Finite Group Databasebeta

← #28519376

Subgroups — #28519376

Order 15104.

1390 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837313C211yesmaximal, characteristic
2C2×#16837309C211yesmaximal, characteristic, complemented
3C2×#16837291C211yesmaximal, characteristic
4#21422360C211yesmaximal, characteristic, complemented
5#21422354C212yesmaximal, complemented
6#21422351C212yesmaximal, complemented
7#21422344C212yesmaximal
8#21422342C211yesmaximal, characteristic
9#21422340C212yesmaximal
10#21422321C211yesmaximal, characteristic, complemented
11C59×#1446C211yesmaximal, characteristic, complemented
12C22×C4⋊Dic59C2211yescharacteristic
13C22×Dic59⋊C4C2211yescharacteristic
14C22×Dic59⋊C4C2212yes—
15C22×(C2×D59)⋊C4C2211yescharacteristic
16C22×(C2×D59)⋊C4C2211yescharacteristic
17C22×(C2×D59)⋊C4C2211yescharacteristic
18C22×(C2×D59)⋊C4C2212yes—
19C22×(C2×D59)⋊C4C2212yes—
20C22×C4×Dic59C2211yescharacteristic
21#16837313C418yescomplemented
22#16837309—24no—
23#16837291—24no—
24C2×C59⋊#95C2212yes—
25C2×C59⋊#95C2212yes—
26C2×C59⋊#95C2211yescharacteristic
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2212yes—
32C2×C59⋊#95C2211yescharacteristic
33C23×Q8×C59C2211yescharacteristic
34C2×#95×C59C2211yescharacteristic
35C2×#95×C59C2211yescharacteristic
36C2×#95×C59C2212yes—
37C2×#95×C59C2212yes—
38C24×D59C2311yescharacteristic
39C23×Dic59C2312yes—
40C23×Dic59C2311yescharacteristic
41C23×Dic59C2312yes—
42C23×Dic59C2311yescharacteristic
43C23×Dic59C2311yescharacteristic
44C2×C4⋊Dic59—22no—
45C2×C4⋊Dic59—42no—
46C2×Dic59⋊C4—22no—
47C2×Dic59⋊C4—42no—
48C2×Dic59⋊C4—42no—
49C2×Dic59⋊C4—42no—
50C2×Dic59⋊C4—42no—
51C2×(C2×D59)⋊C4—22no—
52C2×(C2×D59)⋊C4—22no—
53C2×(C2×D59)⋊C4—22no—
54C2×(C2×D59)⋊C4—22no—
55C2×(C2×D59)⋊C4—42no—
56C2×(C2×D59)⋊C4—42no—
57C2×(C2×D59)⋊C4—42no—
58C2×(C2×D59)⋊C4—42no—
59C2×(C2×D59)⋊C4—42no—
60C2×(C2×D59)⋊C4—42no—
61C2×(C2×D59)⋊C4—42no—
62C2×(C2×D59)⋊C4C2×C414yes—
63C2×(C2×D59)⋊C4C2×C418yes—
64C2×C4×Dic59—42no—
65C2×C4×Dic59D414yes—
66C59⋊#95—22no—
67C59⋊#95—24no—
68C59⋊#95—24no—
69C59⋊#95—42no—
70C59⋊#95—42no—
71C59⋊#95—42no—
72C59⋊#95—42no—
73C59⋊#95C2×C418yes—
74C59⋊#95C2×C414yes—
75C22×Q8×C59—22no—
76C22×Q8×C59—22no—
77C22×Q8×C59—42no—
78C22×Q8×C59—42no—
79C22×Q8×C59C2×C414yes—
80C23×C4×C59C2312yes—
81C23×C4×C59C2311yescharacteristic
82C23×C4×C59C2311yescharacteristic
83C23×C4×C59C2312yes—
84C23×C4×C59C2311yescharacteristic
85#95×C59—22no—
86#95×C59—22no—
87#95×C59—42no—
88#95×C59—42no—
89C24×C59C2411yescharacteristic
90C23×D59—21no—
91C23×D59—21no—
92C23×D59—22no—
93C23×D59—22no—
94C23×D59—22no—
95C23×D59—21no—
96C23×D59—21no—
97C23×D59—21no—
98C23×D59—21no—
99C23×D59—22no—
100C23×D59C22×C412yes—
101C2×Q8×C59—28no—
102C2×Q8×C59—48no—
103C2×Q8×C59—44no—
104C2×Q8×C59—44no—
105C2×Q8×C59—48no—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—22no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—21no—
115C22×C4×C59—21no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59C22×C414yes—
132C22×C4×C59C22×C412yes—
133C22×C4×C59C2×D412yes—
134C22×C4×C59#4112yes—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—22no—
141C22×Dic59—22no—
142C22×Dic59—22no—
143C22×Dic59—22no—
144C22×Dic59—22no—
