Finite Group Databasebeta

← #28495055

Subgroups — #28495055

Order 15104.

930 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837420C211yesmaximal, characteristic
2C2×#16837396C211yesmaximal, characteristic
3C2×#16837375C211yesmaximal, characteristic
4C2×#16837358C211yesmaximal, characteristic, complemented
5#21423812C211yesmaximal, characteristic, complemented
6#21423805C211yesmaximal, characteristic, complemented
7C59×#1740C211yesmaximal, characteristic, complemented
8C22×#11745685C2211yescharacteristic
9C22×#11745664C2211yescharacteristic
10C23×C59⋊C8C2211yescharacteristic
11C2×#14273686C2211yescharacteristic
12#16837420—22no—
13#16837420C414yescomplemented
14C2×#14273655—22no—
15#16837396—22no—
16#16837396—21no—
17#16837396—21no—
18C2×#14273650C412yescomplemented
19C2×#14273647C412yescomplemented
20#16837375—22no—
21#16837375C414yescomplemented
22C2×#14273641—22no—
23#16837358—22no—
24#16837358—22no—
25C2×#14273638—21no—
26C2×C4×C59⋊C8—21no—
27#16837164—22no—
28#16837161—22no—
29C2×#117×C59C2211yescharacteristic
30C2×#117×C59C2211yescharacteristic
31C2×#117×C59C2211yescharacteristic
32C59×#342—22no—
33C2×#11745685—21no—
34C2×#11745685—21no—
35C2×#11745685—21no—
36C2×#11745685—21no—
37C2×#11745685—21no—
38C2×#11745685—21no—
39C2×#11745685—21no—
40C2×#11745685—21no—
41C2×#11745685—21no—
42C2×#11745685—21no—
43C2×#11745685C2×C412yes—
44C2×#11745685C2×C412yes—
45C22×Dic118C2×C411yescharacteristic
46C22×D236C2×C411yescharacteristic
47C22×C22⋊D59—21no—
48C22×C4×D59—21no—
49C2×#11745664—21no—
50C2×#11745664—21no—
51C2×#11745664—21no—
52C2×#11745664C2×C412yes—
53C2×#11745664D412yes—
54C2×#11745664D412yes—
55C22×C59⋊C8—21no—
56C22×C59⋊C8—21no—
57C22×C59⋊C8—21no—
58C22×C59⋊C8—21no—
59C22×C59⋊C8—21no—
60C22×C59⋊C8C2×C412yes—
61C22×C59⋊C8D412yes—
62C22×C59⋊C8D411yescharacteristic
63C22×C59⋊C8D411yescharacteristic
64#14273686—21no—
65#14273686—21no—
66#14273686—21no—
67#14273686C2×C412yes—
68C2×C4⋊Dic59—21no—
69C2×C22⋊Dic59—21no—
70C2×C4×Dic59—21no—
71#14273655—28no—
72#14273650—22no—
73#14273650—22no—
74#14273647—22no—
75#14273647—22no—
76#14273641—28no—
77#14273638—24no—
78C4×C59⋊C8—24no—
79C23×C4×C59C2311yescharacteristic
80#117×C59—21no—
81#117×C59—21no—
82#117×C59—21no—
83#117×C59—21no—
84#117×C59—21no—
85#117×C59—21no—
86#117×C59—21no—
87#117×C59—21no—
88#117×C59—21no—
89#117×C59C2×C412yes—
90#117×C59C2×C412yes—
91#117×C59D412yes—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59—21no—
94C2×C4⋊C4×C59C2×C412yes—
95C2×C22⋊C4×C59—21no—
96C2×C22⋊C4×C59—21no—
97C2×C22⋊C4×C59D412yes—
98C2×C42×C59—21no—
99C2×C42×C59—21no—
100C2×C42×C59—21no—
101C24×C59C2×D411yescharacteristic
102C23×D59—21no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—41no—
106C22×C4×C59C22×C411yescharacteristic
107C22×C4×C59C22×C411yescharacteristic
108C22×C4×C59C22×C411yescharacteristic
109C22×C4×C59C2×D411yescharacteristic
110C22×C4×C59C2×D411yescharacteristic
