic
564C22#1683820511yescharacteristic
565C22#1683821011yescharacteristic
566C22#1683821311yescharacteristic
567C22#1683821711yescharacteristic
568C4—24no—
569C4—44no—
570C4—42no—
571C4—42no—
572C4—4721no—
573C4—4721no—
574C4—4722no—
575C4#1683727514yes—
576C4C2×C4×C22⋊D5911yescharacteristic
577C4C2×C4×C22⋊D5911yescharacteristic
578C4C2×#1427370611yescharacteristic
579C4C2×#1427370611yescharacteristic
580C2—24no—
581C2—44no—
582C2C2×#1683727511yescharacteristic
583C2#2143268911yescharacteristic
584C2#2143269411yescharacteristic
585C2#2143340011yescharacteristic
586C2#2143341111yescharacteristic
587C2#2143405411yescharacteristic
588C2#2143406511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#520C2
socleC23×C59C22×D4
prime radicalC22×#41×C59C22
cosocleC22×C4×C59C24
hypercenterC2×#520D59
hypocenterC59#20849
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#16837442C2semidirect
2. C2×#16837438C2semidirect
5. #21424295C2semidirect
6. #21424295C2semidirect
7. #21424289C2semidirect
8. #21424284C2semidirect
9. #21423817C2semidirect
10. C2×#16837164C2semidirect
24. #16837369C4semidirect
49. C2×#11745664C2×C4semidirect
52. C22×C59⋊C8C2×C4semidirect
74. C22⋊(C59⋊C8)C2×C4semidirect
75. C22⋊(C59⋊C8)C2×C4semidirect
340. C59#20849semidirect

Maximal subgroups

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

subgroupindex
C2×#168374422
C2×#168374382
C2×#168373692
#214266882
#214242952
#214242892
#214242842
#214238172
C2×#168371642
C2×C59×#5202
#2084959