Finite Group Databasebeta

← #28589462

Subgroups — #28589462

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×#16837883C211yesmaximal, characteristic
2C2×#16837438C211yesmaximal, characteristic, complemented
3C2×#16837370C211yesmaximal, characteristic
4#21426684C211yesmaximal, characteristic
5#21424294C212yesmaximal, complemented
6#21424292C212yesmaximal, complemented
7#21424287C212yesmaximal, complemented
8#21424282C212yesmaximal, complemented
9#21423816C211yesmaximal, characteristic, complemented
10C2×#16837164C211yesmaximal, characteristic, complemented
11C2×C59×#428C211yesmaximal, characteristic, complemented
12C22×#11745685C2211yescharacteristic
13C23×C59⋊C8C2211yescharacteristic
14#16837883—24no—
15#16837883—24no—
16C2×#14273692—21no—
17C2×#14273689—21no—
18C2×C4×C22⋊D59—22no—
19C2×#14273686C2211yescharacteristic
20C2×#14273685C2212yes—
21C22×C4⋊Dic59C2211yescharacteristic
22C2×C4⋊C4×D59C2212yes—
23#16837438—24no—
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28#16837370C418yescomplemented
29C2×(C2×D59)⋊C8C2212yes—
30C2×(C2×D59)⋊C8C2212yes—
31#16837308—21no—
32#16837304—21no—
33#16837279—22no—
34#16837259C2212yes—
35#16837259C2212yes—
36#16837165C2212yes—
37#16837165C2212yes—
38#16837164C2212yes—
39#16837164C2212yes—
40C22×M16×C59C2211yescharacteristic
41C2×#117×C59C2211yescharacteristic
42C22×C4⋊C4×C59C2211yescharacteristic
43C59×#428—24no—
44C2×#107×C59C2212yes—
45C2×#106×C59C2212yes—
46C2×#11745685—22no—
47C2×#11745685—22no—
48C2×#11745685—22no—
49C2×#11745685—22no—
50C2×#11745685C2×C414yes—
51C2×#11745685Q814yes—
52C22×Dic118D411yescharacteristic
53C22×D236D411yescharacteristic
54C22×C22⋊D59D412yes—
55C23×Dic59—21no—
56C22×C4×D59—22no—
57C22×C4×D59C2312yes—
58C22×C59⋊C8—22no—
59C22×C59⋊C8—22no—
60C22×C59⋊C8C2×C414yescomplemented
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63#14273692—28no—
64#14273689—28no—
65C4×C22⋊D59—216no—
66#14273686—22no—
67#14273686D414yes—
68#14273685—24no—
69#14273685—24no—
70C2×C4⋊Dic59—22no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2312yes—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—22no—
77C4⋊C4×D59—24no—
78C4⋊C4×D59—24no—
79C2×C22⋊Dic59—21no—
80C2×(C2×D59)⋊C4—22no—
81C2×(C2×D59)⋊C4—22no—
82C2×(C2×D59)⋊C4—22no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—22no—
85#14273660—22no—
86#14273660—22no—
87#14273658—22no—
88#14273658—22no—
89(C2×D59)⋊C8C2×C418yescomplemented
90(C2×D59)⋊C8C2×C418yescomplemented
91C59⋊#95—22no—
92C59⋊#95—22no—
93C23×C4×C59—21no—
94C23×C4×C59C2311yescharacteristic
95C2×M16×C59—22no—
96C2×M16×C59—22no—
97C2×M16×C59C2×C414yescomplemented
98C22×C8×C59C2312yes—
99#117×C59—22no—
100#117×C59—22no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—22no—
103C2×C4⋊C4×C59D414yes—
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2312yes—
107C2×C22⋊C4×C59—21no—
108C2×C42×C59—21no—
109#107×C59—24no—
110#106×C59—24no—
111C24×C59C2×D411yescharacteristic
112C23×D59C2×D412yes—
113C22×C4×C59—22no—
114C22×C4×C59—24no—
115C22×C4×C59—21no—
116C22×C4×C59—21no—
117C22×C4×C59—81no—
118C22×C4×C59—81no—
119C22×C4×C59C22×C412yes—
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D412yes—
