Finite Group Databasebeta

← #28656790

Subgroups — #28656790

Order 15104.

748 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#21428439C211yesmaximal, characteristic
2#21427105C211yesmaximal, characteristic
3#21425416C211yesmaximal, characteristic, complemented
4#21425289C211yesmaximal, characteristic
5#21425245C211yesmaximal, characteristic, complemented
6#21425050C211yesmaximal, characteristic, complemented
7#21424774C211yesmaximal, characteristic, complemented
8#21424733C211yesmaximal, characteristic, complemented
9#21424699C211yesmaximal, characteristic
10#21424684C211yesmaximal, characteristic, complemented
11#21424439C211yesmaximal, characteristic, complemented
12#21424404C211yesmaximal, characteristic
13#21424183C211yesmaximal, characteristic
14#103×Dic59C211yesmaximal, characteristic, complemented
15C59×#2001C211yesmaximal, characteristic
16C2×#14273718C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×Dic59⋊C8C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×C22⋊(C59⋊C8)C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27#16837340—21no—
28#16837327—21no—
29#16837309C2211yescharacteristic
30#16837309C2211yescharacteristic
31#16837309C2211yescharacteristic
32#16837301—21no—
33#16837301C2211yescharacteristic
34#16837296—21no—
35#16837288—21no—
36#16837277—21no—
37#16837267—21no—
38#16837262C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C22⋊C4×Dic59—21no—
42C4×C22⋊Dic59C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C22×Q16×C59C2211yescharacteristic
53C2×C22⋊Q8×C59C2211yescharacteristic
54C2×C22⋊Q8×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×#103×C59C2211yescharacteristic
57C2×C22⋊C8×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62#14273718—22no—
63#14273718—41no—
64Q8×Dic59—22no—
65Q8×Dic59—41no—
66C2×C4⋊Dic59—22no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—41no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59D411yescharacteristic
79C2×C22⋊Dic59D411yescharacteristic
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59C2311yescharacteristic
84C2×C4×Dic59C2311yescharacteristic
85#14273658—22no—
86Dic59⋊C8—41no—
87#14273651—22no—
88#14273651—22no—
89#14273651—41no—
90#14273651—41no—
91C22⋊(C59⋊C8)—41no—
92C8×Dic59—41no—
93C59⋊#95—21no—
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⋊#95D411yescharacteristic
104C59⋊#95D411yescharacteristic
105C22×Q8×C59C2311yescharacteristic
106C22×Q8×C59C2311yescharacteristic
107C23×C4×C59C2311yescharacteristic
108C2×Q16×C59—22no—
109C2×Q16×C59—41no—
110C2×Q16×C59—41no—
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C22⋊Q8×C59—22no—
114C22⋊Q8×C59—22no—
115C22⋊Q8×C59—22no—
116C22⋊Q8×C59—41no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59—21no—
123#103×C59—22no—
124#103×C59—22no—
125C22⋊C8×C59—41no—
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—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59#4112yes—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2411yescharacteristic
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—42no—
156C22×Dic59—42no—
157C22×Dic59—81no—
158C22×Dic59C2×D411yescharacteristic
159C22×Dic59C2×D411yescharacteristic
160C22×Dic59C2×D411yescharacteristic
161C22×Dic59C2×D411yescharacteristic
162Q16×C59—44no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C4⋊C4×C59—24no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59#4112yes—
174C4⋊C4×C59#4112yes—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—42no—
177C22⋊Dic59—22no—
178C22⋊Dic59—22no—
179C22⋊Dic59—42no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—42no—
