Finite Group Databasebeta

← #28589412

Subgroups — #28589412

Order 15104.

1590 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D59C211yesmaximal, characteristic
2C2×#16837438C211yesmaximal, characteristic, complemented
3C22×(C2×D59)⋊C8C211yesmaximal, characteristic
4#21426475C211yesmaximal, characteristic
5#21424294C212yesmaximal, complemented
6#21424292C212yesmaximal, complemented
7#21424287C212yesmaximal, complemented
8#21424282C212yesmaximal, complemented
9#21423813C211yesmaximal, characteristic, complemented
10#21423812C211yesmaximal, characteristic, complemented
11C2×C59×#427C211yesmaximal, characteristic, complemented
12C23×C4×D59C2211yescharacteristic
13C23×C59⋊C8C2211yescharacteristic
14C2×#14273686C2211yescharacteristic
15C2×#14273686C2211yescharacteristic
16C2×#14273685C2212yes—
17C2×#14273684—21no—
18C2×#14273683—21no—
19#117×D59—24no—
20#117×D59—24no—
21C2×C4⋊C4×D59C2212yes—
22C2×C22⋊C4×D59—21no—
23C2×C42×D59—21no—
24#16837438—24no—
25C2×#14273660C2212yescomplemented
26C2×#14273658C2212yescomplemented
27C2×(C2×D59)⋊C8C418yescomplemented
28C2×(C2×D59)⋊C8C2211yescharacteristic
29C2×(C2×D59)⋊C8C2211yescharacteristic
30C2×(C2×D59)⋊C8C2211yescharacteristic
31C2×(C2×D59)⋊C8C2211yescharacteristic
32#16837263—21no—
33#16837259C2212yes—
34#16837259C2212yes—
35#16837257—21no—
36C4×(C2×D59)⋊C4—21no—
37C4×(C2×D59)⋊C4—21no—
38#16837165C2212yescomplemented
39#16837165C2212yescomplemented
40#16837164C2212yescomplemented
41#16837164C2212yescomplemented
42C23×C8×C59C2211yescharacteristic
43C2×#117×C59C2211yescharacteristic
44C2×#117×C59C2211yescharacteristic
45C59×#427—24no—
46C2×#107×C59C2212yescomplemented
47C2×#106×C59C2212yescomplemented
48C24×D59D411yescharacteristic
49C23×Dic59D411yescharacteristic
50C22×C4×D59—22no—
51C22×C4×D59—22no—
52C22×C4×D59—22no—
53C22×C4×D59—22no—
54C22×C4×D59—22no—
55C22×C4×D59C2×C414yes—
56C22×C4×D59D411yescharacteristic
57C22×C4×D59D411yescharacteristic
58C22×C4×D59Q814yes—
59C22×C4×D59C2311yescharacteristic
60C22×C4×D59C2311yescharacteristic
61C22×C59⋊C8—22no—
62C22×C59⋊C8—22no—
63C22×C59⋊C8C2×C414yescomplemented
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273686—22no—
67#14273686—22no—
68#14273686—22no—
69#14273686D414yes—
70#14273685—24no—
71#14273685—24no—
72#14273684—28no—
73#14273683—28no—
74C2×C4⋊Dic59C2312yes—
75C2×C4⋊Dic59C2312yes—
76C2×Dic59⋊C4—22no—
77C4⋊C4×D59—24no—
78C4⋊C4×D59—24no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×(C2×D59)⋊C4—22no—
82C2×(C2×D59)⋊C4—22no—
83C2×(C2×D59)⋊C4—22no—
84C22⋊C4×D59—28no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—22no—
87C2×C4×Dic59—21no—
88C42×D59—28no—
89#14273660—24no—
90#14273658—24no—
91(C2×D59)⋊C8C2×C414yes—
92(C2×D59)⋊C8C2×C414yes—
93(C2×D59)⋊C8C2×C414yescomplemented
94(C2×D59)⋊C8C2×C414yescomplemented
95C59⋊#95—22no—
96C59⋊#95—22no—
