Finite Group Databasebeta

← #28629741

Subgroups — #28629741

Order 15104.

780 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837706C211yesmaximal, characteristic, complemented
2C2×#16837502C211yesmaximal, characteristic
3#21427051C211yesmaximal, characteristic, complemented
4#21425515C211yesmaximal, characteristic, complemented
5#21425505C211yesmaximal, characteristic, complemented
6#21425413C211yesmaximal, characteristic
7#21425413C211yesmaximal, characteristic
8#21425066C211yesmaximal, characteristic, complemented
9#21424862C211yesmaximal, characteristic, complemented
10#21424852C211yesmaximal, characteristic, complemented
11#21424820C211yesmaximal, characteristic, complemented
12#21424720C211yesmaximal, characteristic
13#21424713C211yesmaximal, characteristic
14#21424096C211yesmaximal, characteristic
15C59×#2566C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18#16837706—24no—
19#16837502—24no—
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273638C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837308C2211yescharacteristic
35#16837289—21no—
36#16837289—21no—
37#16837272C2211yescharacteristic
38#16837267—21no—
39C4⋊C4×Dic59—21no—
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#125×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C59×#396—21no—
53C59×#396—21no—
54C59×#390C2211yescharacteristic
55C59×#390C2211yescharacteristic
56C59×#385—21no—
57C59×#385—21no—
58C59×#384C2211yescharacteristic
59C59×#382C2211yescharacteristic
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
69C4×Dic118—22no—
70C4×Dic118D414yes—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—22no—
81#14273660—22no—
82#14273658—22no—
83C4⋊(C59⋊C8)—22no—
84C4⋊(C59⋊C8)—22no—
85C4⋊(C59⋊C8)—22no—
86#14273650—22no—
87#14273650—22no—
88#14273650—22no—
89#14273648—22no—
90#14273648—22no—
91#14273648—22no—
92#14273638—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C22×D4×C59C2311yescharacteristic
100C23×C4×C59—21no—
101#125×C59—22no—
102#125×C59—22no—
103C4×D4×C59—22no—
104C4×D4×C59—22no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59D411yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59#4112yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—42no—
137C22×C4×C59—81no—
138C22×C4×C59—81no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2411yescharacteristic
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118C2×D412yes—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—21no—
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—42no—
166C4⋊C4×C59—81no—
167C4⋊C4×C59—81no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—42no—
170C22⋊C4×C59—42no—
171C22⋊C4×C59—81no—
172C22⋊C4×C59—81no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59C2×D412yes—
178C42×C59#4112yes—
179#11745670—28no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—42no—
186C4⋊Dic59—81no—
187C4⋊Dic59C2×D412yes—
188Dic59⋊C4—24no—
189C4×Dic59—24no—
190C4×Dic59—81no—
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—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8#4112yes—
204C23×C59—22no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59C22×D411yescharacteristic
212D4×C59—22no—
213D4×C59—22no—
214D4×C59—41no—
215D4×C59—41no—
216C2×C4×C59—21no—
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—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—84no—
241C2×C4×C59#12112yes—
242C2×C4×C59#12112yes—
243C2×C4×C59#13712yes—
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—22no—
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—
262Dic118—22no—
263Dic118—22no—
264Dic118—41no—
265Dic118—41no—
266C59⋊C8—22no—
267C59⋊C8—41no—
268C59⋊C8—41no—
269C59⋊C8—41no—
270#24689—591nomaximal, Sylow
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59—81no—
277C22×C59—81no—
278C22×C59C2×#12111yescharacteristic
279C22×C59#54711yescharacteristic
280C22×C59#54811yescharacteristic
281C22×C59C2×#13711yescharacteristic
282C22×C59#57511yescharacteristic
283C22×C59#57611yescharacteristic
284C22×C59#57711yescharacteristic
285C4×C59—22no—
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—41no—
291C4×C59—81no—
292C4×C59—81no—
293C4×C59—81no—
294C4×C59#52511yescharacteristic
295C4×C59#53611yescharacteristic
296C4×C59#55211yescharacteristic
297C4×C59#55611yescharacteristic
298Dic59—42no—
299Dic59—42no—
300Dic59—81no—
301Dic59—81no—
302C2×#479—591no—
303C2×#440—591no—
304#2566D5911yescharacteristic
305#2355—591no—
306#2074—591no—
307#2072—591no—
308#2055—591no—
309#2055—591no—
310#1992—591no—
311#1949—591no—
312#1947—591no—
313#1940—591no—
314#1926—591no—
315#1925—591no—
316#1802—591no—
317C2×C59—82no—
318C2×C59#241511yescharacteristic
319C2×C59#303711yescharacteristic
320C2×C59#304511yescharacteristic
321C2×C59#334611yescharacteristic
322C2×C59#335011yescharacteristic
323C2×C59#337011yescharacteristic