188C4⋊Dic59—42no—
189C4⋊Dic59—42no—
190C4⋊Dic59—81no—
191C4⋊Dic59#4112yes—
192Dic59⋊C4—42no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—21no—
196C4×Dic59—21no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C4×Dic59—81no—
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#13512yes—
213C23×C59C22×D411yescharacteristic
214Q8×C59—22no—
215Q8×C59—22no—
216Q8×C59—41no—
217Q8×C59—41no—
218Q8×C59—41no—
219Q8×C59—41no—
220Q8×C59—41no—
221Q8×C59—41no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
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—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C59H(5,𝔽2)11yescharacteristic
246C2×C4×C59H(5,𝔽2)11yescharacteristic
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—22no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—84no—
271C2×Dic59#13712yes—
272C8×C59—41no—
273C8×C59—41no—
274C59⋊C8—41no—
275C59⋊C8—41no—
276#26725—591nomaximal, Sylow
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—22no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59#52511yescharacteristic
285C22×C59#54611yescharacteristic
286C22×C59#54611yescharacteristic
287C22×C59C2×#13511yescharacteristic
288C22×C59C2×#13711yescharacteristic
289C22×C59#57711yescharacteristic
290C22×C59#57811yescharacteristic
291C4×C59—42no—
292C4×C59—42no—
293C4×C59—41no—
294C4×C59—42no—
295C4×C59—81no—
296C4×C59#52911yescharacteristic
297C4×C59#53811yescharacteristic
298C4×C59#55511yescharacteristic
299C4×C59#55911yescharacteristic
300Dic59—22no—
301Dic59—41no—
302Dic59—41no—
303Dic59—42no—
304Dic59—41no—
305Dic59—81no—
306Dic59—81no—
307Dic59—81no—
308#2489—591no—
309#2360—591no—
310#2056—591no—
311#2035—591no—
312#2026—591no—
313#2001D5911yescharacteristic
314#1989—591no—
315#1933—591no—
316#1927—591no—
317#1923—591no—
318#1920—591no—
319#1878—591no—
320#1872—591no—
321#1823—591no—
322C4×#103—591no—
323C2×C59—42no—
324C2×C59#245911yescharacteristic
325C2×C59#310811yescharacteristic
326C2×C59#312011yescharacteristic
327C2×C59#329611yescharacteristic
328C2×C59#330011yescharacteristic
329C2×C59#331211yescharacteristic
330C2×C59#333111yescharacteristic
331C22×Q16C2×D5911yescharacteristic
332C2×#125—591no—
333C2×C22⋊Q8C2×D5911yescharacteristic
334C2×C22⋊Q8C2×D5911yescharacteristic
335C2×C4×Q8—591no—
336C22×C4⋊C4—591no—
337C2×#106—591no—
338C2×C4⋊C8—591no—
339C2×#103—591no—
340C2×#103—591no—
341C2×#103—591no—
342C2×#103—591no—
343C2×#103C2×D5911yescharacteristic
344C2×#103C2×D5911yescharacteristic
345C2×C22⋊C8—591no—
346C2×C22⋊C8C2×D5911yescharacteristic
347C2×C4×C8—591no—
348#400—1181no—
349#397—1181no—
350#391—591no—
351#391—591no—
352#391—591no—
353#389—591no—
354#389—1181no—
355#388—1181no—
356#387—1181no—
357#385—1181no—
358#382—1181no—
359#380—591no—
360C4×C4⋊C4—591no—
361C4×C4⋊C4—591no—
362C4×C22⋊C4—591no—
363C4×C22⋊C4—1181no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340C2×D5911yescharacteristic
372#336—591no—
373#336—591no—
374C59#2672511yescharacteristic, complemented, Sylow
375C22×Q8C22×D5911yescharacteristic
376C22×Q8C22×D5911yescharacteristic
377C23×C4—1181no—
378C23×C4C22×D5911yescharacteristic
379C2×Q16—22no—
380C2×Q16—41no—
381C2×Q16—41no—
382C22×C8—591no—
383C22×C8—591no—
384C22×C8C22×D5911yescharacteristic
385C22×C8C22×D5911yescharacteristic
386#125—1182no—
387#125—2361no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—41no—
392C4×Q8—1182no—
393C4×Q8—2361no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—1182no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—2361no—
