Finite Group Databasebeta

← #28610057

Subgroups — #28610057

Order 15104.

846 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×#16837736C211yesmaximal, characteristic
2#21427097C211yesmaximal, characteristic
3#21426969C211yesmaximal, characteristic
4#21424834C211yesmaximal, characteristic, complemented
5#21424783C211yesmaximal, characteristic, complemented
6#21424732C211yesmaximal, characteristic, complemented
7#21424667C211yesmaximal, characteristic, complemented
8#21424666C211yesmaximal, characteristic, complemented
9#21424516C211yesmaximal, characteristic
10#21424364C211yesmaximal, characteristic, complemented
11#21424290C211yesmaximal, characteristic
12#21423992C211yesmaximal, characteristic, complemented
13#21423873C211yesmaximal, characteristic
14#340×D59C211yesmaximal, characteristic
15C59×#1763C211yesmaximal, characteristic, complemented
16C22×C8×D59C2211yescharacteristic
17C2×#14273703C2211yescharacteristic
18C2×C4⋊C4×D59C2211yescharacteristic
19C2×C4⋊C4×D59C2211yescharacteristic
20#16837736C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×(C2×D59)⋊C8C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29#16837309C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837298C2211yescharacteristic
33#16837297—21no—
34#16837286—21no—
35#16837286—21no—
36#16837284—21no—
37#16837273—21no—
38#16837261—21no—
39#16837259C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×(C2×D59)⋊C4—21no—
43C4×(C2×D59)⋊C4—21no—
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×#103×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×C4×D59—21no—
61C22×C4×D59—21no—
62C22×C4×D59C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C2×C8×D59—22no—
66C2×C8×D59—22no—
67C2×C8×D59—41no—
68#14273703—22no—
69#14273703C2×C414yes—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C4⋊C4×D59—22no—
81C4⋊C4×D59—22no—
82C4⋊C4×D59—41no—
83C4⋊C4×D59C2×C414yes—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4D411yescharacteristic
87C2×(C2×D59)⋊C4D411yescharacteristic
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59C2311yescharacteristic
91#14273661C2×C414yescomplemented
92#14273660C2×C414yescomplemented
93#14273659—41no—
94#14273658—22no—
95#14273651—41no—
96#14273651C2×C414yescomplemented
97(C2×D59)⋊C8C2×C414yes—
98#14273637—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C22×Q8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C4×Q8×C59—22no—
113C4×Q8×C59—41no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C42×C59—21no—
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#103×C59—22no—
122#103×C59C2×C414yescomplemented
123#97×C59—41no—
124#95×C59—21no—
125C23×D59C2×D411yescharacteristic
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
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—24no—
151C2×C4×D59—42no—
152C2×C4×D59—42no—
153C2×C4×D59—81no—
154C2×C4×D59—81no—
155C2×C4×D59—81no—
156C2×C4×D59C22×C412yes—
157C2×C4×D59#4112yes—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C22×C412yes—
163C2×C8×C59#4112yes—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59C22×C412yes—
170C4⋊C4×C59#4112yes—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—42no—
174C42×C59—81no—
175(C2×D59)⋊C4—24no—
176(C2×D59)⋊C4—22no—
177(C2×D59)⋊C4—22no—
178(C2×D59)⋊C4—42no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59C22×C412yes—
186C4⋊Dic59C22×C412yes—
187Dic59⋊C4—24no—
188Dic59⋊C4—24no—
189Dic59⋊C4—42no—
190Dic59⋊C4—81no—
191Dic59⋊C4—81no—
192Dic59⋊C4—81no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—81no—
196C4×Dic59—81no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8C22×C412yes—
203C8×D59—44no—
204C23×C59C22×D411yescharacteristic
205C22×D59—21no—
206C22×D59—21no—
207C22×D59—21no—
208C22×D59—21no—
