Finite Group Databasebeta

← #28598448

Subgroups — #28598448

Order 15104.

1086 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837956C211yesmaximal, characteristic
2C2×#16837458C211yesmaximal, characteristic
3#21426501C211yesmaximal, characteristic
4#21425067C212yesmaximal, complemented
5#21424914C212yesmaximal, complemented
6#21424407C212yesmaximal
7#21424405C212yesmaximal
8#21424102C211yesmaximal, characteristic, complemented
9#21424101C211yesmaximal, characteristic, complemented
10#21424098C211yesmaximal, characteristic, complemented
11C59×#1803C211yesmaximal, characteristic, complemented
12#16837956—24no—
13#16837956C418yescomplemented
14C2×#14273726C2211yescharacteristic
15C2×#14273725C2211yescharacteristic
16C2×#14273717—21no—
17C2×#14273707C2211yescharacteristic
18C2×#14273705—21no—
19C2×#14273703C2211yescharacteristic
20C2×C4×Dic118—21no—
21C2×C4×Dic118C2211yescharacteristic
22C2×#14273685C2212yes—
23C2×#14273684—21no—
24C2×C42×D59C2211yescharacteristic
25#16837458—24no—
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273652C2212yes—
28C2×#14273651C2212yes—
29C2×#14273650C2212yes—
30C2×(C2×D59)⋊C8C2212yes—
31C2×#14273637C2212yes—
32#16837272C2211yescharacteristic
33#16837267C2212yes—
34#16837259—21no—
35#16837259—21no—
36C4⋊C4×Dic59—21no—
37C4⋊C4×Dic59—21no—
38#16837165C2212yes—
39#16837164C2212yes—
40#16837164C2212yes—
41C2×C4×Q8×C59C2211yescharacteristic
42C2×C4⋊C8×C59C2211yescharacteristic
43C2×#103×C59C2212yes—
44C59×#384C2211yescharacteristic
45C59×#340C2212yes—
46C22×Dic118—21no—
47C22×Dic118C2311yescharacteristic
48C22×C4×D59D411yescharacteristic
49C22×C4×D59D411yescharacteristic
50C22×C4×D59C2311yescharacteristic
51C22×C59⋊C8C2312yes—
52#14273726—22no—
53#14273726C2×C414yes—
54#14273725—22no—
55#14273725C2×C414yes—
56#14273717—28no—
57#14273707—22no—
58#14273707C2×C414yes—
59#14273705—28no—
60#14273703—22no—
61#14273703C2×C414yes—
62C4×Dic118—22no—
63C4×Dic118—28no—
64C4×Dic118C2×C414yes—
65#14273685—42no—
66#14273685—42no—
67#14273684—28no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59C2312yes—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—22no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—22no—
83C2×C4×Dic59D411yescharacteristic
84C2×C4×Dic59D411yescharacteristic
85C2×C4×Dic59C2311yescharacteristic
86C42×D59C2×C414yes—
87C42×D59D414yes—
88C4⋊(C59⋊C8)—22no—
89#14273652—42no—
90#14273651—42no—
91#14273650—42no—
92(C2×D59)⋊C8—24no—
93#14273637—24no—
94C59⋊#95—22no—
95C59⋊#95—22no—
96C22×Q8×C59C2311yescharacteristic
97C22×C8×C59C2312yes—
98C4×Q8×C59—22no—
99C4×Q8×C59C2×C414yes—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59C2312yes—
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C42×C59—21no—
106C2×C42×C59C2311yescharacteristic
107C4⋊C8×C59—22no—
108#103×C59—42no—
109C23×D59C2×D411yescharacteristic
110C2×Q8×C59—21no—
111C2×Q8×C59—21no—
112C2×Q8×C59—21no—
113C2×Q8×C59—21no—
114C2×Q8×C59—21no—
115C2×Q8×C59C22×C412yes—
116C22×C4×C59—21no—
117C22×C4×C59—21no—
118C22×C4×C59—22no—
119C22×C4×C59C2×D411yescharacteristic
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2411yescharacteristic
122C22×Dic59—21no—