324C2×C59#337211yescharacteristic
325C22×SD16—591no—
326C2×#125C2×D5911yescharacteristic
327C2×C4×Q8—591no—
328C2×C4×D4C2×D5911yescharacteristic
329#479—1184no—
330#440—1184no—
331C2×#107—591no—
332C2×#107—591no—
333C2×#106—591no—
334C2×C4⋊C8—591no—
335C2×C4⋊C8—591no—
336C2×C4⋊C8—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#103—591no—
340C2×#102—591no—
341C2×#102—591no—
342C2×#102—591no—
343C2×#97—591no—
344#396—21no—
345#396—21no—
346#391—591no—
347#391—591no—
348#390C2×D5911yescharacteristic
349#390C2×D5911yescharacteristic
350#387—1181no—
351#387—1181no—
352#385—21no—
353#385—21no—
354#384—591no—
355#384C2×D5911yescharacteristic
356#382—1181no—
357#382C2×D5911yescharacteristic
358C4×C4⋊C4—591no—
359C4×C4⋊C4—591no—
360C4×C4⋊C4—1181no—
361C4×C4⋊C4C2×D5911yescharacteristic
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#336—591no—
369#336—591no—
370C59#2468911yescharacteristic, complemented, Sylow
371C22×Q8—591no—
372C22×D4C22×D5911yescharacteristic
373C23×C4—21no—
374C2×SD16—1182no—
375C2×SD16—1182no—
376C2×SD16—1182no—
377C22×C8—591no—
378C22×C8—591no—
379C22×C8—591no—
380C22×C8—591no—
381#125—22no—
382#125—22no—
383C4×Q8—594no—
384C4×Q8—1182no—
385C4×D4—22no—
386C4×D4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
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⋊C4C22⋊D5911yescharacteristic
398C2×C4⋊C4C22⋊D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42C22×D5911yescharacteristic
411C2×C42C22×D5911yescharacteristic
412C2×C42C22×D5911yescharacteristic
413#107—1182no—
414#107—1182no—
415#106—1182no—
416C4⋊C8—1182no—
417C4⋊C8—1182no—
418C4⋊C8—1182no—
419#103—1182no—
420#103—1182no—
421#103—1182no—
422#102—1182no—
423#102—1182no—
424#102—1182no—
425#97—1182no—
426#95—21no—
427#95—21no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95C22⋊D5911yescharacteristic
435#95C22⋊D5911yescharacteristic
436C24—21no—
437C2×Q8—592no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C2×Q8—1181no—
442C2×Q8—1181no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4#1174568912yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466SD16—1188no—
467C2×C8—592no—
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—
477C2×C8—1181no—
478C2×C8—1181no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—592no—
492C4⋊C4—1184no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—4721no—
499C4⋊C4—2362no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—1184no—
510C42—4721no—
511C42—4721no—
512C42D4×D5912yes—
513C42#1174568712yes—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C22×C22⋊D5911yescharacteristic
522Q8—1182no—
523Q8—1182no—
524Q8—2361no—
525Q8—2361no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
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—21no—
544C2×C4—21no—
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—41no—
554C2×C4—84no—
555C2×C4—1182no—
556C2×C4—1182no—
557C2×C4—2361no—
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#1427369812yes—
568C2×C4#1427369812yes—
569C2×C4#1427375712yes—
570C2×C4C2×D4×D5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C2×C4C2×#1174568711yescharacteristic
573C2×C4C2×#1174568911yescharacteristic
574C2×C4#1427377411yescharacteristic
575C2×C4#1427377511yescharacteristic
576C8—1182no—
577C8—2361no—
578C8—2361no—
579C8—2361no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×#1427369811yescharacteristic
588C22#1683805111yescharacteristic
589C22#1683806311yescharacteristic
590C22C2×#1427375711yescharacteristic
591C22#1683820011yescharacteristic
592C22#1683820511yescharacteristic
593C22#1683821311yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—4721no—
604C4—4721no—
605C4—2362no—
606C4—2362no—
607C4#1683791511yescharacteristic
608C4#1683797511yescharacteristic
609C4#1683809411yescharacteristic
610C4#1683812311yescharacteristic
611C2—82no—
612C2#2142752111yescharacteristic
613C2#2143356411yescharacteristic
614C2#2143363111yescharacteristic
615C2#2143658111yescharacteristic
616C2#2143663511yescharacteristic
617C2#2143684911yescharacteristic
618C2#2143687711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2566C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2566D59
hypocenterC59#24689
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2566

Complements

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

NH ≅ G/Ntype
1. C2×#16837706C2semidirect
3. #21427051C2semidirect
4. #21425515C2semidirect
5. #21425505C2semidirect
8. #21425066C2semidirect
9. #21424862C2semidirect
10. #21424852C2semidirect
11. #21424820C2semidirect
370. C59#24689semidirect

Maximal subgroups

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

subgroupindex
C2×#168377062
C2×#168375022
#214270512
#214255152
#214255052
#214254132
#214250662
#214248622
#214248522
#214248202
#214247202
#214247132
#214240962
C59×#25662
#2468959