Finite Group Databasebeta

← #28651601

Subgroups — #28651601

Order 15104.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837711C211yesmaximal, characteristic, complemented
2C2×#16837444C211yesmaximal, characteristic
3#21427821C211yesmaximal, characteristic
4#21427127C211yesmaximal, characteristic
5#21425646C211yesmaximal, characteristic, complemented
6#21425596C211yesmaximal, characteristic, complemented
7#21425475C211yesmaximal, characteristic, complemented
8#21425433C211yesmaximal, characteristic, complemented
9#21424867C211yesmaximal, characteristic, complemented
10#21424687C211yesmaximal, characteristic, complemented
11#21424404C211yesmaximal, characteristic
12#21424352C211yesmaximal, characteristic, complemented
13#21424293C211yesmaximal, characteristic
14#21424279C211yesmaximal, characteristic
15C59×#1894C211yesmaximal, characteristic
16C2×#14273718C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837711—24no—
21#16837444—24no—
22C2×#14273661C2211yescharacteristic
23C2×#14273661C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273659C2211yescharacteristic
26C2×#14273659C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×Dic59⋊C8C2211yescharacteristic
29C2×Dic59⋊C8C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273651C2211yescharacteristic
32C2×C22⋊(C59⋊C8)C2211yescharacteristic
33#16837344—21no—
34#16837334—21no—
35#16837309C2211yescharacteristic
36#16837301C2211yescharacteristic
37#16837301C2211yescharacteristic
38#16837298C2211yescharacteristic
39#16837295—21no—
40#16837295—21no—
41#16837288—21no—
42#16837277—21no—
43#16837267C2211yescharacteristic
44#16837260—21no—
45C4⋊C4×Dic59—21no—
46C4⋊C4×Dic59C2211yescharacteristic
47C4×C22⋊Dic59C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53#16837156C2211yescharacteristic
54C22×M16×C59C2211yescharacteristic
55C2×C22⋊Q8×C59C2211yescharacteristic
56C2×#103×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C59×#380C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C23×Dic59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273718—22no—
65#14273718—41no—
66Q8×Dic59—22no—
67Q8×Dic59—41no—
68#14273686—22no—
69#14273686—41no—
70C2×C4⋊Dic59—22no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—41no—
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59D411yescharacteristic
85C2×C22⋊Dic59D411yescharacteristic
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59C2311yescharacteristic
90C2×C4×Dic59C2311yescharacteristic
91#14273661—22no—
92#14273661D414yes—
93#14273660—41no—
94#14273659—22no—
95#14273659—22no—
96#14273658—22no—
97Dic59⋊C8—22no—
98Dic59⋊C8—22no—
99#14273651—22no—
100#14273651—41no—
101C22⋊(C59⋊C8)—22no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95D411yescharacteristic
111C59⋊#95D411yescharacteristic
112C22×Q8×C59C2311yescharacteristic
113C23×C4×C59C2311yescharacteristic
114C2×M16×C59—22no—
115C2×M16×C59—22no—
116C2×M16×C59—22no—
117C22×C8×C59C2311yescharacteristic
118C22×C8×C59C2311yescharacteristic
119C22⋊Q8×C59—22no—
120C22⋊Q8×C59Q814yes—
121C2×C4⋊C4×C59—21no—
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C22⋊C4×C59—21no—
126C2×C22⋊C4×C59—21no—
127#103×C59—22no—
128#103×C59—22no—
129#95×C59—21no—
130C24×C59C2×D411yescharacteristic
131C2×Q8×C59—21no—
132C2×Q8×C59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59C2×Q812yes—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
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×C59C2×D411yescharacteristic
146C22×C4×C59C2×Q812yes—
147C22×C4×C59#4112yes—
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—42no—
154C22×Dic59—42no—
155C22×Dic59—81no—
156C22×Dic59C2×D411yescharacteristic
157C22×Dic59C2×D411yescharacteristic
158C22×Dic59C2×D411yescharacteristic
159C22×Dic59C2×D411yescharacteristic
