Finite Group Databasebeta

← #28593032

Subgroups — #28593032

Order 15104.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426574C211yesmaximal, characteristic, complemented
2#21426561C211yesmaximal, characteristic
3#21424979C211yesmaximal, characteristic
4#21424952C211yesmaximal, characteristic
5#21424835C211yesmaximal, characteristic, complemented
6#21424830C211yesmaximal, characteristic, complemented
7#21424795C211yesmaximal, characteristic
8#21424788C211yesmaximal, characteristic, complemented
9#21424781C211yesmaximal, characteristic
10#21424774C211yesmaximal, characteristic, complemented
11#21423993C211yesmaximal, characteristic, complemented
12#21423926C211yesmaximal, characteristic, complemented
13#21423921C211yesmaximal, characteristic
14#97×Dic59C211yesmaximal, characteristic, complemented
15C2×C59×#440C211yesmaximal, characteristic
16C2×#14273686—21no—
17C2×#14273661C2211yescharacteristic
18C2×#14273659C2211yescharacteristic
19C2×#14273651C2211yescharacteristic
20C2×#14273651C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273638C2211yescharacteristic
24C2×#14273637C2211yescharacteristic
25C2×#14273637C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27C2×C8×Dic59C2211yescharacteristic
28#16837309C2211yescharacteristic
29#16837309C2211yescharacteristic
30#16837305C2211yescharacteristic
31#16837305C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837301C2211yescharacteristic
34#16837301C2211yescharacteristic
35#16837301C2211yescharacteristic
36#16837295—21no—
37#16837295—21no—
38#16837288—21no—
39#16837288—21no—
40C4×C4⋊Dic59—21no—
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×C22⋊Dic59—21no—
44#16837248—21no—
45C42×Dic59C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52C22×SD16×C59C2211yescharacteristic
53C2×C4×Q8×C59C2211yescharacteristic
54C2×C4×D4×C59C2211yescharacteristic
55C59×#440C418yescomplemented
56C2×#107×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#102×C59C2211yescharacteristic
59C2×#97×C59C2211yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62#14273686—44no—
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C22⋊Dic59—21no—
72C2×C22⋊Dic59—21no—
73C2×C22⋊Dic59—21no—
74C2×C22⋊Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59D411yescharacteristic
78C2×C4×Dic59D411yescharacteristic
79C2×C4×Dic59D411yescharacteristic
80C2×C4×Dic59D411yescharacteristic
81C2×C4×Dic59C2311yescharacteristic
82C2×C4×Dic59C2311yescharacteristic
83#14273661—22no—
84#14273659—22no—
85#14273651—22no—
86#14273651—22no—
87#14273648—22no—
88#14273648—22no—
89#14273638—22no—
90#14273637—22no—
91#14273637—22no—
92C8×Dic59—22no—
93C8×Dic59—22no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C22×Q8×C59C2311yescharacteristic
105C22×D4×C59C2311yescharacteristic
106C23×C4×C59—21no—
107C2×SD16×C59—22no—
108C2×SD16×C59—22no—
109C2×SD16×C59C2×C414yes—
110C22×C8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C4×Q8×C59—22no—
113C4×Q8×C59C2×C414yes—
114C4×D4×C59—22no—
115C4×D4×C59C2×C414yes—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C42×C59—21no—
121C2×C42×C59C2311yescharacteristic
122#107×C59C2×C414yes—
123#103×C59C2×C414yes—
124#102×C59C2×C414yes—
125#97×C59C2×C414yes—
126C24×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59C22×C412yes—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59C22×C412yes—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—24no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2411yescharacteristic
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156SD16×C59—28no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59C22×C412yes—
160C2×C8×C59C22×C412yes—
161C2×C8×C59#4112yes—
162C2×C8×C59#4112yes—
