Finite Group Databasebeta

← #28647557

Subgroups — #28647557

Order 15104.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837443C211yesmaximal, characteristic
2#21427816C211yesmaximal, characteristic
3#21427783C211yesmaximal, characteristic
4#21425581C211yesmaximal, characteristic, complemented
5#21425580C211yesmaximal, characteristic, complemented
6#21425431C211yesmaximal, characteristic, complemented
7#21425421C211yesmaximal, characteristic, complemented
8#21424834C211yesmaximal, characteristic, complemented
9#21424830C211yesmaximal, characteristic, complemented
10#21424294C211yesmaximal, characteristic
11#21424282C211yesmaximal, characteristic
12#21424257C211yesmaximal, characteristic, complemented
13#21424126C211yesmaximal, characteristic
14#21423881C211yesmaximal, characteristic
15C2×C59×#501C211yesmaximal, characteristic, complemented
16C22×#11745659C2211yescharacteristic
17C2×#14273685C2211yescharacteristic
18C2×C4⋊C4×D59C2211yescharacteristic
19#16837443—24no—
20C2×#14273661C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×(C2×D59)⋊C8C2211yescharacteristic
27C2×C8×Dic59C2211yescharacteristic
28#16837340—21no—
29#16837335—21no—
30#16837324—21no—
31#16837313—21no—
32#16837309—21no—
33#16837301C2211yescharacteristic
34#16837301C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837283—21no—
38#16837273—21no—
39#16837267C2211yescharacteristic
40#16837267C2211yescharacteristic
41#16837259—21no—
42#16837259C2211yescharacteristic
43#16837259C2211yescharacteristic
44C4×(C2×D59)⋊C4C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4⋊Q8×C59C2211yescharacteristic
54C2×#125×C59C2211yescharacteristic
55C59×#501—24no—
56C2×#107×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×#106×C59C2211yescharacteristic
59C2×#106×C59C2211yescharacteristic
60C2×#97×C59C2211yescharacteristic
61C22×C4×D59—21no—
62C22×C4×D59C2311yescharacteristic
63C2×#11745659—22no—
64C2×#11745659—22no—
65C2×#11745659Q814yescomplemented
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273685—22no—
69#14273685—41no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C4⋊C4×D59—22no—
80C4⋊C4×D59—41no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4D411yescharacteristic
85C2×(C2×D59)⋊C4D411yescharacteristic
86C2×C4×Dic59—21no—
87C2×C4×Dic59C2311yescharacteristic
88#14273661—22no—
89#14273660—22no—
90#14273660—41no—
91#14273660—41no—
92#14273659—22no—
93#14273658—22no—
94(C2×D59)⋊C8Q814yescomplemented
95C8×Dic59—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C22×Q8×C59—21no—
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C4⋊Q8×C59—22no—
112C4⋊Q8×C59—41no—
113#125×C59—22no—
114#125×C59—41no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123#107×C59—22no—
124#107×C59D414yes—
125#106×C59—22no—
126#106×C59—22no—
127#97×C59—22no—
128C23×D59C2×D411yescharacteristic
129C2×Q8×C59—42no—
130C2×Q8×C59—42no—
131C2×Q8×C59—81no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59C2×D411yescharacteristic
144C22×Dic59C2×D411yescharacteristic
145C22×Dic59C2×D411yescharacteristic
146C2×C4×D59—21no—
147C2×C4×D59—21no—
148C2×C4×D59—21no—
149C2×C4×D59—21no—
150C2×C4×D59—42no—
151C2×C4×D59—42no—
152C2×C4×D59—81no—
153C2×C4×D59C2×Q812yes—
154C2×C4×D59C2×Q812yes—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59C2×D412yes—
158C2×C8×C59C2×Q812yes—
159C2×C8×C59C2×Q812yes—
160C2×C8×C59#4112yes—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59—42no—
170C4⋊C4×C59C2×D412yes—
171C4⋊C4×C59C2×D412yes—
172C42×C59—21no—
173C42×C59—21no—
174(C2×D59)⋊C4—41no—
175(C2×D59)⋊C4—41no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—81no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—81no—
