Finite Group Databasebeta

← #28559315

Subgroups — #28559315

Order 15104.

622 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434692C211yesmaximal, characteristic
2#21433171C211yesmaximal, characteristic, complemented
3#21423008C211yesmaximal, characteristic, complemented
4#21422996C211yesmaximal, characteristic, complemented
5#21422986C211yesmaximal, characteristic, complemented
6#21422979C211yesmaximal, characteristic, complemented
7#21422918C211yesmaximal, characteristic, complemented
8#21422906C211yesmaximal, characteristic, complemented
9C8×#11745671C211yesmaximal, characteristic, complemented
10C8×#11745669C211yesmaximal, characteristic, complemented
11#21422861C211yesmaximal, characteristic, complemented
12#21422556C211yesmaximal, characteristic, complemented
13#21422546C211yesmaximal, characteristic, complemented
14#21422377C211yesmaximal, characteristic, complemented
15C59×#2993C211yesmaximal, characteristic, complemented
16#16837952C2211yescharacteristic
17#16837812C411yescharacteristic
18#16837792C411yescharacteristic
19#16837768C411yescharacteristic
20#16837761C411yescharacteristic
21#16837653C411yescharacteristic
22#16837653C411yescharacteristic
23#16837650C411yescharacteristic
24#16837650C411yescharacteristic
25C4×#11745671C2211yescharacteristic
26C4×#11745671C2211yescharacteristic
27C4×#11745669C2211yescharacteristic
28C4×#11745669C2211yescharacteristic
29#16837481C2211yescharacteristic
30#16837481C2211yescharacteristic
31#16837472—21no—
32#16837472—21no—
33C8×D236C2211yescharacteristic
34C8×D236C2211yescharacteristic
35#16837459C2211yescharacteristic
36#16837367—21no—
37#16837367—21no—
38C4×#11745664C2211yescharacteristic
39C2×C4×C59⋊C8C2211yescharacteristic
40#16837154C2211yescharacteristic
41#16837154C2211yescharacteristic
42#16837152C2211yescharacteristic
43#16837152C2211yescharacteristic
44Q8⋊(C59⋊C8)C2211yescharacteristic
45Q8⋊(C59⋊C8)C2211yescharacteristic
46D4⋊(C59⋊C8)C2211yescharacteristic
47D4⋊(C59⋊C8)C2211yescharacteristic
48#16837129C2211yescharacteristic
49#16837129C2211yescharacteristic
50#16837129C2211yescharacteristic
51#16837129C2211yescharacteristic
52#16837124C2211yescharacteristic
53#16837124C2211yescharacteristic
54C8×C59⋊C8C2211yescharacteristic
55C8×C59⋊C8C2211yescharacteristic
56C59×#529C2211yescharacteristic
57C59×#446C2211yescharacteristic
58C8×Q8×C59C2211yescharacteristic
59C59×#436—21no—
60C59×#436C2211yescharacteristic
61C59×#435—21no—
62C8×D4×C59C2211yescharacteristic
63C59×#433—21no—
64C59×#431C2211yescharacteristic
65C59×#424—21no—
66C59×#423C2211yescharacteristic
67C2×#11745685—21no—
68C2×#11745671C2×C411yescharacteristic
69C2×#11745671C2×C411yescharacteristic
70C2×#11745669C2×C411yescharacteristic
71C2×#11745669C2×C411yescharacteristic
72C2×#11745664C2×C411yescharacteristic
73C2×#11745664C2×C411yescharacteristic
74C2×#11745659—21no—
75C2×#11745659—21no—
76C22×C59⋊C8—21no—
