Finite Group Databasebeta

← #28651619

Subgroups — #28651619

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×#16837709C211yesmaximal, characteristic, complemented
2C2×#16837445C211yesmaximal, characteristic
3#21427820C211yesmaximal, characteristic
4#21425644C211yesmaximal, characteristic, complemented
5#21425592C211yesmaximal, characteristic, complemented
6#21425491C211yesmaximal, characteristic, complemented
7#21425434C211yesmaximal, characteristic, complemented
8#21424864C211yesmaximal, characteristic, complemented
9#21424674C211yesmaximal, characteristic, complemented
10#21424530C211yesmaximal, characteristic
11#21424404C211yesmaximal, characteristic
12#21424364C211yesmaximal, characteristic, complemented
13#21424295C211yesmaximal, characteristic
14#21424281C211yesmaximal, characteristic
15C59×#2362C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19#16837709—24no—
20#16837445—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×C4⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273651C2211yescharacteristic
32C2×#14273651C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34#16837344—21no—
35#16837336—21no—
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837274—21no—
39#16837268C2211yescharacteristic
40#16837262C2211yescharacteristic
41#16837260—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C22⋊Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#388—21no—
57C59×#387—21no—
58C4×C4⋊C4×C59—21no—
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C23×Dic59—21no—
62C2×#11745664—22no—
63C2×#11745664—22no—
64C2×#11745664—22no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273719—22no—
70#14273719—41no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—41no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83#14273660—22no—
84#14273660—22no—
85#14273660—22no—
86#14273658—22no—
87#14273658—41no—
88#14273658D414yes—
89C4⋊(C59⋊C8)—22no—
90C4⋊(C59⋊C8)—22no—
91#14273651—22no—
92#14273651—22no—
93#14273651—22no—
94#14273651—41no—
95C22⋊(C59⋊C8)—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×Q8×C59C2311yescharacteristic
102C23×C4×C59C2311yescharacteristic
103C22⋊Q8×C59—22no—
104C22⋊Q8×C59Q814yes—
105C4×Q8×C59—22no—
106C4×Q8×C59—41no—
107#117×C59—22no—
108#117×C59—41no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C22⋊C4×C59D411yescharacteristic
119C2×C42×C59—21no—
120C2×C42×C59—21no—
121C2×C42×C59—21no—
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59D411yescharacteristic
129#95×C59D411yescharacteristic
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×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×Q812yes—
151C22×C4×C59#4112yes—
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—42no—
158C22×Dic59—42no—
159C22×Dic59—81no—
160C4⋊C4×C59—24no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—42no—
166C4⋊C4×C59—42no—
167C4⋊C4×C59C2×D412yes—
168C4⋊C4×C59C2×Q812yes—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—41no—
171C22⋊C4×C59—41no—
172C22⋊C4×C59—42no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—42no—
178C42×C59—42no—
179C42×C59—42no—
180C22⋊Dic59—42no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—42no—
189C4⋊Dic59C2×D412yes—
190Dic59⋊C4—42no—
191Dic59⋊C4—42no—
192Dic59⋊C4—81no—
193Dic59⋊C4—81no—
194#11745664—28no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8C2×D412yes—
206C2×C59⋊C8#4112yes—
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—21no—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59C22⋊Q812yes—
250C2×C4×C59C22×D411yescharacteristic
251C2×C4×C59C22×Q811yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59H(5,𝔽2)11yescharacteristic
255C2×C4×C592−1+411yescharacteristic
256C2×Dic59—22no—
257C2×Dic59—41no—
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—
269C59⋊C8—22no—
270C59⋊C8—22no—
271C59⋊C8—41no—
272C59⋊C8—41no—
273#26367—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—41no—
291C4×C59—41no—
292C4×C59—42no—
293C4×C59—41no—
294C4×C59—42no—
295C4×C59—41no—
296C4×C59—81no—
297C4×C59#52511yescharacteristic
298C4×C59#53911yescharacteristic
299C4×C59D4×Q811yescharacteristic
300C4×C59#55411yescharacteristic
301Dic59—42no—
302Dic59—81no—
303Dic59—81no—
304Dic59—81no—
305C2×#479—591no—
306C2×#428—591no—
307#2442—591no—
308#2362D5911yescharacteristic
309#2097—591no—
310#2087—591no—
311#2068—591no—
312#2059—591no—
313#1949—591no—
314#1919—591no—
315#1894—591no—
316#1872—591no—
317#1865—591no—
318#1854—591no—
319#1851—591no—
320C2×C59—42no—
321C2×C59#244911yescharacteristic
322C2×C59#308011yescharacteristic
323C2×C59#308211yescharacteristic
324C2×C59#326911yescharacteristic
325C2×C59#327511yescharacteristic
326C2×C59#334911yescharacteristic