97C23×C4×C59C2311yescharacteristic
98C22×C8×C59—22no—
99C22×C8×C59—22no—
100C22×C8×C59C2×C414yescomplemented
101C22×C8×C59C2311yescharacteristic
102C22×C8×C59C2311yescharacteristic
103#117×C59—22no—
104#117×C59—22no—
105#117×C59—22no—
106#117×C59D414yes—
107C2×C4⋊C4×C59C2312yes—
108C2×C4⋊C4×C59C2312yes—
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C42×C59—21no—
112C2×C42×C59—21no—
113#107×C59—24no—
114#106×C59—24no—
115C24×C59C2×D411yescharacteristic
116C23×D59—22no—
117C23×D59—22no—
118C23×D59—24no—
119C23×D59—42no—
120C23×D59C4⋊C414yes—
121C23×D59C2×D411yescharacteristic
122C23×D59C2×D411yescharacteristic
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59C22×C412yes—
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D412yes—
128C22×C4×C59C2×D412yes—
129C22×C4×C59C2×Q812yes—
130C22×C4×C59#4112yes—
131C22×C4×C59#4112yes—
132C22×C4×C59C2411yescharacteristic
133C22×Dic59—22no—
134C22×Dic59—22no—
135C22×Dic59—22no—
136C22×Dic59—22no—
137C22×Dic59C4⋊C414yes—
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C2×C4×D59—28no—
141C2×C4×D59—28no—
142C2×C4×D59—22no—
143C2×C4×D59—22no—
144C2×C4×D59—28no—
145C2×C4×D59—28no—
146C2×C4×D59—22no—
147C2×C4×D59—22no—
148C2×C4×D59—21no—
149C2×C4×D59—21no—
150C2×C4×D59—21no—
151C2×C4×D59—21no—
152C2×C4×D59—21no—
153C2×C4×D59—21no—
154C2×C4×D59—21no—
155C2×C4×D59—21no—
156C2×C4×D59—28no—
157C2×C4×D59—44no—
158C2×C4×D59—44no—
159C2×C4×D59C4⋊C414yes—
160C2×C4×D59C4⋊C414yes—
161C2×C4×D59C22×C412yes—
162C2×C4×D59C22×C412yes—
163C2×C4×D59C2×Q812yes—
164C2×C4×D59C2×Q812yes—
165C2×C8×C59—28no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59C22×C412yes—
171C2×C8×C59C22×C412yes—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59C2×D414yes—
176C22⋊C4×C59—24no—
177C22⋊C4×C59—42no—
178C42×C59—24no—
179C42×C59—42no—
180C22⋊Dic59—24no—
181C22⋊Dic59—42no—
182(C2×D59)⋊C4—28no—
183(C2×D59)⋊C4—44no—
184(C2×D59)⋊C4—44no—
185C4⋊Dic59—22no—
186C4⋊Dic59—22no—
187C4⋊Dic59—22no—
188C4⋊Dic59C2×D414yes—
189Dic59⋊C4—28no—
190C4×Dic59—28no—
191C4×Dic59—24no—
192C4×Dic59—42no—
193C2×C59⋊C8—28no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C22×C412yes—
199C2×C59⋊C8C22×C412yes—
200C23×C59—22no—
201C23×C59—22no—
202C23×C59C2×C4⋊C412yes—
203C23×C59#12012yes—
204C23×C59#12312yes—
205C23×C59C22×D411yescharacteristic
206C22×D59—21no—
207C22×D59—21no—
208C22×D59—22no—
209C22×D59—21no—
210C22×D59—21no—
211C22×D59—22no—
212C22×D59—22no—
213C22×D59—28no—
214C22×D59—28no—
215C22×D59—44no—
216C22×D59—48no—
217C22×D59—48no—
218C22×D59C2×C4⋊C412yes—
219C22×D59C2×C4⋊C412yes—
220C22×D59#13514yes—
221C2×C4×C59—24no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59C2×C4⋊C412yes—
228C2×C4×C59C4×D414yes—
229C2×C4×C59C4×D414yes—
230C2×C4×C59#12012yes—
231C2×C4×C59C22⋊Q814yes—
232C2×C4×C59C22⋊Q814yes—
233C2×C4×C59#12312yes—
234C2×C4×C59C23×C411yescharacteristic