122C22×C4×C59C2×D412yes—
123C22×C4×C59C2×Q812yes—
124C22×C4×C59#4112yes—
125C22×C4×C59#4112yes—
126C22×C4×C59C2411yescharacteristic
127C2×C22⋊D59—24no—
128C2×C22⋊D59—24no—
129C2×C22⋊D59C4⋊C418yes—
130C22×Dic59—21no—
131C22×Dic59—21no—
132C22×Dic59—22no—
133C22×Dic59—42no—
134C22×Dic59—81no—
135C22×Dic59—81no—
136C22×Dic59C2×D412yes—
137#11745685—28no—
138#11745685—28no—
139#11745685—28no—
140#11745685—28no—
141C2×D236—22no—
142C2×D236—22no—
143C2×D236C4⋊C414yes—
144C2×C4×D59—22no—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—22no—
148C2×C4×D59—28no—
149C2×C4×D59—82no—
150C2×C4×D59—82no—
151C2×C4×D59C22×C414yes—
152C2×C4×D59C2×Q814yes—
153C2×Dic118—22no—
154C2×Dic118—22no—
155C2×Dic118C4⋊C414yes—
156M16×C59—28no—
157C2×C8×C59—22no—
158C2×C8×C59—22no—
159C2×C8×C59C22×C414yes—
160C4⋊C4×C59—24no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59C2×D414yes—
167C22⋊C4×C59—42no—
168C42×C59—42no—
169C22⋊Dic59—24no—
170(C2×D59)⋊C4—28no—
171(C2×D59)⋊C4—82no—
172(C2×D59)⋊C4—82no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—22no—
176C4⋊Dic59—22no—
177C4⋊Dic59—42no—
178C4⋊Dic59C2×D412yes—
179C4⋊Dic59C2×D412yes—
180Dic59⋊C4—28no—
181C4×Dic59—28no—
182C4×Dic59—24no—
183C2×C59⋊C8—28no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8C22×C412yes—
189C2×C59⋊C8C22×C412yes—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59C2×C4⋊C412yes—
195C23×C59#12012yes—
196C23×C59#12312yes—
197C23×C59C22×D411yescharacteristic
198C22×D59—22no—
199C22×D59—22no—
200C22×D59—22no—
201C22×D59—22no—
202C22×D59—22no—
203C22×D59—22no—
204C22×D59C2×C4⋊C414yes—
205C2×C4×C59—24no—
206C2×C4×C59—42no—
207C2×C4×C59—42no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—84no—
215C2×C4×C59C2×C4⋊C412yes—
216C2×C4×C59C4×D414yes—
217C2×C4×C59C4×D414yes—
218C2×C4×C59#12012yes—
219C2×C4×C59C22⋊Q814yes—
220C2×C4×C59C22⋊Q814yes—
221C2×C4×C59#12312yes—
222C2×C4×C59C23×C411yescharacteristic
223C2×C4×C59C22×D411yescharacteristic
224C2×C4×C59C22×D411yescharacteristic
225C2×C4×C59C22×Q811yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C22⋊D59—216no—
229C2×Dic59—22no—
230C2×Dic59—22no—
231C2×Dic59—22no—
232C2×Dic59—22no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—42no—
238C2×Dic59—42no—
239C2×Dic59—42no—
240C2×Dic59—84no—
241C2×Dic59C2×C4⋊C414yes—
242D236—28no—
243C4×D59—24no—
244C4×D59—24no—
245C4×D59—24no—
246C4×D59—24no—
247C4×D59—88no—
248Dic118—28no—
249C8×C59—24no—
250C59⋊C8—22no—
251C59⋊C8—22no—
252#20849—591nomaximal, Sylow
253C22×C59—22no—
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—22no—
259C22×C59#38514yes—
260C22×C59C22×C4⋊C411yescharacteristic
261C22×C59C2×#12011yescharacteristic
262C22×C59C2×#12311yescharacteristic
263C22×C59#57611yescharacteristic
264C22×C59#57711yescharacteristic
265C22×C59#57711yescharacteristic
266C22×C59#57811yescharacteristic
267C2×D59—24no—
268C2×D59—24no—
269C2×D59—24no—
270C2×D59—42no—
271C2×D59—42no—
272C2×D59—42no—
273C2×D59—42no—
274C4×C59—44no—
275C4×C59—81no—
276C4×C59—81no—
277C4×C59#38514yes—
278C4×C59C2×C4×D411yescharacteristic
279C4×C59C2×C4×D411yescharacteristic
280C4×C59C2×C22⋊Q811yescharacteristic
281C4×C59C2×C22⋊Q811yescharacteristic
282Dic59—24no—
