Finite Group Databasebeta

← #28610086

Subgroups — #28610086

Order 15104.

874 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837732C211yesmaximal, characteristic
2#21427099C211yesmaximal, characteristic
3#21424836C211yesmaximal, characteristic, complemented
4#21424785C211yesmaximal, characteristic, complemented
5#21424737C211yesmaximal, characteristic, complemented
6#21424678C211yesmaximal, characteristic, complemented
7#21424674C211yesmaximal, characteristic, complemented
8#21424517C211yesmaximal, characteristic
9#21424349C211yesmaximal, characteristic, complemented
10#21424283C211yesmaximal, characteristic
11#21423994C211yesmaximal, characteristic, complemented
12#21423929C211yesmaximal, characteristic, complemented
13#21423874C211yesmaximal, characteristic
14C2×#16837164C211yesmaximal, characteristic
15C59×#2344C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837732C418yescomplemented
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×#14273638C2211yescharacteristic
32#16837308C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837299—21no—
35#16837289—21no—
36#16837274—21no—
37C4×C4⋊Dic59C2211yescharacteristic
38C22⋊C4×Dic59—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×#117×C59—21no—
50C22×C4⋊C4×C59C2211yescharacteristic
51C59×#389C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#387—21no—
54C59×#387—21no—
55C59×#381—21no—
56C59×#380C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C23×Dic59—21no—
62C22×C59⋊C8—22no—
63C22×C59⋊C8—22no—
64C22×C59⋊C8—41no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273706—22no—
70#14273706C2×C414yes—
71C4×Dic118—22no—
72C4×Dic118—41no—
73C2×C4⋊Dic59—22no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—41no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85#14273660—22no—
86#14273660—41no—
87#14273658C2×C414yescomplemented
88#14273658C2×C414yescomplemented
89#14273650—22no—
90#14273650—41no—
91#14273650C2×C414yescomplemented
92#14273650C2×C414yescomplemented
93C22⋊(C59⋊C8)C2×C414yes—
94#14273638—22no—
95#14273638—41no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C23×C4×C59—21no—
101C23×C4×C59C2311yescharacteristic
102#117×C59—44no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—22no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59C2×C414yes—
108C2×C4⋊C4×C59C2311yescharacteristic
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×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59C2×D411yescharacteristic
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—24no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—42no—
137C22×C4×C59—42no—
138C22×C4×C59C22×C412yes—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59#4112yes—
145C22×C4×C59C2411yescharacteristic
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—42no—
151C22×Dic59—42no—
152C22×Dic59—81no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118C22×C412yes—
159C4⋊C4×C59—24no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
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×C59C22×C412yes—
169C4⋊C4×C59C22×C412yes—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—42no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—42no—
178C42×C59—42no—
179C22⋊Dic59—24no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—42no—
185C4⋊Dic59C22×C412yes—
186C4⋊Dic59#4112yes—
187Dic59⋊C4—24no—
188Dic59⋊C4—42no—
189Dic59⋊C4—81no—
190C4×Dic59—42no—
191C4×Dic59—81no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
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—44no—
202C2×C59⋊C8C22×C412yes—
203C2×C59⋊C8C22×C412yes—
204C2×C59⋊C8#4112yes—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59C4×D412yes—
210C23×C59AGL(1,ℤ/8)12yes—
211C23×C59#13712yes—
212C23×C59C22×D411yescharacteristic
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—48no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59C4×D412yes—
245C2×C4×C59C23×C411yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59H(5,𝔽2)11yescharacteristic
250C2×C4×C592−1+411yescharacteristic
251C2×Dic59—22no—
252C2×Dic59—22no—
253C2×Dic59—22no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—84no—
264Dic118—22no—
265Dic118—22no—
266Dic118—41no—
267Dic118—41no—
268C59⋊C8—22no—
269C59⋊C8—22no—
270C59⋊C8—22no—
271C59⋊C8—41no—
272#22934—591nomaximal, Sylow
273C22×C59—22no—
274C22×C59—22no—
275C22×C59—22no—
276C22×C59—22no—
277C22×C59—22no—
278C22×C59—22no—
279C22×C59#44014yes—
280C22×C59C2×C4×D411yescharacteristic
281C22×C59#54611yescharacteristic
282C22×C59#54711yescharacteristic
283C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C59C2×#13711yescharacteristic
285C22×C59#57711yescharacteristic
286C22×C59#57811yescharacteristic
287C4×C59—22no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—42no—
291C4×C59—42no—
292C4×C59—41no—
293C4×C59—41no—
294C4×C59—81no—
295C4×C59—81no—
296C4×C59C4×#4111yescharacteristic
297C4×C59#52011yescharacteristic
298C4×C59#55111yescharacteristic
299C4×C59#55211yescharacteristic
300Dic59—42no—
301Dic59—42no—
302Dic59—42no—
303Dic59—81no—
304C2×#483—591no—
305#2360—591no—
306#2344D5911yescharacteristic
307#1944—591no—
308#1935—591no—
309#1928—591no—
310#1920—591no—
311#1919—591no—
312#1892—591no—
313#1863—591no—
314#1852—591no—
315#1776—591no—
316#1763—591no—
317#1752—591no—
318C2×#340—591no—
319C2×C59—24no—
320C2×C59#235911yescharacteristic
321C2×C59C2×#44011yescharacteristic