163C4⋊C4×C59—24no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59C22×C412yes—
167C4⋊C4×C59C22×C412yes—
168C22⋊C4×C59—24no—
169C42×C59—24no—
170C42×C59C22×C412yes—
171C42×C59#4112yes—
172C22⋊Dic59—42no—
173C22⋊Dic59—42no—
174C22⋊Dic59—42no—
175C22⋊Dic59—42no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—42no—
185C4⋊Dic59—42no—
186C4⋊Dic59—42no—
187C4⋊Dic59—42no—
188C4×Dic59—21no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—22no—
193C4×Dic59—22no—
194C4×Dic59—41no—
195C4×Dic59—41no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8#4112yes—
203C2×C59⋊C8#4112yes—
204C23×C59—22no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59C22×D411yescharacteristic
212Q8×C59—22no—
213Q8×C59—22no—
214Q8×C59—41no—
215Q8×C59—41no—
216D4×C59—22no—
217D4×C59—22no—
218D4×C59—41no—
219D4×C59—41no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—48no—
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×C59C4×D412yes—
235C2×C4×C59C4×D412yes—
236C2×C4×C59C23×C411yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
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×Dic59AGL(1,ℤ/8)12yes—
265C2×Dic59#13712yes—
266C8×C59—22no—
267C8×C59#11714yescomplemented
268C59⋊C8—22no—
269C59⋊C8—22no—
270#21288—591nomaximal, Sylow
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59—81no—
277C22×C59—81no—
278C22×C59C2×C4×D411yescharacteristic
279C22×C59#53211yescharacteristic
280C22×C59#53211yescharacteristic
281C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C59C2×#13711yescharacteristic
283C22×C59#57511yescharacteristic
284C22×C59#57511yescharacteristic
285C4×C59—22no—
286C4×C59—22no—
287C4×C59—42no—
288C4×C59—42no—
289C4×C59—42no—
290C4×C59C2×#11711yescharacteristic
291C4×C59C4×#4111yescharacteristic
292C4×C59#52911yescharacteristic
293C4×C59#52911yescharacteristic
294Dic59—22no—
295Dic59—22no—
296Dic59—41no—
297Dic59—41no—
298Dic59—81no—
299Dic59—81no—
300Dic59—81no—
301Dic59—81no—
302C2×#440D5911yescharacteristic
303#2295—591no—
304#2293—591no—
305#1973—591no—
306#1968—591no—
307#1943—591no—
308#1942—591no—
309#1936—591no—
310#1935—591no—
311#1934—591no—
312#1933—591no—
313#1776—591no—
314#1763—591no—
315#1762—591no—
316C4×#97—591no—
317C2×C59—82no—
318C2×C59#229311yescharacteristic
319C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
320C2×C59C4×#13711yescharacteristic
321C2×C59#318011yescharacteristic
322C2×C59#318111yescharacteristic
323C2×C59#318211yescharacteristic
324C2×C59#318311yescharacteristic
325C22×SD16C2×D5911yescharacteristic
326C2×C4×Q8C2×D5911yescharacteristic
327C2×C4×D4C2×D5911yescharacteristic
328C2×#117—1181no—
329#440Dic5918yescomplemented
330C2×#107—591no—
331C2×#107C2×D5911yescharacteristic
332C2×#106—591no—
333C2×#103—591no—
334C2×#103—591no—
335C2×#103C2×D5911yescharacteristic
336C2×#102—591no—
337C2×#102—591no—
338C2×#102C2×D5911yescharacteristic
339C2×#97—591no—
340C2×#97—591no—
341C2×#97—591no—
342C2×#97C2×D5911yescharacteristic
343C2×C4×C8—591no—
344C2×C4×C8—591no—
345#391—591no—
346#391—591no—
347#390—591no—
348#390—591no—
349#389—591no—
350#389—591no—
351#389—591no—
352#389—591no—
353#388—1181no—
354#388—1181no—
355#387—1181no—
356#387—1181no—
357C4×C4⋊C4—591no—
358C4×C4⋊C4—591no—
359C4×C4⋊C4—1181no—
360C4×C22⋊C4—1181no—
361#376—1181no—
362C43—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369C59#2128811yescharacteristic, complemented, Sylow
370C22×Q8C22×D5911yescharacteristic
371C22×D4C22×D5911yescharacteristic
372C23×C4—21no—
373C2×SD16—22no—
374C2×SD16—22no—
375C2×SD16C2×Dic5914yes—
376C22×C8—591no—
377C22×C8—591no—
378C22×C8C22×D5911yescharacteristic
379C22×C8C22×D5911yescharacteristic
380C4×Q8—22no—
381C4×Q8C2×Dic5914yes—
382C4×D4—22no—
383C4×D4C2×Dic5914yes—