160M16×C59—28no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59C2×D412yes—
166C2×C8×C59#4112yes—
167C4⋊C4×C59—24no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59C2×Q812yes—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—42no—
176C22⋊Dic59—41no—
177C22⋊Dic59—41no—
178C22⋊Dic59—42no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—42no—
186C4⋊Dic59—42no—
187C4⋊Dic59—42no—
188C4⋊Dic59—42no—
189C4⋊Dic59—81no—
190C4⋊Dic59C2×D412yes—
191C4⋊Dic59C2×D412yes—
192Dic59⋊C4—42no—
193Dic59⋊C4—81no—
194C4×Dic59—21no—
195C4×Dic59—21no—
196C4×Dic59—21no—
197C4×Dic59—21no—
198C4×Dic59—42no—
199C4×Dic59—42no—
200C4×Dic59—42no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59C22⋊Q812yes—
214C23×C59C22×D411yescharacteristic
215Q8×C59—22no—
216Q8×C59—22no—
217Q8×C59—41no—
218Q8×C59—41no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—41no—
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×C59C22⋊Q812yes—
238C2×C4×C59C22×D411yescharacteristic
239C2×C4×C59C22×Q811yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59H(5,𝔽2)11yescharacteristic
243C2×C4×C592−1+411yescharacteristic
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—22no—
247C2×Dic59—22no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—41no—
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—84no—
269C8×C59—22no—
270C8×C59—22no—
271C59⋊C8—41no—
272C59⋊C8—41no—
273#26365—591nomaximal, Sylow
274C22×C59—22no—
275C22×C59—22no—
276C22×C59—22no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59—41no—
280C22×C59—41no—
281C22×C59C2×C22⋊Q811yescharacteristic
282C22×C59D4211yescharacteristic
283C22×C59#54711yescharacteristic
284C22×C59#57511yescharacteristic
285C22×C59#57511yescharacteristic
286C22×C59#57711yescharacteristic
287C22×C59#57811yescharacteristic
288C4×C59—22no—
289C4×C59—42no—
290C4×C59—42no—
291C4×C59—81no—
292C4×C59—81no—
293C4×C59#52511yescharacteristic
294C4×C59#53911yescharacteristic
295C4×C59D4×Q811yescharacteristic
296C4×C59#55411yescharacteristic
297Dic59—41no—
298Dic59—41no—
299Dic59—41no—
300Dic59—42no—
301Dic59—42no—
302Dic59—41no—
303Dic59—81no—
304Dic59—81no—
305C2×#479—591no—
306C2×#428—591no—
307#2442—591no—
308#2362—591no—
309#2097—591no—
310#2088—591no—
311#2066—591no—
312#2059—591no—
313#1949—591no—
314#1921—591no—
315#1894D5911yescharacteristic
316#1872—591no—
317#1863—591no—
318#1854—591no—
319#1851—591no—
320C2×C59—42no—
321C2×C59#244911yescharacteristic
322C2×C59#308011yescharacteristic
323C2×C59#308211yescharacteristic
324C2×C59#326711yescharacteristic
325C2×C59#327711yescharacteristic
326C2×C59#335011yescharacteristic
327C2×C59#335211yescharacteristic
328C22×M16C2×D5911yescharacteristic
329C2×#125—591no—
330C2×C22⋊Q8C2×D5911yescharacteristic
331C2×C4×Q8—591no—
332C2×#117—591no—
333C22×C4⋊C4—591no—
334#479—1184no—
335#428—1184no—
336C2×#107—591no—
337C2×#107—591no—
338C2×#107—591no—
339C2×#106—591no—
340C2×#106—591no—
341C2×#106—591no—
342C2×C4⋊C8—591no—
343C2×C4⋊C8—591no—
344C2×#103—591no—
345C2×#103—591no—
346C2×#103C2×D5911yescharacteristic
347C2×#103C2×D5911yescharacteristic
348C2×C22⋊C8—591no—
349#400—1181no—
350#398—1181no—
351#391—591no—
352#389—591no—
353#389—591no—
354#389—591no—
355#388—1181no—
356#388—1181no—
357#387—1181no—
358#385—1181no—
359#382—591no—
360#380—1181no—
361#380C2×D5911yescharacteristic
362C4×C4⋊C4—591no—
363C4×C4⋊C4—1181no—
364C4×C22⋊C4—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340C2×D5911yescharacteristic
370#340C2×D5911yescharacteristic
371#336—591no—
372#336—591no—
373C59#2636511yescharacteristic, complemented, Sylow
374C22×Q8C22×D5911yescharacteristic
375C23×C4—1181no—
376C23×C4C22×D5911yescharacteristic
377C2×M16—22no—
378C2×M16—22no—
379C2×M16—22no—
380C22×C8—591no—
381C22×C8—591no—
382C22×C8C22×D5911yescharacteristic
