Finite Group Databasebeta

← #28661370

Subgroups — #28661370

Order 15104.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428529C211yesmaximal, characteristic, complemented
2#21425499C211yesmaximal, characteristic, complemented
3#21425470C211yesmaximal, characteristic, complemented
4#21425282C211yesmaximal, characteristic
5#21425282C211yesmaximal, characteristic
6#21424859C211yesmaximal, characteristic, complemented
7#21424821C211yesmaximal, characteristic, complemented
8#21424704C211yesmaximal, characteristic, complemented
9#21424681C211yesmaximal, characteristic, complemented
10#21424671C211yesmaximal, characteristic, complemented
11#21424357C211yesmaximal, characteristic, complemented
12#21424289C211yesmaximal, characteristic, complemented
13#21424182C211yesmaximal, characteristic, complemented
14C4×#14273648C211yesmaximal, characteristic, complemented
15C59×#2361C211yesmaximal, characteristic
16C2×#14273658C2211yescharacteristic
17C2×#14273658C2211yescharacteristic
18C2×#14273658C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×C22⋊(C59⋊C8)C2211yescharacteristic
26C2×C22⋊(C59⋊C8)C2211yescharacteristic
27C2×C4×C59⋊C8C2211yescharacteristic
28#16837344—21no—
29#16837325—21no—
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837295—21no—
33#16837289—21no—
34#16837272C2211yescharacteristic
35#16837262C2211yescharacteristic
36#16837262C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#121×C59C2211yescharacteristic
49C2×C4×D4×C59—21no—
50C2×C4×D4×C59C2211yescharacteristic
51C22×C4⋊C4×C59C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#388—21no—
55C59×#386—21no—
56C59×#385—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×C59⋊C8C2311yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59C2311yescharacteristic
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×Dic59⋊C4—21no—
70C2×Dic59⋊C4—21no—
71C2×C22⋊Dic59—21no—
72C2×C22⋊Dic59—21no—
73C2×C4×Dic59—21no—
74#14273658—22no—
75#14273658—22no—
76#14273658—22no—
77#14273658—41no—
78C4⋊(C59⋊C8)—22no—
79#14273648—22no—
80#14273648—22no—
81#14273648—22no—
82#14273648—41no—
83C22⋊(C59⋊C8)—22no—
84C22⋊(C59⋊C8)—41no—
85C4×C59⋊C8—41no—
86C59⋊#95—21no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C22×D4×C59—21no—
92C22×D4×C59C2311yescharacteristic
93C23×C4×C59—21no—
94C23×C4×C59—21no—
95C23×C4×C59C2311yescharacteristic
96#121×C59—22no—
97#121×C59—22no—
98C4×D4×C59—22no—
99C4×D4×C59—41no—
100C4×D4×C59—44no—
101C2×C4⋊C4×C59—22no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—41no—
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59D411yescharacteristic
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C42×C59—21no—
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117#95×C59—21no—
118#95×C59—21no—
119#95×C59D411yescharacteristic
120#95×C59D411yescharacteristic
121C24×C59—21no—
122C24×C59C2×D411yescharacteristic
123C2×D4×C59—24no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—42no—
130C2×D4×C59—42no—
131C2×D4×C59#4112yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—42no—
142C22×C4×C59—42no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—81no—
146C22×C4×C59—81no—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—42no—
162C4⋊C4×C59—42no—
163C4⋊C4×C59#4112yes—
164C4⋊C4×C59#4112yes—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—41no—
167C22⋊C4×C59—42no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—81no—
170C22⋊C4×C59SD1614yes—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—42no—
176C22⋊Dic59—81no—
177C22⋊Dic59—81no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—81no—
187C4⋊Dic59—81no—
188Dic59⋊C4—42no—
189Dic59⋊C4—81no—
190C4×Dic59—81no—
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⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8#4112yes—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—22no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59C2×SD1612yes—
217C23×C59C22×D411yescharacteristic
218D4×C59—22no—
219D4×C59—22no—
220D4×C59—48no—
221D4×C59—41no—
222D4×C59—41no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—22no—
229C2×C4×C59—21no—
230C2×C4×C59—22no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—84no—
250C2×C4×C59—84no—
251C2×C4×C59C2×SD1612yes—
252C2×C4×C59C2×SD1612yes—
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59H(5,𝔽2)11yescharacteristic
258C2×C4×C592−1+411yescharacteristic
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C59⋊C8—22no—
272C59⋊C8—41no—
273C59⋊C8—41no—
274C59⋊C8—41no—
275#27032—591nomaximal, Sylow
276C22×C59—22no—
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—41no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—42no—
284C22×C59—42no—
285C22×C59—42no—
286C22×C59—81no—
287C22×C59—81no—
288C22×C59—81no—
289C22×C59—81no—
290C22×C59#52511yescharacteristic
291C22×C59#54611yescharacteristic
292C22×C59#54711yescharacteristic
293C22×C59C22×SD1611yescharacteristic
294C22×C59#57511yescharacteristic
295C22×C59#57611yescharacteristic
296C22×C59#57711yescharacteristic
297C4×C59—22no—
298C4×C59—22no—
299C4×C59—41no—
300C4×C59—42no—
301C4×C59—41no—
302C4×C59—42no—
303C4×C59—41no—
304C4×C59—81no—
305C4×C59#52911yescharacteristic
306C4×C59#53911yescharacteristic