111C22×C4×C59C2×D411yescharacteristic
112C22×C4×C59C2×D411yescharacteristic
113C22×C4×C59C2×D411yescharacteristic
114C22×C4×C59#4111yescharacteristic
115C22×C4×C59#4111yescharacteristic
116C22×C4×C59#4111yescharacteristic
117C22×C4×C59#4111yescharacteristic
118C22×C4×C59#4111yescharacteristic
119C22×C4×C59#4111yescharacteristic
120C2×C22⋊D59—22no—
121C2×C22⋊D59—22no—
122C2×C22⋊D59—41no—
123C2×C22⋊D59—41no—
124C2×C22⋊D59—41no—
125C2×C22⋊D59—41no—
126C22×Dic59—21no—
127C22×Dic59—41no—
128#11745685—22no—
129#11745685—22no—
130#11745685—22no—
131#11745685—22no—
132#11745685—22no—
133#11745685—22no—
134#11745685—22no—
135#11745685—22no—
136#11745685—41no—
137#11745685—41no—
138#11745685—41no—
139#11745685—41no—
140#11745685—41no—
141#11745685—41no—
142#11745685—41no—
143#11745685—41no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236C4212yes—
150C2×C4×D59—22no—
151C2×C4×D59—22no—
152C2×C4×D59—41no—
153C2×C4×D59—41no—
154C2×C4×D59—41no—
155C2×C4×D59—41no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118C4212yes—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—41no—
168C22⋊C4×C59—22no—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—41no—
171C22⋊C4×C59—41no—
172C22⋊C4×C59—41no—
173C22⋊C4×C59C22⋊C414yes—
174C42×C59—22no—
175C42×C59—22no—
176C42×C59—22no—
177C42×C59—41no—
178C42×C59—41no—
179C42×C59—41no—
180C22⋊Dic59—22no—
181C22⋊Dic59—41no—
182C4⋊Dic59—22no—
183C4⋊Dic59—41no—
184C4×Dic59—22no—
185C4×Dic59—41no—
186#11745664—22no—
187#11745664—22no—
188#11745664—22no—
189#11745664C22⋊C414yes—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—22no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—22no—
197C2×C59⋊C8—22no—
198C2×C59⋊C8—22no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8C22⋊C412yes—
202C2×C59⋊C8C22⋊C412yes—
203C23×C59—42no—
204C23×C59C2×C22⋊C411yescharacteristic
205C23×C59C4×D411yescharacteristic
206C23×C59C4×D411yescharacteristic
207C23×C59#12111yescharacteristic
208C23×C59#12111yescharacteristic
209C23×C59#12411yescharacteristic
210C23×C59#12711yescharacteristic
211C22×D59—22no—
212C22×D59—41no—
213C22×D59—41no—
214C22×D59—82no—
215C2×C4×C59—24no—
216C2×C4×C59—44no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—82no—
220C2×C4×C59—82no—
221C2×C4×C59C2×C4211yescharacteristic
222C2×C4×C59C2×C22⋊C411yescharacteristic
223C2×C4×C59C2×C22⋊C411yescharacteristic
224C2×C4×C59#11711yescharacteristic
225C2×C4×C59#11711yescharacteristic
226C2×C4×C59#11711yescharacteristic
227C2×C4×C59C4×D411yescharacteristic
228C2×C4×C59C4×D411yescharacteristic
229C2×C4×C59C4×D411yescharacteristic
230C2×C4×C59C4×D411yescharacteristic
231C2×C4×C59C4×D411yescharacteristic
232C2×C4×C59C4×D411yescharacteristic
233C2×C4×C59C4×D411yescharacteristic
234C2×C4×C59C4×D411yescharacteristic
235C2×C4×C59C4×D411yescharacteristic
236C2×C4×C59C4×D411yescharacteristic
237C2×C4×C59#12011yescharacteristic
238C2×C4×C59#12111yescharacteristic
