Finite Group Databasebeta

← #28610539

Subgroups — #28610539

Order 15104.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837700C211yesmaximal, characteristic
2#21427194C211yesmaximal, characteristic, complemented
3#21427001C211yesmaximal, characteristic
4#21425059C211yesmaximal, characteristic, complemented
5#21424867C211yesmaximal, characteristic, complemented
6#21424829C211yesmaximal, characteristic, complemented
7#21424823C211yesmaximal, characteristic, complemented
8#21424687C211yesmaximal, characteristic, complemented
9#21424681C211yesmaximal, characteristic, complemented
10#21424357C211yesmaximal, characteristic
11#21424345C211yesmaximal, characteristic
12C8×Dic59⋊C4C211yesmaximal, characteristic, complemented
13#102×Dic59C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C4×#102×C59C211yesmaximal, characteristic
16C2×#14273718C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18C2×D4×Dic59C2211yescharacteristic
19#16837700C418yescomplemented
20C2×#14273661C2211yescharacteristic
21C2×#14273661C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×C8×Dic59C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837299—21no—
33#16837298C2211yescharacteristic
34#16837286—21no—
35#16837285—21no—
36#16837275—21no—
37#16837274—21no—
38#16837267C2211yescharacteristic
39#16837260—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×Dic59⋊C4C2211yescharacteristic
43C4×C4⋊Dic59—21no—
44C2×C59⋊#95—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C23×Dic59—21no—
61C23×Dic59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273718—22no—
65#14273718C2×C414yescomplemented
66D4×Dic59—22no—
67D4×Dic59—22no—
68D4×Dic59—41no—
69D4×Dic59C2×C414yes—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4D411yescharacteristic
81C2×Dic59⋊C4D411yescharacteristic
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59C2311yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88#14273661—41no—
89#14273661C2×C414yescomplemented
90#14273658—41no—
91#14273658C2×C414yescomplemented
92Dic59⋊C8C2×C414yescomplemented
93#14273648—22no—
94#14273648C2×C414yes—
95C8×Dic59—22no—
96C8×Dic59—41no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C22×D4×C59C2311yescharacteristic
108C23×C4×C59—21no—
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C4×D4×C59—22no—
112C4×D4×C59—41no—
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C22⋊C4×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59C2311yescharacteristic
119#102×C59C2×C414yes—
120#102×C59Q814yes—
121C4×C8×C59—41no—
122#95×C59—21no—
123C24×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59C22×C412yes—
128C2×D4×C59C2×Q812yes—
129C2×D4×C59#4112yes—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—42no—
134C22×C4×C59—42no—
135C22×C4×C59—81no—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—24no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—42no—
145C22×Dic59—42no—
146C22×Dic59—81no—
147C22×Dic59—81no—
148C22×Dic59—81no—
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59C22×C412yes—
158C2×C8×C59C2×Q812yes—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59C22×C412yes—
164C4⋊C4×C59C2×Q812yes—
165C22⋊C4×C59—42no—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59—42no—
169C22⋊Dic59—24no—
170C22⋊Dic59—42no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—24no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—42no—
179C4⋊Dic59C22×C412yes—
180C4⋊Dic59C22×C412yes—
181Dic59⋊C4—22no—
182Dic59⋊C4—24no—
183Dic59⋊C4—22no—
184Dic59⋊C4—81no—
185Dic59⋊C4—81no—
186Dic59⋊C4—81no—
187C4×Dic59—21no—
188C4×Dic59—21no—
189C4×Dic59—81no—
190C4×Dic59—81no—
191C4×Dic59C22×C412yes—
192C4×Dic59#4112yes—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C22×C412yes—