283Dic59—42no—
284Dic59—42no—
285Dic59—82no—
286C2×#520—591no—
287C2×#428D5911yescharacteristic, complemented
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×C59C2×#38511yescharacteristic
299C2×C59#291211yescharacteristic
300C2×C59#291311yescharacteristic
301C2×C59#301211yescharacteristic
302C2×C59#301311yescharacteristic
303C2×C59#309611yescharacteristic
304C2×C59#309711yescharacteristic
305D59—44no—
306C22×#41—591no—
307C22×M16C2×D5911yescharacteristic
308C23×C8—591no—
309#520—1184no—
310#520—1184no—
311C2×C4×Q8—1181no—
312C2×C4×D4—1181no—
313C2×C4×D4—1182no—
314C2×#117—591no—
315C2×#117—592no—
316C2×#117C2×D5911yescharacteristic
317C22×C4⋊C4—591no—
318C22×C4⋊C4—592no—
319C22×C4⋊C4C2×D5911yescharacteristic
320#428—24no—
321#427—1184no—
322C2×#107—591no—
323C2×#107—591no—
324C2×#107C2×D5912yes—
325C2×#106—591no—
326C2×#106—591no—
327C2×#106C2×D5912yes—
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—594no—
342C2×#41—594no—
343C2×#41—1182no—
344C2×#41—1182no—
345C2×#41—1182no—
346C2×#41—1182no—
347C22×Q8—591no—
348C22×D4—592no—
349C22×D4—591no—
350C23×C4—21no—
351C23×C4—592no—
352C23×C4—1182no—
353C23×C4—1181no—
354C23×C4C22×D5911yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D5914yescomplemented
358C22×C8—594no—
359C22×C8—591no—
360C22×C8—591no—
361C22×C8—1182no—
362C22×C8—1182no—
363C22×C8C22×D5912yes—
364C4×Q8—1188no—
365C4×D4—1188no—
366C4×D4—11816no—
367#117—22no—
368#117—22no—
369#117—594no—
370#117—1184no—
371#117—1182no—
372#117—1184no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—592no—
378C2×C4⋊C4—1184no—
379C2×C4⋊C4—1182no—
380C2×C4⋊C4—1184no—
381C2×C4⋊C4—1182no—
382C2×C4⋊C4—1182no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4C22⋊D5914yes—
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C4⋊C4C22×D5912yes—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1182no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1182no—
392C2×C22⋊C4—1182no—
393C2×C42—21no—
394C2×C42—1182no—
395C2×C42—1181no—
396#107—24no—
397#107—1182no—
398#107—1182no—
399#106—24no—
400#106—1182no—
401#106—1182no—
402C22⋊C8—598no—
403C22⋊C8—598no—
404#95—1182no—
405#95—1182no—
406C24—592no—
407C24D4×D5911yescharacteristic
408#41—1188no—
409#41—1188no—
410#41—1188no—
411#41—1188no—
412C2×Q8—594no—
413C2×Q8—1182no—
414C2×Q8—1182no—
415C2×D4—598no—
416C2×D4—594no—
417C2×D4—1184no—
418C2×D4—1184no—
419C2×D4—1182no—
420C2×D4—1182no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—592no—
428C22×C4—594no—
429C22×C4—594no—
430C22×C4—1188no—
431C22×C4—1182no—
432C22×C4—1182no—
433C22×C4—1182no—
434C22×C4—1182no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4—1182no—
438C22×C4—4722no—
439C22×C4—4722no—
440C22×C4—4721no—
441C22×C4—4721no—
442C22×C4—2362no—
443C22×C4C2×C4×D5912yes—
444C22×C4#1174568512yes—
445C22×C4D4×D5911yescharacteristic
446C22×C4D4×D5912yes—
447C22×C4Q8×D5912yes—
448C22×C4#1174568912yes—
449C22×C4C2×C22⋊D5912yes—
450C22×C4C23×D5911yescharacteristic
451M16—28no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—592no—
455C2×C8—592no—
456C2×C8—1188no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8C2×C4×D5914yes—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—592no—