209C22×D59C4×D412yes—
210C22×D59#13712yes—
211C22×D59#13712yes—
212Q8×C59—22no—
213Q8×C59—22no—
214Q8×C59—41no—
215Q8×C59—41no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
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—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59C23×C411yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59H(5,𝔽2)11yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
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—
259C2×Dic59—41no—
260C2×Dic59C4×D412yes—
261C4×D59—22no—
262C4×D59—22no—
263C4×D59—22no—
264C4×D59—22no—
265C4×D59—84no—
266C4×D59—84no—
267C8×C59—22no—
268C8×C59—22no—
269C59⋊C8—22no—
270C59⋊C8—41no—
271#22932—591nomaximal, Sylow
272C22×C59C2×C4×D411yescharacteristic
273C22×C59#54611yescharacteristic
274C22×C59#54711yescharacteristic
275C22×C59C2×#13711yescharacteristic
276C22×C59C2×#13711yescharacteristic
277C22×C59#57611yescharacteristic
278C22×C59#57711yescharacteristic
279C2×D59—22no—
280C2×D59—22no—
281C2×D59—22no—
282C2×D59—22no—
283C2×D59—22no—
284C2×D59—22no—
285C2×D59#44114yes—
286C4×C59—42no—
287C4×C59—42no—
288C4×C59—42no—
289C4×C59—41no—
290C4×C59—41no—
291C4×C59—81no—
292C4×C59C4×#4111yescharacteristic
293C4×C59#52011yescharacteristic
294C4×C59#55111yescharacteristic
295C4×C59#55211yescharacteristic
296Dic59—22no—
297Dic59—41no—
298Dic59—41no—
299Dic59—42no—
300Dic59—42no—
301Dic59—81no—
302Dic59—81no—
303C2×#484—591no—
304#2360—591no—
305#2345—591no—
306#1943—591no—
307#1935—591no—
308#1927—591no—
309#1918—591no—
310#1918—591no—
311#1892—591no—
312#1865—591no—
313#1853—591no—
314#1776—591no—
315#1763D5911yescharacteristic, complemented
316#1752—591no—
317C2×#340—591no—
318C2×C59#235911yescharacteristic
319C2×C59C2×#44111yescharacteristic
320C2×C59#294611yescharacteristic
321C2×C59#331411yescharacteristic
322C2×C59#332011yescharacteristic
323C2×C59#334411yescharacteristic
324C2×C59#334611yescharacteristic
325D59—24no—
326C23×C8—591no—
327C2×C22⋊Q8—591no—
328C2×C4×Q8C2×D5911yescharacteristic
329C22×C4⋊C4—591no—
330C22×C4⋊C4—591no—
331#484—598no—
332C2×#107—591no—
333C2×#107—591no—
334C2×#106—591no—
335C2×#106—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#103C2×D5911yescharacteristic
339C2×#103C2×D5911yescharacteristic
340C2×C22⋊C8—591no—
341C2×#97—591no—
342C2×#97C2×D5911yescharacteristic
343#391—591no—
344#389—591no—
345#389—591no—
346#389—591no—
347#389—1181no—
348#387—1181no—
349#387—1181no—
350#387—1181no—
351#385—1181no—
352#380—591no—
353#380—1181no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4C2×D5911yescharacteristic
357C4×C22⋊C4—591no—
358C4×C22⋊C4—1181no—
359C4×C22⋊C4—1181no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340C2×D5911yescharacteristic
367#340C2×D5911yescharacteristic
368#336—591no—
369#336—591no—
370C59#2293211yescharacteristic, complemented, Sylow
371C22×Q8C22×D5911yescharacteristic
372C23×C4—591no—
373C23×C4—1181no—
374C23×C4—1181no—
375C22×C8—591no—
376C22×C8—591no—
377C22×C8—1182no—
378C22×C8—1182no—
379C22×C8—2361no—
380C22×C8C22×D5911yescharacteristic
381C22×C8C22×D5911yescharacteristic
382C22⋊Q8—594no—
383C22⋊Q8—1182no—
384C4×Q8—22no—
385C4×Q8—41no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—594no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—1182no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
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⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C22⋊C4—591no—
405C2×C22⋊C4—591no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—591no—
411C2×C42—1181no—
412C2×C42—1181no—
413C2×C42C22×D5911yescharacteristic
414#107—594no—
415#107—594no—
416#106—1182no—
417#106—2361no—
418#103—22no—
419#103—594no—
420#103—2361no—
