Finite Group Databasebeta

← #28661790

Subgroups — #28661790

Order 15104.

680 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428529C211yesmaximal, characteristic
2#21425504C211yesmaximal, characteristic, complemented
3#21425491C211yesmaximal, characteristic, complemented
4#21425289C211yesmaximal, characteristic
5#21425289C211yesmaximal, characteristic
6#21424859C211yesmaximal, characteristic, complemented
7#21424836C211yesmaximal, characteristic, complemented
8#21424711C211yesmaximal, characteristic
9#21424681C211yesmaximal, characteristic, complemented
10#21424678C211yesmaximal, characteristic, complemented
11#21424364C211yesmaximal, characteristic, complemented
12#21424295C211yesmaximal, characteristic
13#21424191C211yesmaximal, characteristic
14#21423926C211yesmaximal, characteristic, complemented
15C59×#2362C211yesmaximal, characteristic
16C2×#14273660C2211yescharacteristic
17C2×#14273658C2211yescharacteristic
18C2×#14273658C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×C22⋊(C59⋊C8)C2211yescharacteristic
26C2×C22⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
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×C22⋊Q8×C59C2211yescharacteristic
49C2×C4×Q8×C59C2211yescharacteristic
50C2×#117×C59C2211yescharacteristic
51C59×#391C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#388—21no—
54C59×#388—21no—
55C59×#387—21no—
56C4×C4⋊C4×C59—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#14273660—41no—
75#14273658—22no—
76#14273658—22no—
77#14273658—22no—
78C4⋊(C59⋊C8)—22no—
79#14273651—22no—
80#14273651—22no—
81#14273651—22no—
82#14273651—41no—
83C22⋊(C59⋊C8)—22no—
84C22⋊(C59⋊C8)—41no—
85#14273638—41no—
86C59⋊#95—21no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C22×Q8×C59C2311yescharacteristic
92C23×C4×C59C2311yescharacteristic
93C22⋊Q8×C59—22no—
94C22⋊Q8×C59—22no—
95C4×Q8×C59—22no—
96C4×Q8×C59—41no—
97#117×C59—22no—
98#117×C59—41no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C22⋊C4×C59D411yescharacteristic
108C2×C22⋊C4×C59D411yescharacteristic
109C2×C42×C59—21no—
110C2×C42×C59—21no—
111C2×C42×C59—21no—
112C2×C42×C59C2311yescharacteristic
113C2×C42×C59C2311yescharacteristic
114#95×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—21no—
118#95×C59D411yescharacteristic
119#95×C59D411yescharacteristic
120C24×C59C2×D411yescharacteristic
121C2×Q8×C59—21no—
122C2×Q8×C59—21no—
123C2×Q8×C59—21no—
124C2×Q8×C59—21no—
125C2×Q8×C59—21no—
126C2×Q8×C59#4112yes—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
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—
147C4⋊C4×C59—24no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—42no—
153C4⋊C4×C59—42no—
154C4⋊C4×C59#4112yes—
155C4⋊C4×C59#4112yes—
156C22⋊C4×C59—24no—
157C22⋊C4×C59—22no—
158C22⋊C4×C59—41no—
159C22⋊C4×C59—42no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—42no—
165C42×C59—42no—
166C42×C59—42no—
167C22⋊Dic59—81no—
168C22⋊Dic59—81no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—81no—
178C4⋊Dic59—81no—
179Dic59⋊C4—42no—
180Dic59⋊C4—81no—
181C4×Dic59—81no—
182C2×C59⋊C8—21no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
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#4112yes—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59C22×D411yescharacteristic
202Q8×C59—22no—
203Q8×C59—22no—
204Q8×C59—41no—
205Q8×C59—41no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—22no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
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×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59#13712yes—
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59H(5,𝔽2)11yescharacteristic
242C2×C4×C592−1+411yescharacteristic
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C59⋊C8—22no—
256C59⋊C8—41no—
257C59⋊C8—41no—
258C59⋊C8—41no—
259#27060—591nomaximal, Sylow
260C22×C59—41no—
261C22×C59—41no—
262C22×C59—41no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59—41no—
266C22×C59—41no—
267C22×C59#52511yescharacteristic
268C22×C59#54611yescharacteristic
269C22×C59#54711yescharacteristic
270C22×C59C2×#13711yescharacteristic
271C22×C59#57511yescharacteristic
272C22×C59#57711yescharacteristic
273C22×C59#57711yescharacteristic
274C4×C59—22no—
275C4×C59—41no—
276C4×C59—42no—
277C4×C59—41no—
278C4×C59—42no—
279C4×C59—41no—
280C4×C59—42no—
281C4×C59—41no—
282C4×C59—81no—
283C4×C59#52911yescharacteristic
284C4×C59#53911yescharacteristic
285C4×C59#55311yescharacteristic
286C4×C59#55611yescharacteristic
287Dic59—81no—
288Dic59—81no—
289Dic59—81no—
290Dic59—81no—
291#2497—591no—
292#2362D5911yescharacteristic
293#2072—591no—
294#2068—591no—
295#2035—591no—
296#2035—591no—
297#1948—591no—
298#1944—591no—
