Finite Group Databasebeta

← #28589692

Subgroups — #28589692

Order 15104.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837882C211yesmaximal, characteristic
2C2×#16837445C211yesmaximal, characteristic, complemented
3C2×#16837401C211yesmaximal, characteristic
4C2×#16837396C211yesmaximal, characteristic
5#21424363C212yesmaximal, complemented
6#21424356C212yesmaximal, complemented
7#21424349C212yesmaximal, complemented
8#21424342C212yesmaximal, complemented
9C2×#16837164C211yesmaximal, characteristic, complemented
10#21423793C211yesmaximal, characteristic, complemented
11C59×#2316C211yesmaximal, characteristic, complemented
12C22×#11745685C2211yescharacteristic
13C22×#11745664C2211yescharacteristic
14C23×C59⋊C8C2211yescharacteristic
15#16837882—24no—
16#16837882—24no—
17C2×C4×Dic118C2212yes—
18C2×C4×D236C2212yes—
19C2×(C22⋊D59)⋊C4—22no—
20C2×#14273685—21no—
21C22×C4⋊Dic59C2211yescharacteristic
22C2×C4⋊C4×D59—21no—
23#16837445—24no—
24C2×#14273660C2212yes—
25C2×#14273658C2212yes—
26#16837401C418yescomplemented
27#16837396—24no—
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2212yes—
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×#14273647C2212yes—
34#16837164C2212yes—
35#16837164C2212yes—
36#16837157C2212yes—
37#16837157C2212yes—
38C2×#117×C59C2211yescharacteristic
39C2×#117×C59C2211yescharacteristic
40C22×C4⋊C4×C59C2211yescharacteristic
41C59×#389—21no—
42C59×#389—21no—
43C59×#389C2212yes—
44C59×#389C2212yes—
45C59×#380—22no—
46C2×#11745685—22no—
47C2×#11745685—22no—
48C2×#11745685—22no—
49C2×#11745685C2×C414yes—
50C2×#11745685D414yes—
51C2×#11745685Q814yes—
52C22×Dic118C2311yescharacteristic
53C22×D236C2311yescharacteristic
54C22×C22⋊D59—21no—
55C23×Dic59—21no—
56C22×C4×D59—21no—
57C22×C4×D59—22no—
58C2×#11745664—22no—
59C2×#11745664—22no—
60C2×#11745664C2×C414yescomplemented
61C22×C59⋊C8—22no—
62C22×C59⋊C8—22no—
63C22×C59⋊C8—22no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2312yes—
67C4×Dic118—24no—
68C4×Dic118—24no—
69C4×D236—24no—
70C4×D236—24no—
71(C22⋊D59)⋊C4—216no—
72#14273685—28no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—22no—
75C2×C4⋊Dic59D414yes—
76C2×C4⋊Dic59C2312yes—
77C2×Dic59⋊C4—22no—
78C4⋊C4×D59—28no—
79C2×(C2×D59)⋊C4—22no—
80C2×C4×Dic59—22no—
81#14273660—24no—
82#14273658—24no—
83#14273650—22no—
84#14273650—22no—
85#14273650C2×C418yescomplemented
86#14273647—22no—
87#14273647—22no—
88#14273647C2×C418yescomplemented
89C23×C4×C59—21no—
90C23×C4×C59C2311yescharacteristic
91#117×C59—22no—
92#117×C59—22no—
93#117×C59—22no—
94#117×C59—22no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59—22no—
97C2×C4⋊C4×C59—22no—
98C2×C4⋊C4×C59—22no—
99C2×C4⋊C4×C59C2×C414yescomplemented
100C2×C4⋊C4×C59D411yescharacteristic
101C2×C4⋊C4×C59D411yescharacteristic
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C4⋊C4×C59C2312yes—
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59D412yes—
107C2×C42×C59—21no—
108C2×C42×C59C2312yes—
109#95×C59—22no—
110#95×C59—22no—
111C24×C59C2×D411yescharacteristic
112C23×D59—21no—
113C22×C4×C59—22no—
114C22×C4×C59—24no—
115C22×C4×C59—21no—
116C22×C4×C59—21no—
117C22×C4×C59—81no—
118C22×C4×C59—81no—
119C22×C4×C59C22×C412yes—
120C22×C4×C59C2×D412yes—