199C23×C59—22no—
200C23×C59—22no—
201C23×C59—22no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59C22×D411yescharacteristic
207D4×C59—22no—
208D4×C59—22no—
209D4×C59—22no—
210D4×C59C4×Q814yes—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—22no—
214C2×C4×C59—21no—
215C2×C4×C59—22no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
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—84no—
226C2×C4×C59#13512yes—
227C2×C4×C59C23×C411yescharacteristic
228C2×C4×C59C22×Q811yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59H(5,𝔽2)11yescharacteristic
232C2×C4×C592−1+411yescharacteristic
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—22no—
240C2×Dic59—22no—
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—84no—
255C2×Dic59—84no—
256C2×Dic59C4×D412yes—
257C2×Dic59C4×D412yes—
258C2×Dic59#13512yes—
259C8×C59—22no—
260C8×C59—22no—
261C59⋊C8—22no—
262C59⋊C8—41no—
263#22966—591nomaximal, Sylow
264C22×C59—24no—
265C22×C59—42no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59C2×C4×D411yescharacteristic
272C22×C59#54611yescharacteristic
273C22×C59D4×Q811yescharacteristic
274C22×C59C2×#13511yescharacteristic
275C22×C59C2×#13511yescharacteristic
276C22×C59#57611yescharacteristic
277C22×C59#57711yescharacteristic
278C4×C59—22no—
279C4×C59—42no—
280C4×C59—42no—
281C4×C59—41no—
282C4×C59—81no—
283C4×C59C2×C4×Q811yescharacteristic
284C4×C59#52011yescharacteristic
285C4×C59#55211yescharacteristic
286C4×C59#55411yescharacteristic
287Dic59—22no—
288Dic59—22no—
289Dic59—22no—
290Dic59—42no—
291Dic59—42no—
292Dic59—41no—
293Dic59—81no—
294Dic59—81no—
295C2×#478—591no—
296#2370—591no—
297#2349—591no—
298#1991—591no—
299#1949—591no—
300#1942—591no—
301#1941—591no—
302#1921—591no—
303#1920—591no—
304#1864—591no—
305#1862—591no—
306C8×C4⋊C4—591no—
307C4×#102—591no—
308C4×#102—591no—
309C4×#102D5911yescharacteristic
310C2×C59—44no—
311C2×C59#236011yescharacteristic
312C2×C59C4×#13511yescharacteristic
313C2×C59#294611yescharacteristic
314C2×C59#329611yescharacteristic
315C2×C59#329711yescharacteristic
316C2×C59#338011yescharacteristic
317C2×C59#338111yescharacteristic
318C2×#125—591no—
319C2×C4×D4—591no—
320C2×C4×D4—591no—
321C2×C4×D4C2×D5911yescharacteristic
322#478—598no—
323C2×#107—591no—
324C2×#107—591no—
325C2×#106—591no—
326C2×#106—591no—
327C2×C4⋊C8—591no—
328C2×#102—591no—
329C2×#102—591no—
330C2×#102C2×D5911yescharacteristic
331C2×#102C2×D5911yescharacteristic
332C2×C4×C8—591no—
333C2×C4×C8—591no—
334C2×C4×C8C2×D5911yescharacteristic
335#389—591no—
336#389—591no—
337#389—591no—
338#389—591no—
339#389—1181no—
340#387—1181no—
341#387—1181no—
342#385—1181no—
343#385—1181no—
344#382—591no—
345#380—1181no—
346C4×C4⋊C4—591no—
347C4×C4⋊C4—591no—
348C4×C4⋊C4—591no—
349C4×C4⋊C4—1181no—
350C4×C4⋊C4C2×D5911yescharacteristic
351C2×#95—1181no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340C2×D5911yescharacteristic
359#340C2×D5911yescharacteristic
360#336—591no—
361#336—591no—
362C59#2296611yescharacteristic, complemented, Sylow
363C22×D4C22×D5911yescharacteristic
364C23×C4—21no—
365C23×C4—1181no—
366C23×C4—1181no—
367C22×C8—591no—
368C22×C8—591no—
369C22×C8C22×D5911yescharacteristic
370C22×C8C22×D5911yescharacteristic
371#125—594no—
372#125—1182no—
373C4×D4—22no—
374C4×D4—41no—
375C4×D4—594no—
376C4×D4—1182no—
377C4×D4—1182no—
378C4×D4—2361no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4C22×D5911yescharacteristic
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—1181no—
396C2×C22⋊C4—1181no—
397C2×C42—21no—
398C2×C42—591no—
399C2×C42—591no—
400C2×C42—1181no—
401C2×C42—1181no—
402C2×C42C22×D5911yescharacteristic
403#107—594no—
404#107—2361no—
405#106—594no—
406#106—2361no—
407C4⋊C8—594no—
