Finite Group Databasebeta

← #28596393

Subgroups — #28596393

Order 15104.

868 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×#16837699C211yesmaximal, characteristic
2#21426458C211yesmaximal, characteristic
3#21425054C211yesmaximal, characteristic, complemented
4#21424857C211yesmaximal, characteristic, complemented
5#21424671C211yesmaximal, characteristic, complemented
6#21424664C211yesmaximal, characteristic, complemented
7#21424356C211yesmaximal, characteristic
8#21424356C211yesmaximal, characteristic
9#21424093C211yesmaximal, characteristic
10#21423994C211yesmaximal, characteristic, complemented
11C4×#14273658C211yesmaximal, characteristic, complemented
12#21423953C211yesmaximal, characteristic, complemented
13#21423924C211yesmaximal, characteristic
14C4×#14273647C211yesmaximal, characteristic
15C59×#2330C211yesmaximal, characteristic, complemented
16C2×C4×D236C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19#16837699C418yescomplemented
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32C4×C4⋊Dic59C2211yescharacteristic
33C4×C4⋊Dic59C2211yescharacteristic
34C4×C4⋊Dic59C2211yescharacteristic
35#16837249—21no—
36#16837249—21no—
37#16837249—21no—
38#95×D59—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×#125×C59C2211yescharacteristic
50C59×#389—21no—
51C59×#389—21no—
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#387—21no—
55C59×#387—21no—
56C59×#384C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C59×#376C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C4×D236—22no—
69C4×D236—22no—
70C4×D236—22no—
71C4×D236—22no—
72C4×D236C2×C414yes—
73C4×D236Q814yes—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83#14273660C2×C414yescomplemented
84#14273658—22no—
85#14273658—22no—
86#14273658C2×C414yescomplemented
87C4⋊(C59⋊C8)—22no—
88C4⋊(C59⋊C8)C2×C414yescomplemented
89#14273647—22no—
90#14273647—22no—
91#14273647C2×C414yes—
92#14273647C2×C414yes—
93#14273638—22no—
94C4×C59⋊C8—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98#125×C59—22no—
99#125×C59C2×C414yescomplemented
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C42×C59—21no—
111C2×C42×C59—21no—
112C2×C42×C59C2311yescharacteristic
113C2×C42×C59C2311yescharacteristic
114C2×C42×C59C2311yescharacteristic
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—21no—
118#95×C59—21no—
119#95×C59D411yescharacteristic
120#95×C59D411yescharacteristic
121#95×C59D411yescharacteristic
122#95×C59D411yescharacteristic
123C23×D59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2411yescharacteristic
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C2×D236—21no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236C22×C412yes—
142C2×D236C2×Q812yes—
143C2×D236#4112yes—
144C2×C4×D59—24no—
145C2×C4×D59—24no—
146C2×C4×D59—24no—
147C2×C4×D59—81no—
148C2×C4×D59—81no—
149C2×C4×D59—81no—
150C2×C4×D59—81no—
151C2×C4×D59—81no—
152C2×C4×D59—81no—
153C4⋊C4×C59—24no—
154C4⋊C4×C59—24no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—81no—
162C4⋊C4×C59—81no—
163C4⋊C4×C59—81no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59C22×C412yes—
166C4⋊C4×C59C22×C412yes—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—81no—
170C42×C59—81no—
171C42×C59C22×C412yes—
172C42×C59C2×Q812yes—
173C42×C59#4112yes—
174C42×C59#4112yes—
175(C2×D59)⋊C4—24no—
176(C2×D59)⋊C4—24no—
177(C2×D59)⋊C4—24no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C412yes—
182C4⋊Dic59C2×Q812yes—
183C4⋊Dic59#4112yes—
184C4×Dic59—81no—
185C4×Dic59—81no—
186C4×Dic59—81no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8C22×C412yes—
198C2×C59⋊C8C22×C412yes—
199C23×C59C22×D411yescharacteristic
200C22×D59—22no—
201C22×D59—22no—
202C22×D59—22no—