121C22×C4×C59C2×D412yes—
122C22×C4×C59C2×D412yes—
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D412yes—
125C22×C4×C59C2×Q812yes—
126C22×C4×C59#4112yes—
127C22×C4×C59#4112yes—
128C22×C4×C59C2411yescharacteristic
129C2×C22⋊D59—24no—
130C2×C22⋊D59—24no—
131C2×C22⋊D59—24no—
132C22×Dic59—21no—
133C22×Dic59—22no—
134C22×Dic59—24no—
135C22×Dic59—81no—
136C22×Dic59—81no—
137#11745685—28no—
138#11745685—28no—
139#11745685—28no—
140#11745685C4⋊C4116yescomplemented
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236C22×C412yes—
145C2×D236C2×D412yes—
146C2×D236C2×Q812yes—
147C2×C4×D59—24no—
148C2×C4×D59—24no—
149C2×C4×D59—24no—
150C2×C4×D59—28no—
151C2×C4×D59—82no—
152C2×C4×D59—82no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118C22×C412yes—
157C2×Dic118C2×D412yes—
158C2×Dic118C2×Q812yes—
159C4⋊C4×C59—24no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—82no—
168C4⋊C4×C59—82no—
169C4⋊C4×C59C22×C414yes—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—42no—
172C42×C59—22no—
173C42×C59—22no—
174C42×C59—42no—
175(C2×D59)⋊C4—28no—
176C4⋊Dic59—24no—
177C4⋊Dic59—22no—
178C4⋊Dic59C2×D414yes—
179Dic59⋊C4—28no—
180C4×Dic59—28no—
181#11745664—28no—
182C2×C59⋊C8—22no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—44no—
191C2×C59⋊C8C22×C414yes—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59C4×D412yes—
197C23×C59#12012yes—
198C23×C59C22⋊Q812yes—
199C23×C59C22×D411yescharacteristic
200C22×D59—22no—
201C22×D59—22no—
202C22×D59—22no—
203C22×D59—41no—
204C22×D59—41no—
205C22×D59—41no—
206C22×D59—41no—
207C2×C4×C59—24no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—84no—
223C2×C4×C59C2×C4⋊C414yes—
224C2×C4×C59C4×D412yes—
225C2×C4×C59C4×D414yes—
226C2×C4×C59#12012yes—
227C2×C4×C59C22⋊Q812yes—
228C2×C4×C59C22⋊Q814yes—
229C2×C4×C59#12314yes—
230C2×C4×C59C23×C411yescharacteristic
231C2×C4×C59C22×D411yescharacteristic
232C2×C4×C59C22×D411yescharacteristic
233C2×C4×C59C22×Q811yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C22⋊D59—216no—
237C2×Dic59—22no—
238C2×Dic59—22no—
239C2×Dic59—22no—
240C2×Dic59—24no—
241C2×Dic59—42no—
242C2×Dic59—42no—
243C2×Dic59—84no—
244D236—22no—
245D236—22no—
246D236—22no—
247D236C2×C4⋊C414yes—
248C4×D59—216no—
249C4×D59—88no—
250Dic118—22no—
251Dic118—22no—
252Dic118—22no—
253Dic118C2×C4⋊C414yes—
254C59⋊C8—24no—
255C59⋊C8—41no—
256C59⋊C8—41no—
257#20878—591nomaximal, Sylow
258C22×C59—22no—
259C22×C59—22no—
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—22no—
264C22×C59#38514yes—
265C22×C59C2×C4×D411yescharacteristic
266C22×C59C2×#12011yescharacteristic
267C22×C59C2×C22⋊Q811yescharacteristic
268C22×C59#57611yescharacteristic
269C22×C59#57711yescharacteristic
270C22×C59#57711yescharacteristic
271C22×C59#57811yescharacteristic
272C2×D59—24no—
273C2×D59—42no—
274C2×D59—42no—
275C2×D59—42no—
276C2×D59—42no—
277C2×D59—42no—
278C2×D59—42no—
279C4×C59—44no—
280C4×C59—42no—
281C4×C59—42no—
282C4×C59—81no—
283C4×C59—81no—
284C4×C59#38514yes—
285C4×C59C22×C4⋊C411yescharacteristic
286C4×C59C2×C4×D411yescharacteristic
287C4×C59C2×C22⋊Q811yescharacteristic
288C4×C59C2×#12311yescharacteristic
289Dic59—24no—
290Dic59—44no—
291C2×#520—591no—
292C2×#428—591no—
293C2×#417—591no—
294C2×#416—591no—
