Finite Group Databasebeta

← #28646352

Subgroups — #28646352

Order 15104.

774 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427792C211yesmaximal, characteristic
2#21426973C211yesmaximal, characteristic
3#21425657C211yesmaximal, characteristic, complemented
4#21425599C211yesmaximal, characteristic, complemented
5#21425596C211yesmaximal, characteristic, complemented
6#21425428C211yesmaximal, characteristic, complemented
7#21425046C211yesmaximal, characteristic, complemented
8#21424858C211yesmaximal, characteristic, complemented
9#21424677C211yesmaximal, characteristic, complemented
10#21424673C211yesmaximal, characteristic, complemented
11#21424289C211yesmaximal, characteristic
12#21424274C211yesmaximal, characteristic
13#21424186C211yesmaximal, characteristic
14#21424121C211yesmaximal, characteristic
15C2×C59×#340C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×#14273706—21no—
18C2×#14273659C2211yescharacteristic
19C2×#14273659C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2211yescharacteristic
25#16837335—21no—
26#16837322—21no—
27#16837301C2211yescharacteristic
28#16837301C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837298C2211yescharacteristic
31#16837288—21no—
32#16837288—21no—
33#16837275—21no—
34#16837271C2211yescharacteristic
35#16837262—21no—
36#16837262C2211yescharacteristic
37#16837262C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4×C22⋊Dic59—21no—
41C4×C22⋊Dic59C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C23×C8×C59C2211yescharacteristic
53C22×C4⋊C4×C59C2211yescharacteristic
54C22×C4⋊C4×C59C2211yescharacteristic
55C59×#340C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C22×Dic118—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62#14273726—22no—
63#14273726—41no—
64#14273706—44no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×C22⋊Dic59—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59D411yescharacteristic
79C2×C22⋊Dic59D411yescharacteristic
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59C2311yescharacteristic
83C2×C4×Dic59C2311yescharacteristic
84#14273659—22no—
85#14273659—41no—
86#14273658—22no—
87#14273658—41no—
88Dic59⋊C8—22no—
89Dic59⋊C8—41no—
90C22⋊(C59⋊C8)—22no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95D411yescharacteristic
100C59⋊#95D411yescharacteristic
101C23×C4×C59—21no—
102C23×C4×C59—21no—
103C23×C4×C59C2311yescharacteristic
104C22×C8×C59—22no—
105C22×C8×C59—22no—
106C22×C8×C59—41no—
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59—22no—
111C2×C4⋊C4×C59—22no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—41no—
114C2×C4⋊C4×C59Q814yescomplemented
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C24×C59C2×D411yescharacteristic
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—24no—
128C22×C4×C59—21no—
129C22×C4×C59—42no—
130C22×C4×C59—42no—
131C22×C4×C59—81no—
132C22×C4×C59—81no—
133C22×C4×C59—81no—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×Q812yes—
136C22×C4×C59#4112yes—
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59C2×D411yescharacteristic
143C22×Dic59C2×D411yescharacteristic
144C22×Dic59C2×D411yescharacteristic
145C22×Dic59C2×D411yescharacteristic
146C2×Dic118—42no—
147C2×Dic118—42no—
148C2×Dic118—81no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—44no—
155C2×C8×C59#4112yes—
156C4⋊C4×C59—24no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—42no—
163C4⋊C4×C59C2×Q812yes—
164C4⋊C4×C59C2×Q812yes—
165C4⋊C4×C59#4112yes—
166C22⋊Dic59—22no—
167C22⋊Dic59—42no—
168C22⋊Dic59—42no—
