Finite Group Databasebeta

← #28661850

Subgroups — #28661850

Order 15104.

848 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428671C211yesmaximal, characteristic, complemented
2#21425629C211yesmaximal, characteristic, complemented
3#21425413C211yesmaximal, characteristic, complemented
4#21425327C211yesmaximal, characteristic
5#21425184C211yesmaximal, characteristic
6#21424949C211yesmaximal, characteristic, complemented
7#21424730C211yesmaximal, characteristic, complemented
8#21424720C211yesmaximal, characteristic, complemented
9#21424681C211yesmaximal, characteristic, complemented
10#21424655C211yesmaximal, characteristic, complemented
11#21424430C211yesmaximal, characteristic, complemented
12#21424430C211yesmaximal, characteristic, complemented
13#21424096C211yesmaximal, characteristic, complemented
14C4×#14273648C211yesmaximal, characteristic, complemented
15C59×#2366C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C22×#11745670C2211yescharacteristic
18C2×#14273658C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273650C2211yescharacteristic
22C2×#14273650C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837344—21no—
31#16837341—21no—
32#16837308C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837308C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837288—21no—
37#16837285—21no—
38#16837272C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#127×C59C2211yescharacteristic
49C2×C4×D4×C59—21no—
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C59×#390—21no—
53C59×#390C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#386—21no—
56C59×#386—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C2×#11745670—22no—
61C2×#11745670—22no—
62C2×#11745670—22no—
63C2×#11745670—41no—
64C2×#11745670—41no—
65C2×#11745670—41no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×C4×Dic59—21no—
77#14273658—22no—
78#14273658—41no—
79C4⋊(C59⋊C8)—41no—
80#14273650—22no—
81#14273650—41no—
82#14273648—22no—
83#14273648—22no—
84#14273648—22no—
85#14273648—41no—
86#14273648—41no—
87#14273648—41no—
88C4×C59⋊C8—41no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×D4×C59—21no—
98C22×D4×C59—21no—
99C22×D4×C59C2311yescharacteristic
100C22×D4×C59C2311yescharacteristic
101C23×C4×C59—21no—
102C23×C4×C59—21no—
103#127×C59—22no—
104#127×C59—22no—
105C4×D4×C59—22no—
106C4×D4×C59—22no—
107C4×D4×C59—41no—
108C4×D4×C59—41no—
109C4×D4×C59—44no—
110C2×C4⋊C4×C59D411yescharacteristic
111C2×C4⋊C4×C59D411yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C42×C59—21no—
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#95×C59—21no—
125#95×C59D411yescharacteristic
126#95×C59D411yescharacteristic
127C24×C59—21no—
128C24×C59—21no—
129C2×D4×C59—24no—
130C2×D4×C59—24no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59—21no—
141C2×D4×C59—42no—
142C2×D4×C59—42no—
143C2×D4×C59—81no—
144C2×D4×C59—81no—
145C2×D4×C59#4112yes—
146C2×D4×C59#4112yes—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—42no—
150C22×C4×C59—42no—
151C22×C4×C59—42no—
152C22×C4×C59—42no—
153C22×C4×C59—81no—
154C22×C4×C59—81no—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2411yescharacteristic
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C22×Dic59—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—41no—
176C4⋊C4×C59SD1614yescomplemented
177C4⋊C4×C59#4112yes—
178C22⋊C4×C59—42no—
179C22⋊C4×C59—42no—
180C22⋊C4×C59—42no—
181C22⋊C4×C59—42no—
182C22⋊C4×C59—81no—
183C22⋊C4×C59—81no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—42no—
191#11745670—44no—
192#11745670—44no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—42no—
200C4⋊Dic59—81no—
201Dic59⋊C4—81no—
202C4×Dic59—81no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8#4112yes—
215C23×C59—22no—