322C2×C59#294711yescharacteristic
323C2×C59#330611yescharacteristic
324C2×C59#331211yescharacteristic
325C2×C59#334311yescharacteristic
326C2×C59#334411yescharacteristic
327C23×C8—591no—
328C2×C22⋊Q8—591no—
329C2×C4×Q8—591no—
330C2×#117—21no—
331C22×C4⋊C4—591no—
332C22×C4⋊C4C2×D5911yescharacteristic
333#483—598no—
334C2×#107—591no—
335C2×#107—591no—
336C2×#106—591no—
337C2×#106—591no—
338C2×#103—591no—
339C2×#103—591no—
340C2×#103—591no—
341C2×#103—591no—
342C2×C22⋊C8—591no—
343C2×#97—591no—
344C2×#97—591no—
345#391—591no—
346#389—591no—
347#389—1181no—
348#389C2×D5911yescharacteristic
349#389C2×D5911yescharacteristic
350#387—21no—
351#387—21no—
352#387—1181no—
353#385—1181no—
354#381—21no—
355#380C2×D5911yescharacteristic
356C4×C4⋊C4—591no—
357C4×C4⋊C4C2×D5911yescharacteristic
358C4×C4⋊C4C2×D5911yescharacteristic
359C4×C22⋊C4—1181no—
360C4×C22⋊C4C2×D5911yescharacteristic
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#336—591no—
370#336—591no—
371C59#2293411yescharacteristic, complemented, Sylow
372C22×Q8—591no—
373C23×C4—21no—
374C23×C4—1181no—
375C23×C4C22×D5911yescharacteristic
376C22×C8—591no—
377C22×C8—591no—
378C22×C8—591no—
379C22×C8—591no—
380C22×C8—1182no—
381C22×C8—1182no—
382C22×C8—2361no—
383C22⋊Q8—594no—
384C22⋊Q8—1182no—
385C4×Q8—1182no—
386C4×Q8—2361no—
387#117—44no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—1182no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—2361no—
401C2×C4⋊C4C4×D5914yes—
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1181no—
408C2×C22⋊C4C22⋊D5911yescharacteristic
409C2×C22⋊C4C22⋊D5911yescharacteristic
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—1181no—
413C2×C42—1181no—
414C2×C42C22×D5911yescharacteristic
415C2×C42C22×D5911yescharacteristic
416#107—1182no—
417#107—2361no—
418#106—594no—
419#106—594no—
420#103—594no—
421#103—594no—
422#103—1182no—
423#103—2361no—
424C22⋊C8—594no—
425#97—1182no—
426#97—2361no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95C22⋊D5911yescharacteristic
437#95C22⋊D5911yescharacteristic
438C24C2×C22⋊D5911yescharacteristic
439C2×Q8—592no—
440C2×Q8—1181no—
441C2×Q8—1181no—
442C2×Q8—1181no—
443C2×Q8—1181no—
444C2×Q8—1181no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—4721no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4C2×C4×D5912yes—
464C22×C4#1174568512yes—
465C22×C4C2×C22⋊D5911yescharacteristic
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C2×C22⋊D5911yescharacteristic
468C22×C4C2×C22⋊D5911yescharacteristic
469C22×C4C2×C22⋊D5911yescharacteristic
470C22×C4C23×D5911yescharacteristic
471C2×C8—592no—
472C2×C8—592no—
473C2×C8—592no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—2364no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—592no—
494C4⋊C4—592no—
495C4⋊C4—1184no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—4721no—
501C4⋊C4—2362no—
502C4⋊C4—2362no—
503C4⋊C4C2×C4×D5912yes—
504C4⋊C4C2×C4×D5912yes—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—42no—
508C22⋊C4—1184no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—42no—
515C42—4721no—
516C42—2362no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23C4×C22⋊D5912yes—
522C23#1427375312yes—
523C23#1427375712yes—
524C23C22×C22⋊D5911yescharacteristic
525Q8—1182no—
526Q8—1182no—
527Q8—2361no—
528Q8—2361no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—48no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—1182no—
561C2×C4—1182no—
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×C4C4×C22⋊D5912yes—
574C2×C4C22×C4×D5911yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C2×C4#41×D5911yescharacteristic
577C2×C4C2×#1174568511yescharacteristic
578C2×C4#1427377311yescharacteristic
579C2×C4#1427377411yescharacteristic
580C8—1182no—
581C8—1182no—
582C8—1182no—
583C8—2361no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22#1683750214yes—
591C22C2×C4×C22⋊D5911yescharacteristic
592C22#1683804011yescharacteristic
593C22#1683805111yescharacteristic
594C22C2×#1427375311yescharacteristic
595C22C2×#1427375711yescharacteristic
596C22#1683821011yescharacteristic
597C22#1683821711yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—4721no—
608C4—2362no—
609C4—2362no—
610C4—2362no—
611C4#1683787011yescharacteristic
612C4#1683788211yescharacteristic
613C4#1683807611yescharacteristic
614C4#1683808511yescharacteristic
615C2—24no—
616C2#2142708811yescharacteristic
617C2C2×#1683750211yescharacteristic
618C2#2143293911yescharacteristic
619C2#2143605111yescharacteristic
620C2#2143614111yescharacteristic
621C2#2143653011yescharacteristic
622C2#2143655111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2344C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#2344D59
hypocenterC59#22934
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
3. #21424836C2semidirect
4. #21424785C2semidirect
5. #21424737C2semidirect
6. #21424678C2semidirect
7. #21424674C2semidirect
9. #21424349C2semidirect
11. #21423994C2semidirect
12. #21423929C2semidirect
20. #16837732C4semidirect
87. #14273658C2×C4semidirect
88. #14273658C2×C4semidirect
91. #14273650C2×C4semidirect
92. #14273650C2×C4semidirect
371. C59#22934semidirect

Maximal subgroups

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

subgroupindex
C2×#168377322
#214270992
#214248362
#214247852
#214247372
#214246782
#214246742
#214245172
#214243492
#214242832
#214239942
#214239292
#214238742
C2×#168371642
C59×#23442
#2293459