123C22×Dic59—21no—
124C22×Dic59—22no—
125C22×Dic59C2×D411yescharacteristic
126C22×Dic59C2×D411yescharacteristic
127C22×Dic59C2×D411yescharacteristic
128C2×C4×D59—22no—
129C2×C4×D59—22no—
130C2×C4×D59—21no—
131C2×C4×D59—21no—
132C2×C4×D59—21no—
133C2×C4×D59—21no—
134C2×C4×D59—41no—
135C2×C4×D59—41no—
136C2×C4×D59C22⋊C414yes—
137C2×C4×D59C22⋊C414yes—
138C2×C4×D59C22×C412yes—
139C2×C4×D59C2×D412yes—
140C2×Dic118—21no—
141C2×Dic118—21no—
142C2×Dic118—21no—
143C2×Dic118—21no—
144C2×Dic118—21no—
145C2×Dic118—24no—
146C2×Dic118—42no—
147C2×Dic118—42no—
148C2×Dic118C22×C412yes—
149C2×C8×C59—22no—
150C2×C8×C59—22no—
151C2×C8×C59—22no—
152C4⋊C4×C59—24no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—42no—
157C4⋊C4×C59—42no—
158C4⋊C4×C59C22×C412yes—
159C42×C59—24no—
160C42×C59C22×C412yes—
161C42×C59C2×D412yes—
162(C2×D59)⋊C4—24no—
163(C2×D59)⋊C4—24no—
164(C2×D59)⋊C4—44no—
165C4⋊Dic59—21no—
166C4⋊Dic59—24no—
167C4⋊Dic59—22no—
168C4⋊Dic59—22no—
169C4⋊Dic59—42no—
170C4⋊Dic59—42no—
171C4⋊Dic59C22×C412yes—
172Dic59⋊C4—24no—
173Dic59⋊C4—24no—
174Dic59⋊C4—24no—
175Dic59⋊C4—24no—
176Dic59⋊C4—24no—
177C4×Dic59—24no—
178C4×Dic59—44no—
179C4×Dic59C22⋊C414yes—
180C4×Dic59C22⋊C414yes—
181C4×Dic59C22×C412yes—
182C4×Dic59C2×D412yes—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—22no—
185C2×C59⋊C8—22no—
186C23×C59C22×D411yescharacteristic
187C22×D59—21no—
188C22×D59—21no—
189C22×D59—21no—
190C22×D59—21no—
191C22×D59—21no—
192C22×D59—21no—
193C22×D59C2×C22⋊C412yes—
194Q8×C59—22no—
195Q8×C59—22no—
196Q8×C59—41no—
197Q8×C59—41no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—22no—
201C2×C4×C59—22no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—42no—
207C2×C4×C59—42no—
208C2×C4×C59—42no—
209C2×C4×C59C2×C22⋊C412yes—
210C2×C4×C59C2×C22⋊C412yes—
211C2×C4×C59#13712yes—
212C2×C4×C59#13712yes—
213C2×C4×C59C23×C411yescharacteristic
214C2×C4×C59C22×D411yescharacteristic
215C2×C4×C59H(5,𝔽2)11yescharacteristic
216C2×C4×C59H(5,𝔽2)11yescharacteristic
217C2×C4×C592−1+411yescharacteristic
218C2×C4×C592−1+411yescharacteristic
219C2×Dic59—21no—
220C2×Dic59—21no—
221C2×Dic59—21no—
222C2×Dic59—21no—
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—22no—
226C2×Dic59—22no—
227C2×Dic59—41no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231C2×Dic59—42no—
232C2×Dic59—42no—
233C2×Dic59—42no—
234C2×Dic59C2×C22⋊C412yes—
235C2×Dic59C2×C22⋊C412yes—
236C2×Dic59C2×C22⋊C412yes—
237C4×D59—22no—
238C4×D59—22no—
239C4×D59—22no—
240C4×D59—22no—
241C4×D59—28no—
242C4×D59—44no—
243Dic118—22no—
244Dic118—22no—
245Dic118—48no—
246Dic118—41no—
247Dic118—41no—
248C8×C59—42no—
249C59⋊C8—42no—
250#21870—591nomaximal, Sylow
251C22×C59C22×C22⋊C411yescharacteristic
252C22×C59#53511yescharacteristic
253C22×C59#53611yescharacteristic
254C22×C59C2×#13711yescharacteristic
255C22×C59C2×#13711yescharacteristic
256C22×C59#57511yescharacteristic
257C22×C59#57511yescharacteristic
258C2×D59—22no—
259C2×D59—22no—
260C2×D59—22no—
261C2×D59—41no—
262C2×D59—41no—
263C2×D59—41no—
264C2×D59—41no—
265C4×C59—22no—
266C4×C59—42no—
267C4×C59—42no—
268C4×C59—82no—
269C4×C59#41914yes—
270C4×C59#52011yescharacteristic
271C4×C59#52011yescharacteristic
