Finite Group Databasebeta

← #28588221

Subgroups — #28588221

Order 15104.

1462 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×#16837883C211yesmaximal, characteristic
2C2×#16837428C211yesmaximal, characteristic, complemented
3C2×#16837401C211yesmaximal, characteristic
4C2×#16837399C211yesmaximal, characteristic
5#21426720C211yesmaximal, characteristic, complemented
6#21424110C212yesmaximal, complemented
7#21424105C212yesmaximal, complemented
8#21424100C212yesmaximal, complemented
9#21424095C212yesmaximal, complemented
10#21423815C211yesmaximal, characteristic, complemented
11C2×C59×#340C211yesmaximal, characteristic, complemented
12C22×#11745685C2211yescharacteristic
13C22×#11745664C2211yescharacteristic
14#16837883—24no—
15#16837883C418yescomplemented
16C2×#14273692C2212yes—
17C2×#14273689C2212yes—
18C2×C4×C22⋊D59—22no—
19C2×#14273686C2211yescharacteristic
20C2×#14273686C2211yescharacteristic
21C2×#14273685—21no—
22C2×C4⋊C4×D59—21no—
23#16837428—24no—
24C2×Dic59⋊C8C2212yes—
25C2×Dic59⋊C8C2212yes—
26#16837401—24no—
27#16837399—24no—
28C2×#14273652C2211yescharacteristic
29C2×#14273652C2211yescharacteristic
30C2×#14273650C2212yes—
31C2×#14273649C2211yescharacteristic
32C2×#14273649C2211yescharacteristic
33C2×#14273647C2212yes—
34#16837298—21no—
35#16837298—21no—
36#16837275—22no—
37#16837271C2212yes—
38#16837267C2212yes—
39#16837164C2212yes—
40#16837164C2212yes—
41C23×C8×C59C2211yescharacteristic
42C22×C4⋊C4×C59C2211yescharacteristic
43C22×C4⋊C4×C59C2211yescharacteristic
44C59×#340C2212yes—
45C59×#340C2212yes—
46C2×#11745685—22no—
47C2×#11745685—22no—
48C2×#11745685—22no—
49C2×#11745685C2×C414yes—
50C2×#11745685D414yes—
51C2×#11745685Q814yes—
52C22×Dic118C2311yescharacteristic
53C22×D236C2311yescharacteristic
54C22×C22⋊D59—21no—
55C22×C4×D59—21no—
56C22×C4×D59—22no—
57C2×#11745664—22no—
58C2×#11745664—22no—
59C2×#11745664—22no—
60C22×C59⋊C8C2312yes—
61#14273692—24no—
62#14273692C2×C418yescomplemented
63#14273689—24no—
64#14273689C2×C418yescomplemented
65C4×C22⋊D59—216no—
66#14273686—22no—
67#14273686—22no—
68#14273686C2×C414yes—
69#14273686D414yes—
70#14273685—28no—
71C2×C4⋊Dic59D411yescharacteristic
72C2×C4⋊Dic59D411yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—22no—
76C2×Dic59⋊C4—22no—
77C2×Dic59⋊C4—22no—
78C4⋊C4×D59—28no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59D412yes—
81C2×(C2×D59)⋊C4—22no—
82C2×C4×Dic59—22no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59C2312yes—
85Dic59⋊C8—24no—
86Dic59⋊C8—24no—
87#14273652—22no—
88#14273652—22no—
89#14273650—24no—
90#14273649—22no—
91#14273649—22no—
92#14273647—24no—
93C59⋊#95—22no—
94C59⋊#95—22no—
95C23×C4×C59—21no—
96C23×C4×C59—21no—
97C23×C4×C59C2311yescharacteristic
98C22×C8×C59—22no—
99C22×C8×C59—22no—
100C22×C8×C59—22no—
101C22×C8×C59C2311yescharacteristic
102C22×C8×C59C2311yescharacteristic
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—22no—
105C2×C4⋊C4×C59—22no—
106C2×C4⋊C4×C59—22no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59C2×C414yes—
109C2×C4⋊C4×C59C2312yes—
110C2×C4⋊C4×C59C2312yes—
111C24×C59C2×D411yescharacteristic
112C23×D59—21no—
113C22×C4×C59—22no—
114C22×C4×C59—24no—
115C22×C4×C59—22no—
116C22×C4×C59—42no—
117C22×C4×C59—81no—
118C22×C4×C59—81no—
119C22×C4×C59—81no—
120C22×C4×C59—81no—
121C22×C4×C59C22×C412yes—
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D412yes—
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—21no—
134C22×Dic59—21no—
135C22×Dic59C2×D412yes—
136C22×Dic59C2×D412yes—
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—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—44no—
166C4⋊C4×C59—24no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59C22×C414yes—
172C22⋊Dic59—42no—
173C22⋊Dic59C22⋊C418yes—
174(C2×D59)⋊C4—28no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59C22⋊C414yes—
180C4⋊Dic59C22⋊C414yes—