77C2×C8×D59—21no—
78C2×C8×D59—21no—
79#14273727C2×C411yescharacteristic
80#14273724C2×C411yescharacteristic
81#14273715—21no—
82#14273701—21no—
83#14273701C2×C411yescharacteristic
84#14273701C2×C411yescharacteristic
85C4×D236—21no—
86C4×D236C2311yescharacteristic
87C4×D236C2311yescharacteristic
88#14273660C2×C411yescharacteristic
89#14273660C2×C411yescharacteristic
90#14273658C2×C411yescharacteristic
91#14273658C2×C411yescharacteristic
92C4⋊(C59⋊C8)C2311yescharacteristic
93C4⋊(C59⋊C8)C2311yescharacteristic
94#14273651C2×C411yescharacteristic
95#14273651C2×C411yescharacteristic
96#14273648C2×C411yescharacteristic
97#14273648C2×C411yescharacteristic
98#14273647C2×C411yescharacteristic
99#14273647C2×C411yescharacteristic
100#14273647C2×C411yescharacteristic
101#14273647C2×C411yescharacteristic
102C22⋊(C59⋊C8)—21no—
103(C2×D59)⋊C8—21no—
104(C2×D59)⋊C8—21no—
105(C2×D59)⋊C8—21no—
106(C2×D59)⋊C8—21no—
107#14273638—21no—
108#14273638C2×C411yescharacteristic
109#14273638C2×C411yescharacteristic
110C4×C59⋊C8—21no—
111C4×C59⋊C8C2311yescharacteristic
112C4×C59⋊C8C2311yescharacteristic
113C4×C59⋊C8C2311yescharacteristic
114C4×C59⋊C8C2311yescharacteristic
115C2×M16×C59—21no—
116C2×M16×C59—21no—
117C22×C8×C59—21no—
118#126×C59—21no—
119#125×C59C2×C411yescharacteristic
120#124×C59C2×C411yescharacteristic
121#123×C59—21no—
122C22⋊Q8×C59C2×C411yescharacteristic
123#121×C59C2×C411yescharacteristic
124C4×Q8×C59C2311yescharacteristic
125C4×D4×C59—21no—
126C4×D4×C59C2311yescharacteristic
127#117×C59—21no—
128C2×C42×C59C2311yescharacteristic
129C4⋊C8×C59—21no—
130C4⋊C8×C59—21no—
131C4⋊C8×C59—21no—
132C4⋊C8×C59D411yescharacteristic
133C4⋊C8×C59D411yescharacteristic
134C4⋊C8×C59C2311yescharacteristic
135C4⋊C8×C59C2311yescharacteristic
136C22⋊C8×C59—21no—
137C22⋊C8×C59—21no—
138C22⋊C8×C59D411yescharacteristic
139C22⋊C8×C59D411yescharacteristic
140#97×C59—21no—
141C4×C8×C59—21no—
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59C22×C411yescharacteristic
145C2×D4×C59—21no—
146C2×D4×C59C22×C411yescharacteristic
147C22×C4×C59—21no—
148C22×C4×C59C22×C411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59#4111yescharacteristic
151C2×C22⋊D59—21no—
152C2×C22⋊D59—21no—
153#11745685—81no—
154#11745685—81no—
155C2×D236—21no—
156C2×D236—21no—
157C2×D236C22×C411yescharacteristic
158C2×D236C22×C411yescharacteristic
159C2×C4×D59—21no—
160C2×C4×D59—21no—
161C2×Dic118—21no—
162M16×C59—81no—
163M16×C59—81no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—81no—
167C2×C8×C59C2×D411yescharacteristic
168C2×C8×C59C2×D411yescharacteristic
169C2×C8×C59C2×D411yescharacteristic
170C2×C8×C59C2×D411yescharacteristic
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59C22×C411yescharacteristic
176C4⋊C4×C59C22×C411yescharacteristic
177C22⋊C4×C59—21no—
178C22⋊C4×C59—21no—
179C22⋊C4×C59—21no—
180C22⋊C4×C59—21no—
181C22⋊C4×C59—21no—
182C42×C59—21no—
