Finite Group Databasebeta

← #28553835

Subgroups — #28553835

Order 15104.

1426 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21422357C212yesmaximal, complemented
2#21422348C212yesmaximal, complemented
3#21422341C212yesmaximal, complemented
4#21422341C212yesmaximal, complemented
5#21422339C212yesmaximal, complemented
6#21422339C212yesmaximal, complemented
7#21422337C211yesmaximal, characteristic
8#21422337C211yesmaximal, characteristic
9C59×#1438C211yesmaximal, characteristic
10C22×C4⋊Dic59C2212yes—
11C22×Dic59⋊C4C2212yes—
12C22×C22⋊Dic59C2212yes—
13C22×C22⋊Dic59C2212yes—
14C22×C22⋊Dic59C2211yescharacteristic
15C22×C22⋊Dic59C2212yes—
16C22×C22⋊Dic59C2211yescharacteristic
17C2×C59⋊#95C2212yes—
18C2×C59⋊#95C2211yescharacteristic
19C2×C59⋊#95C2212yes—
20C2×C59⋊#95C2211yescharacteristic
21C2×C59⋊#95C2212yes—
22C2×C59⋊#95C2212yes—
23C2×C59⋊#95C2212yes—
24C2×C59⋊#95C2212yes—
25C2×C59⋊#95C2212yes—
26C23×D4×C59C2211yescharacteristic
27C22×C22⋊C4×C59C2211yescharacteristic
28C22×C22⋊C4×C59C2212yes—
29C22×C22⋊C4×C59C2211yescharacteristic
30C2×#95×C59C2212yes—
31C23×Dic59C2312yes—
32C23×Dic59C2312yes—
33C23×Dic59C2312yes—
34C23×Dic59C2312yes—
35C2×C4⋊Dic59—24no—
36C2×C4⋊Dic59—42no—
37C2×C4⋊Dic59—42no—
38C2×Dic59⋊C4—42no—
39C2×Dic59⋊C4—42no—
40C2×Dic59⋊C4—42no—
41C2×C22⋊Dic59—24no—
42C2×C22⋊Dic59—24no—
43C2×C22⋊Dic59—42no—
44C2×C22⋊Dic59—42no—
45C2×C22⋊Dic59—42no—
46C2×C22⋊Dic59—42no—
47C2×C22⋊Dic59—41no—
48C2×C22⋊Dic59—42no—
49C2×C22⋊Dic59—42no—
50C2×C22⋊Dic59—42no—
51C2×C22⋊Dic59—42no—
52C2×C22⋊Dic59—42no—
53C2×C22⋊Dic59—41no—
54C59⋊#95—24no—
55C59⋊#95—42no—
56C59⋊#95—42no—
57C59⋊#95—41no—
58C59⋊#95—41no—
59C59⋊#95—42no—
60C59⋊#95—42no—
61C59⋊#95—42no—
62C59⋊#95—42no—
63C25×C59C2311yescharacteristic
64C25×C59C2311yescharacteristic
65C22×D4×C59—42no—
66C22×D4×C59—41no—
67C22×D4×C59—42no—
68C22×D4×C59—41no—
69C22×D4×C59Q814yes—
70C23×C4×C59C2312yes—
71C23×C4×C59C2311yescharacteristic
72C23×C4×C59C2312yes—
73C2×C22⋊C4×C59—24no—
74C2×C22⋊C4×C59—24no—
75C2×C22⋊C4×C59—41no—
76C2×C22⋊C4×C59—42no—
77C2×C22⋊C4×C59—42no—
78C2×C22⋊C4×C59—42no—
79C2×C22⋊C4×C59—41no—
80#95×C59—42no—
81C24×C59—22no—
82C24×C59—22no—
83C24×C59—22no—
84C24×C59—22no—
85C24×C59—22no—
86C24×C59—21no—
87C24×C59—21no—
88C24×C59—22no—
89C24×C59—22no—
90C24×C59—22no—
91C24×C59—22no—
92C24×C59—21no—
93C24×C59—22no—
94C24×C59—21no—
95C24×C59—22no—
96C24×C59—22no—
97C24×C59C2×Q812yes—
98C24×C59C2×Q812yes—
99C24×C59C2411yescharacteristic
100C2×D4×C59—44no—
101C2×D4×C59—44no—
102C2×D4×C59—44no—
103C2×D4×C59—48no—
104C2×D4×C59—48no—
105C22×C4×C59—22no—
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—21no—
113C22×C4×C59—22no—
114C22×C4×C59—21no—
115C22×C4×C59—22no—
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×C59C2×Q812yes—
123C22×C4×C59#4114yes—
124C22×Dic59—22no—