181Dic59⋊C4—28no—
182Dic59⋊C4—82no—
183Dic59⋊C4—82no—
184C4×Dic59—28no—
185C4×Dic59—42no—
186C4×Dic59C22×C414yes—
187C4×Dic59C2×D414yes—
188#11745664—44no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—22no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59C2×C22⋊C412yes—
197C23×C59#12112yes—
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—42no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59—42no—
212C2×C4×C59—42no—
213C2×C4×C59—84no—
214C2×C4×C59—84no—
215C2×C4×C59C2×C22⋊C412yes—
216C2×C4×C59C2×C4⋊C414yes—
217C2×C4×C59#11714yes—
218C2×C4×C59#12114yes—
219C2×C4×C59#12112yes—
220C2×C4×C59C22⋊Q814yes—
221C2×C4×C59C22⋊Q812yes—
222C2×C4×C59C23×C411yescharacteristic
223C2×C4×C59C22×D411yescharacteristic
224C2×C4×C59C22×D411yescharacteristic
225C2×C4×C59C22×Q811yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C22⋊D59—216no—
229C2×Dic59—22no—
230C2×Dic59—22no—
231C2×Dic59—22no—
232C2×Dic59—22no—
233C2×Dic59—22no—
234C2×Dic59—22no—
235C2×Dic59—22no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59C2×C22⋊C414yes—
243C2×Dic59C2×C22⋊C414yes—
244D236—22no—
245D236—22no—
246D236—22no—
247D236C2×C4⋊C414yes—
248C4×D59—216no—
249C4×D59—88no—
250Dic118—22no—
251Dic118—22no—
252Dic118—22no—
253Dic118C2×C4⋊C414yes—
254C8×C59—41no—
255C8×C59—41no—
256C59⋊C8—42no—
257#20652—591nomaximal, Sylow
258C22×C59—22no—
259C22×C59—22no—
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—22no—
264C22×C59#38014yes—
265C22×C59C22×C22⋊C411yescharacteristic
266C22×C59C2×#12111yescharacteristic
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—82no—
281C4×C59#38014yes—
282C4×C59C22×C4⋊C411yescharacteristic
283C4×C59C2×#11711yescharacteristic
284C4×C59C2×#12111yescharacteristic
285C4×C59C2×C22⋊Q811yescharacteristic
286Dic59—24no—
287Dic59—24no—
288Dic59—24no—
289Dic59—81no—
290Dic59—81no—
291C2×#520—591no—
292C2×#424—591no—
293C2×#417—591no—
294C2×#416—591no—
295#2317—591no—
296#1805—592no—
297#1804—592no—
298#1803—592no—
299#1802—592no—
300#1736—591no—
301C2×#340D5911yescharacteristic, complemented
302C2×C59—24no—
303C2×C59C2×#38011yescharacteristic
304C2×C59#289711yescharacteristic
305C2×C59#289811yescharacteristic
306C2×C59#304511yescharacteristic
307C2×C59#304611yescharacteristic
308C2×C59#308311yescharacteristic
309C2×C59#308411yescharacteristic
310D59—44no—
311C22×#41—591no—
312C22×M16—591no—
313C23×C8C2×D5911yescharacteristic
314#520—598no—
315#520—1184no—
316C2×C4×Q8—592no—
317C2×C4×D4—592no—
318C2×C4×D4—1182no—
319C2×#117—591no—
320C2×#117—591no—
321C2×#117—1181no—
322C22×C4⋊C4—1181no—
323C22×C4⋊C4C2×D5911yescharacteristic
324C22×C4⋊C4C2×D5911yescharacteristic
325#424—1184no—
326C2×C4⋊C8—592no—
327C2×C4⋊C8—592no—
328#417—1184no—
329#416—1184no—
330C2×#103—591no—
331C2×#103—591no—
332C2×#103—592no—
333C2×#102—591no—
334C2×#102—591no—
335C2×#102—592no—
336#389—1181no—
337#389—1181no—
338#385—1182no—
339#384—592no—
340#382—592no—
341#340—592no—
342#340—592no—
343#340C2×D5912yes—
344#340C2×D5912yes—
345C59#2065211yescharacteristic, 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—21no—
357C23×C4—1182no—
358C23×C4—1181no—
359C23×C4C22×D5911yescharacteristic
360C2×M16—1182no—
361C2×M16—1182no—
362C2×M16—1182no—
363C22×C8—22no—
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—592no—
367C22×C8C22×D5911yescharacteristic
368C22×C8C22×D5911yescharacteristic
369C4×Q8—598no—
370C4×Q8—1184no—
371C4×D4—598no—
372C4×D4—1184no—
373C4×D4—11816no—
374#117—594no—
375#117—594no—
376#117—1188no—
377#117—1182no—
378#117—1182no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—1188no—
389C2×C4⋊C4—1182no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1182no—
392C2×C4⋊C4C4×D5914yes—
393C2×C4⋊C4C22×D5912yes—
394C2×C4⋊C4C22×D5912yes—
395C2×C22⋊C4—592no—
396C2×C22⋊C4—1182no—
397C2×C22⋊C4—1181no—
398C2×C42—592no—
399C2×C42—1182no—
400C2×C42—1181no—