169C22⋊Dic59SD1614yes—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—42no—
179C4⋊Dic59—81no—
180C4⋊Dic59—81no—
181Dic59⋊C4—42no—
182Dic59⋊C4—42no—
183Dic59⋊C4—42no—
184Dic59⋊C4—81no—
185C4×Dic59—21no—
186C4×Dic59—21no—
187C4×Dic59—21no—
188C4×Dic59—21no—
189C4×Dic59—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—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59C22⋊Q812yes—
202C23×C59C2×SD1612yes—
203C23×C59#13712yes—
204C23×C59C22×D411yescharacteristic
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—22no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—84no—
224C2×C4×C59—84no—
225C2×C4×C59C22⋊Q812yes—
226C2×C4×C59C22×Q811yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59H(5,𝔽2)11yescharacteristic
231C2×C4×C592−1+411yescharacteristic
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—41no—
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×Dic59C2×SD1612yes—
255C2×Dic59C2×SD1612yes—
256Dic118—84no—
257C8×C59—22no—
258C8×C59—41no—
259C59⋊C8—41no—
260C59⋊C8—41no—
261#25992—591nomaximal, Sylow
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59#47514yes—
269C22×C59C2×C22⋊Q811yescharacteristic
270C22×C59#54611yescharacteristic
271C22×C59#54711yescharacteristic
272C22×C59C22×SD1611yescharacteristic
273C22×C59C2×#13711yescharacteristic
274C22×C59#57611yescharacteristic
275C22×C59#57711yescharacteristic
276C4×C59—22no—
277C4×C59—42no—
278C4×C59—42no—
279C4×C59—42no—
280C4×C59—81no—
281C4×C59#53311yescharacteristic
282C4×C59#54411yescharacteristic
283C4×C59#55111yescharacteristic
284C4×C59#55211yescharacteristic
285Dic59—22no—
286Dic59—22no—
287Dic59—41no—
288Dic59—41no—
289Dic59—81no—
290Dic59—81no—
291Dic59—81no—
292Dic59—81no—
293#2440—591no—
294#2345—591no—
295#2099—591no—
296#2088—591no—
297#2088—591no—
298#2058—591no—
299#1988—591no—
300#1948—591no—
301#1919—591no—
302#1919—591no—
303#1853—591no—
304#1850—591no—
305#1824—591no—
306#1807—591no—
307C2×#340D5911yescharacteristic
308C2×C59—24no—
309C2×C59#243811yescharacteristic
310C2×C59C2×#47511yescharacteristic
311C2×C59#308111yescharacteristic
312C2×C59#329511yescharacteristic
313C2×C59#332011yescharacteristic
314C2×C59#333511yescharacteristic
315C2×C59#334611yescharacteristic
316C23×C8C2×D5911yescharacteristic
317C2×C4⋊Q8—591no—
318C2×C22⋊Q8—1181no—
319C22×C4⋊C4C2×D5911yescharacteristic
320C22×C4⋊C4C2×D5911yescharacteristic
321C2×#106—591no—
322C2×#106—591no—
323C2×#106—591no—
324C2×#106—591no—
325C2×C4⋊C8—591no—
326C2×C4⋊C8—591no—
327C2×C22⋊C8—591no—
328#398—1181no—
329#395—1181no—
330#389—591no—
331#389—591no—
332#389—591no—
333#389—591no—
334#387—1181no—
335#387—1181no—
336#385—1181no—
337#384—591no—
338#380—591no—
339#380—591no—
340#380—1181no—
341C4×C4⋊C4—591no—
342C4×C4⋊C4—591no—
343C4×C22⋊C4—591no—
344C4×C22⋊C4—1181no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340C2×D5911yescharacteristic
354#340C2×D5911yescharacteristic
355#340C2×D5911yescharacteristic
356#340C2×D5911yescharacteristic
357#336—591no—
358#336—591no—
359C59#2599211yescharacteristic, complemented, Sylow
360C22×Q8—1181no—
361C23×C4—21no—
362C23×C4—21no—
363C23×C4C22×D5911yescharacteristic
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—41no—
367C22×C8—591no—
368C22×C8—591no—
369C22×C8C22×D5911yescharacteristic
370C22×C8C22×D5911yescharacteristic
371C4⋊Q8—1182no—
372C4⋊Q8—2361no—
373C22⋊Q8—2364no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4Dic11814yescomplemented
391C2×C4⋊C4C22×D5911yescharacteristic
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C22⋊C4—591no—