408#102—594no—
409#102—1182no—
410#102Dic11814yes—
411#102C4×D5914yes—
412C4×C8—41no—
413C4×C8—1182no—
414C4×C8—2361no—
415#95—21no—
416#95—591no—
417#95—591no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426C24—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×Dic11812yes—
431C2×D4C2×C4×D5912yes—
432C2×D4#1174568512yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—81no—
439C22×C4—591no—
440C22×C4—591no—
441C22×C4—591no—
442C22×C4—591no—
443C22×C4—1184no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—4721no—
449C22×C4—4721no—
450C22×C4—4721no—
451C22×C4—2362no—
452C22×C4—2362no—
453C22×C4D4×D5911yescharacteristic
454C22×C4D4×D5911yescharacteristic
455C22×C4C23×D5911yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—592no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8C2×Dic11812yes—
467C2×C8C2×C4×D5912yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—592no—
473C4⋊C4—592no—
474C4⋊C4—1182no—
475C4⋊C4—1184no—
476C4⋊C4—1184no—
477C4⋊C4—1182no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—4721no—
485C4⋊C4—4721no—
486C4⋊C4—4721no—
487C4⋊C4—2362no—
488C4⋊C4C2×Dic11812yes—
489C4⋊C4C2×C4×D5912yes—
490C22⋊C4—42no—
491C22⋊C4—1184no—
492C22⋊C4—2362no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—592no—
497C42—592no—
498C42—1181no—
499C42—1181no—
500C42—4721no—
501C42—4721no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×D4×D5911yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4C4×Dic11814yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—592no—
530C2×C4—592no—
531C2×C4—592no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1182no—
542C2×C4—1182no—
543C2×C4—4724no—
544C2×C4—4724no—
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#1427374612yes—
556C2×C4C22×C4×D5911yescharacteristic
557C2×C4C22×Dic11811yescharacteristic
558C2×C4#41×D5911yescharacteristic
559C2×C4C2×#1174568511yescharacteristic
560C2×C4#1427377211yescharacteristic
561C2×C4#1427377411yescharacteristic
562C8—22no—
563C8—22no—
564C8—1182no—
565C8—2361no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C4×D4×D5911yescharacteristic
574C22#1683804211yescharacteristic
575C22D4×Dic11811yescharacteristic
576C22#135×D5911yescharacteristic
577C22C2×#1427374611yescharacteristic
578C22#1683820411yescharacteristic
579C22#1683820811yescharacteristic
580C4—22no—
581C4—42no—
582C4—42no—
583C4—41no—
584C4—81no—
585C4—1182no—
586C4—1182no—
587C4—1182no—
588C4—4721no—
589C4—4721no—
590C4—2361no—
591C4—2362no—
592C4—2362no—
593C4C2×C4×Dic11811yescharacteristic
594C4#1683788111yescharacteristic
595C4#1683808511yescharacteristic
596C4#1683811111yescharacteristic
597C2—44no—
598C2#2142710211yescharacteristic
599C2C4×#1427374611yescharacteristic
600C2#2143292311yescharacteristic
601C2#2143592711yescharacteristic
602C2#2143594711yescharacteristic
603C2#2143696111yescharacteristic
604C2#2143697211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#102×C59C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenterC4×#102D59
hypocenterC59#22966
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #21427194C2semidirect
4. #21425059C2semidirect
5. #21424867C2semidirect
6. #21424829C2semidirect
7. #21424823C2semidirect
8. #21424687C2semidirect
9. #21424681C2semidirect
12. C8×Dic59⋊C4C2semidirect
19. #16837700C4semidirect
65. #14273718C2×C4semidirect
89. #14273661C2×C4semidirect
91. #14273658C2×C4semidirect
92. Dic59⋊C8C2×C4semidirect
362. C59#22966semidirect

Maximal subgroups

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

subgroupindex
C2×#168377002
#214271942
#214270012
#214250592
#214248672
#214248292
#214248232
#214246872
#214246812
#214243572
#214243452
C8×Dic59⋊C42
#102×Dic592
C4×#102×C592
#2296659