Finite Group Databasebeta

← #28629679

Subgroups — #28629679

Order 15104.

778 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837709C211yesmaximal, characteristic, complemented
2C2×C4×#11745671C211yesmaximal, characteristic
3#21429296C211yesmaximal, characteristic
4#21425514C211yesmaximal, characteristic, complemented
5#21425504C211yesmaximal, characteristic, complemented
6#21425411C211yesmaximal, characteristic
7#21425411C211yesmaximal, characteristic
8#21425060C211yesmaximal, characteristic, complemented
9#21424864C211yesmaximal, characteristic, complemented
10#21424859C211yesmaximal, characteristic, complemented
11#21424815C211yesmaximal, characteristic, complemented
12#21424723C211yesmaximal, characteristic
13#21424716C211yesmaximal, characteristic
14#21424103C211yesmaximal, characteristic
15C59×#2356C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19#16837709—24no—
20C4×#11745671—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273647C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837324—21no—
35#16837324—21no—
36#16837304C2211yescharacteristic
37#16837304C2211yescharacteristic
38#16837276—21no—
39#16837276—21no—
40#16837272C2211yescharacteristic
41#16837268C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C59×#391C2211yescharacteristic
52C59×#391C2211yescharacteristic
53C59×#387—21no—
54C59×#387—21no—
55C59×#383—21no—
56C59×#382C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C59×#376—21no—
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C2×#11745671—22no—
64C2×#11745671—22no—
65C2×#11745671—22no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70#14273719—22no—
71#14273719—22no—
72C4×D236—22no—
73C4×D236—22no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86#14273660—22no—
87#14273658—22no—
88#14273658—22no—
89C4⋊(C59⋊C8)—22no—
90C4⋊(C59⋊C8)—22no—
91C4⋊(C59⋊C8)—22no—
92#14273651—22no—
93#14273651—22no—
94#14273651—22no—
95#14273647—22no—
96#14273647—22no—
97#14273647—22no—
98C4×C59⋊C8—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×Q8×C59C2311yescharacteristic
102C4×Q8×C59—22no—
103C4×Q8×C59D414yes—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59D411yescharacteristic
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C42×C59—21no—
111C2×C42×C59—21no—
112C2×C42×C59—21no—
113C2×C42×C59C2311yescharacteristic
114C2×C42×C59C2311yescharacteristic
115C2×C42×C59C2311yescharacteristic
116#95×C59—21no—
117#95×C59—21no—
118#95×C59—21no—
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
126C23×D59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59C2×D412yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236#4112yes—
153C2×C4×D59—42no—
154C2×C4×D59—81no—
155C2×C4×D59—81no—
156C4⋊C4×C59—24no—
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×C59C2×D412yes—
165C42×C59—24no—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59—42no—
171C42×C59—42no—
172C42×C59C2×D412yes—
173C42×C59#4112yes—
174#11745671—28no—
175(C2×D59)⋊C4—42no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—81no—
179(C2×D59)⋊C4—81no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—81no—
187C4⋊Dic59—81no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
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#4112yes—
202C23×C59C22×D411yescharacteristic
203C22×D59—22no—
204C22×D59—41no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
210Q8×C59—22no—
211Q8×C59—22no—
212Q8×C59—41no—
213Q8×C59—41no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
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—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
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#12112yes—
242C2×C4×C59#12112yes—
243C2×C4×C59#13512yes—
244C2×C4×C59C22×D411yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C592−1+411yescharacteristic
249C2×C4×C592−1+411yescharacteristic
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259D236—22no—
260D236—22no—
261D236—41no—
262D236—41no—
263C4×D59—84no—
264C59⋊C8—22no—
265C59⋊C8—41no—
266C59⋊C8—41no—
267C59⋊C8—41no—
268#24685—591nomaximal, Sylow
269C22×C59C2×#12111yescharacteristic
270C22×C59#54711yescharacteristic
271C22×C59#54811yescharacteristic
272C22×C59C2×#13511yescharacteristic
273C22×C59#57511yescharacteristic
274C22×C59#57711yescharacteristic
275C22×C59#57711yescharacteristic
276C2×D59—42no—
277C2×D59—42no—
278C2×D59—42no—
279C2×D59—81no—
280C2×D59—81no—
281C2×D59—81no—
282C2×D59—81no—
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—22no—
286C4×C59—41no—
287C4×C59—41no—
288C4×C59—41no—
289C4×C59—42no—
290C4×C59—42no—
291C4×C59—81no—
292C4×C59—81no—
293C4×C59#52511yescharacteristic
294C4×C59#53611yescharacteristic
295C4×C59#55211yescharacteristic
296C4×C59#55611yescharacteristic
297Dic59—81no—
298Dic59—81no—
299Dic59—81no—
300C2×#479—591no—
301C2×C4×SD16—591no—
302#2566—591no—
303#2356D5911yescharacteristic, complemented
304#2074—591no—
305#2072—591no—
306#2055—591no—
307#2055—591no—
308#1991—591no—
309#1949—591no—
310#1948—591no—
311#1939—591no—
312#1926—591no—
313#1925—591no—
314#1804—591no—
315C2×C59#241511yescharacteristic
316C2×C59#303511yescharacteristic
317C2×C59#304311yescharacteristic