125C22×Dic59—22no—
126C22×Dic59—22no—
127C22×Dic59—22no—
128C22×Dic59—22no—
129C22×Dic59—22no—
130C22×Dic59—22no—
131C22×Dic59—22no—
132C22×Dic59—22no—
133C22×Dic59—22no—
134C22×Dic59—22no—
135C22×Dic59—22no—
136C22×Dic59—22no—
137C22×Dic59—22no—
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—22no—
150C22×Dic59—22no—
151C22×Dic59#4114yes—
152C22⋊C4×C59—44no—
153C22⋊C4×C59—48no—
154C22⋊C4×C59—44no—
155C22⋊Dic59—48no—
156C22⋊Dic59—48no—
157C22⋊Dic59—48no—
158C22⋊Dic59—44no—
159C22⋊Dic59—44no—
160C4⋊Dic59—48no—
161Dic59⋊C4—48no—
162C23×C59—22no—
163C23×C59—22no—
164C23×C59—22no—
165C23×C59—22no—
166C23×C59—22no—
167C23×C59—22no—
168C23×C59—24no—
169C23×C59—24no—
170C23×C59—24no—
171C23×C59—24no—
172C23×C59—42no—
173C23×C59—42no—
174C23×C59—42no—
175C23×C59—42no—
176C23×C59—42no—
177C23×C59—42no—
178C23×C59—42no—
179C23×C59—42no—
180C23×C59—42no—
181C23×C59—42no—
182C23×C59—42no—
183C23×C59—42no—
184C23×C59—41no—
185C23×C59—41no—
186C23×C59—42no—
187C23×C59—42no—
188C23×C59—42no—
189C23×C59—42no—
190C23×C59—41no—
191C23×C59—41no—
192C23×C59—42no—
193C23×C59—41no—
194C23×C59—42no—
195C23×C59—42no—
196C23×C59—42no—
197C23×C59—42no—
198C23×C59—42no—
199C23×C59—42no—
200C23×C59—42no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59C22×Q811yescharacteristic
205C23×C59C2×#4112yes—
206C23×C59C2×#4112yes—
207C23×C59H(5,𝔽2)12yes—
208C23×C59H(5,𝔽2)11yescharacteristic
209C23×C59H(5,𝔽2)12yes—
210C23×C59H(5,𝔽2)12yes—
211C23×C59H(5,𝔽2)11yescharacteristic
212C23×C59H(5,𝔽2)12yes—
213D4×C59—416no—
214C2×C4×C59—24no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—24no—
219C2×C4×C59—24no—
220C2×C4×C59—42no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
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×Dic59—24no—
234C2×Dic59—24no—
235C2×Dic59—24no—
236C2×Dic59—42no—
237C2×Dic59—42no—
238C2×Dic59—42no—
239C2×Dic59—42no—
240C2×Dic59—42no—
241C2×Dic59—42no—
242C2×Dic59—42no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248C2×Dic59—42no—
249C2×Dic59—42no—
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—
261#17272—591nomaximal, Sylow
262C22×C59—42no—
263C22×C59—42no—
264C22×C59—42no—
265C22×C59—42no—
266C22×C59—44no—
267C22×C59—44no—
268C22×C59—44no—
269C22×C59—44no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—82no—
273C22×C59—82no—
274C22×C59—82no—
275C22×C59—82no—
276C22×C59—82no—
277C22×C59—82no—
278C22×C59—82no—
279C22×C59—82no—
280C22×C59#53812yes—
281C22×C59#53812yes—
282C22×C59#53812yes—
283C22×C59#53812yes—
284C22×C59#54012yes—
285C22×C59#54012yes—
286C22×C59#54311yescharacteristic
287C22×C59#54311yescharacteristic
288C22×C59#54311yescharacteristic
289C22×C59#55111yescharacteristic
290C22×C59#55312yes—
291C22×C59#55412yes—
292C22×C59#55412yes—
293C22×C59#55512yes—
294C22×C59#55911yescharacteristic