188C4⋊Dic59—81no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191Dic59⋊C4—81no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—42no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8C2×Q812yes—
200C2×C59⋊C8C2×Q812yes—
201#11745659—28no—
202C23×C59C22×D411yescharacteristic
203C22×D59—21no—
204C22×D59—21no—
205C22×D59—21no—
206C22×D59—21no—
207C22×D59—21no—
208C22×D59—21no—
209C22×D59C4⋊Q812yes—
210Q8×C59—84no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59C22×D411yescharacteristic
230C2×C4×C59C22×Q811yescharacteristic
231C2×C4×C59C22×Q811yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59H(5,𝔽2)11yescharacteristic
234C2×C4×C592−1+411yescharacteristic
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59C4⋊Q812yes—
256C4×D59—22no—
257C4×D59—22no—
258C4×D59—22no—
259C4×D59—22no—
260C4×D59—84no—
261C8×C59—22no—
262C8×C59C22⋊Q814yes—
263C59⋊C8—22no—
264C59⋊C8—22no—
265#26079—591nomaximal, Sylow
266C22×C59C2×C4⋊Q811yescharacteristic
267C22×C59D4211yescharacteristic
268C22×C59#54711yescharacteristic
269C22×C59#57511yescharacteristic
270C22×C59#57511yescharacteristic
271C22×C59#57611yescharacteristic
272C22×C59#57811yescharacteristic
273C2×D59—22no—
274C2×D59—22no—
275C2×D59—22no—
276C2×D59—41no—
277C2×D59—41no—
278C2×D59—41no—
279C2×D59—41no—
280C4×C59—42no—
281C4×C59—41no—
282C4×C59—42no—
283C4×C59—41no—
284C4×C59—81no—
285C4×C59—81no—
286C4×C59C2×C22⋊Q811yescharacteristic
287C4×C59#54411yescharacteristic
288C4×C59D4×Q811yescharacteristic
289C4×C59#55411yescharacteristic
290Dic59—22no—
291Dic59—41no—
292Dic59—41no—
293Dic59—81no—
294Dic59—81no—
295Dic59—81no—
296Dic59—81no—
297C2×#501D5911yescharacteristic, complemented
298C2×#428—591no—
299#2442—591no—
300#2439—591no—
301#2085—591no—
302#2085—591no—
303#2059—591no—
304#2057—591no—
305#1943—591no—
306#1942—591no—
307#1854—591no—
308#1852—591no—
309#1846—591no—
310#1809—591no—
311#1753—591no—
312C2×C59#244011yescharacteristic
313C2×C59#316411yescharacteristic
314C2×C59#316611yescharacteristic
315C2×C59#327311yescharacteristic
316C2×C59#327711yescharacteristic
317C2×C59#334711yescharacteristic
318C2×C59#334911yescharacteristic
319D59—42no—
320C22×M16—591no—
321C2×C4⋊Q8C2×D5911yescharacteristic
322C2×#125C2×D5911yescharacteristic
323C2×#117—591no—
324C22×C4⋊C4—591no—
325#501—24no—
326#428—1184no—
327C2×#107—591no—
328C2×#107—591no—
329C2×#107—591no—
330C2×#107—591no—
331C2×#107C2×D5911yescharacteristic
332C2×#107C2×D5911yescharacteristic
333C2×#106—591no—
334C2×#106—591no—
335C2×#106C2×D5911yescharacteristic
336C2×#106C2×D5911yescharacteristic
337C2×C22⋊C8—591no—
338C2×#97C2×D5911yescharacteristic
339C2×C4×C8—591no—
340#400—1181no—
341#398—1181no—
342#396—1181no—
343#393—1181no—
344#391—1181no—
345#389—591no—
346#389—591no—
347#389—591no—
348#389—591no—
349#387—1181no—
350#385—1181no—
351#382—591no—
352#382—591no—
353#380—591no—
354#380—591no—
355#380—1181no—
356C4×C22⋊C4—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#336—591no—
364#336—591no—
365C59#2607911yescharacteristic, complemented, Sylow
366C22×Q8—21no—
367C23×C4—591no—
368C23×C4—1181no—
369C2×M16—594no—
370C2×M16—1182no—
371C2×M16—1182no—
372C22×C8—591no—
373C22×C8—591no—
374C22×C8C22×D5911yescharacteristic
375C22×C8C22×D5911yescharacteristic
376C4⋊Q8—22no—
377C4⋊Q8—41no—
378#125—22no—
379#125—41no—
380#117—1182no—
381#117—2361no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1182no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—2361no—
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C22⋊C4—591no—
401C2×C22⋊C4—591no—
402C2×C22⋊C4—1181no—
403C2×C22⋊C4—1181no—
404C2×C22⋊C4—1181no—
405C2×C42—591no—