318C2×C59#333711yescharacteristic
319C2×C59#335111yescharacteristic
320C2×C59#336011yescharacteristic
321C2×C59#337611yescharacteristic
322D59—82no—
323C22×SD16—591no—
324C2×#125—591no—
325C2×C4×Q8C2×D5911yescharacteristic
326C2×C4×D4—591no—
327#479—1184no—
328C4×SD16—1184no—
329C2×#107—591no—
330C2×#106—591no—
331C2×#106—591no—
332C2×C4⋊C8—591no—
333C2×C4⋊C8—591no—
334C2×C4⋊C8—591no—
335C2×#103—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#102—591no—
339C2×#102—591no—
340C2×#102—591no—
341C2×C4×C8—591no—
342#396—1181no—
343#396—1181no—
344#391C2×D5911yescharacteristic
345#391C2×D5911yescharacteristic
346#390—591no—
347#390—591no—
348#387—21no—
349#387—21no—
350#385—1181no—
351#385—1181no—
352#384—591no—
353#383—21no—
354#382—591no—
355#382C2×D5911yescharacteristic
356C4×C4⋊C4C2×D5911yescharacteristic
357C4×C4⋊C4C2×D5911yescharacteristic
358C4×C4⋊C4C2×D5911yescharacteristic
359#376—21no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#336—591no—
367#336—591no—
368C59#2468511yescharacteristic, complemented, Sylow
369C22×Q8C22×D5911yescharacteristic
370C22×D4—591no—
371C23×C4—1181no—
372C2×SD16—1182no—
373C2×SD16—1182no—
374C2×SD16—1182no—
375C22×C8—591no—
376C22×C8—591no—
377C22×C8—591no—
378C22×C8—591no—
379#125—1182no—
380#125—1182no—
381C4×Q8—22no—
382C4×Q8D23614yes—
383C4×D4—1182no—
384C4×D4—1182no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4C22⋊D5911yescharacteristic
394C2×C4⋊C4C22⋊D5911yescharacteristic
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#107—1182no—
410#106—1182no—
411#106—1182no—
412C4⋊C8—1182no—
413C4⋊C8—1182no—
414C4⋊C8—1182no—
415#103—1182no—
416#103—1182no—
417#103—1182no—
418#102—1182no—
419#102—1182no—
420#102—1182no—
421C4×C8—1182no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—1181no—
431#95—1181no—
432#95C22⋊D5911yescharacteristic
433#95C22⋊D5911yescharacteristic
434C24—1181no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×D23612yes—
441C2×D4—592no—
442C2×D4—1181no—
443C2×D4—1181no—
444C2×D4—1181no—
445C2×D4—1181no—
446C2×D4—1181no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—4721no—
455C22×C4—4721no—
456C22×C4—2362no—
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C23×D5911yescharacteristic
464SD16—1188no—
465C2×C8—592no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—41no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—4721no—
492C4⋊C4—4721no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4C2×D23612yes—
496C22⋊C4—4721no—
497C22⋊C4—4721no—
498C22⋊C4—2362no—
499C22⋊C4—2362no—
500C22⋊C4—2362no—
501C42—24no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—42no—
508C42C2×D23612yes—
509C42#1174568512yes—
510C23—1182no—
511C23—2361no—
512C23—2361no—
513C23—2361no—
514C23—2361no—
515C23—2361no—
516C23—2361no—
517C23C22×C22⋊D5911yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—1182no—
523D4—1182no—
524D4—2361no—
525D4—2361no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
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—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—4724no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4#1427370112yes—
564C2×C4#1427370112yes—
565C2×C4#1427374712yes—
566C2×C4C22×D23611yescharacteristic
567C2×C4#41×D5911yescharacteristic
568C2×C4#41×D5911yescharacteristic
569C2×C4C2×#1174568511yescharacteristic
570C2×C4#1427377411yescharacteristic
571C2×C4#1427377411yescharacteristic
572C8—1182no—
573C8—2361no—
574C8—2361no—
575C8—2361no—
576C22—4721no—
577C22—4721no—
578C22—4721no—
579C22—4721no—
580C22—2362no—
581C22—2362no—
582C22—2362no—
583C22C2×#1427370111yescharacteristic
584C22#1683805111yescharacteristic
585C22#1683805911yescharacteristic
586C22C2×#1427374711yescharacteristic
587C22#1683820011yescharacteristic
588C22#1683821011yescharacteristic
589C22#1683821011yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—42no—
595C4—42no—
596C4—41no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—4721no—
601C4—4721no—
602C4—4721no—
603C4#1683791411yescharacteristic
604C4#1683797811yescharacteristic
605C4#1683809111yescharacteristic
606C4#1683812011yescharacteristic
607C2—4722no—
608C2#2142751911yescharacteristic
609C2#2143355111yescharacteristic
610C2#2143361711yescharacteristic
611C2#2143647511yescharacteristic
612C2#2143665611yescharacteristic
613C2#2143674111yescharacteristic
614C2#2143691711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2356C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2356D59
hypocenterC59#24685
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2356

Complements

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

NH ≅ G/Ntype
1. C2×#16837709C2semidirect
4. #21425514C2semidirect
5. #21425504C2semidirect
8. #21425060C2semidirect
9. #21424864C2semidirect
10. #21424859C2semidirect
11. #21424815C2semidirect
15. C59×#2356C2semidirect
303. #2356D59semidirect
368. C59#24685semidirect

Maximal subgroups

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

subgroupindex
C2×#168377092
C2×C4×#117456712
#214292962
#214255142
#214255042
#214254112
#214250602
#214248642
#214248592
#214248152
#214247232
#214247162
#214241032
C59×#23562
#2468559