295#2316D5911yescharacteristic, complemented
296#1865—592no—
297#1864—592no—
298#1863—592no—
299#1862—592no—
300C2×#340—591no—
301#1729—591no—
302C2×C59—24no—
303C2×C59C2×#38511yescharacteristic
304C2×C59#294611yescharacteristic
305C2×C59#294711yescharacteristic
306C2×C59#301211yescharacteristic
307C2×C59#301311yescharacteristic
308C2×C59#308311yescharacteristic
309C2×C59#308411yescharacteristic
310D59—44no—
311C22×#41—591no—
312C22×M16—591no—
313C23×C8—591no—
314#520—1184no—
315#520—1184no—
316C2×C4×Q8—592no—
317C2×C4×D4—592no—
318C2×C4×D4—1182no—
319C2×#117—1181no—
320C2×#117C2×D5911yescharacteristic
321C2×#117C2×D5911yescharacteristic
322C22×C4⋊C4—591no—
323C22×C4⋊C4—1181no—
324C22×C4⋊C4C2×D5911yescharacteristic
325#428—1184no—
326C2×#107—592no—
327C2×#106—592no—
328#417—598no—
329#416—1184no—
330C2×#103—592no—
331C2×#103—591no—
332C2×#103—591no—
333C2×#102—592no—
334C2×#102—591no—
335C2×#102—591no—
336#389—21no—
337#389—21no—
338#389C2×D5912yes—
339#389C2×D5912yes—
340#380—22no—
341#340—592no—
342#340—592no—
343#336—592no—
344#336—592no—
345C59#2087811yescharacteristic, complemented, Sylow
346C2×#41—594no—
347C2×#41—594no—
348C2×#41—594no—
349C2×#41—1182no—
350C2×#41—1182no—
351C2×#41—1182no—
352C22×Q8—591no—
353C22×D4—591no—
354C22×D4—1181no—
355C23×C4—21no—
356C23×C4—1181no—
357C23×C4—1181no—
358C23×C4—1182no—
359C23×C4C22×D5911yescharacteristic
360C2×M16—594no—
361C2×M16—1182no—
362C2×M16—1182no—
363C22×C8—592no—
364C22×C8—591no—
365C22×C8—591no—
366C22×C8—1182no—
367C22×C8—1182no—
368C22×C8—1182no—
369C4×Q8—1184no—
370C4×Q8—1184no—
371C4×D4—1184no—
372C4×D4—1184no—
373C4×D4—11816no—
374#117—22no—
375#117—22no—
376#117—22no—
377#117—22no—
378#117—1188no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—594no—
384C2×C4⋊C4—592no—
385C2×C4⋊C4—1188no—
386C2×C4⋊C4—1182no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1182no—
389C2×C4⋊C4C4×D5914yescomplemented
390C2×C4⋊C4C22⋊D5911yescharacteristic
391C2×C4⋊C4C22⋊D5911yescharacteristic
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C4⋊C4C22×D5912yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1182no—
397C2×C22⋊C4C22⋊D5912yes—
398C2×C42—21no—
399C2×C42—1182no—
400C2×C42C22×D5912yes—
401#107—1184no—
402#106—1184no—
403#103—598no—
404#103—1182no—
405#103—1182no—
406#102—598no—
407#102—1182no—
408#102—1182no—
409#95—22no—
410#95—22no—
411C24—1181no—
412C24C2×C22⋊D5911yescharacteristic
413#41—5916no—
414#41—1188no—
415#41—1188no—
416#41—1188no—
417C2×Q8—592no—
418C2×Q8—592no—
419C2×Q8—592no—
420C2×Q8—1181no—
421C2×Q8—1181no—
422C2×Q8—1181no—
423C2×D4—592no—
424C2×D4—592no—
425C2×D4—592no—
426C2×D4—1184no—
427C2×D4—1184no—
428C2×D4—1184no—
429C2×D4—1181no—
430C2×D4—1181no—
431C2×D4—1181no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—1188no—
439C22×C4—1181no—
440C22×C4—1184no—
441C22×C4—1184no—
442C22×C4—1184no—
443C22×C4—1184no—
444C22×C4—1182no—
445C22×C4—4722no—
446C22×C4—4722no—
447C22×C4—4721no—
448C22×C4—4721no—
449C22×C4C2×C4×D5912yes—
450C22×C4#1174568512yes—
451C22×C4D4×D5912yes—
452C22×C4D4×D5912yes—
453C22×C4Q8×D5912yes—
454C22×C4#1174568912yes—
455C22×C4C2×C22⋊D5912yes—
456C22×C4C2×C22⋊D5912yes—
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C23×D5911yescharacteristic
