Finite Group Databasebeta

← #28656805

Subgroups — #28656805

Order 15104.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428472C211yesmaximal, characteristic
2#21427105C211yesmaximal, characteristic
3#21425416C211yesmaximal, characteristic, complemented
4#21425296C211yesmaximal, characteristic
5#21425235C211yesmaximal, characteristic, complemented
6#21425042C211yesmaximal, characteristic, complemented
7#21424774C211yesmaximal, characteristic, complemented
8#21424740C211yesmaximal, characteristic, complemented
9#21424699C211yesmaximal, characteristic
10#21424691C211yesmaximal, characteristic, complemented
11#21424439C211yesmaximal, characteristic, complemented
12#21424404C211yesmaximal, characteristic
13#21424187C211yesmaximal, characteristic
14#103×Dic59C211yesmaximal, characteristic, complemented
15C59×#2001C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×Dic59⋊C8C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×C22⋊(C59⋊C8)C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27#16837340—21no—
28#16837314—21no—
29#16837309C2211yescharacteristic
30#16837309C2211yescharacteristic
31#16837309C2211yescharacteristic
32#16837307—21no—
33#16837301—21no—
34#16837301C2211yescharacteristic
35#16837296—21no—
36#16837288—21no—
37#16837277—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
59C22×Dic118—21no—
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63#14273726—22no—
64#14273726—41no—
65Q8×Dic59—22no—
66Q8×Dic59—41no—
67C2×C4⋊Dic59—22no—
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#14273660—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
162C2×Dic118—42no—
163C2×Dic118—42no—
164C2×Dic118—81no—
165Q16×C59—44no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59—21no—
172C4⋊C4×C59—24no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—42no—
176C4⋊C4×C59#4112yes—
177C4⋊C4×C59#4112yes—
178C22⋊C4×C59—24no—
179C22⋊C4×C59—42no—
180C22⋊Dic59—41no—
181C22⋊Dic59—81no—
182C22⋊Dic59Q1614yescomplemented
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—42no—
191C4⋊Dic59—42no—
192C4⋊Dic59—81no—
193C4⋊Dic59#4112yes—
194Dic59⋊C4—42no—
195C4×Dic59—21no—
196C4×Dic59—21no—
197C4×Dic59—21no—
198C4×Dic59—21no—
199C4×Dic59—42no—
200C4×Dic59—81no—
201C4×Dic59—81no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59C2×Q1612yes—
215C23×C59C22×D411yescharacteristic
216Q8×C59—22no—
217Q8×C59—22no—
218Q8×C59—41no—
219Q8×C59—41no—
220Q8×C59—41no—
221Q8×C59—41no—
222Q8×C59—41no—
223Q8×C59—41no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
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×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59H(5,𝔽2)11yescharacteristic
248C2×C4×C59H(5,𝔽2)11yescharacteristic
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—22no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
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—41no—
271C2×Dic59—84no—
272C2×Dic59C2×Q1612yes—
273C2×Dic59C2×Q1612yes—
274Dic118—84no—
275C8×C59—41no—
276C8×C59—41no—
277C59⋊C8—41no—
278C59⋊C8—41no—
279#26726—591nomaximal, Sylow
280C22×C59—22no—
281C22×C59—22no—
282C22×C59—22no—
283C22×C59—41no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59#52511yescharacteristic
288C22×C59#54611yescharacteristic
289C22×C59#54611yescharacteristic
290C22×C59C22×Q1611yescharacteristic
291C22×C59#57511yescharacteristic
292C22×C59#57711yescharacteristic
293C22×C59#57711yescharacteristic
294C4×C59—42no—
295C4×C59—41no—
296C4×C59—42no—
297C4×C59—42no—
298C4×C59—81no—
299C4×C59#52911yescharacteristic
300C4×C59#53811yescharacteristic
301C4×C59#55511yescharacteristic
302C4×C59#55911yescharacteristic
303Dic59—22no—
304Dic59—22no—
305Dic59—41no—
306Dic59—42no—
307Dic59—41no—
308Dic59—81no—
309Dic59—81no—
310Dic59—81no—
311#2492—591no—
312#2360—591no—
313#2056—591no—
314#2036—591no—
315#2024—591no—
316#2001D5911yescharacteristic
317#1987—591no—
318#1933—591no—
319#1928—591no—
320#1923—591no—
321#1921—591no—
322#1878—591no—
323#1872—591no—
324#1824—591no—
325C4×#103—591no—
326C2×C59—42no—
327C2×C59#245911yescharacteristic
328C2×C59#310411yescharacteristic
329C2×C59#312511yescharacteristic
330C2×C59#329911yescharacteristic
331C2×C59#329911yescharacteristic
332C2×C59#331111yescharacteristic
333C2×C59#332311yescharacteristic
334C22×Q16C2×D5911yescharacteristic
335C2×C4⋊Q8—591no—
336C2×C22⋊Q8C2×D5911yescharacteristic
337C2×C22⋊Q8C2×D5911yescharacteristic
338C2×C4×Q8—591no—
339C22×C4⋊C4—591no—
340C2×#107—591no—
341C2×C4⋊C8—591no—
342C2×#103—591no—
343C2×#103—591no—
344C2×#103—591no—
345C2×#103—591no—
346C2×#103C2×D5911yescharacteristic
347C2×#103C2×D5911yescharacteristic
348C2×C22⋊C8—591no—
349C2×C22⋊C8C2×D5911yescharacteristic
350C2×C4×C8—591no—
351#400—1181no—
352#393—1181no—
353#391—591no—