145C22×Dic59—22no—
146C22×Dic59—22no—
147C22×Dic59—22no—
148C22×Dic59—22no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—22no—
152C22×Dic59—21no—
153C22×Dic59—22no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—22no—
158C22×Dic59—22no—
159C22×Dic59—22no—
160C22×Dic59C22×C412yes—
161C22×Dic59C22×C414yes—
162C22×Dic59C2×D412yes—
163C22×Dic59C2×D412yes—
164(C2×D59)⋊C4—28no—
165(C2×D59)⋊C4—48no—
166(C2×D59)⋊C4—44no—
167(C2×D59)⋊C4—44no—
168(C2×D59)⋊C4—48no—
169C4⋊Dic59—44no—
170Dic59⋊C4—44no—
171Dic59⋊C4—48no—
172C4×Dic59—44no—
173C23×C59C23×C411yescharacteristic
174C23×C59C22×D411yescharacteristic
175C23×C59C2×#4111yescharacteristic
176C23×C59H(5,𝔽2)12yes—
177C23×C59H(5,𝔽2)12yes—
178C23×C59H(5,𝔽2)12yes—
179C23×C59H(5,𝔽2)11yescharacteristic
180C23×C592−1+411yescharacteristic
181C23×C592−1+411yescharacteristic
182C23×C592−1+411yescharacteristic
183C23×C592−1+412yes—
184C22×D59—22no—
185C22×D59—24no—
186C22×D59—22no—
187C22×D59—22no—
188C22×D59—24no—
189C22×D59—41no—
190C22×D59—41no—
191C22×D59—41no—
192C22×D59—41no—
193C22×D59—41no—
194C22×D59—41no—
195C22×D59—42no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—42no—
199C22×D59—42no—
200C22×D59—42no—
201C22×D59—42no—
202C22×D59—42no—
203C22×D59—42no—
204C22×D59—42no—
205C22×D59—42no—
206C22×D59—42no—
207Q8×C59—416no—
208C2×C4×C59—24no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—24no—
212C2×C4×C59—24no—
213C2×C4×C59—24no—
214C2×C4×C59—24no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
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—42no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59C4×D414yes—
237C2×Dic59—22no—
238C2×Dic59—24no—
239C2×Dic59—24no—
240C2×Dic59—24no—
241C2×Dic59—24no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—22no—
246C2×Dic59—22no—
247C2×Dic59—22no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—42no—
251C2×Dic59—42no—
252C2×Dic59—42no—
253C2×Dic59—42no—
254C2×Dic59—42no—
255C2×Dic59—42no—
256C2×Dic59—42no—
257C2×Dic59—42no—
258C2×Dic59—42no—
259C2×Dic59—42no—
260C2×Dic59—42no—
261C2×Dic59—42no—
262C2×Dic59—42no—
263C2×Dic59—42no—
264C2×Dic59—42no—
265C2×Dic59—42no—
266#14625—591nomaximal, Sylow
267C22×C59C2×C4×D411yescharacteristic
268C22×C59#51812yes—
269C22×C59#51911yescharacteristic
270C22×C59#52011yescharacteristic
271C22×C59#52012yes—
272C22×C59#53411yescharacteristic
273C22×C59#53511yescharacteristic
274C22×C59#53511yescharacteristic
275C22×C59#53611yescharacteristic
276C22×C59#53611yescharacteristic
277C22×C59#53711yescharacteristic
278C22×C59#53812yes—
279C22×C59#54111yescharacteristic
280C22×C59#54211yescharacteristic
281C22×C59#54212yes—
282C22×C59H(3,𝔽4)12yes—
283C22×C59#56212yes—
284C22×C59#56212yes—
285C22×C59#56212yes—
286C22×C59#56212yes—
287C22×C59#56312yes—
288C22×C59#56312yes—
289C22×C59#56312yes—
290C2×D59—42no—
291C2×D59—42no—
292C2×D59—42no—
293C2×D59—44no—
294C2×D59—44no—
295C2×D59—81no—
296C2×D59—81no—
297C2×D59—81no—
298C2×D59—82no—
299C2×D59—82no—
300C2×D59—81no—
301C4×C59—24no—
302C4×C59—44no—
303C4×C59—81no—
304C4×C59—81no—
305C4×C59—82no—
306Dic59—42no—
307Dic59—42no—
308Dic59—44no—
309Dic59—42no—
310Dic59—82no—
311C2×#393—591no—
312C2×#391—591no—
313C2×#388—591no—
