Finite Group Databasebeta

← #28652294

Subgroups — #28652294

Order 15104.

818 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#21427798C211yesmaximal, characteristic
2#21425654C211yesmaximal, characteristic, complemented
3#21425598C211yesmaximal, characteristic, complemented
4#21425477C211yesmaximal, characteristic, complemented
5#21425237C211yesmaximal, characteristic, complemented
6#21425048C211yesmaximal, characteristic, complemented
7#21424674C211yesmaximal, characteristic, complemented
8#21424520C211yesmaximal, characteristic
9#21424404C211yesmaximal, characteristic
10#21424364C211yesmaximal, characteristic, complemented
11#21424289C211yesmaximal, characteristic
12#21424276C211yesmaximal, characteristic
13#21424101C211yesmaximal, characteristic, complemented
14C2×#16837164C211yesmaximal, characteristic
15C59×#2360C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C2×#14273727C2211yescharacteristic
18C2×#14273704—21no—
19C22×C4⋊Dic59C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29#16837334—21no—
30#16837323—21no—
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837277—21no—
34#16837272C2211yescharacteristic
35#16837262C2211yescharacteristic
36#16837164C2211yescharacteristic
37#16837164C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C22⋊Q8×C59C2211yescharacteristic
49C2×C4×Q8×C59C2211yescharacteristic
50C22×C4⋊C4×C59C2211yescharacteristic
51C59×#391C2211yescharacteristic
52C59×#389—21no—
53C59×#389C2211yescharacteristic
54C59×#387—21no—
55C59×#385—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C22⋊C4×C59—21no—
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×Dic118—21no—
60C23×Dic59—21no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8—41no—
63C22×C59⋊C8—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273727—22no—
69#14273727—41no—
70#14273704—44no—
71C2×C4⋊Dic59—22no—
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—
84#14273658—22no—
85#14273658—41no—
86C4⋊(C59⋊C8)—41no—
87C4⋊(C59⋊C8)—41no—
88#14273651—22no—
89#14273651—41no—
90#14273651—41no—
91#14273651—41no—
92C22⋊(C59⋊C8)—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×Q8×C59C2311yescharacteristic
97C23×C4×C59—21no—
98C23×C4×C59C2311yescharacteristic
99C22⋊Q8×C59—22no—
100C22⋊Q8×C59Q814yescomplemented
101C4×Q8×C59—22no—
102C4×Q8×C59—41no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—22no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—41no—
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C2×Q812yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—42no—
141C22×C4×C59—42no—
142C22×C4×C59—81no—
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×Q812yes—
149C22×C4×C59#4112yes—
150C22×C4×C59C2411yescharacteristic
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—42no—
156C22×Dic59—42no—
157C22×Dic59—81no—
158C2×Dic118—42no—
159C2×Dic118—42no—
160C2×Dic118—81no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—42no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—81no—
169C4⋊C4×C59C2×Q812yes—
170C4⋊C4×C59#4112yes—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59SD1614yes—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—42no—
179C42×C59—81no—
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⋊Dic59—42no—
190C4⋊Dic59—42no—
191C4⋊Dic59—81no—
192C4⋊Dic59#4112yes—
193Dic59⋊C4—42no—
194Dic59⋊C4—42no—
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⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—44no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59C22⋊Q812yes—
213C23×C59C2×SD1612yes—
214C23×C59#13712yes—
215C23×C59C22×D411yescharacteristic
216Q8×C59—22no—
217Q8×C59—22no—
218Q8×C59—41no—
219Q8×C59—41no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—41no—
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—84no—
249C2×C4×C59C22⋊Q812yes—
250C2×C4×C59C2×SD1612yes—
251C2×C4×C59C2×SD1612yes—
252C2×C4×C59C22×Q811yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59H(5,𝔽2)11yescharacteristic
257C2×C4×C59H(5,𝔽2)11yescharacteristic
258C2×Dic59—22no—
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—
271Dic118—84no—
272C59⋊C8—41no—
273C59⋊C8—41no—
274C59⋊C8—41no—
275C59⋊C8—41no—
276#26412—591nomaximal, Sylow
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—22no—
280C22×C59—22no—
281C22×C59—22no—
282C22×C59—22no—
283C22×C59#47514yes—
284C22×C59C2×C22⋊Q811yescharacteristic
285C22×C59#54611yescharacteristic
286C22×C59#54611yescharacteristic
287C22×C59C22×SD1611yescharacteristic
288C22×C59C2×#13711yescharacteristic
289C22×C59#57611yescharacteristic
290C22×C59#57811yescharacteristic
291C4×C59—22no—
292C4×C59—22no—
293C4×C59—22no—
294C4×C59—42no—
295C4×C59—42no—
296C4×C59—41no—
297C4×C59—41no—
298C4×C59—42no—
299C4×C59—81no—
300C4×C59#52911yescharacteristic
301C4×C59#53811yescharacteristic
302C4×C59#55411yescharacteristic
303C4×C59#55411yescharacteristic
304Dic59—42no—
305Dic59—81no—
306Dic59—81no—
307Dic59—81no—
308#2440—591no—
309#2360D5911yescharacteristic
310#2099—591no—
311#2088—591no—
312#2066—591no—
313#2024—591no—
314#1988—591no—
315#1919—591no—
316#1892—591no—
317#1872—591no—
318#1865—591no—
319#1853—591no—
320#1850—591no—
321#1803—591no—
322C2×#340—591no—
323C2×C59—24no—
324C2×C59#245011yescharacteristic
325C2×C59C2×#47511yescharacteristic