295C22×C59#56012yes—
296C22×C59#56012yes—
297C22×C59#56012yes—
298C22×C59#56012yes—
299C22×C59#56011yescharacteristic
300C22×C59H(3,𝔽4)11yescharacteristic
301C4×C59—42no—
302C4×C59—44no—
303C4×C59—82no—
304Dic59—44no—
305Dic59—82no—
306Dic59—82no—
307Dic59—82no—
308#1443—592no—
309#1441—592no—
310#1439—592no—
311#1439—592no—
312#1439—592no—
313#1439—592no—
314#1438—591no—
315#1438—591no—
316#1438D5911yescharacteristic
317C2×C59—82no—
318C2×C59—82no—
319C2×C59#274811yescharacteristic
320C2×C59#275612yes—
321C2×C59#278012yes—
322C2×C59#278412yes—
323C2×C59#280312yes—
324C2×C59#280411yescharacteristic
325C2×C59#282512yes—
326C2×C59#282812yes—
327C2×C59#284211yescharacteristic
328C23×D4C2×D5911yescharacteristic
329C22×C4⋊C4—592no—
330C22×C4⋊C4—592no—
331C22×C22⋊C4—592no—
332C22×C22⋊C4—592no—
333C22×C22⋊C4—591no—
334C22×C22⋊C4—592no—
335C22×C22⋊C4—591no—
336C22×C22⋊C4C2×D5911yescharacteristic
337C22×C22⋊C4C2×D5912yes—
338C22×C22⋊C4C2×D5911yescharacteristic
339C2×#95—592no—
340C2×#95—591no—
341C2×#95—592no—
342C2×#95—591no—
343C2×#95—592no—
344C2×#95—592no—
345C2×#95—592no—
346C2×#95—592no—
347C2×#95—592no—
348C2×#95C2×D5912yes—
349C59#1727211yescharacteristic, complemented, Sylow
350C25C22×D5911yescharacteristic
351C25C22×D5911yescharacteristic
352C22×D4—42no—
353C22×D4—41no—
354C22×D4—41no—
355C22×D4—42no—
356C22×D4Dic11814yes—
357C23×C4—592no—
358C23×C4—592no—
359C23×C4—592no—
360C23×C4—592no—
361C23×C4C22×D5912yes—
362C23×C4C22×D5911yescharacteristic
363C23×C4C22×D5912yes—
364C2×C4⋊C4—1184no—
365C2×C4⋊C4—2362no—
366C2×C4⋊C4—2362no—
367C2×C4⋊C4—2362no—
368C2×C4⋊C4—2362no—
369C2×C4⋊C4—2362no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—41no—
373C2×C22⋊C4—42no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4—41no—
376C2×C22⋊C4—42no—
377C2×C22⋊C4—1184no—
378C2×C22⋊C4—1184no—
379C2×C22⋊C4—2362no—
380C2×C22⋊C4—2362no—
381C2×C22⋊C4—2362no—
382C2×C22⋊C4—2362no—
383C2×C22⋊C4—2361no—
384C2×C22⋊C4—2362no—
385C2×C22⋊C4—2362no—
386C2×C22⋊C4—2362no—
387C2×C22⋊C4—2361no—
388C2×C22⋊C4—2362no—
389C2×C22⋊C4—2362no—
390#95—42no—
391#95—1184no—
392#95—2362no—
393#95—2362no—
394#95—2361no—
395#95—2361no—
396#95—2362no—
397#95—2362no—
398#95—2362no—
399#95—2362no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24—22no—
409C24—21no—
410C24—21no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24C2×Dic11812yes—
417C24C2×Dic11812yes—
418C24C23×D5911yescharacteristic
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—48no—
423C2×D4—48no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—594no—
442C22×C4—1182no—
443C22×C4—1182no—
444C22×C4—1182no—
445C22×C4—1182no—
446C22×C4—1182no—
447C22×C4—1182no—
448C22×C4—1182no—
449C22×C4—1182no—
450C22×C4—1182no—
451C22×C4—1182no—
452C22×C4—1182no—
453C22×C4—1182no—
454C22×C4—1182no—
