Finite Group Databasebeta

← #28629754

Subgroups — #28629754

Order 15104.

808 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837702C211yesmaximal, characteristic
2C2×#16837502C211yesmaximal, characteristic
3#21429287C211yesmaximal, characteristic, complemented
4#21425509C211yesmaximal, characteristic
5#21425499C211yesmaximal, characteristic
6#21425413C211yesmaximal, characteristic
7#21425413C211yesmaximal, characteristic
8#21425064C211yesmaximal, characteristic, complemented
9#21424864C211yesmaximal, characteristic, complemented
10#21424854C211yesmaximal, characteristic, complemented
11#21424820C211yesmaximal, characteristic, complemented
12#21424737C211yesmaximal, characteristic, complemented
13#21424730C211yesmaximal, characteristic, complemented
14#21424098C211yesmaximal, characteristic, complemented
15C59×#2354C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19#16837702—24no—
20#16837502—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×#14273638C2211yescharacteristic
34#16837332—21no—
35#16837332—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837287—21no—
39#16837287—21no—
40#16837272C2211yescharacteristic
41#16837268C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×#117×C59—21no—
52C59×#390C2211yescharacteristic
53C59×#390C2211yescharacteristic
54C59×#388—21no—
55C59×#388—21no—
56C59×#384C2211yescharacteristic
57C59×#380—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C2×#11745670—22no—
63C2×#11745670—22no—
64C2×#11745670—22no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273719—22no—
70#14273719—22no—
71C4×Dic118—22no—
72C4×Dic118—22no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C4×Dic59—21no—
82#14273660—22no—
83#14273660—22no—
84#14273658—22no—
85C4⋊(C59⋊C8)—22no—
86C4⋊(C59⋊C8)—22no—
87C4⋊(C59⋊C8)—22no—
88#14273650—22no—
89#14273650—22no—
90#14273650—22no—
91#14273648—22no—
92#14273648—22no—
93#14273648—22no—
94#14273638—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×D4×C59C2311yescharacteristic
102C23×C4×C59—21no—
103C4×D4×C59—22no—
104C4×D4×C59D414yes—
105#117×C59—44no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C42×C59—21no—
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123#95×C59—21no—
124#95×C59—21no—
125#95×C59D411yescharacteristic
126#95×C59D411yescharacteristic
127C24×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59C2×D412yes—
134C22×C4×C59—21no—
135C22×C4×C59—24no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—42no—
139C22×C4×C59—42no—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2411yescharacteristic
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118#4112yes—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—41no—
164C4⋊C4×C59—42no—
165C4⋊C4×C59—81no—
166C4⋊C4×C59C2×D412yes—
167C22⋊C4×C59—24no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—42no—
170C22⋊C4×C59—81no—
171C22⋊C4×C59—81no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—42no—
177C42×C59—42no—
178C42×C59C2×D412yes—
179C42×C59#4112yes—
180#11745670—28no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—42no—
188C4⋊Dic59—42no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191Dic59⋊C4—42no—
192C4×Dic59—42no—
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—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8#4112yes—
205C23×C59—22no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59C22×D411yescharacteristic
213D4×C59—22no—
214D4×C59—22no—
215D4×C59—41no—
216D4×C59—41no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—22no—
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—48no—
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#12112yes—
243C2×C4×C59#12112yes—
244C2×C4×C59AGL(1,ℤ/8)12yes—
245C2×C4×C59C22×D411yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C592−1+411yescharacteristic
250C2×C4×C592−1+411yescharacteristic
251C2×Dic59—22no—
252C2×Dic59—41no—
253C2×Dic59—41no—
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—
263Dic118—22no—
264Dic118—22no—
265Dic118—41no—
266Dic118—41no—
267C59⋊C8—22no—
268C59⋊C8—41no—
269C59⋊C8—41no—
270C59⋊C8—41no—
271#24690—591nomaximal, Sylow
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—42no—
275C22×C59—81no—
276C22×C59—81no—
277C22×C59—81no—
278C22×C59—81no—
279C22×C59C2×#12111yescharacteristic
280C22×C59#54711yescharacteristic
281C22×C59#54811yescharacteristic
282C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C59#57511yescharacteristic
284C22×C59#57711yescharacteristic
285C22×C59#57811yescharacteristic
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—22no—
289C4×C59—41no—
290C4×C59—41no—
291C4×C59—41no—
292C4×C59—42no—
293C4×C59—81no—
294C4×C59—81no—
295C4×C59#52511yescharacteristic
296C4×C59#53611yescharacteristic
297C4×C59#55211yescharacteristic
298C4×C59#55611yescharacteristic
299Dic59—42no—
300Dic59—81no—
301Dic59—81no—
302Dic59—81no—
303C2×#478—591no—
304C2×#440—591no—
305#2565—591no—
306#2354D5911yescharacteristic
307#2073—591no—
308#2071—591no—
309#2055—591no—