383C22×C8C22×D5911yescharacteristic
384#125—1182no—
385#125—2361no—
386C22⋊Q8—22no—
387C22⋊Q8Dic11814yes—
388C4×Q8—1182no—
389C4×Q8—2361no—
390#117—1182no—
391#117—2361no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—1182no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—2361no—
406C2×C4⋊C4C22×D5911yescharacteristic
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#107—594no—
420#107—1182no—
421#107—2361no—
422#106—1182no—
423#106—1182no—
424#106—1182no—
425C4⋊C8—1182no—
426C4⋊C8—1182no—
427#103—22no—
428#103—22no—
429#103—1182no—
430#103—2361no—
431C22⋊C8—1182no—
432#95—21no—
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—
443C24D4×D5911yescharacteristic
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8C2×Dic11812yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—591no—
459C22×C4—591no—
460C22×C4—591no—
461C22×C4—591no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—1181no—
466C22×C4—4721no—
467C22×C4—2362no—
468C22×C4—2362no—
469C22×C4C2×Dic11812yes—
470C22×C4D4×D5911yescharacteristic
471C22×C4#1174568712yes—
472C22×C4C23×D5911yescharacteristic
473M16—28no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8D4×D5912yes—
485C2×C8#1174568712yes—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—592no—
493C4⋊C4—592no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—4721no—
501C4⋊C4—4721no—
502C4⋊C4—2362no—
503C4⋊C4—2362no—
504C4⋊C4—2362no—
505C4⋊C4—2362no—
506C4⋊C4—2362no—
507C4⋊C4C2×Dic11812yes—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—2361no—
511C22⋊C4—2361no—
512C22⋊C4—2362no—
513C42—1181no—
514C42—1181no—
515C42—1181no—
516C42—1181no—
517C42—2362no—
518C42—2362no—
519C42—2362no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1427370412yes—
527C23C2×D4×D5911yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—1181no—
560C2×C4—1182no—
561C2×C4—1182no—
562C2×C4—1181no—
563C2×C4—1181no—
564C2×C4—4724no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4#1427370412yes—
576C2×C4C2×D4×D5911yescharacteristic
577C2×C4C22×Dic11811yescharacteristic
578C2×C4C2×#1174568711yescharacteristic
579C2×C4C2×#1174568711yescharacteristic
580C2×C4#1427377311yescharacteristic
581C2×C4#1427377411yescharacteristic
582C8—22no—
583C8—22no—
584C8—2361no—
585C8—2361no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1427370411yescharacteristic
594C22#1683803611yescharacteristic
595C22#1683805211yescharacteristic
596C22#1683819811yescharacteristic
597C22#1683820111yescharacteristic
598C22#1683821111yescharacteristic
599C22#1683821811yescharacteristic
600C4—22no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—4721no—
606C4—4721no—
607C4—2361no—
608C4—2361no—
609C4—2361no—
610C4—2362no—
611C4—2362no—
612C4—2361no—
613C4#1683790811yescharacteristic
614C4#1683799611yescharacteristic
615C4D4×Dic11811yescharacteristic
616C4#1683810911yescharacteristic
617C2—42no—
618C2#2142791111yescharacteristic
619C2#2143391611yescharacteristic
620C2#2143394011yescharacteristic
621C2#2143562811yescharacteristic
622C2#2143575011yescharacteristic
623C2#2143663611yescharacteristic
624C2#2143666611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1894C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1894D59
hypocenterC59#26365
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
1. C2×#16837711C2semidirect
5. #21425646C2semidirect
6. #21425596C2semidirect
7. #21425475C2semidirect
8. #21425433C2semidirect
9. #21424867C2semidirect
10. #21424687C2semidirect
12. #21424352C2semidirect
373. C59#26365semidirect

Maximal subgroups

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

subgroupindex
C2×#168377112
C2×#168374442
#214278212
#214271272
#214256462
#214255962
#214254752
#214254332
#214248672
#214246872
#214244042
#214243522
#214242932
#214242792
C59×#18942
#2636559