239C2×C4×C59#12111yescharacteristic
240C2×C4×C59#12111yescharacteristic
241C2×C4×C59#12111yescharacteristic
242C2×C4×C59#12311yescharacteristic
243C2×C4×C59#12311yescharacteristic
244C2×C4×C59#12311yescharacteristic
245C2×C4×C59#12411yescharacteristic
246C2×C4×C59#12411yescharacteristic
247C2×C4×C59#12611yescharacteristic
248C2×C4×C59#12611yescharacteristic
249C22⋊D59—42no—
250C22⋊D59—42no—
251C22⋊D59—42no—
252C22⋊D59—42no—
253C2×Dic59—22no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—42no—
257C2×Dic59—82no—
258D236—22no—
259D236—22no—
260D236—41no—
261D236—41no—
262C4×D59—42no—
263C4×D59—42no—
264C4×D59—42no—
265C4×D59—42no—
266Dic118—22no—
267Dic118—22no—
268Dic118—41no—
269Dic118—41no—
270C59⋊C8—22no—
271C59⋊C8—22no—
272C59⋊C8—24no—
273#10944—591nomaximal, Sylow
274C22×C59—22no—
275C22×C59—22no—
276C22×C59—22no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59—41no—
280C22×C59—41no—
281C22×C59C4×C22⋊C411yescharacteristic
282C22×C59#38611yescharacteristic
283C22×C59#38811yescharacteristic
284C22×C59#38811yescharacteristic
285C22×C59#39011yescharacteristic
286C22×C59#39011yescharacteristic
287C22×C59#39011yescharacteristic
288C2×D59—42no—
289C2×D59—82no—
290C2×D59—82no—
291C2×D59—82no—
292C4×C59—44no—
293C4×C59—82no—
294C4×C59—82no—
295C4×C59C4×C22⋊C411yescharacteristic
296C4×C59C4×C22⋊C411yescharacteristic
297C4×C59#38611yescharacteristic
298C4×C59#38611yescharacteristic
299C4×C59#38811yescharacteristic
300C4×C59#38811yescharacteristic
301C4×C59#38811yescharacteristic
302C4×C59#38811yescharacteristic
303Dic59—42no—
304Dic59—82no—
305C2×#421—591no—
306C2×#416—591no—
307C2×#410—591no—
308C2×#405—591no—
309#1740D5911yescharacteristic, complemented
310#1735—591no—
311#1733—591no—
312C2×C59—21no—
313C2×C59—21no—
314C2×C59—21no—
315C2×C59—21no—
316C2×C59—21no—
317C2×C59—21no—
318C2×C59#143711yescharacteristic
319C2×C59#197612yes—
320D59—82no—
321C22×#41—591no—
322C22×M16—591no—
323C23×C8—591no—
324C2×#117—591no—
325C2×#117C2×D5911yescharacteristic
326C2×#117C2×D5911yescharacteristic
327C2×#117C2×D5911yescharacteristic
328#421—594no—
329#421—1182no—
330C2×#104—1182no—
331#416—1182no—
332#416—1181no—
333#416—1181no—
334C2×#103—592no—
335C2×#102—592no—
336#410—594no—
337#410—1182no—
338C2×#99—1182no—
339#405—1182no—
340#405—1182no—
341C2×#97—1181no—
342C2×C4×C8—1181no—
343#342—22no—
344#340—1182no—
345#339—1182no—
346C59#1094411yescharacteristic, complemented, Sylow
347C2×#41—592no—
348C2×#41—592no—
349C2×#41—1181no—
350C2×#41—1181no—
351C2×#41—1181no—
352C2×#41—1181no—
353C2×#41—1181no—
354C2×#41—1181no—
355C2×#41—1181no—
356C2×#41—1181no—
357C2×#41—1181no—
358C2×#41—1181no—
359C22×Q8—591no—
360C22×D4—591no—
361C22×D4—1181no—
362C23×C4—1181no—
363C23×C4C22×D5911yescharacteristic
364C2×M16—592no—
365C2×M16—592no—
366C2×M16—592no—
367C2×M16—1181no—
368C2×M16—1181no—
369C2×M16—1181no—
370C22×C8—592no—
371C22×C8—592no—