354#391—591no—
355#391—591no—
356#391—1181no—
357#389—591no—
358#389—1181no—
359#388—1181no—
360#387—1181no—
361#385—1181no—
362#380—591no—
363C4×C4⋊C4—591no—
364C4×C4⋊C4—591no—
365C4×C22⋊C4—591no—
366C4×C22⋊C4—1181no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#340—591no—
373#340—591no—
374#340C2×D5911yescharacteristic
375#336—591no—
376#336—591no—
377C59#2672611yescharacteristic, complemented, Sylow
378C22×Q8—1181no—
379C22×Q8C22×D5911yescharacteristic
380C22×Q8C22×D5911yescharacteristic
381C23×C4—1181no—
382C23×C4C22×D5911yescharacteristic
383C2×Q16—22no—
384C2×Q16—41no—
385C2×Q16—41no—
386C22×C8—591no—
387C22×C8—591no—
388C22×C8C22×D5911yescharacteristic
389C22×C8C22×D5911yescharacteristic
390C4⋊Q8—1182no—
391C4⋊Q8—2361no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—41no—
396C4×Q8—1182no—
397C4×Q8—2361no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—1182no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—2361no—
410C2×C4⋊C4C22×D5911yescharacteristic
411C2×C4⋊C4C22×D5911yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—591no—
415C2×C22⋊C4—591no—
416C2×C22⋊C4—1181no—
417C2×C42—591no—
418C2×C42—591no—
419C2×C42—1181no—
420C2×C42—1181no—
421C2×C42—1181no—
422#107—1182no—
423C4⋊C8—2361no—
424#103—22no—
425#103—22no—
426#103—1182no—
427#103—1182no—
428#103—2361no—
429#103—2361no—
430C22⋊C8—41no—
431C22⋊C8—2361no—
432C4×C8—2361no—
433#95—591no—
434#95—591no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442#95—1181no—
443#95—1181no—
444#95—1181no—
445C24D4×D5911yescharacteristic
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—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—4721no—
458C2×Q8—2362no—
459C2×Q8—2362no—
460C2×Q8#1174568712yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—591no—
472C22×C4—591no—
473C22×C4—591no—
474C22×C4—591no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—4721no—
480C22×C4—2362no—
481C22×C4—2362no—
482C22×C4D4×D5911yescharacteristic
483C22×C4C23×D5911yescharacteristic
484Q16—44no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—592no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—4721no—
510C4⋊C4—2362no—
511C4⋊C4—2362no—
512C4⋊C4—2362no—
513C4⋊C4#1174568712yes—
514C4⋊C4#1174568712yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—594no—
518C22⋊C4—4721no—
519C22⋊C4—2361no—
520C42—1181no—
521C42—1181no—
522C42—1181no—
523C42—1181no—
524C42—4721no—
525C42—4721no—
526C42—2362no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23Q16×D5912yes—
534C23C2×D4×D5911yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—4724no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—592no—
564C2×C4—592no—
565C2×C4—1181no—
566C2×C4—1181no—
567C2×C4—1181no—
568C2×C4—1181no—
569C2×C4—1181no—
570C2×C4—1181no—
571C2×C4—1181no—
572C2×C4—1181no—
573C2×C4—1181no—
574C2×C4—1181no—
575C2×C4—1182no—
576C2×C4—4724no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4#41×D5911yescharacteristic
589C2×C4C2×#1174568711yescharacteristic
590C2×C4C2×#1174568711yescharacteristic
591C2×C4C2×#1174568711yescharacteristic
592C2×C4#1427377211yescharacteristic
593C2×C4#1427377211yescharacteristic
594C8—41no—
595C8—41no—
596C8—2361no—
597C8—2361no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1683790811yescharacteristic
606C22#1683803911yescharacteristic
607C22#1683804111yescharacteristic
608C22C2×Q16×D5911yescharacteristic
609C22#1683820111yescharacteristic
610C22#1683820911yescharacteristic
611C22#1683821111yescharacteristic
612C4—41no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—81no—
617C4—1182no—
618C4—1182no—
619C4—4721no—
620C4—4721no—
621C4—4721no—
622C4—2361no—
623C4—2361no—
624C4—2362no—
625C4#1683793311yescharacteristic
626C4#1683798711yescharacteristic
627C4#1683811611yescharacteristic
628C4#1683813811yescharacteristic
629C2—42no—
630C2#2142802811yescharacteristic
631C2#2143413011yescharacteristic
632C2#2143436611yescharacteristic
633C2#2143596511yescharacteristic
634C2#2143596711yescharacteristic
635C2#2143612711yescharacteristic
636C2#2143630511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2001C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2001D59
hypocenterC59#26726
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. #21425235C2semidirect
6. #21425042C2semidirect
7. #21424774C2semidirect
8. #21424740C2semidirect
10. #21424691C2semidirect
11. #21424439C2semidirect
14. #103×Dic59C2semidirect
182. C22⋊Dic59Q16semidirect
377. C59#26726semidirect

Maximal subgroups

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

subgroupindex
#214284722
#214271052
#214254162
#214252962
#214252352
#214250422
#214247742
#214247402
#214246992
#214246912
#214244392
#214244042
#214241872
#103×Dic592
C59×#20012
#2672659