216C23×C59—22no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59C22×D411yescharacteristic
230D4×C59—22no—
231D4×C59—22no—
232D4×C59—22no—
233D4×C59—22no—
234D4×C59—48no—
235D4×C59—41no—
236D4×C59—41no—
237D4×C59—41no—
238D4×C59—41no—
239D4×C59—84no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—22no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—84no—
262C2×C4×C59—84no—
263C2×C4×C59C2×SD1612yes—
264C2×C4×C59C2×SD1612yes—
265C2×C4×C59C2×SD1612yes—
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59C2×#4111yescharacteristic
270C2×C4×C59H(5,𝔽2)11yescharacteristic
271C2×C4×C59H(5,𝔽2)11yescharacteristic
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284Dic118—41no—
285Dic118—41no—
286Dic118—41no—
287Dic118—41no—
288C59⋊C8—22no—
289C59⋊C8—41no—
290C59⋊C8—41no—
291C59⋊C8—41no—
292#27064—591nomaximal, Sylow
293C22×C59—42no—
294C22×C59—42no—
295C22×C59—42no—
296C22×C59—42no—
297C22×C59—42no—
298C22×C59—42no—
299C22×C59—81no—
300C22×C59—81no—
301C22×C59—81no—
302C22×C59—81no—
303C22×C59—81no—
304C22×C59—81no—
305C22×C59—81no—
306C22×C59—81no—
307C22×C59#52511yescharacteristic
308C22×C59#54611yescharacteristic
309C22×C59#55011yescharacteristic
310C22×C59C22×SD1611yescharacteristic
311C22×C59#57511yescharacteristic
312C22×C59#57611yescharacteristic
313C22×C59#57711yescharacteristic
314C4×C59—22no—
315C4×C59—22no—
316C4×C59—22no—
317C4×C59—41no—
318C4×C59—42no—
319C4×C59—41no—
320C4×C59—41no—
321C4×C59—81no—
322C4×C59#52911yescharacteristic
323C4×C59#53811yescharacteristic
324C4×C59#55111yescharacteristic
325C4×C59#55311yescharacteristic
326Dic59—81no—
327Dic59—81no—
328Dic59—81no—
329Dic59—81no—
330#2511—591no—
331#2366D5911yescharacteristic
332#2094—591no—
333#2055—591no—
334#2041—591no—
335#2014—591no—
336#1968—591no—
337#1927—591no—
338#1926—591no—
339#1920—591no—
340#1916—591no—
341#1877—591no—
342#1877—591no—
343#1802—591no—
344C4×#102—591no—
345C2×C59—82no—
346C2×C59—82no—
347C2×C59#246711yescharacteristic
348C2×C59#310111yescharacteristic
349C2×C59#311811yescharacteristic
350C2×C59#329511yescharacteristic
351C2×C59#331711yescharacteristic
352C2×C59#339211yescharacteristic
353C2×C59#340311yescharacteristic
354C22×SD16—591no—
355C22×SD16—591no—
356C2×#127C2×D5911yescharacteristic
357C2×C4×D4—21no—
358C2×C4×D4C2×D5911yescharacteristic
359C2×C4×D4C2×D5911yescharacteristic
360C2×#106—591no—
361C2×#106—591no—
362C2×C4⋊C8—591no—
363C2×#103—591no—
364C2×#103—591no—
365C2×#102—591no—
366C2×#102—591no—
367C2×#102—591no—
368C2×#102—591no—
369C2×#102—591no—
370C2×#102—591no—
371C2×C4×C8—591no—
372#400—1181no—
373#400—1181no—
374#391—591no—
375#391—591no—
376#391—591no—
377#390—21no—
378#390C2×D5911yescharacteristic
379#390C2×D5911yescharacteristic
380#389—591no—
381#387—1181no—
382#387—1181no—
383#386—21no—
384#386—21no—
385#384—591no—
386C4×C4⋊C4—591no—
387C4×C4⋊C4C2×D5911yescharacteristic
388C4×C4⋊C4C2×D5911yescharacteristic
389#340—591no—
390#340—591no—
391#340—591no—
392#340—591no—
393#336—591no—
394#336—591no—
395#336—591no—
396#336—591no—
397C59#2706411yescharacteristic, complemented, Sylow
398C22×Q8—591no—
399C22×D4—21no—
400C22×D4—21no—
401C22×D4C22×D5911yescharacteristic
402C22×D4C22×D5911yescharacteristic
403C23×C4—21no—
404C23×C4—21no—
405C2×SD16—1182no—
406C2×SD16—1182no—
407C2×SD16—1182no—
408C2×SD16—2361no—
409C2×SD16—2361no—
410C2×SD16—2361no—
411C22×C8—591no—
412C22×C8—591no—
413C22×C8—591no—
414C22×C8—591no—
415#127—22no—
416#127—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—41no—
420C4×D4—41no—
421C4×D4—44no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4C22⋊D5911yescharacteristic
429C2×C4⋊C4C22⋊D5911yescharacteristic
430C2×C4⋊C4C22×D5911yescharacteristic
431C2×C4⋊C4C22×D5911yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C42—21no—
439C2×C42—1181no—
440C2×C42C22×D5911yescharacteristic
