Finite Group Databasebeta

← #28646759

Subgroups — #28646759

Order 15104.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427821C211yesmaximal, characteristic
2#21426981C211yesmaximal, characteristic
3#21425669C211yesmaximal, characteristic, complemented
4#21425600C211yesmaximal, characteristic, complemented
5#21425596C211yesmaximal, characteristic, complemented
6#21425433C211yesmaximal, characteristic, complemented
7#21425052C211yesmaximal, characteristic, complemented
8#21424863C211yesmaximal, characteristic, complemented
9#21424677C211yesmaximal, characteristic, complemented
10#21424673C211yesmaximal, characteristic, complemented
11#21424291C211yesmaximal, characteristic
12#21424278C211yesmaximal, characteristic
13#21424127C211yesmaximal, characteristic
14#21423817C211yesmaximal, characteristic
15C59×#1825C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273718C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273659C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25#16837344—21no—
26#16837334—21no—
27#16837301C2211yescharacteristic
28#16837301C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837298C2211yescharacteristic
31#16837295—21no—
32#16837295—21no—
33#16837277—21no—
34#16837268—21no—
35#16837267C2211yescharacteristic
36#16837262C2211yescharacteristic
37#16837260—21no—
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4×C22⋊Dic59C2211yescharacteristic
41#16837248—21no—
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C2×C22⋊C8×C59C2211yescharacteristic
54C59×#380C2211yescharacteristic
55C59×#340C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C23×Dic59—21no—
60C2×#11745664—22no—
61C2×#11745664—22no—
62C2×#11745664—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273718—22no—
66#14273718—41no—
67C2×C4⋊Dic59—22no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—41no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59D411yescharacteristic
82C2×C22⋊Dic59D411yescharacteristic
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59C2311yescharacteristic
86C2×C4×Dic59C2311yescharacteristic
87#14273660—41no—
88#14273659—22no—
89#14273659—41no—
90#14273658—22no—
91Dic59⋊C8—22no—
92Dic59⋊C8—41no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C23×C4×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107#117×C59—22no—
108#117×C59Q814yes—
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C42×C59—21no—
116C22⋊C8×C59—22no—
117#95×C59—21no—
118C24×C59C2×D411yescharacteristic
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×Q812yes—
129C22×C4×C59#4112yes—
130C22×C4×C59C2411yescharacteristic
131C22×Dic59—21no—
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—42no—
136C22×Dic59—42no—
137C22×Dic59—81no—
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59C2×Q812yes—
155C4⋊C4×C59C2×Q812yes—
156C22⋊C4×C59—24no—
157C22⋊C4×C59—42no—
158C42×C59—24no—
159C22⋊Dic59—41no—
160C22⋊Dic59—41no—
161C22⋊Dic59—81no—
162C4⋊Dic59—21no—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—42no—
170C4⋊Dic59—42no—
171C4⋊Dic59—81no—
172C4⋊Dic59—81no—
173C4⋊Dic59—81no—
174C4⋊Dic59#4112yes—
175Dic59⋊C4—42no—
176Dic59⋊C4—81no—
177C4×Dic59—21no—
178C4×Dic59—21no—
179C4×Dic59—21no—
180C4×Dic59—21no—
181C4×Dic59—42no—
182C4×Dic59—42no—
183#11745664—44no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8#4112yes—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59C22⋊Q812yes—
197C23×C59C22×D411yescharacteristic
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—22no—
201C2×C4×C59—22no—
202C2×C4×C59—22no—
203C2×C4×C59—22no—
204C2×C4×C59—22no—
205C2×C4×C59—22no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59C22⋊Q812yes—
217C2×C4×C59C22×Q811yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C59H(5,𝔽2)11yescharacteristic
222C2×C4×C592−1+411yescharacteristic
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—22no—
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—84no—
248C8×C59—41no—
249C8×C59—41no—
250C59⋊C8—22no—
251C59⋊C8—41no—
252#26019—591nomaximal, Sylow
253C22×C59—22no—
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—41no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59C2×C22⋊Q811yescharacteristic
261C22×C59#54611yescharacteristic
262C22×C59#54711yescharacteristic
263C22×C59#57511yescharacteristic
264C22×C59#57511yescharacteristic
265C22×C59#57711yescharacteristic
266C22×C59#57711yescharacteristic
267C4×C59—22no—
268C4×C59—42no—
269C4×C59—42no—
270C4×C59—81no—
271C4×C59—81no—
272C4×C59#53311yescharacteristic
273C4×C59#54411yescharacteristic
274C4×C59#55111yescharacteristic
275C4×C59#55211yescharacteristic
276Dic59—41no—
277Dic59—41no—
278Dic59—41no—
279Dic59—41no—
280Dic59—42no—
281Dic59—81no—
282Dic59—81no—
283Dic59—81no—
284#2442—591no—
285#2346—591no—
286#2101—591no—
287#2088—591no—
288#2088—591no—
289#2059—591no—
290#1989—591no—
291#1949—591no—
292#1919—591no—
293#1919—591no—
294#1853—591no—
295#1851—591no—
296#1825D5911yescharacteristic
297#1809—591no—
298#1736—591no—
299C2×C59—42no—
300C2×C59#243811yescharacteristic
301C2×C59#308011yescharacteristic
302C2×C59#308411yescharacteristic