314#1446D5911yescharacteristic, complemented
315#1444—591no—
316#1442—592no—
317#1442—592no—
318#1440—592no—
319#1439—592no—
320#1439—591no—
321#1434—591no—
322C2×C59#232111yescharacteristic
323C2×C59#232311yescharacteristic
324C2×C59#232611yescharacteristic
325C2×C59#237812yes—
326C2×C59#237912yes—
327C2×C59#281411yescharacteristic
328C2×C59#281611yescharacteristic
329C2×C59#281911yescharacteristic
330C2×C59#282111yescharacteristic
331C2×C59#283112yes—
332C2×C59#283212yes—
333D59—82no—
334C23×Q8C2×D5911yescharacteristic
335C22×C4⋊C4—591no—
336C22×C4⋊C4—591no—
337C22×C4⋊C4—592no—
338C22×C22⋊C4—591no—
339C22×C22⋊C4—592no—
340C22×C22⋊C4—591no—
341C22×C22⋊C4—591no—
342C22×C22⋊C4—592no—
343C22×C42—591no—
344#393—598no—
345#391—1184no—
346#388—1184no—
347C2×#95—591no—
348C2×#95—592no—
349C2×#95—592no—
350C2×#95—592no—
351C2×#95—591no—
352C2×#95—592no—
353C2×#95—592no—
354C2×#95—592no—
355C2×#95—592no—
356C2×#95C2×D5911yescharacteristic
357C2×#95C2×D5911yescharacteristic
358C2×#95C2×D5912yes—
359C2×#95C2×D5912yes—
360C59#1462511yescharacteristic, complemented, Sylow
361C25—591no—
362C22×Q8—22no—
363C22×Q8—22no—
364C22×Q8—42no—
365C22×Q8—42no—
366C22×Q8C4×D5914yes—
367C23×C4—591no—
368C23×C4—592no—
369C23×C4—591no—
370C23×C4—591no—
371C23×C4—592no—
372C23×C4C22×D5911yescharacteristic
373C23×C4C22×D5911yescharacteristic
374C23×C4C22×D5912yes—
375C23×C4C22×D5912yes—
376C23×C4C22×D5911yescharacteristic
377C2×C4⋊C4—1182no—
378C2×C4⋊C4—1182no—
379C2×C4⋊C4—2362no—
380C2×C4⋊C4—2362no—
381C2×C4⋊C4—2362no—
382C2×C4⋊C4—2362no—
383C2×C4⋊C4—2362no—
384C2×C22⋊C4—594no—
385C2×C22⋊C4—598no—
386C2×C22⋊C4—1182no—
387C2×C22⋊C4—1182no—
388C2×C22⋊C4—1182no—
389C2×C22⋊C4—1182no—
390C2×C22⋊C4—2362no—
391C2×C22⋊C4—2362no—
392C2×C22⋊C4—2362no—
393C2×C22⋊C4—2362no—
394C2×C22⋊C4—2362no—
395C2×C22⋊C4—2362no—
396C2×C22⋊C4—2362no—
397C2×C42—594no—
398C2×C42—2362no—
399#95—22no—
400#95—22no—
401#95—42no—
402#95—42no—
403#95—598no—
404#95—594no—
405#95—1182no—
406#95—1184no—
407#95—1184no—
408#95—2362no—
409#95—2362no—
410#95—2362no—
411#95—2362no—
412C24—592no—
413C24—1181no—
414C24—1181no—
415C24—1182no—
416C24—1182no—
417C24—1181no—
418C24—1182no—
419C24—1181no—
420C24—1181no—
421C24—1181no—
422C24—1182no—
423C24C23×D5911yescharacteristic
424C2×Q8—28no—
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—48no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—592no—
455C22×C4—594no—
456C22×C4—592no—
457C22×C4—592no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1182no—
461C22×C4—1182no—
462C22×C4—1182no—
463C22×C4—1182no—
464C22×C4—1181no—
465C22×C4—1182no—
466C22×C4—1182no—
467C22×C4—1182no—
468C22×C4—1182no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1182no—
472C22×C4—1181no—
473C22×C4—1182no—
474C22×C4—1182no—
475C22×C4—1182no—
476C22×C4—1182no—
477C22×C4—1182no—
478C22×C4—1182no—
479C22×C4—1182no—
480C22×C4—1182no—
481C22×C4—1182no—
482C22×C4—1182no—
483C22×C4C2×C4×D5914yes—
484C22×C4C2×C4×D5912yes—
485C22×C4D4×D5912yes—
486C22×C4#1174568712yes—
487C4⋊C4—2364no—
488C4⋊C4—2364no—
489C4⋊C4—2368no—
490C22⋊C4—1188no—
491C22⋊C4—2368no—
492C22⋊C4—2364no—
493C22⋊C4—2364no—
494C22⋊C4—2368no—
495C42—2364no—
496C23—1182no—
497C23—1184no—