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—591no—
412C2×C22⋊C4—591no—
413C2×C22⋊C4—1181no—
414C2×C42—591no—
415C2×C42—591no—
416C2×C42—1181no—
417C2×C42—1181no—
418C2×C42—1181no—
419#106—1182no—
420C4⋊C8—2361no—
421#103—22no—
422#103—22no—
423#103—1182no—
424#103—1182no—
425#103—2361no—
426#103—2361no—
427C22⋊C8—41no—
428C22⋊C8—2361no—
429C4×C8—2361no—
430#95—591no—
431#95—591no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442C24D4×D5911yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8#1174568712yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—591no—
466C22×C4—591no—
467C22×C4—591no—
468C22×C4—591no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—4721no—
474C22×C4—2362no—
475C22×C4—2362no—
476C22×C4D4×D5911yescharacteristic
477C22×C4C23×D5911yescharacteristic
478Q16—44no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
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—42no—
495C4⋊C4—592no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—4721no—
504C4⋊C4—2362no—
505C4⋊C4—2362no—
506C4⋊C4—2362no—
507C4⋊C4—2362no—
508C4⋊C4#1174568712yes—
509C4⋊C4#1174568712yes—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—1182no—
513C22⋊C4—1182no—
514C22⋊C4—2362no—
515C42—1181no—
516C42—1181no—
517C42—1181no—
518C42—1181no—
519C42—4721no—
520C42—2362no—
521C42—2362no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1427375012yes—
529C23C2×D4×D5911yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—592no—
558C2×C4—1181no—
559C2×C4—1181no—
560C2×C4—1181no—
561C2×C4—1181no—
562C2×C4—1181no—
563C2×C4—1181no—
564C2×C4—1181no—
565C2×C4—1181no—
566C2×C4—1181no—
567C2×C4—1181no—
568C2×C4—1181no—
569C2×C4—1182no—
570C2×C4—4724no—
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—2361no—
582C2×C4#41×D5911yescharacteristic
583C2×C4C2×#1174568711yescharacteristic
584C2×C4C2×#1174568711yescharacteristic
585C2×C4C2×#1174568711yescharacteristic
586C2×C4#1427377211yescharacteristic
587C2×C4#1427377211yescharacteristic
588C8—41no—
589C8—41no—
590C8—2361no—
591C8—2361no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1683790811yescharacteristic
600C22#1683803911yescharacteristic
601C22#1683804111yescharacteristic
602C22C2×#1427375011yescharacteristic
603C22#137×D5911yescharacteristic
604C22#1683820911yescharacteristic
605C22#1683821811yescharacteristic
606C4—41no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—81no—
611C4—1182no—
612C4—4721no—
613C4—4721no—
614C4—4721no—
615C4—2361no—
616C4—2362no—
617C4—2361no—
618C4—2361no—
619C4#1683793311yescharacteristic
620C4#1683798711yescharacteristic
621C4#1683811611yescharacteristic
622C4#1683813811yescharacteristic
623C2—42no—
624C2#2142802811yescharacteristic
625C2#2143417711yescharacteristic
626C2#2143430311yescharacteristic
627C2#2143593711yescharacteristic
628C2#2143598711yescharacteristic
629C2#2143614211yescharacteristic
630C2#2143640511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2001C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2001D59
hypocenterC59#26725
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
3. #21425416C2semidirect
5. #21425245C2semidirect
6. #21425050C2semidirect
7. #21424774C2semidirect
8. #21424733C2semidirect
10. #21424684C2semidirect
11. #21424439C2semidirect
14. #103×Dic59C2semidirect
374. C59#26725semidirect

Maximal subgroups

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

subgroupindex
#214284392
#214271052
#214254162
#214252892
#214252452
#214250502
#214247742
#214247332
#214246992
#214246842
#214244392
#214244042
#214241832
#103×Dic592
C59×#20012
#2672559