326C2×C59#308311yescharacteristic
327C2×C59#329511yescharacteristic
328C2×C59#329811yescharacteristic
329C2×C59#331211yescharacteristic
330C2×C59#331411yescharacteristic
331C23×C8—591no—
332C2×C4⋊Q8—591no—
333C2×C22⋊Q8—1181no—
334C2×C22⋊Q8C2×D5911yescharacteristic
335C2×C4×Q8C2×D5911yescharacteristic
336C22×C4⋊C4—591no—
337C22×C4⋊C4C2×D5911yescharacteristic
338C2×#106—591no—
339C2×#106—591no—
340C2×C4⋊C8—591no—
341C2×C4⋊C8—591no—
342C2×#103—591no—
343C2×#103—591no—
344C2×#103—591no—
345C2×#103—591no—
346C2×C22⋊C8—591no—
347#398—1181no—
348#395—1181no—
349#391C2×D5911yescharacteristic
350#389—21no—
351#389—591no—
352#389—591no—
353#389C2×D5911yescharacteristic
354#387—21no—
355#385—21no—
356#385—1181no—
357#384—591no—
358#380—591no—
359C4×C4⋊C4C2×D5911yescharacteristic
360C4×C22⋊C4—21no—
361C4×C22⋊C4C2×D5911yescharacteristic
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#336—591no—
373#336—591no—
374C59#2641211yescharacteristic, complemented, Sylow
375C22×Q8—1181no—
376C22×Q8C22×D5911yescharacteristic
377C23×C4—21no—
378C23×C4—1181no—
379C23×C4C22×D5911yescharacteristic
380C22×C8—591no—
381C22×C8—591no—
382C22×C8—591no—
383C22×C8—591no—
384C22×C8—1182no—
385C22×C8—2361no—
386C22×C8—2361no—
387C4⋊Q8—1182no—
388C4⋊Q8—2361no—
389C22⋊Q8—22no—
390C22⋊Q8—2364no—
391C22⋊Q8Dic11814yescomplemented
392C4×Q8—22no—
393C4×Q8—41no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
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—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—2361no—
412C2×C4⋊C4C22×D5911yescharacteristic
413C2×C4⋊C4C22×D5911yescharacteristic
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1181no—
417C2×C22⋊C4C22⋊D5911yescharacteristic
418C2×C22⋊C4C22⋊D5911yescharacteristic
419C2×C42—21no—
420C2×C42—21no—
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423#106—1182no—
424#106—2361no—
425C4⋊C8—2361no—
426C4⋊C8—2361no—
427#103—1182no—
428#103—2361no—
429#103—2361no—
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#95C22⋊D5911yescharacteristic
440#95C22⋊D5911yescharacteristic
441C24C2×C22⋊D5911yescharacteristic
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—4721no—
448C2×Q8—2362no—
449C2×Q8—2362no—
450C2×Q8C2×Dic11812yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—81no—
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×C4#1174568512yes—
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C23×D5911yescharacteristic
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—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—2364no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—592no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—4721no—
507C4⋊C4—2362no—
508C4⋊C4—2362no—
509C4⋊C4—2362no—
510C4⋊C4—2362no—
511C4⋊C4—2362no—
512C4⋊C4C2×Dic11812yes—
513C4⋊C4#1174568712yes—
514C22⋊C4—24no—
515C22⋊C4—22no—
516C22⋊C4—2362no—
517C22⋊C4#1174567114yes—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—42no—
523C42—81no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1427370612yes—
529C23C2×#1174567112yes—
530C23#1427375712yes—
531C23C22×C22⋊D5911yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—4724no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4—1182no—
567C2×C4—4724no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
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#1427370612yes—
580C2×C4C2×#1174567112yes—
581C2×C4C2×#1174567112yes—
582C2×C4C22×Dic11811yescharacteristic
583C2×C4#41×D5911yescharacteristic
584C2×C4C2×#1174568711yescharacteristic
585C2×C4C2×#1174568511yescharacteristic
586C2×C4#1427377211yescharacteristic
587C2×C4#1427377311yescharacteristic
588C8—2361no—
589C8—2361no—
590C8—2361no—
591C8—2361no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1683768714yes—
599C22C2×#1427370611yescharacteristic
600C22#1683804011yescharacteristic
601C22#1683804611yescharacteristic
602C22C22×#1174567111yescharacteristic
603C22C2×#1427375711yescharacteristic
604C22#1683820511yescharacteristic
605C22#1683821711yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—41no—
614C4—81no—
615C4—4721no—
616C4—4721no—
617C4—4721no—
618C4—2362no—
619C4#1683793211yescharacteristic
620C4#1683798511yescharacteristic
621C4#1683810911yescharacteristic
622C4#1683811111yescharacteristic
623C2—24no—
624C2#2142791411yescharacteristic
625C2C2×#1683768711yescharacteristic
626C2#2143394711yescharacteristic
627C2#2143591411yescharacteristic
628C2#2143596111yescharacteristic
629C2#2143614111yescharacteristic
630C2#2143617711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2360C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2360D59
hypocenterC59#26412
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #21425654C2semidirect
3. #21425598C2semidirect
4. #21425477C2semidirect
5. #21425237C2semidirect
6. #21425048C2semidirect
7. #21424674C2semidirect
10. #21424364C2semidirect
13. #21424101C2semidirect
100. C22⋊Q8×C59Q8semidirect
374. C59#26412semidirect
391. C22⋊Q8Dic118semidirect

Maximal subgroups

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

subgroupindex
#214277982
#214256542
#214255982
#214254772
#214252372
#214250482
#214246742
#214245202
#214244042
#214243642
#214242892
#214242762
#214241012
C2×#168371642
C59×#23602
#2641259