203C22×D59—41no—
204C22×D59—41no—
205C22×D59—41no—
206C22×D59—41no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
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×C59C4×D412yes—
235C2×C4×C59C22⋊Q812yes—
236C2×C4×C59C22⋊Q812yes—
237C2×C4×C59C23×C411yescharacteristic
238C2×C4×C59C22×Q811yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—22no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252D236—22no—
253D236—22no—
254D236—22no—
255D236C4×Q814yes—
256C4×D59—84no—
257C4×D59—84no—
258C4×D59—84no—
259C59⋊C8—22no—
260C59⋊C8—22no—
261C59⋊C8—41no—
262C59⋊C8—41no—
263#21619—591nomaximal, Sylow
264C22×C59C2×C4×D411yescharacteristic
265C22×C59C2×C22⋊Q811yescharacteristic
266C22×C59#52511yescharacteristic
267C22×C59#57611yescharacteristic
268C22×C59#57711yescharacteristic
269C22×C59#57711yescharacteristic
270C22×C59#57711yescharacteristic
271C2×D59—24no—
272C2×D59—42no—
273C2×D59—42no—
274C2×D59—42no—
275C2×D59—42no—
276C2×D59—42no—
277C2×D59—42no—
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—22no—
281C4×C59—22no—
282C4×C59—41no—
283C4×C59—41no—
284C4×C59—42no—
285C4×C59—42no—
286C4×C59—81no—
287C4×C59—81no—
288C4×C59C2×C4×Q811yescharacteristic
289C4×C59C4×#4111yescharacteristic
290C4×C59#52911yescharacteristic
291C4×C59#53311yescharacteristic
292Dic59—42no—
293Dic59—42no—
294Dic59—42no—
295C2×#478—591no—
296#2330D5911yescharacteristic, complemented
297#2275—591no—
298#1989—591no—
299#1948—591no—
300#1919—591no—
301#1918—591no—
302#1864—591no—
303#1864—591no—
304#1802—591no—
305#1776—591no—
306C4×#106—591no—
307#1768—591no—
308#1762—591no—
309C4×#102—591no—
310C2×C59C4×C22⋊Q811yescharacteristic
311C2×C59#294611yescharacteristic
312C2×C59#295211yescharacteristic
313C2×C59#308111yescharacteristic
314C2×C59#308411yescharacteristic
315C2×C59#311911yescharacteristic
316C2×C59#312311yescharacteristic
317D59—44no—
318C2×#125C2×D5911yescharacteristic
319C2×C4×D4—591no—
320C2×C4×D4—591no—
321C2×C4×D4—591no—
322#478—598no—
323C2×#107—591no—
324C2×#106—591no—
325C2×#106—591no—
326C2×#106—591no—
327C2×C4⋊C8—591no—
328C2×C4⋊C8—591no—
329C2×#102—591no—
330C2×#102—591no—
331C2×#102—591no—
332C2×#102—591no—
333C2×#97—591no—
334C2×C4×C8—591no—
335#389—21no—
336#389—21no—
337#389C2×D5911yescharacteristic
338#389C2×D5911yescharacteristic
339#387—21no—
340#387—21no—
341#384C2×D5911yescharacteristic
342C4×C4⋊C4—591no—
343C4×C4⋊C4—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4C2×D5911yescharacteristic
346C4×C4⋊C4C2×D5911yescharacteristic
347C4×C22⋊C4—1181no—
348C4×C22⋊C4—1181no—
349C4×C22⋊C4—1181no—
350#376C2×D5911yescharacteristic
351C2×#95—1181no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#336—591no—
361#336—591no—
362C59#2161911yescharacteristic, complemented, Sylow
363C22×D4—591no—
364C23×C4—1181no—
365C23×C4—1181no—
366C23×C4—1181no—
367C22×C8—591no—
368C22×C8—591no—
369C22×C8—591no—
370C22×C8—591no—
371#125—22no—
372#125C4×D5914yescomplemented
373C4×D4—594no—
374C4×D4—594no—
375C4×D4—1182no—
376C4×D4—1182no—
377C4×D4—1182no—
378C4×D4—1182no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C4⋊C4C22×D5911yescharacteristic
390C2×C4⋊C4C22×D5911yescharacteristic
391C2×C4⋊C4C22×D5911yescharacteristic
392C2×C22⋊C4—1181no—
393C2×C22⋊C4—1181no—
394C2×C22⋊C4—1181no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—1181no—
398C2×C42—1181no—
399C2×C42—1181no—
400C2×C42C22×D5911yescharacteristic
401C2×C42C22×D5911yescharacteristic
402C2×C42C22×D5911yescharacteristic
403#107—594no—
404#106—594no—
405#106—1182no—
406#106—1182no—
407C4⋊C8—594no—
408C4⋊C8—1182no—
409#102—594no—
410#102—594no—
411#102—1182no—
412#102—1182no—
413#97—1182no—
414C4×C8—1182no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95C22⋊D5911yescharacteristic