372C22×C8—591no—
373C22×C8—591no—
374C22×C8—1181no—
375C22×C8—1181no—
376C22×C8—1181no—
377C22×C8—1181no—
378C22×C8—1181no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117—592no—
389#117—1181no—
390#117—1181no—
391#117—1181no—
392#117C4×D5912yes—
393#117C4×D5912yes—
394#117D23612yes—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4C4×D5912yes—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4C22⋊D5912yes—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—1181no—
407#104—1188no—
408#103—1182no—
409#103—1182no—
410#102—1182no—
411#102—1182no—
412#99—1188no—
413#97—1184no—
414C4×C8—1184no—
415C24—1181no—
416C24C2×C22⋊D5911yescharacteristic
417#41—1182no—
418#41—1182no—
419#41—1182no—
420#41—1182no—
421#41—1182no—
422#41—1182no—
423#41—1182no—
424#41—1182no—
425#41—2361no—
426#41—2361no—
427#41—2361no—
428#41—2361no—
429#41—2361no—
430#41—2361no—
431#41—2361no—
432#41—2361no—
433C2×Q8—592no—
434C2×Q8—1181no—
435C2×Q8—1181no—
436C2×Q8—1181no—
437C2×Q8—1181no—
438C2×Q8—1181no—
439C2×D4—592no—
440C2×D4—1182no—
441C2×D4—1182no—
442C2×D4—1181no—
443C2×D4—1181no—
444C2×D4—1181no—
445C2×D4—1181no—
446C2×D4—1181no—
447C2×D4—2361no—
448C2×D4—2361no—
449C2×D4—2361no—
450C2×D4—2361no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—1181no—
455C22×C4—1182no—
456C22×C4—1182no—
457C22×C4—2361no—
458C22×C4—2361no—
459C22×C4—2361no—
460C22×C4—2361no—
461C22×C4—2361no—
462C22×C4C2×C4×D5911yescharacteristic
463C22×C4C2×C4×D5911yescharacteristic
464C22×C4C2×C4×D5911yescharacteristic
465C22×C4C2×D23611yescharacteristic
466C22×C4#1174568511yescharacteristic
467C22×C4#1174568511yescharacteristic
468C22×C4D4×D5911yescharacteristic
469C22×C4D4×D5911yescharacteristic
470C22×C4D4×D5911yescharacteristic
471C22×C4D4×D5911yescharacteristic
472C22×C4#1174568711yescharacteristic
473C22×C4#1174568711yescharacteristic
474C22×C4#1174568911yescharacteristic
475C22×C4#1174568911yescharacteristic
476M16—594no—
477M16—1182no—
478M16—1182no—
479M16—1182no—
480C2×C8—592no—
481C2×C8—592no—
482C2×C8—1182no—
483C2×C8—1182no—
484C2×C8—1182no—
485C2×C8—1181no—
486C2×C8—1182no—
487C2×C8—1181no—
488C2×C8—1182no—
489C2×C8—1182no—
490C2×C8—1182no—
491C2×C8—1181no—
492C2×C8—1181no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—41no—
499C4⋊C4—1182no—
500C4⋊C4—2361no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—1182no—
507C22⋊C4—2361no—
508C22⋊C4(C2×D59)⋊C414yes—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—1182no—
516C42—2361no—
517C23—42no—
518C23—1182no—
519C23—4722no—
520C23—2361no—
521C23—2361no—
522C23C2×(C2×D59)⋊C411yescharacteristic
523C23C4×C22⋊D5911yescharacteristic
524C23C4×C22⋊D5911yescharacteristic
525C23#1427369911yescharacteristic
526C23#1427369911yescharacteristic
527C23#1427371611yescharacteristic
528C23#1427372311yescharacteristic
529Q8—1182no—
530Q8—1182no—
531Q8—2361no—
532Q8—2361no—
533D4—1182no—
534D4—1182no—
535D4—2362no—