421#103C4×D5914yescomplemented
422C22⋊C8—594no—
423#97—41no—
424#97—1182no—
425#95—21no—
426#95—591no—
427#95—591no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436C24—591no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D5912yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—591no—
448C22×C4—592no—
449C22×C4—592no—
450C22×C4—591no—
451C22×C4—591no—
452C22×C4—1184no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—4721no—
462C22×C4—4721no—
463C22×C4—4721no—
464C22×C4—2362no—
465C22×C4—2362no—
466C22×C4D4×D5911yescharacteristic
467C22×C4D4×D5911yescharacteristic
468C22×C4C23×D5911yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—592no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—2364no—
480C2×C8C2×C4×D5912yes—
481C2×C8#1174568512yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—592no—
488C4⋊C4—592no—
489C4⋊C4—1184no—
490C4⋊C4—1184no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—4721no—
498C4⋊C4—4721no—
499C4⋊C4—4721no—
500C4⋊C4—2362no—
501C4⋊C4C2×C4×D5912yes—
502C4⋊C4#1174568512yes—
503C22⋊C4—1182no—
504C22⋊C4—1182no—
505C22⋊C4—1184no—
506C22⋊C4—2362no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—81no—
511C42—1181no—
512C42—1181no—
513C42—4721no—
514C42—4721no—
515C23—592no—
516C23—592no—
517C23—592no—
518C23—1181no—
519C23—1181no—
520C23—1181no—
521C23—1181no—
522C23C2×D4×D5911yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—592no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1182no—
549C2×C4—1182no—
550C2×C4—1182no—
551C2×C4—1182no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1182no—
559C2×C4—1182no—
560C2×C4—4724no—
561C2×C4—4724no—
562C2×C4—2361no—
563C2×C4—2361no—
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×C4C22×C4×D5911yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C2×C4C2×#1174568511yescharacteristic
575C2×C4C2×#1174568511yescharacteristic
576C2×C4#1427377211yescharacteristic
577C2×C4#1427377411yescharacteristic
578C8—22no—
579C8—22no—
580C8—1182no—
581C8—2361no—
582C22—594no—
583C22—1182no—
584C22—1182no—
585C22—1182no—
586C22—1182no—
587C22—1182no—
588C22—1182no—
589C22C4×D4×D5911yescharacteristic
590C22#1683804211yescharacteristic
591C22#1683805711yescharacteristic
592C22#137×D5911yescharacteristic
593C22#137×D5911yescharacteristic
594C22#1683820611yescharacteristic
595C22#1683821111yescharacteristic
596C4—41no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—42no—
601C4—81no—
602C4—1182no—
603C4—4721no—
604C4—4721no—
605C4—2362no—
606C4—2362no—
607C4—2361no—
608C4—2361no—
609C4C4×#1174568511yescharacteristic
610C4#1683788111yescharacteristic
611C4#1683808111yescharacteristic
612C4#1683809111yescharacteristic
613C2—1184no—
614C2#2142708911yescharacteristic
615C2#441×D5911yescharacteristic
616C2#2143292711yescharacteristic
617C2#2143617811yescharacteristic
618C2#2143626811yescharacteristic
619C2#2143655611yescharacteristic
620C2#2143658611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1763C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1763D59
hypocenterC59#22932
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
4. #21424834C2semidirect
5. #21424783C2semidirect
6. #21424732C2semidirect
7. #21424667C2semidirect
8. #21424666C2semidirect
10. #21424364C2semidirect
12. #21423992C2semidirect
15. C59×#1763C2semidirect
20. #16837736C4semidirect
91. #14273661C2×C4semidirect
92. #14273660C2×C4semidirect
96. #14273651C2×C4semidirect
122. #103×C59C2×C4semidirect
315. #1763D59semidirect
370. C59#22932semidirect
421. #103C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377362
#214270972
#214269692
#214248342
#214247832
#214247322
#214246672
#214246662
#214245162
#214243642
#214242902
#214239922
#214238732
#340×D592
C59×#17632
#2293259