441C2×C42C22×D5911yescharacteristic
442C2×C42C22×D5911yescharacteristic
443#106—1182no—
444#106—2361no—
445C4⋊C8—2361no—
446#103—1182no—
447#103—2361no—
448#102—1182no—
449#102—1182no—
450#102—1182no—
451#102—2361no—
452#102—2361no—
453#102—2361no—
454C4×C8—2361no—
455#95—21no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464#95C22⋊D5911yescharacteristic
465#95C22⋊D5911yescharacteristic
466C24—21no—
467C24—21no—
468C2×Q8—1181no—
469C2×Q8—1181no—
470C2×Q8—1181no—
471C2×Q8—1181no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—42no—
487C2×D4—42no—
488C2×D4—81no—
489C2×D4—81no—
490C2×D4#1174568512yes—
491C2×D4#1174568512yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—1181no—
504C22×C4C2×C22⋊D5911yescharacteristic
505C22×C4C2×C22⋊D5911yescharacteristic
506C22×C4C2×C22⋊D5911yescharacteristic
507C22×C4C2×C22⋊D5911yescharacteristic
508C22×C4C2×C22⋊D5911yescharacteristic
509C22×C4C2×C22⋊D5911yescharacteristic
510C22×C4C23×D5911yescharacteristic
511SD16—2364no—
512SD16—2364no—
513C2×C8—592no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—41no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—1181no—
534C4⋊C4—1181no—
535C4⋊C4—4721no—
536C4⋊C4—4721no—
537C4⋊C4—2362no—
538C4⋊C4#1174567014yescomplemented
539C4⋊C4#1174568712yes—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—81no—
545C22⋊C4—81no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—42no—
553C42—4721no—
554C23—22no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23C22×C22⋊D5911yescharacteristic
569Q8—2361no—
570Q8—2361no—
571Q8—2361no—
572Q8—2361no—
573D4—22no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4—48no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—84no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4—2361no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4C2×#1174567012yes—
619C2×C4C2×#1174567012yes—
620C2×C4C2×#1174567012yes—
621C2×C4#41×D5911yescharacteristic
622C2×C4C2×#1174568711yescharacteristic
623C2×C4C2×#1174568511yescharacteristic
624C2×C4C2×#1174568511yescharacteristic
625C2×C4#1427377211yescharacteristic
626C2×C4#1427377311yescharacteristic
627C8—1182no—
628C8—2361no—
629C8—2361no—
630C8—2361no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22#1683791311yescharacteristic
646C22#1683804611yescharacteristic
647C22#1683807111yescharacteristic
648C22C22×#1174567011yescharacteristic
649C22#1683820011yescharacteristic
650C22#1683820511yescharacteristic
651C22#1683821311yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—4721no—
661C4—4721no—
662C4—4721no—
663C4—4721no—
664C4#1683793211yescharacteristic
665C4#1683798511yescharacteristic
666C4#1683808111yescharacteristic
667C4#1683809811yescharacteristic
668C2—82no—
669C2—82no—
670C2#2142814111yescharacteristic
671C2#2143410411yescharacteristic
672C2#2143428011yescharacteristic
673C2#2143592011yescharacteristic
674C2#2143622211yescharacteristic
675C2#2143707711yescharacteristic
676C2#2143718011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2366C2
socleC23×C59C22×D4
prime radicalC2×#127×C59C22
hypercenter#2366D59
hypocenterC59#27064
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. #21428671C2semidirect
2. #21425629C2semidirect
3. #21425413C2semidirect
6. #21424949C2semidirect
7. #21424730C2semidirect
8. #21424720C2semidirect
9. #21424681C2semidirect
10. #21424655C2semidirect
11. #21424430C2semidirect
12. #21424430C2semidirect
13. #21424096C2semidirect
14. C4×#14273648C2semidirect
176. C4⋊C4×C59SD16semidirect
397. C59#27064semidirect
538. C4⋊C4#11745670semidirect

Maximal subgroups

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

subgroupindex
#214286712
#214256292
#214254132
#214253272
#214251842
#214249492
#214247302
#214247202
#214246812
#214246552
#214244302
#214240962
C4×#142736482
C59×#23662
#2706459