235C2×C4×C59C22×D411yescharacteristic
236C2×C4×C59C22×D411yescharacteristic
237C2×C4×C59C22×Q811yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—28no—
245C2×Dic59—24no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248C2×Dic59—42no—
249C2×Dic59—42no—
250C2×Dic59—42no—
251C2×Dic59C2×C4⋊C412yes—
252C2×Dic59C2×C4⋊C412yes—
253C4×D59—22no—
254C4×D59—22no—
255C4×D59—22no—
256C4×D59—22no—
257C4×D59—22no—
258C4×D59—22no—
259C4×D59—22no—
260C4×D59—22no—
261C4×D59—28no—
262C4×D59—28no—
263C4×D59—416no—
264C8×C59—22no—
265C8×C59—22no—
266C59⋊C8—22no—
267C59⋊C8—22no—
268#20843—591nomaximal, Sylow
269C22×C59—24no—
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—24no—
273C22×C59#38514yes—
274C22×C59C22×C4⋊C411yescharacteristic
275C22×C59C2×#12011yescharacteristic
276C22×C59C2×#12311yescharacteristic
277C22×C59C2×#13512yes—
278C22×C59#57512yes—
279C2×D59—24no—
280C2×D59—22no—
281C2×D59—24no—
282C2×D59—22no—
283C2×D59—22no—
284C2×D59—22no—
285C2×D59—22no—
286C2×D59—416no—
287C2×D59—42no—
288C2×D59—42no—
289C2×D59#42714yescomplemented
290C4×C59—44no—
291C4×C59—82no—
292C4×C59#38514yes—
293C4×C59C2×C4×D411yescharacteristic
294C4×C59C2×C4×D411yescharacteristic
295C4×C59C2×C22⋊Q811yescharacteristic
296C4×C59C2×C22⋊Q811yescharacteristic
297Dic59—22no—
298Dic59—22no—
299Dic59—44no—
300Dic59—82no—
301C22×#117—591no—
302C2×#427—591no—
303C2×#427D5911yescharacteristic, complemented
304C22×C22⋊C8—591no—
305#2280—591no—
306#1854—592no—
307#1854—592no—
308#1853—592no—
309#1852—592no—
310#1735—591no—
311#1735—591no—
312C2×C59—24no—
313C2×C59C2×#38511yescharacteristic
314C2×C59C2×#42711yescharacteristic
315C2×C59#291111yescharacteristic
316C2×C59#300212yes—
317C2×C59#309112yes—
318D59—24no—
319D59—42no—
320C24×C4—591no—
321C23×C8—591no—
322C23×C8C2×D5911yescharacteristic
323C2×#117—591no—
324C2×#117—591no—
325C2×#117—592no—
326C2×#117—1181no—
327C2×#117—1181no—
328C2×#117—1184no—
329C2×#117—1184no—
330C2×#117C2×D5911yescharacteristic
331C2×#117C2×D5911yescharacteristic
332C22×C4⋊C4—592no—
333C22×C22⋊C4—1181no—
334C22×C42—1181no—
335#427—24no—
336#427—1184no—
337C2×#107—592no—
338C2×#107C2×D5912yescomplemented
339C2×#106—592no—
340C2×#106C2×D5912yescomplemented
341C2×C22⋊C8—598no—
342C2×C22⋊C8—591no—
343C2×C22⋊C8—591no—
344C2×C22⋊C8—591no—
345C2×C22⋊C8—591no—
346#381—1181no—
347#380—592no—
348#380—592no—
349#379—1181no—
350C4×C22⋊C4—1181no—
351C4×C22⋊C4—1181no—
352#340—592no—
353#340—592no—
354#340—592no—
355#340—592no—
356C59#2084311yescharacteristic, complemented, Sylow
357C25—591no—
358C23×C4—591no—
359C23×C4—594no—
360C23×C4—591no—
361C23×C4—591no—
362C23×C4—594no—
363C23×C4—591no—
364C23×C4—591no—
365C23×C4—1182no—
366C23×C4—1182no—
367C23×C4—1182no—
368C23×C4—1182no—
369C23×C4—1182no—
370C23×C4C22×D5911yescharacteristic
371C22×C8—22no—
372C22×C8—22no—