183C42×C59C22×C411yescharacteristic
184C42×C59C2×D411yescharacteristic
185C42×C59#4111yescharacteristic
186C42×C59C2411yescharacteristic
187#11745671—22no—
188#11745671—22no—
189#11745669—22no—
190#11745669—22no—
191(C2×D59)⋊C4—21no—
192(C2×D59)⋊C4—21no—
193(C2×D59)⋊C4—21no—
194(C2×D59)⋊C4—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59C22×C411yescharacteristic
197C4⋊Dic59C22×C411yescharacteristic
198#11745664—22no—
199#11745664—22no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—42no—
205C2×C59⋊C8C22×C411yescharacteristic
206C2×C59⋊C8C22×C411yescharacteristic
207C2×C59⋊C8C22×C411yescharacteristic
208C2×C59⋊C8C22×C411yescharacteristic
209#11745659—81no—
210#11745659—81no—
211C8×D59—81no—
212C8×D59—81no—
213C23×C59—21no—
214C23×C59C4×D411yescharacteristic
215C22×D59—21no—
216C22×D59—21no—
217Q8×C59—21no—
218Q8×C59—21no—
219D4×C59—21no—
220D4×C59—21no—
221D4×C59—81no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59C4×D411yescharacteristic
233C2×C4×C59C4×D411yescharacteristic
234C2×C4×C59C4×D411yescharacteristic
235C2×C4×C59C23×C411yescharacteristic
236C2×C4×C59C22×D411yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C22⋊D59—81no—
239C22⋊D59—81no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242D236—21no—
243D236—21no—
244D236—21no—
245D236—21no—
246D236—81no—
247D236—81no—
248C4×D59—41no—
249C4×D59—41no—
250C4×D59—41no—
251C4×D59—41no—
252Dic118—81no—
253C8×C59—21no—
254C8×C59—21no—
255C8×C59—21no—
256C8×C59—21no—
257C8×C59—41no—
258C8×C59—41no—
259C59⋊C8—22no—
260C59⋊C8—21no—
261C59⋊C8—21no—
262C59⋊C8—41no—
263C59⋊C8—41no—
264C59⋊C8—41no—
265C59⋊C8#13112yes—
266C59⋊C8#13112yes—
267#17968—591nomaximal, Sylow
268C22×C59—21no—
269C22×C59—21no—
270C22×C59—21no—
271C22×C59—41no—
272C22×C59—41no—
273C22×C59—41no—
274C22×C59C2×C4×D411yescharacteristic
275C2×D59—41no—
276C2×D59—41no—
277C2×D59—41no—
278C2×D59—41no—
279C2×D59—41no—
280C2×D59—41no—
281C4×C59—21no—
282C4×C59—21no—
283C4×C59—21no—
284C4×C59—21no—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59—41no—
288C4×C59C2×#13111yescharacteristic
289C4×C59C2×#13111yescharacteristic
290C4×C59#57611yescharacteristic
291C4×C59#57711yescharacteristic
292Dic59—41no—
293Dic59—41no—
294#3171—591no—
295#2993D5911yescharacteristic, complemented
296#2981—591no—
297#1594—591no—
298#1592—591no—
299#1590—591no—
300#1589—591no—
301#1580—591no—
302SD16⋊C8—591no—
303C8×SD16—591no—
304C8×D8—591no—
305#1569—591no—
306#1494—591no—
307#1492—591no—
308C8×M16—591no—
309C2×C59—41no—
310C2×C59—81no—
311C2×C59#293111yescharacteristic
312C2×C59#294611yescharacteristic
313C2×C59#295611yescharacteristic
314D59—81no—
315D59—81no—
316#532—591no—
317#529C2×D5911yescharacteristic
318#501—591no—
319#496—591no—
320#490—591no—
321#489—591no—
322#469—591no—
323#469—591no—
324#468—591no—
325#468—591no—
326#446C2×D5911yescharacteristic
327C8×Q8C2×D5911yescharacteristic
328C4×SD16—591no—
329C4×SD16—591no—