384#117—2364no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C42—21no—
402C2×C42—591no—
403C2×C42—591no—
404C2×C42—591no—
405C2×C42—591no—
406C2×C42—591no—
407C2×C42—591no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42C22×D5911yescharacteristic
411#107—1182no—
412#107C2×Dic5914yes—
413#106—1182no—
414#103—1182no—
415#103—1182no—
416#103C2×Dic5914yes—
417#102—1182no—
418#102—1182no—
419#102C2×Dic5914yes—
420#97—1182no—
421#97—1182no—
422#97—1182no—
423#97C2×Dic5914yes—
424C4×C8—1182no—
425C4×C8—1182no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C22×Dic5912yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C22×Dic5912yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—591no—
458C22×C4—591no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4D4×D5911yescharacteristic
464C22×C4D4×D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466SD16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—592no—
470C2×C8—592no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8#1174568512yes—
476C2×C8#1174568512yes—
477C2×C8C22×Dic5912yes—
478C2×C8C22×Dic5912yes—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—2362no—
491C4⋊C4—2362no—
492C4⋊C4—2362no—
493C4⋊C4—2362no—
494C4⋊C4C22×Dic5912yes—
495C4⋊C4C22×Dic5912yes—
496C22⋊C4—24no—
497C22⋊C4—2362no—
498C22⋊C4—2362no—
499C22⋊C4—2362no—
500C22⋊C4—2362no—
501C42—24no—
502C42—1182no—
503C42—1182no—
504C42—1181no—
505C42—1181no—
506C42—1181no—
507C42—1181no—
508C42—2361no—
509C42—2361no—
510C42—2362no—
511C42—2362no—
512C42#1174568712yes—
513C42C22×Dic5912yes—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D5911yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—592no—
545C2×C4—592no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4D4×Dic5912yes—
569C2×C4D4×Dic5912yes—
570C2×C4C23×Dic5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C2×C4#41×D5911yescharacteristic
573C2×C4C2×#1174568711yescharacteristic
574C2×C4C2×#1174568511yescharacteristic
575C2×C4C2×#1174568511yescharacteristic
576C8—22no—
577C8—1182no—
578C8—1182no—
579C8#1427368614yescomplemented
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×D4×Dic5911yescharacteristic
588C22#1683794911yescharacteristic
589C22#1683794911yescharacteristic
590C22AGL(1,ℤ/8)×D5911yescharacteristic
591C22#137×D5911yescharacteristic
592C22#1683819811yescharacteristic
593C22#1683820111yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—1182no—
600C4—1182no—
601C4—4721no—
602C4—4721no—
603C4—4721no—
604C4—4721no—
605C4—2361no—
606C4—2361no—
607C4C2×#1427368611yescharacteristic
608C4#41×Dic5911yescharacteristic
609C4#1683793411yescharacteristic
610C4#1683793411yescharacteristic
611C2—82no—
612C2#2142656111yescharacteristic
613C2AGL(1,ℤ/8)×Dic5911yescharacteristic
614C2#137×Dic5911yescharacteristic
615C2#2143479011yescharacteristic
616C2#2143480111yescharacteristic
617C2#2143481111yescharacteristic
618C2#2143482111yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#440C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenterC2×#440D59
hypocenterC59#21288
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #21426574C2semidirect
5. #21424835C2semidirect
6. #21424830C2semidirect
8. #21424788C2semidirect
10. #21424774C2semidirect
11. #21423993C2semidirect
12. #21423926C2semidirect
14. #97×Dic59C2semidirect
55. C59×#440C4semidirect
267. C8×C59#117semidirect
329. #440Dic59semidirect
369. C59#21288semidirect
579. C8#14273686semidirect

Maximal subgroups

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

subgroupindex
#214265742
#214265612
#214249792
#214249522
#214248352
#214248302
#214247952
#214247882
#214247812
#214247742
#214239932
#214239262
#214239212
#97×Dic592
C2×C59×#4402
#2128859