536D4—2362no—
537D4—2362no—
538D4—2362no—
539D4—2361no—
540D4—2361no—
541C2×C4—24no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—1182no—
548C2×C4—4722no—
549C2×C4—2362no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2362no—
553C2×C4—2362no—
554C2×C4—2362no—
555C2×C4—2362no—
556C2×C4C42×D5911yescharacteristic
557C2×C4C22⋊C4×D5911yescharacteristic
558C2×C4C22⋊C4×D5911yescharacteristic
559C2×C4#1427368411yescharacteristic
560C2×C4#1427368511yescharacteristic
561C2×C4#1427368511yescharacteristic
562C2×C4(C22⋊D59)⋊C411yescharacteristic
563C2×C4(C22⋊D59)⋊C411yescharacteristic
564C2×C4(C22⋊D59)⋊C411yescharacteristic
565C2×C4(C22⋊D59)⋊C411yescharacteristic
566C2×C4#1427368911yescharacteristic
567C2×C4#1427368911yescharacteristic
568C2×C4#1427368911yescharacteristic
569C2×C4#1427368911yescharacteristic
570C2×C4C4×D23611yescharacteristic
571C2×C4C4×D23611yescharacteristic
572C2×C4#1427369711yescharacteristic
573C2×C4#1427370011yescharacteristic
574C2×C4#1427370011yescharacteristic
575C2×C4#1427370211yescharacteristic
576C2×C4#1427370211yescharacteristic
577C2×C4#1427371111yescharacteristic
578C2×C4#1427371211yescharacteristic
579C2×C4#1427371211yescharacteristic
580C2×C4#1427371311yescharacteristic
581C2×C4#1427371311yescharacteristic
582C2×C4#1427372211yescharacteristic
583C2×C4#1427372211yescharacteristic
584C8—1182no—
585C8—1182no—
586C8—1184no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—4722no—
595C22—4722no—
596C22—4722no—
597C22—2362no—
598C22C4×(C2×D59)⋊C411yescharacteristic
599C22#1683727911yescharacteristic
600C22#1683729211yescharacteristic
601C22#1683729211yescharacteristic
602C22#1683730211yescharacteristic
603C22#1683730211yescharacteristic
604C22#1683730311yescharacteristic
605C4—44no—
606C4—82no—
607C4—82no—
608C4—4722no—
609C4—2362no—
610C4#1683724911yescharacteristic
611C4#1683724911yescharacteristic
612C4#1683728011yescharacteristic
613C4#1683728011yescharacteristic
614C4#1683729111yescharacteristic
615C4#1683729111yescharacteristic
616C4#1683729411yescharacteristic
617C4#1683729411yescharacteristic
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—4722no—
625C2#2142233211yescharacteristic
626C2#2142499112yes—
subgroupG/H
centerC2×C4#14273700
commutatorC2×C4×C59C2×C42
FrattiniC23×C4C22×D59
FittingC59×#1740C2
socleC22×C59#390
prime radicalC22×#11745685C22
cosocleC23×C4×C59C23
hypercenter#1740D59
hypocenterC59#10944
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
4. C2×#16837358C2semidirect
5. #21423812C2semidirect
6. #21423805C2semidirect
7. C59×#1740C2semidirect
13. #16837420C4semidirect
18. C2×#14273650C4semidirect
19. C2×#14273647C4semidirect
21. #16837375C4semidirect
309. #1740D59semidirect
346. C59#10944semidirect

Maximal subgroups

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

subgroupindex
C2×#168374202
C2×#168373962
C2×#168373752
C2×#168373582
#214238122
#214238052
C59×#17402
#1094459