272C4×C59#53711yescharacteristic
273C4×C59#53711yescharacteristic
274Dic59—22no—
275Dic59—22no—
276Dic59—22no—
277Dic59—42no—
278Dic59—42no—
279Dic59—82no—
280C2×#533—591no—
281C2×#432—591no—
282#2285—591no—
283#1993—592no—
284#1960—592no—
285#1873—592no—
286#1873—592no—
287#1803—591no—
288#1803—591no—
289#1803—591no—
290#1803D5911yescharacteristic, complemented
291C2×C59#231111yescharacteristic
292C2×C59C2×#41911yescharacteristic
293C2×C59#289611yescharacteristic
294C2×C59#321611yescharacteristic
295C2×C59#321611yescharacteristic
296C2×C59#323711yescharacteristic
297C2×C59#323711yescharacteristic
298D59—42no—
299#533—598no—
300#533—1184no—
301C2×C4⋊Q8—591no—
302C2×C4⋊Q8—591no—
303C2×#125—1181no—
304C2×C22⋊Q8—591no—
305C2×C22⋊Q8—591no—
306C2×C22⋊Q8—1181no—
307C2×C4×Q8—591no—
308C2×C4×Q8—1181no—
309C2×C4×Q8C2×D5911yescharacteristic
310C2×#117—592no—
311C2×#117—1181no—
312C22×C42—591no—
313#432—1184no—
314C2×C4⋊C8—591no—
315C2×C4⋊C8C2×D5911yescharacteristic
316C2×#103—592no—
317C2×#103—592no—
318C2×#103—592no—
319C2×#103C2×D5912yes—
320C2×C22⋊C8—592no—
321C2×#97—592no—
322#384—591no—
323#384C2×D5911yescharacteristic
324#382—592no—
325#380—1181no—
326#380—1181no—
327C4×C4⋊C4—1181no—
328C4×C4⋊C4—1181no—
329#340—592no—
330#340—592no—
331#340—592no—
332#340C2×D5912yes—
333C59#2187011yescharacteristic, complemented, Sylow
334C22×Q8—591no—
335C22×Q8—1181no—
336C22×Q8C22×D5911yescharacteristic
337C23×C4—591no—
338C23×C4—591no—
339C23×C4—591no—
340C22×C8—592no—
341C22×C8C22×D5912yes—
342C4⋊Q8—594no—
343C4⋊Q8—594no—
344C4⋊Q8—1182no—
345C4⋊Q8—1182no—
346#125—1188no—
347C22⋊Q8—594no—
348C22⋊Q8—594no—
349C22⋊Q8—1182no—
350C22⋊Q8—1188no—
351C22⋊Q8—1182no—
352C4×Q8—22no—
353C4×Q8—594no—
354C4×Q8—1188no—
355C4×Q8—1182no—
356C4×Q8C4×D5914yes—
357#117—1188no—
358#117—2362no—
359#117—2362no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—591no—
364C2×C4⋊C4—592no—
365C2×C4⋊C4—1181no—
366C2×C4⋊C4—1181no—
367C2×C4⋊C4—1181no—
368C2×C4⋊C4—1181no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4C22×D5912yes—
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C22⋊C4—1181no—
376C2×C22⋊C4—1181no—
377C2×C22⋊C4—1182no—
378C2×C42—21no—
379C2×C42—591no—
380C2×C42—591no—
381C2×C42—591no—
382C2×C42—594no—
383C2×C42—594no—
384C2×C42—1181no—
385C2×C42—1182no—
386C2×C42C22×D5911yescharacteristic
387C4⋊C8—22no—
388C4⋊C8—1182no—
389#103—42no—
390#103—2362no—
391#103—2362no—
392#103—2362no—
393C22⋊C8—1184no—
394#97—1184no—
395#95—1182no—
396#95—1182no—
397C24—591no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—592no—
404C2×Q8—1184no—
405C2×Q8—1181no—
406C2×Q8—1181no—
407C2×Q8—1181no—
408C2×Q8—1181no—
409C2×Q8—1181no—
410C2×Q8—2362no—
411C2×Q8—2362no—
412C2×Q8C2×C4×D5912yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—591no—
417C22×C4—591no—
418C22×C4—591no—
419C22×C4—594no—
420C22×C4—592no—
421C22×C4—592no—
422C22×C4—594no—
423C22×C4—1182no—
424C22×C4—1182no—
425C22×C4—1181no—
426C22×C4—1181no—
427C22×C4—1181no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1182no—
432C22×C4—2361no—
433C22×C4—2361no—
434C22×C4D4×D5911yescharacteristic
435C22×C4D4×D5911yescharacteristic