401C4⋊C8—1184no—
402C4⋊C8—1184no—
403#103—1184no—
404#103—1182no—
405#103—1182no—
406#102—1184no—
407#102—1182no—
408#102—1182no—
409#95—1182no—
410#95—1182no—
411C24—1181no—
412C24D4×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—22no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—592no—
441C22×C4—592no—
442C22×C4—1188no—
443C22×C4—1181no—
444C22×C4—1184no—
445C22×C4—1184no—
446C22×C4—1184no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—4722no—
450C22×C4—4722no—
451C22×C4C2×C4×D5912yes—
452C22×C4#1174568512yes—
453C22×C4#1174568512yes—
454C22×C4D4×D5911yescharacteristic
455C22×C4D4×D5912yes—
456C22×C4D4×D5912yes—
457C22×C4Q8×D5912yes—
458C22×C4C23×D5911yescharacteristic
459M16—2364no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—44no—
467C2×C8—1182no—
468C2×C8—1182no—
469C2×C8—1182no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4—594no—
476C4⋊C4—594no—
477C4⋊C4—1188no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—4722no—
483C4⋊C4—4722no—
484C4⋊C4C2×C4×D5914yes—
485C22⋊C4—598no—
486C22⋊C4—1188no—
487C22⋊C4—2362no—
488C42—594no—
489C42—594no—
490C42—1188no—
491C42—2362no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—1182no—
497C23—1182no—
498C23—1182no—
499C23—2361no—
500C23—2361no—
501C23—2361no—
502C23—2361no—
503C23C22⋊C4×D5912yes—
504C23#1427370012yes—
505C23#1427370512yes—
506C23C2×D4×D5911yescharacteristic
507Q8—594no—
508Q8—1182no—
509Q8—1182no—
510Q8—1182no—
511D4—594no—
512D4—11816no—
513D4—1182no—
514D4—1182no—
515D4—1182no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—594no—
525C2×C4—594no—
526C2×C4—1182no—
527C2×C4—1182no—
528C2×C4—1182no—
529C2×C4—1182no—
530C2×C4—1182no—
531C2×C4—1182no—
532C2×C4—1182no—
533C2×C4—11816no—
534C2×C4—4728no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4C22⋊C4×D5912yes—
542C2×C4C4⋊C4×D5914yes—
543C2×C4#1427368414yes—
544C2×C4#1427370014yes—
545C2×C4#1427370012yes—
546C2×C4#1427370514yes—
547C2×C4#1427370512yes—
548C2×C4C22×C4×D5911yescharacteristic
549C2×C4C2×D4×D5911yescharacteristic
550C2×C4C2×D4×D5911yescharacteristic
551C2×C4C2×Q8×D5911yescharacteristic
552C2×C4C2×#1174568511yescharacteristic
553C2×C4C2×#1174568511yescharacteristic
554C8—41no—
555C8—41no—
556C8—2362no—
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#1683726114yes—
571C22C2×C22⋊C4×D5911yescharacteristic
572C22C2×#1427370011yescharacteristic
573C22C2×#1427370511yescharacteristic
574C22#1683820311yescharacteristic
575C22#1683820811yescharacteristic
576C22#1683820811yescharacteristic
577C22#1683821511yescharacteristic
578C4—44no—
579C4—82no—
580C4—1184no—
581C4—1184no—
582C4—1184no—
583C4—4721no—
584C4—4721no—
585C4#1683726114yes—
586C4C2×C4⋊C4×D5911yescharacteristic
587C4C2×#1427368411yescharacteristic
588C4C2×#1427370011yescharacteristic
589C4C2×#1427370511yescharacteristic
590C2—24no—
591C2—2364no—
592C2C2×#1683726111yescharacteristic
593C2#2143261711yescharacteristic
594C2#2143262411yescharacteristic
595C2#2143364011yescharacteristic
596C2#2143365111yescharacteristic
597C2#2143395211yescharacteristic
598C2#2143396311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#340C2
socleC23×C59C22×D4
prime radicalC22×#11745685C22
cosocleC22×C4×C59C24
hypercenterC2×#340D59
hypocenterC59#20652
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C2×#16837428C2semidirect
5. #21426720C2semidirect
6. #21424110C2semidirect
7. #21424105C2semidirect
8. #21424100C2semidirect
9. #21424095C2semidirect
10. #21423815C2semidirect
11. C2×C59×#340C2semidirect
15. #16837883C4semidirect
62. #14273692C2×C4semidirect
64. #14273689C2×C4semidirect
140. #11745685C4⋊C4semidirect
301. C2×#340D59semidirect
345. C59#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#168378832
C2×#168374282
C2×#168374012
C2×#168373992
#214267202
#214241102
#214241052
#214241002
#214240952
#214238152
C2×C59×#3402
#2065259