307C4×C59#55311yescharacteristic
308C4×C59#55611yescharacteristic
309Dic59—81no—
310Dic59—81no—
311Dic59—81no—
312Dic59—81no—
313#2497—591no—
314#2361D5911yescharacteristic
315#2071—591no—
316#2065—591no—
317#2034—591no—
318#2034—591no—
319#1948—591no—
320#1941—591no—
321#1923—591no—
322#1920—591no—
323#1919—591no—
324#1864—591no—
325#1853—591no—
326#1823—591no—
327C4×#102—591no—
328C2×C59—42no—
329C2×C59—82no—
330C2×C59#246611yescharacteristic
331C2×C59#310111yescharacteristic
332C2×C59#311911yescharacteristic
333C2×C59#329511yescharacteristic
334C2×C59#332111yescharacteristic
335C2×C59#333511yescharacteristic
336C2×C59#335511yescharacteristic
337C2×#121C2×D5911yescharacteristic
338C2×C4×D4—21no—
339C2×C4×D4C2×D5911yescharacteristic
340C22×C4⋊C4C2×D5911yescharacteristic
341C2×#106—591no—
342C2×#106—591no—
343C2×#106—591no—
344C2×#106—591no—
345C2×C4⋊C8—591no—
346C2×#102—591no—
347C2×#102—591no—
348C2×#102—591no—
349C2×#102—591no—
350C2×C22⋊C8—591no—
351C2×C22⋊C8—591no—
352C2×C4×C8—591no—
353#400—1181no—
354#396—1181no—
355#390C2×D5911yescharacteristic
356#389—591no—
357#389—591no—
358#389C2×D5911yescharacteristic
359#388—21no—
360#388—1181no—
361#387—1181no—
362#386—21no—
363#385—21no—
364#384—591no—
365#380—591no—
366#380—591no—
367C4×C4⋊C4—591no—
368C4×C4⋊C4C2×D5911yescharacteristic
369C4×C22⋊C4C2×D5911yescharacteristic
370#340—591no—
371#340—591no—
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#336—591no—
379#336—591no—
380C59#2703211yescharacteristic, complemented, Sylow
381C22×D4—21no—
382C22×D4C22×D5911yescharacteristic
383C23×C4—21no—
384C23×C4—21no—
385C23×C4C22×D5911yescharacteristic
386C22×C8—591no—
387C22×C8—591no—
388C22×C8—591no—
389C22×C8—591no—
390#121—22no—
391#121—22no—
392C4×D4—22no—
393C4×D4—41no—
394C4×D4—44no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—41no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C4⋊C4C22×D5911yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1181no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4C22⋊D5911yescharacteristic
417C2×C22⋊C4C22⋊D5911yescharacteristic
418C2×C42—21no—
419C2×C42—1181no—
420C2×C42C22×D5911yescharacteristic
421C2×C42C22×D5911yescharacteristic
422#106—1182no—
423#106—1182no—
424#106—1182no—
425#106—2361no—
426C4⋊C8—1182no—
427#102—1182no—
428#102—1182no—
429#102—1182no—
430#102—2361no—
431C22⋊C8—1182no—
432C22⋊C8—2361no—
433C4×C8—2361no—
434#95—21no—
435#95—21no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95C22⋊D5911yescharacteristic
442#95C22⋊D5911yescharacteristic
443C24—21no—
444C24C2×C22⋊D5911yescharacteristic
445C2×D4—24no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4#1174568512yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C23×D5911yescharacteristic
479C2×C8—592no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—4721no—
506C4⋊C4—4721no—
507C4⋊C4—4721no—
508C4⋊C4—2362no—
509C4⋊C4#1174568512yes—
510C4⋊C4#1174568712yes—
511C22⋊C4—24no—
512C22⋊C4—41no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—4721no—
517C22⋊C4—4721no—
518C22⋊C4#1174567014yes—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—4721no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23C2×#1174567012yes—
539C23C22×C22⋊D5911yescharacteristic
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4C2×#1174567012yes—
586C2×C4C2×#1174567012yes—
587C2×C4#41×D5911yescharacteristic
588C2×C4C2×#1174568711yescharacteristic
589C2×C4C2×#1174568511yescharacteristic
590C2×C4C2×#1174568511yescharacteristic
591C2×C4#1427377211yescharacteristic
592C2×C4#1427377411yescharacteristic
593C8—1182no—
594C8—2361no—
595C8—2361no—
596C8—2361no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1683791311yescharacteristic
612C22#1683804611yescharacteristic
613C22#1683805111yescharacteristic
614C22C22×#1174567011yescharacteristic
615C22#1683820011yescharacteristic
616C22#1683820511yescharacteristic
617C22#1683821011yescharacteristic
618C4—22no—
619C4—22no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—81no—
626C4—4721no—
627C4—4721no—
628C4—4721no—
629C4—4721no—
630C4#1683793211yescharacteristic
631C4#1683799411yescharacteristic
632C4#1683810211yescharacteristic
633C4#1683812211yescharacteristic
634C2—42no—
635C2—82no—
636C2#2142812611yescharacteristic
637C2#2143410411yescharacteristic
638C2#2143429511yescharacteristic
639C2#2143592011yescharacteristic
640C2#2143628211yescharacteristic
641C2#2143645311yescharacteristic
642C2#2143669611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2361C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2361D59
hypocenterC59#27032
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #21428529C2semidirect
2. #21425499C2semidirect
3. #21425470C2semidirect
6. #21424859C2semidirect
7. #21424821C2semidirect
8. #21424704C2semidirect
9. #21424681C2semidirect
10. #21424671C2semidirect
11. #21424357C2semidirect
12. #21424289C2semidirect
13. #21424182C2semidirect
14. C4×#14273648C2semidirect
380. C59#27032semidirect

Maximal subgroups

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

subgroupindex
#214285292
#214254992
#214254702
#214252822
#214248592
#214248212
#214247042
#214246812
#214246712
#214243572
#214242892
#214241822
C4×#142736482
C59×#23612
#2703259