436C22×C4C23×D5911yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—1182no—
441C2×C8—1182no—
442C2×C8—1182no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—592no—
450C4⋊C4—1184no—
451C4⋊C4—1184no—
452C4⋊C4—1184no—
453C4⋊C4—1184no—
454C4⋊C4—1184no—
455C4⋊C4—1184no—
456C4⋊C4—1181no—
457C4⋊C4—1182no—
458C4⋊C4—1182no—
459C4⋊C4—2362no—
460C4⋊C4—2362no—
461C4⋊C4C2×C4×D5912yes—
462C22⋊C4—1184no—
463C22⋊C4—1184no—
464C22⋊C4—2364no—
465C42—24no—
466C42—594no—
467C42—594no—
468C42—592no—
469C42—592no—
470C42—1184no—
471C42—2364no—
472C42C2×C4×D5912yes—
473C42D4×D5912yes—
474C23—592no—
475C23—1181no—
476C23—1181no—
477C23—1181no—
478C23—1181no—
479C23—1181no—
480C23—1181no—
481C23C2×D4×D5911yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—41no—
485Q8—41no—
486Q8—1182no—
487Q8—1182no—
488Q8—2368no—
489Q8—2361no—
490Q8—2361no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—592no—
503C2×C4—592no—
504C2×C4—592no—
505C2×C4—1181no—
506C2×C4—1181no—
507C2×C4—1181no—
508C2×C4—1181no—
509C2×C4—1181no—
510C2×C4—1181no—
511C2×C4—1188no—
512C2×C4—1182no—
513C2×C4—1182no—
514C2×C4—1182no—
515C2×C4—1182no—
516C2×C4—1182no—
517C2×C4—1182no—
518C2×C4—2364no—
519C2×C4—2362no—
520C2×C4—2362no—
521C2×C4—2362no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4C22⋊C4×D5912yes—
527C2×C4C22⋊C4×D5912yes—
528C2×C4#1427375612yes—
529C2×C4#1427375912yes—
530C2×C4C22×C4×D5911yescharacteristic
531C2×C4C2×D4×D5911yescharacteristic
532C2×C4#1427377211yescharacteristic
533C2×C4#1427377311yescharacteristic
534C2×C4#1427377411yescharacteristic
535C2×C4#1427377511yescharacteristic
536C8—42no—
537C8—2362no—
538C22—1182no—
539C22—1182no—
540C22—1182no—
541C22—2361no—
542C22—2361no—
543C22—2361no—
544C22—2361no—
545C22C2×C22⋊C4×D5911yescharacteristic
546C22#1683796811yescharacteristic
547C22#1683797511yescharacteristic
548C22C2×#1427375611yescharacteristic
549C22C2×#1427375911yescharacteristic
550C22#1683819811yescharacteristic
551C22#1683820111yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—1182no—
557C4—1182no—
558C4—1182no—
559C4—4722no—
560C4—2362no—
561C4—2362no—
562C4#1683740914yes—
563C4#1683788211yescharacteristic
564C4#1683788311yescharacteristic
565C4#1683798111yescharacteristic
566C4#1683798211yescharacteristic
567C2—2362no—
568C2#2142668511yescharacteristic
569C2C2×#1683740911yescharacteristic
570C2#2143261211yescharacteristic
571C2#2143510511yescharacteristic
572C2#2143510611yescharacteristic
573C2#2143530811yescharacteristic
574C2#2143530911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1803C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1803D59
hypocenterC59#21870
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #21425067C2semidirect
5. #21424914C2semidirect
8. #21424102C2semidirect
9. #21424101C2semidirect
10. #21424098C2semidirect
11. C59×#1803C2semidirect
13. #16837956C4semidirect
290. #1803D59semidirect
333. C59#21870semidirect

Maximal subgroups

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

subgroupindex
C2×#168379562
C2×#168374582
#214265012
#214250672
#214249142
#214244072
#214244052
#214241022
#214241012
#214240982
C59×#18032
#2187059