299#1924—591no—
300#1920—591no—
301#1920—591no—
302#1865—591no—
303#1854—591no—
304#1825—591no—
305#1763—591no—
306C2×C59—81no—
307C2×C59#246611yescharacteristic
308C2×C59#311611yescharacteristic
309C2×C59#312611yescharacteristic
310C2×C59#332111yescharacteristic
311C2×C59#333111yescharacteristic
312C2×C59#334411yescharacteristic
313C2×C59#335011yescharacteristic
314C2×C22⋊Q8C2×D5911yescharacteristic
315C2×C4×Q8C2×D5911yescharacteristic
316C2×#117C2×D5911yescharacteristic
317C2×#107—591no—
318C2×#106—591no—
319C2×#106—591no—
320C2×#106—591no—
321C2×C4⋊C8—591no—
322C2×#103—591no—
323C2×#103—591no—
324C2×#103—591no—
325C2×#103—591no—
326C2×C22⋊C8—591no—
327C2×C22⋊C8—591no—
328C2×#97—591no—
329#400—1181no—
330#396—1181no—
331#391C2×D5911yescharacteristic
332#389—591no—
333#389—591no—
334#389C2×D5911yescharacteristic
335#388—21no—
336#388—21no—
337#388—1181no—
338#387—21no—
339#387—1181no—
340#384—591no—
341#380—591no—
342#380—591no—
343C4×C4⋊C4—21no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4C2×D5911yescharacteristic
346C4×C22⋊C4C2×D5911yescharacteristic
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#336—591no—
356#336—591no—
357C59#2706011yescharacteristic, complemented, Sylow
358C22×Q8C22×D5911yescharacteristic
359C23×C4C22×D5911yescharacteristic
360C22×C8—591no—
361C22×C8—591no—
362C22×C8—591no—
363C22×C8—591no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C4×Q8—22no—
367C4×Q8—41no—
368#117—22no—
369#117—41no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1181no—
387C2×C22⋊C4—1181no—
388C2×C22⋊C4C22⋊D5911yescharacteristic
389C2×C22⋊C4C22⋊D5911yescharacteristic
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—1181no—
394C2×C42C22×D5911yescharacteristic
395C2×C42C22×D5911yescharacteristic
396#107—2361no—
397#106—1182no—
398#106—1182no—
399#106—1182no—
400C4⋊C8—1182no—
401#103—1182no—
402#103—1182no—
403#103—1182no—
404#103—2361no—
405C22⋊C8—1182no—
406C22⋊C8—2361no—
407#97—2361no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95C22⋊D5911yescharacteristic
418#95C22⋊D5911yescharacteristic
419C24C2×C22⋊D5911yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8#1174568512yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4C2×C22⋊D5911yescharacteristic
441C22×C4C2×C22⋊D5911yescharacteristic
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C2×C22⋊D5911yescharacteristic
445C22×C4C23×D5911yescharacteristic
446C2×C8—592no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—4721no—
474C4⋊C4—4721no—
475C4⋊C4—4721no—
476C4⋊C4—2362no—
477C4⋊C4#1174568512yes—
478C4⋊C4#1174568712yes—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—41no—
482C22⋊C4—42no—
483C22⋊C4—4721no—
484C22⋊C4—4721no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—4721no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C22×C22⋊D5911yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4#1427375712yes—
548C2×C4#41×D5911yescharacteristic
549C2×C4C2×#1174568711yescharacteristic
550C2×C4C2×#1174568511yescharacteristic
551C2×C4C2×#1174568511yescharacteristic
552C2×C4#1427377211yescharacteristic
553C2×C4#1427377411yescharacteristic
554C8—1182no—
555C8—2361no—
556C8—2361no—
557C8—2361no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22#1683791311yescharacteristic
566C22#1683804611yescharacteristic
567C22#1683805111yescharacteristic
568C22C2×#1427375711yescharacteristic
569C22#1683820011yescharacteristic
570C22#1683821011yescharacteristic
571C22#1683821011yescharacteristic
572C4—22no—
573C4—41no—
574C4—42no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—41no—
579C4—42no—
580C4—81no—
581C4—4721no—
582C4—4721no—
583C4—4721no—
584C4—4721no—
585C4#1683793211yescharacteristic
586C4#1683799411yescharacteristic
587C4#1683810211yescharacteristic
588C4#1683812211yescharacteristic
589C2—81no—
590C2#2142812611yescharacteristic
591C2#2143425811yescharacteristic
592C2#2143438711yescharacteristic
593C2#2143628211yescharacteristic
594C2#2143641211yescharacteristic
595C2#2143655111yescharacteristic
596C2#2143664111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2362C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2362D59
hypocenterC59#27060
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #21425504C2semidirect
3. #21425491C2semidirect
6. #21424859C2semidirect
7. #21424836C2semidirect
9. #21424681C2semidirect
10. #21424678C2semidirect
11. #21424364C2semidirect
14. #21423926C2semidirect
357. C59#27060semidirect

Maximal subgroups

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

subgroupindex
#214285292
#214255042
#214254912
#214252892
#214248592
#214248362
#214247112
#214246812
#214246782
#214243642
#214242952
#214241912
#214239262
C59×#23622
#2706059