330C4×D8—591no—
331C4×D8—591no—
332#436—21no—
333#436—591no—
334#436—591no—
335#436C2×D5911yescharacteristic
336#435—21no—
337#435—1181no—
338#435—1181no—
339C8×D4—591no—
340C8×D4—591no—
341C8×D4C2×D5911yescharacteristic
342#433—21no—
343#432—591no—
344#431C2×D5911yescharacteristic
345#424—21no—
346#423C2×D5911yescharacteristic
347#408—1181no—
348#408—1181no—
349C4×M16—591no—
350C2×C4×C8—591no—
351#335—591no—
352#335—591no—
353#334—591no—
354#334—591no—
355Q8⋊C8—591no—
356Q8⋊C8—591no—
357D4⋊C8—591no—
358D4⋊C8—591no—
359D4⋊C8—591no—
360D4⋊C8—591no—
361D4⋊C8—591no—
362D4⋊C8—591no—
363#322—591no—
364#322—591no—
365C82—591no—
366C82—591no—
367C59#1796811yescharacteristic, complemented, Sylow
368C2×#41—1181no—
369C2×SD16—591no—
370C2×SD16—591no—
371C2×D8—591no—
372C2×D8—591no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—591no—
376C2×M16—591no—
377C2×M16—1181no—
378C2×M16—1181no—
379C22×C8—21no—
380C22×C8—1181no—
381C22×C8—1181no—
382C22×C8—1181no—
383C4⋊Q8—591no—
384#127—591no—
385#126—21no—
386#125C4×D5911yescharacteristic
387#124—1181no—
388#124C4×D5911yescharacteristic
389#123—21no—
390C22⋊Q8C4×D5911yescharacteristic
391#121—591no—
392#121—591no—
393#121—1181no—
394#121C4×D5911yescharacteristic
395C4×Q8C22×D5911yescharacteristic
396C4×D4—21no—
397C4×D4—591no—
398C4×D4—591no—
399C4×D4—1181no—
400C4×D4C22×D5911yescharacteristic
401#117—21no—
402C2×C42C22×D5911yescharacteristic
403#107—591no—
404#107—591no—
405#106—591no—
406#106—591no—
407C4⋊C8—21no—
408C4⋊C8—21no—
409C4⋊C8—21no—
410C4⋊C8—591no—
411C4⋊C8—591no—
412C4⋊C8C22⋊D5911yescharacteristic
413C4⋊C8C22⋊D5911yescharacteristic
414C4⋊C8C22×D5911yescharacteristic
415C4⋊C8C22×D5911yescharacteristic
416#103—591no—
417#103—591no—
418#102—591no—
419#102—591no—
420#102—591no—
421#102—591no—
422#102—591no—
423#102—591no—
424C22⋊C8—21no—
425C22⋊C8—21no—
426C22⋊C8—1181no—
427C22⋊C8—1181no—
428C22⋊C8—1181no—
429C22⋊C8—1181no—
430C22⋊C8—1181no—
431C22⋊C8C22⋊D5911yescharacteristic
432C22⋊C8C22⋊D5911yescharacteristic
433#97—21no—
434#97—591no—
435#97—591no—
436#97—1181no—
437C4×C8—21no—
438C4×C8—591no—
439C4×C8—591no—
440C4×C8—591no—
441C4×C8—591no—
442C4×C8—1181no—
443C4×C8C22×D5911yescharacteristic
444C4×C8C22×D5911yescharacteristic
445#41—4721no—
446#41—4721no—
447C2×Q8—1181no—
448C2×Q8C2×C4×D5911yescharacteristic
449C2×D4—21no—
450C2×D4—591no—
451C2×D4—591no—
452C2×D4—1181no—
453C2×D4—1181no—
454C2×D4—1181no—
455C2×D4—1181no—
456C2×D4C2×C4×D5911yescharacteristic
457C22×C4—21no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4C2×C4×D5911yescharacteristic
461C22×C4#1174568511yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463SD16—1182no—
464SD16—1182no—
465D8—1182no—
466D8—1182no—
467M16—81no—
468M16—81no—
469M16—1182no—
470M16—1182no—
471M16—4721no—
472M16—4721no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—81no—
476C2×C8—591no—
477C2×C8—591no—
478C2×C8—591no—