373C22×C8—594no—
374C22×C8—591no—
375C22×C8—591no—
376C22×C8—1182no—
377C22×C8—1182no—
378C22×C8C4×D5914yescomplemented
379C22×C8C22×D5911yescharacteristic
380C22×C8C22×D5911yescharacteristic
381#117—22no—
382#117—22no—
383#117—22no—
384#117—594no—
385#117—1188no—
386#117—1188no—
387#117—1184no—
388#117—1182no—
389#117—1184no—
390#117—1182no—
391#117—1182no—
392#117C22⋊D5914yes—
393C2×C4⋊C4—592no—
394C2×C4⋊C4—592no—
395C2×C4⋊C4—1184no—
396C2×C4⋊C4—1182no—
397C2×C4⋊C4—1184no—
398C2×C4⋊C4C22×D5912yes—
399C2×C4⋊C4C22×D5912yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1188no—
403C2×C22⋊C4—1182no—
404C2×C22⋊C4—1181no—
405C2×C22⋊C4—1182no—
406C2×C22⋊C4—1182no—
407C2×C22⋊C4—1181no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—1188no—
411C2×C42—1182no—
412C2×C42—1181no—
413C2×C42—1181no—
414#107—24no—
415#107—1184no—
416#106—24no—
417#106—1184no—
418C22⋊C8—594no—
419C22⋊C8—594no—
420C22⋊C8—594no—
421C22⋊C8—594no—
422#95—1182no—
423#95—1182no—
424C24—591no—
425C24—591no—
426C24—594no—
427C24—1184no—
428C24—1182no—
429C24—1182no—
430C24—2362no—
431C24D4×D5911yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—594no—
435C22×C4—591no—
436C22×C4—591no—
437C22×C4—594no—
438C22×C4—594no—
439C22×C4—592no—
440C22×C4—592no—
441C22×C4—592no—
442C22×C4—592no—
443C22×C4—1188no—
444C22×C4—1182no—
445C22×C4—1182no—
446C22×C4—1188no—
447C22×C4—1188no—
448C22×C4—1182no—
449C22×C4—1182no—
450C22×C4—1188no—
451C22×C4—1188no—
452C22×C4—1182no—
453C22×C4—1182no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1182no—
463C22×C4—1182no—
464C22×C4—2364no—
465C22×C4—2364no—
466C22×C4C2×C4×D5912yes—
467C22×C4#1174568512yes—
468C22×C4D4×D5912yes—
469C22×C4D4×D5911yescharacteristic
470C22×C4Q8×D5912yes—
471C22×C4#1174568912yes—
472C22×C4C2×C22⋊D5912yes—
473C22×C4C23×D5911yescharacteristic
474C2×C8—28no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—592no—
480C2×C8—592no—
481C2×C8—1188no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8C2×C4×D5912yes—
487C2×C8C2×C4×D5912yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—594no—
492C4⋊C4—1188no—
493C4⋊C4—1182no—
494C4⋊C4—1182no—
495C4⋊C4—1182no—
496C4⋊C4C2×C22⋊D5914yes—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—1188no—
500C22⋊C4—1184no—
501C22⋊C4—2364no—
502C22⋊C4—2364no—
503C22⋊C4—2362no—
504C42—24no—
505C42—42no—
506C42—1188no—
507C42—1184no—
508C42—2362no—
509C23—22no—
510C23—22no—
511C23—594no—
512C23—592no—
513C23—592no—
514C23—1188no—
515C23—1181no—
516C23—1182no—
517C23—1181no—
518C23—1181no—
519C23—1182no—
520C23—1182no—
521C23—1188no—
522C23—1181no—
523C23—2364no—
524C23—2368no—
525C23—2368no—
526C23C4⋊C4×D5912yes—
527C23#1427369512yes—
528C23#1427371212yes—
529C23C2×D4×D5911yescharacteristic
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—592no—
