Finite Group Databasebeta

← #28640372

Subgroups — #28640372

Order 15104.

754 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837801C211yesmaximal, characteristic, complemented
2#21429292C211yesmaximal, characteristic
3#21427196C211yesmaximal, characteristic, complemented
4#21425581C211yesmaximal, characteristic, complemented
5#21425570C211yesmaximal, characteristic, complemented
6#21425466C211yesmaximal, characteristic
7#21425466C211yesmaximal, characteristic
8#21425051C211yesmaximal, characteristic, complemented
9#21424861C211yesmaximal, characteristic, complemented
10#21424856C211yesmaximal, characteristic, complemented
11#21424812C211yesmaximal, characteristic, complemented
12#21424719C211yesmaximal, characteristic
13#21424719C211yesmaximal, characteristic
14C4×#14273648C211yesmaximal, characteristic
15C2×C59×#478C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17#16837801—24no—
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×C4×C59⋊C8C2211yescharacteristic
27#16837325—21no—
28#16837325—21no—
29#16837305C2211yescharacteristic
30#16837305C2211yescharacteristic
31#16837301—21no—
32#16837301C2211yescharacteristic
33#16837301C2211yescharacteristic
34#16837288—21no—
35#16837288—21no—
36#16837275—21no—
37#16837275—21no—
38#16837271C2211yescharacteristic
39#16837267C2211yescharacteristic
40C4⋊C4×Dic59—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#125×C59C2211yescharacteristic
54C2×C4×D4×C59C2211yescharacteristic
55C59×#478—24no—
56C2×#107×C59C2211yescharacteristic
57C2×#106×C59C2211yescharacteristic
58C2×C4⋊C8×C59C2211yescharacteristic
59C2×#102×C59C2211yescharacteristic
60C2×#102×C59C2211yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63#14273719—22no—
64#14273719—22no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59D411yescharacteristic
68C2×C4⋊Dic59D411yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59C2311yescharacteristic
85C2×C4×Dic59C2311yescharacteristic
86#14273660—22no—
87#14273660—22no—
88#14273658—22no—
89#14273658—22no—
90Dic59⋊C8—22no—
91Dic59⋊C8—22no—
92#14273648—22no—
93#14273648Q814yes—
94C4×C59⋊C8—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59—21no—
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108#125×C59—22no—
109#125×C59D414yes—
110C4×D4×C59—22no—
111C4×D4×C59—22no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C42×C59C2311yescharacteristic
119#107×C59—22no—
120#106×C59—22no—
121C4⋊C8×C59—22no—
122#102×C59—22no—
123#102×C59—22no—
124C24×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59C2×Q812yes—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—42no—
135C22×C4×C59—81no—
136C22×C4×C59—81no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2411yescharacteristic
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59C2×D411yescharacteristic
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C4⋊C4×C59—24no—
155C4⋊C4×C59—24no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59C2×D412yes—
161C4⋊C4×C59C2×D412yes—
162C22⋊C4×C59—42no—
163C42×C59C2×D412yes—
164C42×C59#4112yes—
165C22⋊Dic59—42no—
166C22⋊Dic59—42no—
167C22⋊Dic59—81no—
168C22⋊Dic59—81no—
169C4⋊Dic59—21no—
170C4⋊Dic59—22no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—41no—
178C4⋊Dic59—42no—
179C4⋊Dic59—81no—
180C4⋊Dic59C2×Q812yes—
181Dic59⋊C4—42no—
182Dic59⋊C4—42no—
183Dic59⋊C4—81no—
184Dic59⋊C4—81no—
185C4×Dic59—21no—
186C4×Dic59—21no—
187C4×Dic59—21no—
188C4×Dic59—21no—
189C4×Dic59—42no—
190C4×Dic59—81no—
191C4×Dic59—81no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8C2×Q812yes—
197C2×C59⋊C8#4112yes—
