Finite Group Databasebeta

← #28657164

Subgroups — #28657164

Order 15104.

736 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428470C211yesmaximal, characteristic
2#21425418C211yesmaximal, characteristic, complemented
3#21425289C211yesmaximal, characteristic
4#21425237C211yesmaximal, characteristic, complemented
5#21425104C211yesmaximal, characteristic
6#21425044C211yesmaximal, characteristic, complemented
7#21424785C211yesmaximal, characteristic, complemented
8#21424730C211yesmaximal, characteristic, complemented
9#21424711C211yesmaximal, characteristic
10#21424678C211yesmaximal, characteristic, complemented
11#21424437C211yesmaximal, characteristic, complemented
12#21424408C211yesmaximal, characteristic
13#21424191C211yesmaximal, characteristic
14#21423926C211yesmaximal, characteristic, complemented
15C59×#2362C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C2×#14273727C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30#16837344—21no—
31#16837314—21no—
32#16837308C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837306—21no—
35#16837295—21no—
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×Dic118—21no—
60C22×Dic118C2311yescharacteristic
61C2×#11745672—22no—
62C2×#11745672—41no—
63C2×#11745672—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273727—22no—
69#14273727—41no—
70#14273706—22no—
71#14273706—41no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C4×Dic59—21no—
80#14273658—22no—
81C4⋊(C59⋊C8)—41no—
82#14273651—22no—
83#14273651—22no—
84#14273651—41no—
85#14273650—22no—
86#14273650—22no—
87#14273650—41no—
88C22⋊(C59⋊C8)—41no—
89C22⋊(C59⋊C8)—41no—
90#14273638—41no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×Q8×C59C2311yescharacteristic
98C23×C4×C59C2311yescharacteristic
99C22⋊Q8×C59—22no—
100C22⋊Q8×C59—22no—
101C4×Q8×C59—22no—
102C4×Q8×C59—41no—
103#117×C59—22no—
104#117×C59—41no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59—21no—
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×C59C2×D411yescharacteristic
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59#4112yes—
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—21no—
142C22×C4×C59—21no—
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—42no—
160C2×Dic118—42no—
161C2×Dic118—81no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59#4112yes—
170C4⋊C4×C59#4112yes—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—41no—
174C22⋊C4×C59—42no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—42no—
180C42×C59—42no—
181C42×C59—42no—
182C22⋊Dic59—42no—
183C22⋊Dic59—81no—
184#11745672—44no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—42no—
191C4⋊Dic59#4112yes—
192Dic59⋊C4—42no—
193C4×Dic59—81no—
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—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59C22×D411yescharacteristic
214Q8×C59—22no—
215Q8×C59—22no—
216Q8×C59—41no—
217Q8×C59—41no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
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—21no—
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—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59#13712yes—
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59H(5,𝔽2)11yescharacteristic
254C2×C4×C59H(5,𝔽2)11yescharacteristic
255C2×Dic59—22no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267Dic118—41no—
268Dic118—41no—
269Dic118—41no—
270Dic118—41no—
271Dic118—84no—
272C59⋊C8—41no—
273C59⋊C8—41no—
274C59⋊C8—41no—
275C59⋊C8—41no—
276#26750—591nomaximal, Sylow
277C22×C59—41no—
278C22×C59—41no—
279C22×C59—41no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59#52511yescharacteristic
285C22×C59#54611yescharacteristic
286C22×C59#54611yescharacteristic
287C22×C59C2×#13711yescharacteristic
288C22×C59#57511yescharacteristic
289C22×C59#57711yescharacteristic
290C22×C59#57811yescharacteristic
291C4×C59—22no—
292C4×C59—41no—
293C4×C59—42no—
294C4×C59—42no—
295C4×C59—41no—
296C4×C59—42no—
297C4×C59—41no—
298C4×C59—41no—
299C4×C59—81no—
300C4×C59#52911yescharacteristic
301C4×C59#53811yescharacteristic
302C4×C59#55511yescharacteristic
303C4×C59#55911yescharacteristic
304Dic59—42no—
305Dic59—81no—
306Dic59—81no—
307Dic59—81no—
308#2492—591no—
309#2362D5911yescharacteristic
310#2056—591no—
311#2035—591no—
312#2024—591no—
313#2001—591no—
314#1987—591no—
315#1935—591no—
316#1927—591no—
317#1924—591no—
318#1920—591no—
319#1878—591no—
320#1873—591no—
321#1825—591no—
322#1763—591no—
323C2×C59—81no—
324C2×C59#245911yescharacteristic
325C2×C59#311611yescharacteristic
326C2×C59#312111yescharacteristic
327C2×C59#331111yescharacteristic
328C2×C59#331211yescharacteristic
329C2×C59#331311yescharacteristic
330C2×C59#333111yescharacteristic
331C22×Q16—591no—
332C2×C4⋊Q8—591no—
333C2×C22⋊Q8—591no—
334C2×C22⋊Q8C2×D5911yescharacteristic