537C2×C4—592no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1188no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1182no—
544C2×C4—1182no—
545C2×C4—1182no—
546C2×C4—1182no—
547C2×C4—1188no—
548C2×C4—1188no—
549C2×C4—1182no—
550C2×C4—1182no—
551C2×C4—1182no—
552C2×C4—1182no—
553C2×C4—1184no—
554C2×C4—23616no—
555C2×C4—2362no—
556C2×C4—2362no—
557C2×C4—2362no—
558C2×C4—2362no—
559C2×C4—2362no—
560C2×C4C4⋊C4×D5912yes—
561C2×C4C4×C22⋊D5914yes—
562C2×C4#1427368914yes—
563C2×C4#1427369512yes—
564C2×C4#1427370314yes—
565C2×C4#1427370514yes—
566C2×C4#1427371212yes—
567C2×C4C22×C4×D5911yescharacteristic
568C2×C4C2×D4×D5911yescharacteristic
569C2×C4C22×C22⋊D5911yescharacteristic
570C2×C4C2×Q8×D5911yescharacteristic
571C2×C4C2×#1174568511yescharacteristic
572C2×C4C2×#1174568911yescharacteristic
573C8—22no—
574C8—22no—
575C8—1182no—
576C8—1182no—
577C22—22no—
578C22—22no—
579C22—24no—
580C22—24no—
581C22—594no—
582C22—1182no—
583C22—1182no—
584C22—1184no—
585C22—1184no—
586C22—1182no—
587C22—1182no—
588C22—1182no—
589C22—23616no—
590C22—2362no—
591C22—2362no—
592C22#1683727314yes—
593C22C2×C4⋊C4×D5911yescharacteristic
594C22C2×#1427369511yescharacteristic
595C22C2×#1427371211yescharacteristic
596C22#135×D5912yes—
597C22#1683819912yes—
598C4—44no—
599C4—82no—
600C4—1182no—
601C4—1182no—
602C4—4722no—
603C4—2364no—
604C4#1683727314yes—
605C4C2×C4×C22⋊D5911yescharacteristic
606C4C2×#1427368911yescharacteristic
607C4C2×#1427370311yescharacteristic
608C4C2×#1427370511yescharacteristic
609C2—24no—
610C2—1184no—
611C2—2362no—
612C2C2×#1683727311yescharacteristic
613C2#427×D5911yescharacteristic
614C2#2143268411yescharacteristic
615C2#2143331012yes—
616C2#2143401312yes—
subgroupG/H
centerC22×C4D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#427C2
socleC23×C59C22×D4
prime radicalC24×D59D4
cosocleC22×C4×C59C24
hypercenterC2×#427D59
hypocenterC59#20843
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. C2×#16837438C2semidirect
5. #21424294C2semidirect
6. #21424292C2semidirect
7. #21424287C2semidirect
8. #21424282C2semidirect
9. #21423813C2semidirect
10. #21423812C2semidirect
11. C2×C59×#427C2semidirect
25. C2×#14273660C22semidirect
26. C2×#14273658C22semidirect
27. C2×(C2×D59)⋊C8C4semidirect
38. #16837165C22semidirect
39. #16837165C22semidirect
40. #16837164C22semidirect
41. #16837164C22semidirect
46. C2×#107×C59C22semidirect
47. C2×#106×C59C22semidirect
63. C22×C59⋊C8C2×C4semidirect
93. (C2×D59)⋊C8C2×C4semidirect
94. (C2×D59)⋊C8C2×C4semidirect
100. C22×C8×C59C2×C4semidirect
289. C2×D59#427semidirect
303. C2×#427D59semidirect
338. C2×#107C2×D59semidirect
340. C2×#106C2×D59semidirect
356. C59#20843semidirect
378. C22×C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D592
C2×#168374382
C22×(C2×D59)⋊C82
#214264752
#214242942
#214242922
#214242872
#214242822
#214238132
#214238122
C2×C59×#4272
#2084359