469C4⋊C4—592no—
470C4⋊C4—1188no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1182no—
474C4⋊C4—1182no—
475C4⋊C4—2362no—
476C4⋊C4C2×C22⋊D5914yes—
477C22⋊C4—42no—
478C22⋊C4—1188no—
479C22⋊C4—1184no—
480C22⋊C4—4722no—
481C22⋊C4—4722no—
482C42—42no—
483C42—1188no—
484C42—1184no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—594no—
490C23—1182no—
491C23—1182no—
492C23—1182no—
493C23—1182no—
494C23—1182no—
495C23—1182no—
496C23C4⋊C4×D5912yes—
497C23#1427369512yes—
498C23#1427371212yes—
499C23C2×D4×D5911yescharacteristic
500Q8—1188no—
501D4—11816no—
502D4—1188no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—84no—
513C2×C4—594no—
514C2×C4—1182no—
515C2×C4—1182no—
516C2×C4—1184no—
517C2×C4—1184no—
518C2×C4—1184no—
519C2×C4—1184no—
520C2×C4—1182no—
521C2×C4—1182no—
522C2×C4—4728no—
523C2×C4—4724no—
524C2×C4—2362no—
525C2×C4—2362no—
526C2×C4—2362no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4C4⋊C4×D5912yes—
532C2×C4C4×C22⋊D5914yes—
533C2×C4#1427368914yes—
534C2×C4#1427369512yes—
535C2×C4#1427370314yes—
536C2×C4#1427370514yes—
537C2×C4#1427371212yes—
538C2×C4C22×C4×D5911yescharacteristic
539C2×C4C2×D4×D5911yescharacteristic
540C2×C4C22×C22⋊D5911yescharacteristic
541C2×C4C2×Q8×D5911yescharacteristic
542C2×C4C2×#1174568511yescharacteristic
543C2×C4C2×#1174568911yescharacteristic
544C8—24no—
545C8—1182no—
546C8—1182no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—1184no—
554C22—1184no—
555C22—1184no—
556C22—2362no—
557C22—2362no—
558C22—2362no—
559C22—2362no—
560C22#1683727314yes—
561C22C2×C4⋊C4×D5911yescharacteristic
562C22C2×#1427369511yescharacteristic
563C22C2×#1427371211yescharacteristic
564C22#1683820611yescharacteristic
565C22#1683821111yescharacteristic
566C22#1683821211yescharacteristic
567C22#1683821811yescharacteristic
568C4—44no—
569C4—81no—
570C4—81no—
571C4—1184no—
572C4—4722no—
573C4—2362no—
574C4—2362no—
575C4#1683727314yes—
576C4C2×C4×C22⋊D5911yescharacteristic
577C4C2×#1427368911yescharacteristic
578C4C2×#1427370311yescharacteristic
579C4C2×#1427370511yescharacteristic
580C2—24no—
581C2—2364no—
582C2C2×#1683727311yescharacteristic
583C2#2143269011yescharacteristic
584C2#2143269511yescharacteristic
585C2#2143339911yescharacteristic
586C2#2143341011yescharacteristic
587C2#2143405611yescharacteristic
588C2#2143406711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#428C2
socleC23×C59C22×D4
prime radicalC22×#11745685C22
cosocleC22×C4×C59C24
hypercenterC2×#428D59
hypocenterC59#20849
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#16837438C2semidirect
5. #21424294C2semidirect
6. #21424292C2semidirect
7. #21424287C2semidirect
8. #21424282C2semidirect
9. #21423816C2semidirect
10. C2×#16837164C2semidirect
11. C2×C59×#428C2semidirect
28. #16837370C4semidirect
60. C22×C59⋊C8C2×C4semidirect
89. (C2×D59)⋊C8C2×C4semidirect
90. (C2×D59)⋊C8C2×C4semidirect
97. C2×M16×C59C2×C4semidirect
287. C2×#428D59semidirect
340. C59#20849semidirect
357. C2×M16C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378832
C2×#168374382
C2×#168373702
#214266842
#214242942
#214242922
#214242872
#214242822
#214238162
C2×#168371642
C2×C59×#4282
#2084959