Finite Group Databasebeta

← #28661927

Subgroups — #28661927

Order 15104.

734 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427152C211yesmaximal, characteristic, complemented
2#21425647C211yesmaximal, characteristic, complemented
3#21425413C211yesmaximal, characteristic
4#21425342C211yesmaximal, characteristic, complemented
5#21425178C211yesmaximal, characteristic
6#21424949C211yesmaximal, characteristic
7#21424737C211yesmaximal, characteristic, complemented
8#21424713C211yesmaximal, characteristic
9#21424678C211yesmaximal, characteristic, complemented
10#21424655C211yesmaximal, characteristic, complemented
11#21424454C211yesmaximal, characteristic
12#21424454C211yesmaximal, characteristic
13#21424108C211yesmaximal, characteristic, complemented
14C4×#14273650C211yesmaximal, characteristic, complemented
15C59×#2513C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C22×#11745670C2211yescharacteristic
18C2×#14273727C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20C2×C4×Dic118C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×C4×C59⋊C8C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837308C2211yescharacteristic
35#16837298—21no—
36#16837285—21no—
37#16837285—21no—
38C4⋊C4×Dic59—21no—
39C4×C4⋊Dic59C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C59×#397—21no—
49C59×#397—21no—
50C59×#390C2211yescharacteristic
51C59×#390C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#388—21no—
55C59×#388—21no—
56C59×#382C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C22×Dic118C2311yescharacteristic
61C2×#11745670—22no—
62C2×#11745670—22no—
63C2×#11745670—22no—
64C2×#11745670—41no—
65C2×#11745670—41no—
66C2×#11745670—41no—
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71#14273727—22no—
72#14273727—22no—
73C4×Dic118—22no—
74C4×Dic118—22no—
75C4×Dic118—41no—
76C4×Dic118—41no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86#14273660—41no—
87#14273658—22no—
88C4⋊(C59⋊C8)—41no—
89#14273650—22no—
90#14273650—22no—
91#14273650—22no—
92#14273650—41no—
93#14273650—41no—
94#14273650—41no—
95#14273648—22no—
96#14273648—41no—
97C4×C59⋊C8—41no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×D4×C59C2311yescharacteristic
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59D411yescharacteristic
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118#4112yes—
158C2×Dic118#4112yes—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—81no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—42no—
170C22⋊C4×C59—81no—
171C22⋊C4×C59—81no—
172C22⋊C4×C59—81no—
173C22⋊C4×C59—81no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—81no—
181#11745670—44no—
182#11745670—44no—
183C4⋊Dic59—24no—
184C4⋊Dic59—24no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59#4112yes—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191C4×Dic59—42no—
192C4×Dic59—42no—
193C4×Dic59—42no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8#4112yes—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59C22×D411yescharacteristic
214D4×C59—41no—
215D4×C59—41no—
216D4×C59—41no—
217D4×C59—41no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59#13512yes—
244C2×C4×C59#13712yes—
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59H(5,𝔽2)11yescharacteristic
250C2×C4×C59H(5,𝔽2)11yescharacteristic
251C2×Dic59—22no—
252C2×Dic59—22no—
253C2×Dic59—22no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263Dic118—22no—
264Dic118—22no—
265Dic118—22no—
266Dic118—22no—
267Dic118—41no—
268Dic118—41no—
269Dic118—41no—
270Dic118—41no—
271C59⋊C8—22no—
272C59⋊C8—41no—
273C59⋊C8—41no—
274C59⋊C8—41no—
275#27069—591nomaximal, Sylow
276C22×C59—81no—
277C22×C59—81no—
278C22×C59—81no—
279C22×C59—81no—
280C22×C59—81no—
281C22×C59—81no—
282C22×C59—81no—
283C22×C59#52511yescharacteristic
284C22×C59#54611yescharacteristic
285C22×C59#55011yescharacteristic
286C22×C59C2×#13511yescharacteristic
287C22×C59C2×#13711yescharacteristic
288C22×C59#57711yescharacteristic
289C22×C59#57711yescharacteristic
290C4×C59—22no—
291C4×C59—22no—
292C4×C59—41no—
293C4×C59—41no—
294C4×C59—41no—
295C4×C59—41no—
296C4×C59—81no—
297C4×C59—81no—
298C4×C59—81no—
299C4×C59#52911yescharacteristic
300C4×C59#53811yescharacteristic
301C4×C59#55111yescharacteristic
302C4×C59#55311yescharacteristic
303Dic59—42no—
304Dic59—42no—
305Dic59—42no—
306Dic59—81no—
307#2513D5911yescharacteristic
308#2365—591no—
309#2097—591no—
310#2055—591no—
311#2043—591no—
312#2013—591no—
313#1968—591no—
314#1928—591no—
315#1925—591no—
316#1920—591no—
317#1916—591no—
318#1880—591no—
319#1880—591no—
320#1805—591no—
321C4×#103—591no—
322C2×C59—161no—
323C2×C59#246711yescharacteristic
324C2×C59#310811yescharacteristic
325C2×C59#312611yescharacteristic
326C2×C59#330311yescharacteristic
327C2×C59#332011yescharacteristic
328C2×C59#339411yescharacteristic
329C2×C59#340011yescharacteristic
330C22×SD16—591no—
331C22×SD16—591no—
332C2×C4⋊Q8—591no—
333C2×C4×Q8—591no—