310#2055—591no—
311#1992—591no—
312#1949—591no—
313#1947—591no—
314#1940—591no—
315#1928—591no—
316#1927—591no—
317#1803—591no—
318C2×C59—82no—
319C2×C59#241511yescharacteristic
320C2×C59#303611yescharacteristic
321C2×C59#304611yescharacteristic
322C2×C59#334311yescharacteristic
323C2×C59#334811yescharacteristic
324C2×C59#336811yescharacteristic
325C2×C59#337711yescharacteristic
326C22×SD16—591no—
327C2×#125—591no—
328C2×C4×Q8—591no—
329C2×C4×D4C2×D5911yescharacteristic
330C2×#117—21no—
331#478—1184no—
332#440—1184no—
333C2×#107—591no—
334C2×#107—591no—
335C2×#106—591no—
336C2×C4⋊C8—591no—
337C2×C4⋊C8—591no—
338C2×C4⋊C8—591no—
339C2×#103—591no—
340C2×#103—591no—
341C2×#103—591no—
342C2×#102—591no—
343C2×#102—591no—
344C2×#102—591no—
345C2×#97—591no—
346#398—1181no—
347#398—1181no—
348#391—591no—
349#391—591no—
350#390C2×D5911yescharacteristic
351#390C2×D5911yescharacteristic
352#388—21no—
353#388—21no—
354#387—1181no—
355#387—1181no—
356#384—591no—
357#384C2×D5911yescharacteristic
358#382—591no—
359#380—21no—
360C4×C4⋊C4C2×D5911yescharacteristic
361C4×C4⋊C4C2×D5911yescharacteristic
362C4×C4⋊C4C2×D5911yescharacteristic
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#2469011yescharacteristic, complemented, Sylow
372C22×Q8—591no—
373C22×D4C22×D5911yescharacteristic
374C23×C4—21no—
375C2×SD16—1182no—
376C2×SD16—1182no—
377C2×SD16—1182no—
378C22×C8—591no—
379C22×C8—591no—
380C22×C8—591no—
381C22×C8—591no—
382#125—1182no—
383#125—1182no—
384C4×Q8—1182no—
385C4×Q8—1182no—
386C4×D4—22no—
387C4×D4D23614yes—
388#117—44no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4C22⋊D5911yescharacteristic
400C2×C4⋊C4C22⋊D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—1181no—
412C2×C42C22×D5911yescharacteristic
413C2×C42C22×D5911yescharacteristic
414C2×C42C22×D5911yescharacteristic
415#107—1182no—
416#107—1182no—
417#106—1182no—
418C4⋊C8—1182no—
419C4⋊C8—1182no—
420C4⋊C8—1182no—
421#103—1182no—
422#103—1182no—
423#103—1182no—
424#102—1182no—
425#102—1182no—
426#102—1182no—
427#97—1182no—
428#95—21no—
429#95—21no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95C22⋊D5911yescharacteristic
437#95C22⋊D5911yescharacteristic
438C24—21no—
439C2×Q8—592no—
440C2×Q8—1181no—
441C2×Q8—1181no—
442C2×Q8—1181no—
443C2×Q8—1181no—
444C2×Q8—1181no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×D23612yes—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C2×C22⋊D5911yescharacteristic
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C23×D5911yescharacteristic
468SD16—1188no—
469C2×C8—592no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—2362no—
497C4⋊C4—2362no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4C2×D23612yes—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—42no—
513C42—2362no—
514C42C2×D23612yes—
515C42#1174568512yes—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C22×C22⋊D5911yescharacteristic
524Q8—1182no—
525Q8—1182no—
526Q8—2361no—
527Q8—2361no—
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
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—21no—
548C2×C4—48no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—1182no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4#1427370112yes—
570C2×C4#1427370112yes—
571C2×C4#1427375312yes—
572C2×C4C22×D23611yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C2×C4#41×D5911yescharacteristic
575C2×C4C2×#1174568511yescharacteristic
576C2×C4#1427377411yescharacteristic
577C2×C4#1427377411yescharacteristic
578C8—1182no—
579C8—2361no—
580C8—2361no—
581C8—2361no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22C2×#1427370111yescharacteristic
590C22#1683805111yescharacteristic
591C22#1683805911yescharacteristic
592C22C2×#1427375311yescharacteristic
593C22#1683820011yescharacteristic
594C22#1683821311yescharacteristic
595C22#1683821711yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—4721no—
606C4—4721no—
607C4—4721no—
608C4—2362no—
609C4#1683791411yescharacteristic
610C4#1683797811yescharacteristic
611C4#1683809111yescharacteristic
612C4#1683812011yescharacteristic
613C2—82no—
614C2#2142751911yescharacteristic
615C2#2143356011yescharacteristic
616C2#2143364611yescharacteristic
617C2#2143653011yescharacteristic
618C2#2143661111yescharacteristic
619C2#2143682111yescharacteristic
620C2#2143693211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2354C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2354D59
hypocenterC59#24690
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
3. #21429287C2semidirect
8. #21425064C2semidirect
9. #21424864C2semidirect
10. #21424854C2semidirect
11. #21424820C2semidirect
12. #21424737C2semidirect
13. #21424730C2semidirect
14. #21424098C2semidirect
371. C59#24690semidirect

Maximal subgroups

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

subgroupindex
C2×#168377022
C2×#168375022
#214292872
#214255092
#214254992
#214254132
#214250642
#214248642
#214248542
#214248202
#214247372
#214247302
#214240982
C59×#23542
#2469059