479C2×C8—591no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—4721no—
485C2×C8—4721no—
486C2×C8—2362no—
487C2×C8C2×C22⋊D5911yescharacteristic
488C2×C8C2×C22⋊D5911yescharacteristic
489C2×C8C2×C22⋊D5911yescharacteristic
490C2×C8C2×C22⋊D5911yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—591no—
496C4⋊C4—591no—
497C4⋊C4—1181no—
498C4⋊C4C2×C4×D5911yescharacteristic
499C4⋊C4C2×C4×D5911yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—1181no—
506C22⋊C4—1181no—
507C22⋊C4—1181no—
508C22⋊C4—1181no—
509C42—21no—
510C42C2×C4×D5911yescharacteristic
511C42#1174568511yescharacteristic
512C42C2×C22⋊D5911yescharacteristic
513C42C23×D5911yescharacteristic
514C23—21no—
515C23—1181no—
516C23—1181no—
517C23C4×C22⋊D5911yescharacteristic
518Q8—21no—
519Q8—21no—
520Q8—4721no—
521D4—21no—
522D4—21no—
523D4—81no—
524D4—1181no—
525D4—1181no—
526D4—1181no—
527D4—1181no—
528D4—4721no—
529D4—4721no—
530D4—4721no—
531D4—4721no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4C4×C22⋊D5911yescharacteristic
549C2×C4C4×C22⋊D5911yescharacteristic
550C2×C4C4×C22⋊D5911yescharacteristic
551C2×C4C22×C4×D5911yescharacteristic
552C2×C4C22×C22⋊D5911yescharacteristic
553C2×C4C2×#1174568511yescharacteristic
554C8—21no—
555C8—21no—
556C8—21no—
557C8—21no—
558C8—41no—
559C8—41no—
560C8—592no—
561C8—592no—
562C8—1182no—
563C8—1181no—
564C8—1181no—
565C8—2361no—
566C8—2361no—
567C8—2361no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—2361no—
575C22—2361no—
576C22—2361no—
577C22—2361no—
578C22—2361no—
579C22—2361no—
580C22C2×C4×C22⋊D5911yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—2361no—
589C4—2361no—
590C4#131×D5911yescharacteristic
591C4#131×D5911yescharacteristic
592C4#1683820511yescharacteristic
593C4#1683821011yescharacteristic
594C2—41no—
595C2—81no—
596C2—4721no—
597C2—4721no—
598C2#2143280611yescharacteristic
599C2#2143292411yescharacteristic
600C2#2143300311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#2993C2
socleC22×C59C2×C4×D4
prime radical#21434692C2
cosocleC42×C59C24
hypercenter#2993D59
hypocenterC59#17968
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2993

Complements

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

NH ≅ G/Ntype
2. #21433171C2semidirect
3. #21423008C2semidirect
4. #21422996C2semidirect
5. #21422986C2semidirect
6. #21422979C2semidirect
7. #21422918C2semidirect
8. #21422906C2semidirect
9. C8×#11745671C2semidirect
10. C8×#11745669C2semidirect
11. #21422861C2semidirect
12. #21422556C2semidirect
13. #21422546C2semidirect
14. #21422377C2semidirect
15. C59×#2993C2semidirect
295. #2993D59semidirect
367. C59#17968semidirect

Maximal subgroups

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

subgroupindex
#214346922
#214331712
#214230082
#214229962
#214229862
#214229792
#214229182
#214229062
C8×#117456712
C8×#117456692
#214228612
#214225562
#214225462
#214223772
C59×#29932
#1796859