335C2×C4×Q8C2×D5911yescharacteristic
336C2×#117C2×D5911yescharacteristic
337C2×#106—591no—
338C2×C4⋊C8—591no—
339C2×#103—591no—
340C2×#103—591no—
341C2×#103—591no—
342C2×#103—591no—
343C2×#103—591no—
344C2×#103—591no—
345C2×C22⋊C8—591no—
346C2×C22⋊C8—591no—
347C2×#97—591no—
348#400—1181no—
349#393—1181no—
350#391—591no—
351#391—591no—
352#391—1181no—
353#391C2×D5911yescharacteristic
354#389C2×D5911yescharacteristic
355#388—21no—
356#388—21no—
357#388—1181no—
358#387—21no—
359#380—591no—
360C4×C4⋊C4—21no—
361C4×C4⋊C4—591no—
362C4×C4⋊C4C2×D5911yescharacteristic
363C4×C22⋊C4C2×D5911yescharacteristic
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#336—591no—
373#336—591no—
374C59#2675011yescharacteristic, complemented, Sylow
375C22×Q8—591no—
376C22×Q8—1181no—
377C22×Q8C22×D5911yescharacteristic
378C23×C4C22×D5911yescharacteristic
379C2×Q16—1182no—
380C2×Q16—2361no—
381C2×Q16—2361no—
382C22×C8—591no—
383C22×C8—591no—
384C22×C8—591no—
385C22×C8—591no—
386C4⋊Q8—1182no—
387C4⋊Q8—2361no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—1182no—
391C22⋊Q8—2361no—
392C4×Q8—22no—
393C4×Q8—41no—
394#117—22no—
395#117—41no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1181no—
410C2×C22⋊C4—1181no—
411C2×C22⋊C4C22⋊D5911yescharacteristic
412C2×C22⋊C4C22⋊D5911yescharacteristic
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—1181no—
417C2×C42C22×D5911yescharacteristic
418C2×C42C22×D5911yescharacteristic
419#106—1182no—
420C4⋊C8—2361no—
421#103—1182no—
422#103—1182no—
423#103—1182no—
424#103—1182no—
425#103—2361no—
426#103—2361no—
427C22⋊C8—2361no—
428C22⋊C8—2361no—
429#97—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#95—1181no—
440#95C22⋊D5911yescharacteristic
441#95C22⋊D5911yescharacteristic
442C24C2×C22⋊D5911yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C2×Q8—1181no—
453C2×Q8—1181no—
454C2×Q8—4721no—
455C2×Q8—2362no—
456C2×Q8—2362no—
457C2×Q8#1174568512yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C23×D5911yescharacteristic
478Q16—2364no—
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—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—592no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—2362no—
505C4⋊C4—2362no—
506C4⋊C4#1174568512yes—
507C4⋊C4#1174568712yes—
508C22⋊C4—24no—
509C22⋊C4—22no—
510C22⋊C4—41no—
511C22⋊C4—42no—
512C22⋊C4—4721no—
513C22⋊C4—2362no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—4721no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23C22×C22⋊D5911yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—4724no—
535Q8—2361no—
536Q8—2361no—
537Q8—2361no—
538Q8—2361no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
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—22no—
555C2×C4—21no—
556C2×C4—41no—
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—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—1182no—
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—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4#1427375712yes—
582C2×C4#41×D5911yescharacteristic
583C2×C4C2×#1174568711yescharacteristic
584C2×C4C2×#1174568511yescharacteristic
585C2×C4C2×#1174568511yescharacteristic
586C2×C4#1427377211yescharacteristic
587C2×C4#1427377311yescharacteristic
588C8—2361no—
589C8—2361no—
590C8—2361no—
591C8—2361no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1683791311yescharacteristic
600C22#1683804011yescharacteristic
601C22#1683804611yescharacteristic
602C22C2×#1427375711yescharacteristic
603C22#1683820011yescharacteristic
604C22#1683821011yescharacteristic
605C22#1683821711yescharacteristic
606C4—22no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—81no—
615C4—4721no—
616C4—4721no—
617C4—4721no—
618C4—2362no—
619C4#1683793211yescharacteristic
620C4#1683798511yescharacteristic
621C4#1683811711yescharacteristic
622C4#1683813511yescharacteristic
623C2—81no—
624C2#2142802711yescharacteristic
625C2#2143425811yescharacteristic
626C2#2143432511yescharacteristic
627C2#2143612611yescharacteristic
628C2#2143614111yescharacteristic
629C2#2143616211yescharacteristic
630C2#2143641211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2362C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2362D59
hypocenterC59#26750
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. #21425418C2semidirect
4. #21425237C2semidirect
6. #21425044C2semidirect
7. #21424785C2semidirect
8. #21424730C2semidirect
10. #21424678C2semidirect
11. #21424437C2semidirect
14. #21423926C2semidirect
374. C59#26750semidirect

Maximal subgroups

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

subgroupindex
#214284702
#214254182
#214252892
#214252372
#214251042
#214250442
#214247852
#214247302
#214247112
#214246782
#214244372
#214244082
#214241912
#214239262
C59×#23622
#2675059