406C2×C42—1181no—
407C2×C42C22×D5911yescharacteristic
408#107—22no—
409#107—1182no—
410#107—1182no—
411#107—2361no—
412#107—2361no—
413#107C22⋊D5914yes—
414#106—22no—
415#106—22no—
416#106—1182no—
417#106—1182no—
418C22⋊C8—594no—
419#97—22no—
420C4×C8—1182no—
421#95—591no—
422#95—591no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433C24—591no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—81no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—591no—
442C22×C4—592no—
443C22×C4—592no—
444C22×C4—591no—
445C22×C4—591no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—4721no—
455C22×C4—2362no—
456C22×C4—2362no—
457C22×C4D4×D5911yescharacteristic
458C22×C4D4×D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460M16—1188no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—592no—
464C2×C8—592no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8Q8×D5912yes—
470C2×C8Q8×D5912yes—
471C2×C8#1174568912yes—
472C2×C8C2×C22⋊D5912yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—4721no—
491C4⋊C4—4721no—
492C4⋊C4—4721no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4C2×C22⋊D5912yes—
496C4⋊C4C2×C22⋊D5912yes—
497C22⋊C4—4721no—
498C22⋊C4—2361no—
499C22⋊C4—2361no—
500C22⋊C4—2362no—
501C22⋊C4—2362no—
502C42—21no—
503C42—21no—
504C42—1181no—
505C42—1181no—
506C42—2362no—
507C23—592no—
508C23—1181no—
509C23—1181no—
510C23—1181no—
511C23—1181no—
512C23—1181no—
513C23—1181no—
514C23C2×D4×D5911yescharacteristic
515Q8—84no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—592no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1182no—
538C2×C4—1182no—
539C2×C4—1182no—
540C2×C4—1182no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—4724no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4C22×C22⋊D5911yescharacteristic
561C2×C4C2×Q8×D5911yescharacteristic
562C2×C4C2×Q8×D5911yescharacteristic
563C2×C4C2×#1174568911yescharacteristic
564C2×C4#1427377211yescharacteristic
565C2×C4#1427377511yescharacteristic
566C8—22no—
567C8—1182no—
568C8—1182no—
569C8#1427370314yes—
570C22—1182no—
571C22—1182no—
572C22—1182no—
573C22—2361no—
574C22—2361no—
575C22—2361no—
576C22—2361no—
577C22C4⋊Q8×D5911yescharacteristic
578C22D4×C22⋊D5911yescharacteristic
579C22#1683805511yescharacteristic
580C22#1683819811yescharacteristic
581C22#1683820111yescharacteristic
582C22#1683820611yescharacteristic
583C22#1683821811yescharacteristic
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—1182no—
591C4—4721no—
592C4—4721no—
593C4—4721no—
594C4—4721no—
595C4—2361no—
596C4—2361no—
597C4C2×#1427370311yescharacteristic
598C4#1683802911yescharacteristic
599C4Q8×C22⋊D5911yescharacteristic
600C4#1683811311yescharacteristic
601C2—2362no—
602C2#2142779411yescharacteristic
603C2#2143463111yescharacteristic
604C2#2143464711yescharacteristic
605C2#2143570111yescharacteristic
606C2#2143574711yescharacteristic
607C2#2143659511yescharacteristic
608C2#2143662511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#501C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
hypercenterC2×#501D59
hypocenterC59#26079
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
4. #21425581C2semidirect
5. #21425580C2semidirect
6. #21425431C2semidirect
7. #21425421C2semidirect
8. #21424834C2semidirect
9. #21424830C2semidirect
12. #21424257C2semidirect
15. C2×C59×#501C2semidirect
65. C2×#11745659Q8semidirect
94. (C2×D59)⋊C8Q8semidirect
297. C2×#501D59semidirect
365. C59#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#168374432
#214278162
#214277832
#214255812
#214255802
#214254312
#214254212
#214248342
#214248302
#214242942
#214242822
#214242572
#214241262
#214238812
C2×C59×#5012
#2607959