423#95C22⋊D5911yescharacteristic
424#95C22⋊D5911yescharacteristic
425#95C22⋊D5911yescharacteristic
426C24—1181no—
427C2×D4—592no—
428C2×D4—592no—
429C2×D4—592no—
430C2×D4—1181no—
431C2×D4—1181no—
432C2×D4—1181no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—1184no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1184no—
442C22×C4—1184no—
443C22×C4—4721no—
444C22×C4—4721no—
445C22×C4—4721no—
446C22×C4—4721no—
447C22×C4—4721no—
448C22×C4—4721no—
449C22×C4C2×C22⋊D5911yescharacteristic
450C22×C4C2×C22⋊D5911yescharacteristic
451C22×C4C2×C22⋊D5911yescharacteristic
452C22×C4C2×C22⋊D5911yescharacteristic
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C23×D5911yescharacteristic
456C2×C8—592no—
457C2×C8—592no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—592no—
481C4⋊C4—592no—
482C4⋊C4—592no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4C2×C4×D5912yes—
487C4⋊C4C2×C4×D5912yes—
488C22⋊C4—1184no—
489C22⋊C4—1184no—
490C22⋊C4—1184no—
491C42—21no—
492C42—21no—
493C42—81no—
494C42—81no—
495C42—4721no—
496C42—4721no—
497C42—4721no—
498C42C2×C4×D5912yes—
499C42#1174568512yes—
500C42Q8×D5912yes—
501C42#1174568912yes—
502C23—1182no—
503C23—1182no—
504C23—1182no—
505C23—2361no—
506C23—2361no—
507C23—2361no—
508C23—2361no—
509C23C22×C22⋊D5911yescharacteristic
510D4—594no—
511D4—1182no—
512D4—1182no—
513D4—1182no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—1182no—
541C2×C4—1182no—
542C2×C4—1182no—
543C2×C4—4724no—
544C2×C4—4724no—
545C2×C4—4724no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4C4×C22⋊D5912yes—
553C2×C4C4×C22⋊D5912yes—
554C2×C4#1427370312yes—
555C2×C4#1427370312yes—
556C2×C4C22×C4×D5911yescharacteristic
557C2×C4C2×Q8×D5911yescharacteristic
558C2×C4#41×D5911yescharacteristic
559C2×C4#41×D5911yescharacteristic
560C2×C4C2×#1174568511yescharacteristic
561C2×C4C2×#1174568911yescharacteristic
562C8—1182no—
563C8—1182no—
564C8—2361no—
565C8—2361no—
566C22—1184no—
567C22—2362no—
568C22—2362no—
569C22—2362no—
570C22—2362no—
571C22—2362no—
572C22—2362no—
573C22C2×C4×C22⋊D5911yescharacteristic
574C22C2×#1427370311yescharacteristic
575C22#1683790911yescharacteristic
576C22#1683820511yescharacteristic
577C22#1683821011yescharacteristic
578C22#1683821011yescharacteristic
579C22#1683821311yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—2362no—
591C4—2362no—
592C4—2362no—
593C4C4×Q8×D5911yescharacteristic
594C4C4×#1174568911yescharacteristic
595C4#1683793911yescharacteristic
596C4#1683795611yescharacteristic
597C2—2364no—
598C2C4×#1427370311yescharacteristic
599C2#2143292411yescharacteristic
600C2#2143297711yescharacteristic
601C2#2143392111yescharacteristic
602C2#2143395411yescharacteristic
603C2#2143428911yescharacteristic
604C2#2143434311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2330C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#2330D59
hypocenterC59#21619
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2330

Complements

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

NH ≅ G/Ntype
3. #21425054C2semidirect
4. #21424857C2semidirect
5. #21424671C2semidirect
6. #21424664C2semidirect
10. #21423994C2semidirect
11. C4×#14273658C2semidirect
12. #21423953C2semidirect
15. C59×#2330C2semidirect
19. #16837699C4semidirect
83. #14273660C2×C4semidirect
86. #14273658C2×C4semidirect
88. C4⋊(C59⋊C8)C2×C4semidirect
99. #125×C59C2×C4semidirect
296. #2330D59semidirect
362. C59#21619semidirect
372. #125C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168376992
#214264582
#214250542
#214248572
#214246712
#214246642
#214243562
#214240932
#214239942
C4×#142736582
#214239532
#214239242
C4×#142736472
C59×#23302
#2161959