334C2×C4×Q8—591no—
335C2×#107—591no—
336C2×#106—591no—
337C2×C4⋊C8—591no—
338C2×#103—591no—
339C2×#103—591no—
340C2×#103—591no—
341C2×#103—591no—
342C2×#103—591no—
343C2×#103—591no—
344C2×#102—591no—
345C2×#102—591no—
346C2×C4×C8—591no—
347#397—21no—
348#397—21no—
349#391—591no—
350#391—591no—
351#390C2×D5911yescharacteristic
352#390C2×D5911yescharacteristic
353#390C2×D5911yescharacteristic
354#389—1181no—
355#389C2×D5911yescharacteristic
356#388—21no—
357#388—21no—
358#387—1181no—
359#387—1181no—
360#382C2×D5911yescharacteristic
361C4×C4⋊C4—591no—
362C4×C4⋊C4—1181no—
363C4×C4⋊C4C2×D5911yescharacteristic
364C4×C4⋊C4C2×D5911yescharacteristic
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#336—591no—
370#336—591no—
371#336—591no—
372#336—591no—
373C59#2706911yescharacteristic, complemented, Sylow
374C22×Q8—591no—
375C22×Q8—591no—
376C22×D4C22×D5911yescharacteristic
377C2×SD16—1182no—
378C2×SD16—1182no—
379C2×SD16—1182no—
380C2×SD16—2361no—
381C2×SD16—2361no—
382C2×SD16—2361no—
383C22×C8—591no—
384C22×C8—591no—
385C22×C8—591no—
386C22×C8—591no—
387C4⋊Q8—1182no—
388C4⋊Q8—1182no—
389C4×Q8—1182no—
390C4×Q8—1182no—
391C4×Q8—2361no—
392C4×Q8—2361no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4C22⋊D5911yescharacteristic
401C2×C4⋊C4C22⋊D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C42—21no—
412C2×C42—1181no—
413C2×C42—1181no—
414C2×C42—1181no—
415C2×C42C22×D5911yescharacteristic
416C2×C42C22×D5911yescharacteristic
417C2×C42C22×D5911yescharacteristic
418#107—2361no—
419#106—1182no—
420C4⋊C8—2361no—
421#103—1182no—
422#103—1182no—
423#103—1182no—
424#103—2361no—
425#103—2361no—
426#103—2361no—
427#102—1182no—
428#102—2361no—
429C4×C8—2361no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95C22⋊D5911yescharacteristic
440#95C22⋊D5911yescharacteristic
441C24—21no—
442C2×Q8—592no—
443C2×Q8—592no—
444C2×Q8—1181no—
445C2×Q8—1181no—
446C2×Q8—1181no—
447C2×Q8—1181no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C2×Q8—1181no—
453C2×Q8—1181no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—1181no—
466C22×C4—1181no—
467C22×C4C2×C22⋊D5911yescharacteristic
468C22×C4C2×C22⋊D5911yescharacteristic
469C22×C4C2×C22⋊D5911yescharacteristic
470C22×C4C2×C22⋊D5911yescharacteristic
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C23×D5911yescharacteristic
474SD16—2364no—
475SD16—2364no—
476C2×C8—592no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—81no—
497C4⋊C4—592no—
498C4⋊C4—1184no—
499C4⋊C4—1184no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—2362no—
504C4⋊C4—2362no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—81no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—81no—
518C42—2362no—
519C42—2362no—
520C42—2362no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C22×C22⋊D5911yescharacteristic
529Q8—1182no—
530Q8—1182no—
531Q8—1182no—
532Q8—1182no—
533Q8—2361no—
534Q8—2361no—
535Q8—2361no—
536Q8—2361no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—1182no—
567C2×C4—1182no—
568C2×C4—1182no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4#1427374712yes—
579C2×C4#1427375712yes—
580C2×C4#41×D5911yescharacteristic
581C2×C4#41×D5911yescharacteristic
582C2×C4#41×D5911yescharacteristic
583C2×C4#41×D5911yescharacteristic
584C2×C4#1427377311yescharacteristic
585C2×C4#1427377311yescharacteristic
586C8—1182no—
587C8—2361no—
588C8—2361no—
589C8—2361no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22#1683790911yescharacteristic
598C22#1683804011yescharacteristic
599C22#1683807111yescharacteristic
600C22C2×#1427374711yescharacteristic
601C22C2×#1427375711yescharacteristic
602C22#1683821311yescharacteristic
603C22#1683821311yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—4721no—
614C4—2362no—
615C4—2362no—
616C4—2362no—
617C4#1683793511yescharacteristic
618C4#1683799111yescharacteristic
619C4#1683807711yescharacteristic
620C4#1683810311yescharacteristic
621C2—161no—
622C2#2142814311yescharacteristic
623C2#2143417211yescharacteristic
624C2#2143438211yescharacteristic
625C2#2143601811yescharacteristic
626C2#2143626111yescharacteristic
627C2#2143708811yescharacteristic
628C2#2143714911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2513C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2513D59
hypocenterC59#27069
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2513

Complements

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

NH ≅ G/Ntype
1. #21427152C2semidirect
2. #21425647C2semidirect
4. #21425342C2semidirect
7. #21424737C2semidirect
9. #21424678C2semidirect
10. #21424655C2semidirect
13. #21424108C2semidirect
14. C4×#14273650C2semidirect
373. C59#27069semidirect

Maximal subgroups

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

subgroupindex
#214271522
#214256472
#214254132
#214253422
#214251782
#214249492
#214247372
#214247132
#214246782
#214246552
#214244542
#214241082
C4×#142736502
C59×#25132
#2706959