303C2×C59#332111yescharacteristic
304C2×C59#332311yescharacteristic
305C2×C59#334811yescharacteristic
306C2×C59#335111yescharacteristic
307C22×M16—591no—
308C2×#125—591no—
309C2×#117C2×D5911yescharacteristic
310C22×C4⋊C4—591no—
311C2×#107—591no—
312C2×#106—591no—
313C2×#106—591no—
314C2×#106—591no—
315C2×C4⋊C8—591no—
316C2×C4⋊C8—591no—
317C2×C22⋊C8C2×D5911yescharacteristic
318#400—1181no—
319#398—1181no—
320#389—591no—
321#389—591no—
322#389—591no—
323#389—591no—
324#388—1181no—
325#388—1181no—
326#385—1181no—
327#382—591no—
328#382—1181no—
329#380—591no—
330#380—1181no—
331#380C2×D5911yescharacteristic
332C4×C4⋊C4—591no—
333C4×C4⋊C4—591no—
334C4×C22⋊C4—591no—
335#376—1181no—
336#340—591no—
337#340—591no—
338#340—591no—
339#340—591no—
340#340—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340C2×D5911yescharacteristic
345#340C2×D5911yescharacteristic
346#340C2×D5911yescharacteristic
347#340C2×D5911yescharacteristic
348#336—591no—
349#336—591no—
350C59#2601911yescharacteristic, complemented, Sylow
351C23×C4—1181no—
352C23×C4C22×D5911yescharacteristic
353C2×M16—1182no—
354C2×M16—1182no—
355C2×M16—2361no—
356C22×C8—591no—
357C22×C8—591no—
358C22×C8C22×D5911yescharacteristic
359C22×C8C22×D5911yescharacteristic
360#125—1182no—
361#125—2361no—
362#117—22no—
363#117Dic11814yes—
364C2×C4⋊C4—591no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—1182no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—2361no—
377C2×C4⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—591no—
384C2×C22⋊C4—591no—
385C2×C22⋊C4—1181no—
386C2×C42—21no—
387C2×C42—591no—
388C2×C42—591no—
389C2×C42—1181no—
390C2×C42—1181no—
391#107—2361no—
392#106—1182no—
393#106—1182no—
394#106—2361no—
395C4⋊C8—1182no—
396C4⋊C8—2361no—
397C22⋊C8—22no—
398#95—21no—
399#95—591no—
400#95—591no—
401#95—1181no—
402#95—1181no—
403#95—1181no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410C24D4×D5911yescharacteristic
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—591no—
420C22×C4—591no—
421C22×C4—591no—
422C22×C4—591no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4—1181no—
427C22×C4—4721no—
428C22×C4—2362no—
429C22×C4—2362no—
430C22×C4C2×Dic11812yes—
431C22×C4D4×D5911yescharacteristic
432C22×C4#1174568712yes—
433C22×C4C23×D5911yescharacteristic
434M16—2364no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—592no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—592no—
454C4⋊C4—1181no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—4721no—
462C4⋊C4—4721no—
463C4⋊C4—4721no—
464C4⋊C4—4721no—
465C4⋊C4—2362no—
466C4⋊C4—2362no—
467C4⋊C4—2362no—
468C4⋊C4C2×Dic11812yes—
469C4⋊C4C2×Dic11812yes—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—4721no—
473C22⋊C4—2361no—
474C22⋊C4—2361no—
475C42—24no—
476C42—1181no—
477C42—1181no—
478C42—1181no—
479C42—1181no—
480C42—2362no—
481C42—2362no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1427370412yes—
489C23C2×D4×D5911yescharacteristic
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—1181no—
509C2×C4—1181no—
510C2×C4—1181no—
511C2×C4—1181no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4—1181no—
517C2×C4—1181no—
518C2×C4—1182no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—4724no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4#1427370412yes—
534C2×C4C22×Dic11811yescharacteristic
535C2×C4#41×D5911yescharacteristic
536C2×C4#41×D5911yescharacteristic
537C2×C4C2×#1174568711yescharacteristic
538C2×C4#1427377311yescharacteristic
539C2×C4#1427377411yescharacteristic
540C8—41no—
541C8—41no—
542C8—1182no—
543C8—2361no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1427370411yescharacteristic
552C22#1683804411yescharacteristic
553C22#1683805011yescharacteristic
554C22#1683819811yescharacteristic
555C22#1683820111yescharacteristic
556C22#1683820911yescharacteristic
557C22#1683820911yescharacteristic
558C4—22no—
559C4—42no—
560C4—42no—
561C4—81no—
562C4—81no—
563C4—4721no—
564C4—4721no—
565C4—4721no—
566C4—2361no—
567C4—2361no—
568C4—2362no—
569C4—2361no—
570C4—2361no—
571C4#1683795511yescharacteristic
572C4#1683803011yescharacteristic
573C4#1683807511yescharacteristic
574C4#1683808911yescharacteristic
575C2—42no—
576C2#2142777411yescharacteristic
577C2#2143391611yescharacteristic
578C2#2143395611yescharacteristic
579C2#2143628511yescharacteristic
580C2#2143631011yescharacteristic
581C2#2143660411yescharacteristic
582C2#2143664911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1825C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1825D59
hypocenterC59#26019
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
3. #21425669C2semidirect
4. #21425600C2semidirect
5. #21425596C2semidirect
6. #21425433C2semidirect
7. #21425052C2semidirect
8. #21424863C2semidirect
9. #21424677C2semidirect
10. #21424673C2semidirect
350. C59#26019semidirect

Maximal subgroups

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

subgroupindex
#214278212
#214269812
#214256692
#214256002
#214255962
#214254332
#214250522
#214248632
#214246772
#214246732
#214242912
#214242782
#214241272
#214238172
C59×#18252
#2601959