459M16—1188no—
460C2×C8—594no—
461C2×C8—1182no—
462C2×C8—1182no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—2364no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—594no—
481C4⋊C4—1188no—
482C4⋊C4—1184no—
483C4⋊C4—1182no—
484C4⋊C4C2×C4×D5914yes—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—1188no—
488C42—22no—
489C42—22no—
490C42—42no—
491C42—1188no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—1182no—
497C23—1182no—
498C23—1182no—
499C23—2361no—
500C23—2361no—
501C23—2361no—
502C23—2361no—
503C23C4×C22⋊D5912yes—
504C23#1427369512yes—
505C23#1427370312yes—
506C23C22×C22⋊D5911yescharacteristic
507Q8—594no—
508Q8—1182no—
509Q8—1182no—
510Q8—1182no—
511D4—594no—
512D4—11816no—
513D4—1182no—
514D4—1182no—
515D4—1182no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—1182no—
533C2×C4—1182no—
534C2×C4—1182no—
535C2×C4—11816no—
536C2×C4—1184no—
537C2×C4—4728no—
538C2×C4—4724no—
539C2×C4—2362no—
540C2×C4—2362no—
541C2×C4C4⋊C4×D5914yes—
542C2×C4C4×C22⋊D5912yes—
543C2×C4#1427368914yes—
544C2×C4#1427369512yes—
545C2×C4#1427370312yes—
546C2×C4#1427370514yes—
547C2×C4#1427371214yes—
548C2×C4C22×C4×D5911yescharacteristic
549C2×C4C2×D4×D5911yescharacteristic
550C2×C4C2×D4×D5911yescharacteristic
551C2×C4C2×Q8×D5911yescharacteristic
552C2×C4C2×#1174568511yescharacteristic
553C2×C4C2×#1174568911yescharacteristic
554C8—1184no—
555C8—2361no—
556C8—2361no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—1184no—
564C22—2362no—
565C22—2362no—
566C22—2362no—
567C22—2362no—
568C22—2362no—
569C22—2362no—
570C22#1683727314yes—
571C22C2×C4×C22⋊D5911yescharacteristic
572C22C2×#1427369511yescharacteristic
573C22C2×#1427370311yescharacteristic
574C22#1683820511yescharacteristic
575C22#1683821011yescharacteristic
576C22#1683821311yescharacteristic
577C22#1683821711yescharacteristic
578C4—42no—
579C4—42no—
580C4—44no—
581C4—81no—
582C4—81no—
583C4—1184no—
584C4—2364no—
585C4#1683727314yes—
586C4C2×C4⋊C4×D5911yescharacteristic
587C4C2×#1427368911yescharacteristic
588C4C2×#1427370511yescharacteristic
589C4C2×#1427371211yescharacteristic
590C2—24no—
591C2—2364no—
592C2C2×#1683727311yescharacteristic
593C2#2143292811yescharacteristic
594C2#2143293911yescharacteristic
595C2#2143339711yescharacteristic
596C2#2143340811yescharacteristic
597C2#2143394311yescharacteristic
598C2#2143395411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2316C2
socleC23×C59C22×D4
prime radicalC22×#11745685C22
cosocleC22×C4×C59C24
hypercenter#2316D59
hypocenterC59#20878
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. C2×#16837445C2semidirect
5. #21424363C2semidirect
6. #21424356C2semidirect
7. #21424349C2semidirect
8. #21424342C2semidirect
9. C2×#16837164C2semidirect
10. #21423793C2semidirect
11. C59×#2316C2semidirect
26. #16837401C4semidirect
60. C2×#11745664C2×C4semidirect
85. #14273650C2×C4semidirect
88. #14273647C2×C4semidirect
99. C2×C4⋊C4×C59C2×C4semidirect
140. #11745685C4⋊C4semidirect
295. #2316D59semidirect
345. C59#20878semidirect
389. C2×C4⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378822
C2×#168374452
C2×#168374012
C2×#168373962
#214243632
#214243562
#214243492
#214243422
C2×#168371642
#214237932
C59×#23162
#2087859