498C23—1182no—
499C23—1184no—
500C23—1182no—
501C23—2362no—
502C23—2362no—
503C23—2362no—
504C23—2362no—
505C23—2362no—
506C23—2362no—
507C23—2362no—
508C23—2362no—
509C23—2361no—
510C23—2361no—
511C23—2361no—
512C23—2361no—
513C23—2361no—
514C23—2361no—
515C23—2362no—
516C23—2361no—
517C23—2361no—
518C23—2362no—
519C23C22×C4×D5911yescharacteristic
520C23C2×D4×D5911yescharacteristic
521C23C2×#1174568711yescharacteristic
522C23#1427377212yes—
523C23#1427377212yes—
524C23#1427377311yescharacteristic
525C23#1427377312yes—
526C23#1427377411yescharacteristic
527C23#1427377511yescharacteristic
528C23#1427377511yescharacteristic
529C23#1427377512yes—
530Q8—416no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—1182no—
560C2×C4—1182no—
561C2×C4—1182no—
562C2×C4—1182no—
563C2×C4—1182no—
564C2×C4—1182no—
565C2×C4—1182no—
566C2×C4—1184no—
567C2×C4—1184no—
568C2×C4—1184no—
569C2×C4—1184no—
570C2×C4—2362no—
571C2×C4—2362no—
572C2×C4—2362no—
573C2×C4—2362no—
574C2×C4—2362no—
575C2×C4—2362no—
576C2×C4—2362no—
577C2×C4—2362no—
578C2×C4—2362no—
579C2×C4—2362no—
580C2×C4—2362no—
581C2×C4—2362no—
582C2×C4—2362no—
583C2×C4—2362no—
584C2×C4—2362no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2362no—
588C2×C4(C22⋊D59)⋊C414yes—
589C22—4721no—
590C22—4721no—
591C22—4721no—
592C22—4721no—
593C22—4722no—
594C22—4722no—
595C22—2364no—
596C22—2362no—
597C22—2364no—
598C22—2362no—
599C22—2362no—
600C22C2×(C22⋊D59)⋊C411yescharacteristic
601C22#1683787512yes—
602C22#1683787811yescharacteristic
603C22#1683788412yes—
604C22#1683788411yescharacteristic
605C22#1683796211yescharacteristic
606C22#1683796711yescharacteristic
607C22#1683797011yescharacteristic
608C22#1683797511yescharacteristic
609C22#1683797711yescharacteristic
610C22#1683798211yescharacteristic
611C22#1683798712yes—
612C22#1683801311yescharacteristic
613C22#1683802111yescharacteristic
614C22#1683802512yes—
615C22#1683814412yes—
616C22#1683814612yes—
617C22#1683814812yes—
618C22#1683814812yes—
619C22#1683814812yes—
620C22#1683815112yes—
621C22#1683815112yes—
622C22#1683815312yes—
623C4—24no—
624C4—44no—
625C4—81no—
626C4—81no—
627C4—82no—
628C4—4722no—
629C4—2362no—
630C4—2362no—
631C4—2364no—
632C4—2362no—
633C2—4722no—
634C2#2142675111yescharacteristic
635C2#2142678111yescharacteristic
636C2#2142681011yescharacteristic
637C2#2142725712yes—
638C2#2142726812yes—
639C2#2143210811yescharacteristic
640C2#2143212211yescharacteristic
641C2#2143214411yescharacteristic
642C2#2143216011yescharacteristic
643C2#2143225612yes—
644C2#2143226612yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1446C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenter#1446D59
hypocenterC59#14625
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
2. C2×#16837309C2semidirect
4. #21422360C2semidirect
5. #21422354C2semidirect
6. #21422351C2semidirect
10. #21422321C2semidirect
11. C59×#1446C2semidirect
21. #16837313C4semidirect
314. #1446D59semidirect
360. C59#14625semidirect

Maximal subgroups

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

subgroupindex
C2×#168373132
C2×#168373092
C2×#168372912
#214223602
#214223542
#214223512
#214223442
#214223422
#214223402
#214223212
C59×#14462
#1462559