396C2×C22⋊C4—591no—
397C2×C22⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C42—591no—
400C2×C42—591no—
401C2×C42—1181no—
402C2×C42—1181no—
403#106—1182no—
404#106—1182no—
405#106—2361no—
406#106—2361no—
407C4⋊C8—1182no—
408C4⋊C8—2361no—
409C22⋊C8—1182no—
410#95—591no—
411#95—591no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420C24D4×D5911yescharacteristic
421C2×Q8—4721no—
422C2×Q8—2362no—
423C2×Q8—2362no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—591no—
439C22×C4—591no—
440C22×C4—591no—
441C22×C4—591no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4C2×Dic11812yes—
447C22×C4D4×D5911yescharacteristic
448C22×C4#1174568712yes—
449C22×C4C23×D5911yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—44no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8#1174568512yes—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—4721no—
479C4⋊C4—4721no—
480C4⋊C4—4721no—
481C4⋊C4—2362no—
482C4⋊C4—2362no—
483C4⋊C4—2362no—
484C4⋊C4—2362no—
485C4⋊C4C2×Dic11812yes—
486C4⋊C4C2×Dic11812yes—
487C4⋊C4#1174568512yes—
488C22⋊C4—594no—
489C22⋊C4—1182no—
490C22⋊C4—2362no—
491C22⋊C4—2362no—
492C42—1181no—
493C42—1181no—
494C42—1181no—
495C42—1181no—
496C42—4721no—
497C42—4721no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23#1427370412yes—
503C23SD16×D5912yes—
504C23#1427375912yes—
505C23C2×D4×D5911yescharacteristic
506Q8—4724no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4—592no—
528C2×C4—592no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
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—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4#1427370412yes—
552C2×C4C22×Dic11811yescharacteristic
553C2×C4C2×#1174568711yescharacteristic
554C2×C4C2×#1174568511yescharacteristic
555C2×C4C2×#1174568511yescharacteristic
556C2×C4#1427377211yescharacteristic
557C2×C4#1427377511yescharacteristic
558C8—22no—
559C8—41no—
560C8—2361no—
561C8—2361no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22#1683768514yes—
569C22C2×#1427370411yescharacteristic
570C22#1683804211yescharacteristic
571C22#1683805311yescharacteristic
572C22C2×SD16×D5911yescharacteristic
573C22C2×#1427375911yescharacteristic
574C22#1683820311yescharacteristic
575C22#1683820811yescharacteristic
576C4—22no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—1182no—
582C4—1182no—
583C4—4721no—
584C4—4721no—
585C4—4721no—
586C4—4721no—
587C4—2361no—
588C4—2361no—
589C4#1683795911yescharacteristic
590C4#1683803311yescharacteristic
591C4#1683807811yescharacteristic
592C4#1683808711yescharacteristic
593C2—24no—
594C2#2142777211yescharacteristic
595C2C2×#1683768511yescharacteristic
596C2#2143392311yescharacteristic
597C2#2143591611yescharacteristic
598C2#2143626311yescharacteristic
599C2#2143644911yescharacteristic
600C2#2143657711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#340C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenterC2×#340D59
hypocenterC59#25992
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #21425657C2semidirect
4. #21425599C2semidirect
5. #21425596C2semidirect
6. #21425428C2semidirect
7. #21425046C2semidirect
8. #21424858C2semidirect
9. #21424677C2semidirect
10. #21424673C2semidirect
114. C2×C4⋊C4×C59Q8semidirect
359. C59#25992semidirect
390. C2×C4⋊C4Dic118semidirect

Maximal subgroups

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

subgroupindex
#214277922
#214269732
#214256572
#214255992
#214255962
#214254282
#214250462
#214248582
#214246772
#214246732
#214242892
#214242742
#214241862
#214241212
C2×C59×#3402
#2599259