327C2×C59#335111yescharacteristic
328C22×M16—591no—
329C2×#125—591no—
330C2×C22⋊Q8C2×D5911yescharacteristic
331C2×C4×Q8C2×D5911yescharacteristic
332C2×#117C2×D5911yescharacteristic
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×#103—591no—
347C2×#103—591no—
348C2×C22⋊C8—591no—
349#400—1181no—
350#398—1181no—
351#391C2×D5911yescharacteristic
352#389—591no—
353#389—591no—
354#389C2×D5911yescharacteristic
355#388—21no—
356#388—21no—
357#387—21no—
358#385—1181no—
359#382—591no—
360#380—591no—
361#380—1181no—
362C4×C4⋊C4—21no—
363C4×C4⋊C4C2×D5911yescharacteristic
364C4×C22⋊C4C2×D5911yescharacteristic
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#336—591no—
372#336—591no—
373C59#2636711yescharacteristic, complemented, Sylow
374C22×Q8C22×D5911yescharacteristic
375C23×C4—1181no—
376C23×C4C22×D5911yescharacteristic
377C2×M16—1182no—
378C2×M16—1182no—
379C2×M16—1182no—
380C22×C8—591no—
381C22×C8—591no—
382C22×C8—591no—
383C22×C8—591no—
384#125—1182no—
385#125—2361no—
386C22⋊Q8—22no—
387C22⋊Q8Dic11814yes—
388C4×Q8—22no—
389C4×Q8—41no—
390#117—22no—
391#117—41no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—1182no—
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—1181no—
412C2×C22⋊C4C22⋊D5911yescharacteristic
413C2×C22⋊C4C22⋊D5911yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42C22×D5911yescharacteristic
418C2×C42C22×D5911yescharacteristic
419#107—1182no—
420#107—1182no—
421#107—1182no—
422#106—594no—
423#106—1182no—
424#106—2361no—
425C4⋊C8—1182no—
426C4⋊C8—1182no—
427#103—1182no—
428#103—1182no—
429#103—1182no—
430#103—2361no—
431C22⋊C8—1182no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95C22⋊D5911yescharacteristic
442#95C22⋊D5911yescharacteristic
443C24C2×C22⋊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—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—4721no—
463C22×C4—2362no—
464C22×C4—2362no—
465C22×C4C2×Dic11812yes—
466C22×C4#1174568512yes—
467C22×C4C2×C22⋊D5911yescharacteristic
468C22×C4C2×C22⋊D5911yescharacteristic
469C22×C4C2×C22⋊D5911yescharacteristic
470C22×C4C2×C22⋊D5911yescharacteristic
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C23×D5911yescharacteristic
473M16—1188no—
474C2×C8—592no—
475C2×C8—592no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—592no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—4721no—
502C4⋊C4—4721no—
503C4⋊C4—2362no—
504C4⋊C4—2362no—
505C4⋊C4—2362no—
506C4⋊C4C2×Dic11812yes—
507C4⋊C4D4×D5912yes—
508C22⋊C4—24no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—42no—
512C22⋊C4—2362no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—42no—
519C42—42no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1427370612yes—
527C23C22×C22⋊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—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
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—1182no—
563C2×C4—4724no—
564C2×C4—2361no—
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#1427370612yes—
576C2×C4C2×D4×D5911yescharacteristic
577C2×C4C22×Dic11811yescharacteristic
578C2×C4#41×D5911yescharacteristic
579C2×C4C2×#1174568511yescharacteristic
580C2×C4#1427377211yescharacteristic
581C2×C4#1427377411yescharacteristic
582C8—1182no—
583C8—1182no—
584C8—2361no—
585C8—2361no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1427370611yescharacteristic
594C22D4×C22⋊D5911yescharacteristic
595C22#1683805111yescharacteristic
596C22#1683820011yescharacteristic
597C22#1683820011yescharacteristic
598C22#1683821011yescharacteristic
599C22#1683821711yescharacteristic
600C4—22no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—4721no—
610C4—4721no—
611C4—4721no—
612C4—2362no—
613C4#1683791111yescharacteristic
614C4#1683799411yescharacteristic
615C4D4×Dic11811yescharacteristic
616C4#1683811111yescharacteristic
617C2—42no—
618C2#2142789911yescharacteristic
619C2#2143391811yescharacteristic
620C2#2143393811yescharacteristic
621C2#2143564711yescharacteristic
622C2#2143572811yescharacteristic
623C2#2143662011yescharacteristic
624C2#2143665611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2362C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2362D59
hypocenterC59#26367
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
1. C2×#16837709C2semidirect
4. #21425644C2semidirect
5. #21425592C2semidirect
6. #21425491C2semidirect
7. #21425434C2semidirect
8. #21424864C2semidirect
9. #21424674C2semidirect
12. #21424364C2semidirect
373. C59#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#168377092
C2×#168374452
#214278202
#214256442
#214255922
#214254912
#214254342
#214248642
#214246742
#214245302
#214244042
#214243642
#214242952
#214242812
C59×#23622
#2636759