455C22×C4—1182no—
456C22×C4—1182no—
457C22×C4—1182no—
458C22×C4—1182no—
459C22×C4—1182no—
460C22×C4—1182no—
461C22×C4—1182no—
462C22×C4—1182no—
463C22×C4—1182no—
464C22×C4—1182no—
465C22×C4—1182no—
466C22×C4—1182no—
467C22×C4—1182no—
468C22×C4—1182no—
469C22×C4C2×Dic11812yes—
470C22×C4#1174568714yes—
471C4⋊C4—2368no—
472C4⋊C4—2368no—
473C22⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C22⋊C4—2368no—
477C22⋊C4—2368no—
478C22⋊C4—2364no—
479C22⋊C4—2368no—
480C22⋊C4—2364no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23C22×Dic11811yescharacteristic
524C23#41×D5912yes—
525C23C2×#1174568712yes—
526C23#1427377212yes—
527C23#1427377211yescharacteristic
528C23#1427377211yescharacteristic
529C23#1427377212yes—
530C23#1427377212yes—
531C23#1427377312yes—
532D4—416no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
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—42no—
551C2×C4—42no—
552C2×C4—1184no—
553C2×C4—1184no—
554C2×C4—1184no—
555C2×C4—2362no—
556C2×C4—2362no—
557C2×C4—2362no—
558C2×C4—2362no—
559C2×C4—2362no—
560C2×C4—2362no—
561C2×C4—2362no—
562C2×C4—2362no—
563C2×C4—2362no—
564C2×C4—2362no—
565C2×C4—2362no—
566C2×C4—2362no—
567C2×C4—2362no—
568C2×C4—2362no—
569C2×C4—2362no—
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—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22#1683798412yes—
599C22#1683798612yes—
600C22#1683798712yes—
601C22#1683798812yes—
602C22#1683800512yes—
603C22#1683800612yes—
604C22#1683802811yescharacteristic
605C22#1683802811yescharacteristic
606C22#1683802811yescharacteristic
607C22#1683808311yescharacteristic
608C22#1683809512yes—
609C22#1683810912yes—
610C22#1683811112yes—
611C22#1683811612yes—
612C22#1683813611yescharacteristic
613C22#1683814012yes—
614C22#1683814012yes—
615C22#1683814112yes—
616C22#1683814112yes—
617C22#1683814211yescharacteristic
618C22#1683814511yescharacteristic
619C4—42no—
620C4—44no—
621C4—82no—
622C4—4722no—
623C4—4722no—
624C4—4722no—
625C4—2364no—
626C2—82no—
627C2—82no—
628C2#2143131111yescharacteristic
629C2#2143142012yes—
630C2#2143174512yes—
631C2#2143179412yes—
632C2#2143202212yes—
633C2#2143203911yescharacteristic
634C2#2143218112yes—
635C2#2143221812yes—
636C2#2143236011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D59
commutator, socle, cosocleC24×C59C24
FittingC59×#1438C2
prime radicalC23×D4×C59C22
hypercenter#1438D59
hypocenterC59#17272
LCSG ▷ C24×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
1. #21422357C2semidirect
2. #21422348C2semidirect
3. #21422341C2semidirect
4. #21422341C2semidirect
5. #21422339C2semidirect
6. #21422339C2semidirect
349. C59#17272semidirect

Maximal subgroups

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

subgroupindex
#214223572
#214223482
#214223412
#214223392
#214223372
C59×#14382
#1727259