198C23×C59—22no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59C22×D411yescharacteristic
206D4×C59—22no—
207D4×C59—22no—
208D4×C59—41no—
209D4×C59—41no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
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—84no—
224C2×C4×C59#12112yes—
225C2×C4×C59#12112yes—
226C2×C4×C59C22×D411yescharacteristic
227C2×C4×C59C22×Q811yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C592−1+411yescharacteristic
231C2×C4×C592−1+411yescharacteristic
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—22no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
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×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59#13512yes—
256C8×C59—41no—
257C8×C59—41no—
258C59⋊C8—22no—
259C59⋊C8—22no—
260#25458—591nomaximal, Sylow
261C22×C59—42no—
262C22×C59—42no—
263C22×C59—42no—
264C22×C59—81no—
265C22×C59—81no—
266C22×C59—81no—
267C22×C59—81no—
268C22×C59C2×#12111yescharacteristic
269C22×C59#54711yescharacteristic
270C22×C59D4×Q811yescharacteristic
271C22×C59C2×#13511yescharacteristic
272C22×C59#57511yescharacteristic
273C22×C59#57611yescharacteristic
274C22×C59#57711yescharacteristic
275C4×C59—22no—
276C4×C59—22no—
277C4×C59—42no—
278C4×C59—42no—
279C4×C59—81no—
280C4×C59#53311yescharacteristic
281C4×C59#54111yescharacteristic
282C4×C59#54811yescharacteristic
283C4×C59D4×Q811yescharacteristic
284Dic59—22no—
285Dic59—41no—
286Dic59—41no—
287Dic59—41no—
288Dic59—42no—
289Dic59—81no—
290Dic59—81no—
291Dic59—81no—
292C2×#499—591no—
293C2×#478D5911yescharacteristic
294#2566—591no—
295#2370—591no—
296#2085—591no—
297#2083—591no—
298#2065—591no—
299#2065—591no—
300#1989—591no—
301#1949—591no—
302#1948—591no—
303#1939—591no—
304#1925—591no—
305#1925—591no—
306C4×#102—591no—
307C2×C59—82no—
308C2×C59#242911yescharacteristic
309C2×C59#303511yescharacteristic
310C2×C59#304511yescharacteristic
311C2×C59#333711yescharacteristic
312C2×C59#335511yescharacteristic
313C2×C59#338011yescharacteristic
314C2×C59#338811yescharacteristic
315C2×#125—591no—
316C2×#125C2×D5911yescharacteristic
317C2×C4×D4C2×D5911yescharacteristic
318#499—1184no—
319#478—24no—
320C2×#107—591no—
321C2×#107—591no—
322C2×#107C2×D5911yescharacteristic
323C2×#106—591no—
324C2×#106—591no—
325C2×#106C2×D5911yescharacteristic
326C2×C4⋊C8—591no—
327C2×C4⋊C8—591no—
328C2×C4⋊C8C2×D5911yescharacteristic
329C2×#102—591no—
330C2×#102—591no—
331C2×#102C2×D5911yescharacteristic
332C2×#102C2×D5911yescharacteristic
333C2×C4×C8—591no—
334#396—1181no—
335#396—1181no—
336#390—591no—
337#390—591no—
338#389—591no—
339#389—591no—
340#389—1181no—
341#387—1181no—
342#387—1181no—
343#385—1181no—
344#385—1181no—
345#384—591no—
346#382—591no—
347C4×C4⋊C4—591no—
348C4×C4⋊C4—591no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—1181no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#336—591no—
359#336—591no—
360C59#2545811yescharacteristic, complemented, Sylow
361C22×D4C22×D5911yescharacteristic
362C23×C4—21no—
363C22×C8—591no—
364C22×C8—591no—
365C22×C8C22×D5911yescharacteristic
366C22×C8C22×D5911yescharacteristic
367#125—22no—
368#125—1182no—
369#125—1182no—
370#125C22⋊D5914yes—
371C4×D4—22no—
372C4×D4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
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⋊C4C22×D5911yescharacteristic
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C4⋊C4C22×D5911yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1181no—
392C2×C22⋊C4—1181no—
393C2×C22⋊C4—1181no—
394C2×C22⋊C4—1181no—
395C2×C42—591no—
396C2×C42—591no—
397C2×C42—1181no—
398C2×C42—1181no—
399C2×C42—1181no—
400C2×C42C22×D5911yescharacteristic
401#107—22no—
402#107—1182no—
403#107—1182no—
404#106—22no—
405#106—1182no—
406#106—1182no—
407C4⋊C8—22no—
408C4⋊C8—1182no—
409C4⋊C8—1182no—
410#102—22no—
411#102—22no—
412#102—594no—
413#102—1182no—
414C4×C8—1182no—
415#95—591no—
416#95—591no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424C24—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4Q8×D5912yes—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—42no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—591no—
438C22×C4—591no—
439C22×C4—591no—
440C22×C4—591no—
441C22×C4—1181no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4D4×D5911yescharacteristic
446C22×C4D4×D5911yescharacteristic
447C22×C4C23×D5911yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—592no—
455C2×C8—592no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—592no—
467C4⋊C4—1182no—
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—2361no—
479C4⋊C4—2362no—
480C4⋊C4—2362no—
481C4⋊C4—2362no—
482C4⋊C4C2×C22⋊D5912yes—
483C4⋊C4C2×C22⋊D5912yes—
484C22⋊C4—42no—
485C22⋊C4—4721no—
486C22⋊C4—4721no—
487C22⋊C4—2362no—
488C22⋊C4—2362no—
489C42—1181no—
490C42—1181no—
491C42—1181no—
492C42—1181no—
493C42—4721no—
494C42—4721no—
495C42—2362no—
496C42#1174568712yes—
497C42C2×C22⋊D5912yes—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C2×D4×D5911yescharacteristic
506D4—22no—
507D4—22no—
508D4—41no—
509D4—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—84no—
524C2×C4—592no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1182no—
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×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4#1427369912yes—
549C2×C4#1427369912yes—
550C2×C4C22×C22⋊D5911yescharacteristic
551C2×C4C2×Q8×D5911yescharacteristic
552C2×C4#41×D5911yescharacteristic
553C2×C4C2×#1174568711yescharacteristic
554C2×C4#1427377411yescharacteristic
555C2×C4#1427377511yescharacteristic
556C8—41no—
557C8—41no—
558C8—1182no—
559C8—1182no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22C2×#1427369911yescharacteristic
568C22#1683805411yescharacteristic
569C22#1683806511yescharacteristic
570C22#135×D5911yescharacteristic
571C22#1683819911yescharacteristic
572C22#1683820611yescharacteristic
573C22#1683821211yescharacteristic
574C4—22no—
575C4—22no—
576C4—42no—
577C4—42no—
578C4—81no—
579C4—1182no—
580C4—4721no—
581C4—4721no—
582C4—4721no—
583C4—2361no—
584C4—2361no—
585C4—2362no—
586C4—2361no—
587C4#1683795411yescharacteristic
588C4#1683801211yescharacteristic
589C4#1683805911yescharacteristic
590C4Q8×C22⋊D5911yescharacteristic
591C2—82no—
592C2#2142769711yescharacteristic
593C2#2143354711yescharacteristic
594C2#2143363911yescharacteristic
595C2#2143648211yescharacteristic
596C2#2143669911yescharacteristic
597C2#2143695811yescharacteristic
598C2#2143704511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#478C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenterC2×#478D59
hypocenterC59#25458
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×#16837801C2semidirect
3. #21427196C2semidirect
4. #21425581C2semidirect
5. #21425570C2semidirect
8. #21425051C2semidirect
9. #21424861C2semidirect
10. #21424856C2semidirect
11. #21424812C2semidirect
360. C59#25458semidirect

Maximal subgroups

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

subgroupindex
C2×#168378012
#214292922
#214271962
#214255812
#214255702
#214254662
#214250512
#214248612
#214248562
#214248122
#214247192
C4×#142736482
C2×C59×#4782
#2545859