Finite Group Databasebeta

← #28598573

Subgroups — #28598573

Order 15104.

1162 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837956C211yesmaximal, characteristic
2C2×C4⋊C8×D59C211yesmaximal, characteristic
3#21426501C211yesmaximal, characteristic
4#21425042C212yesmaximal, complemented
5#21425017C212yesmaximal, complemented
6#21424518C212yesmaximal
7#21424516C212yesmaximal
8#21424102C211yesmaximal, characteristic, complemented
9#21424101C211yesmaximal, characteristic, complemented
10#21424098C211yesmaximal, characteristic, complemented
11C59×#1803C211yesmaximal, characteristic, complemented
12C22×C8×D59C2212yes—
13#16837956—24no—
14#16837956C418yescomplemented
15C2×#14273726C2211yescharacteristic
16C2×#14273725C2211yescharacteristic
17C2×#14273717—21no—
18C2×#14273707C2211yescharacteristic
19C2×#14273705—21no—
20C2×#14273703C2211yescharacteristic
21C2×C4×Dic118—21no—
22C2×C4×Dic118C2211yescharacteristic
23C2×#14273685—21no—
24C2×#14273684—21no—
25C2×C42×D59C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×Dic59⋊C8C2212yes—
28C4⋊C8×D59—24no—
29C2×#14273652C2212yes—
30C2×#14273651C2212yes—
31C2×#14273650C2212yes—
32#16837272C2211yescharacteristic
33#16837267—21no—
34#16837259—21no—
35#16837259C2212yes—
36C4⋊C4×Dic59—21no—
37C4⋊C4×Dic59C2212yes—
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—
52C2×C8×D59—24no—
53C2×C8×D59—42no—
54C2×C8×D59—42no—
55#14273726—22no—
56#14273726C2×C414yes—
57#14273725—22no—
58#14273725C2×C414yes—
59#14273717—28no—
60#14273707—22no—
61#14273707C2×C414yes—
62#14273705—28no—
63#14273703—22no—
64#14273703C2×C414yes—
65C4×Dic118—22no—
66C4×Dic118—28no—
67C4×Dic118C2×C414yes—
68#14273685—44no—
69#14273684—28no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2312yes—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—22no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—22no—
85C2×C4×Dic59D411yescharacteristic
86C2×C4×Dic59D411yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88C42×D59C2×C414yes—
89C42×D59D414yes—
90C4⋊(C59⋊C8)—22no—
91Dic59⋊C8—24no—
92#14273652—42no—
93#14273651—42no—
94#14273650—42no—
95C59⋊#95—22no—
96C59⋊#95—22no—
97C22×Q8×C59C2311yescharacteristic
98C22×C8×C59C2312yes—
99C4×Q8×C59—22no—
100C4×Q8×C59C2×C414yes—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59C2312yes—
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C42×C59—21no—
107C2×C42×C59C2311yescharacteristic
108C4⋊C8×C59—22no—
109#103×C59—42no—
110C23×D59C2×D411yescharacteristic
111C2×Q8×C59—21no—
112C2×Q8×C59—21no—
113C2×Q8×C59—21no—
114C2×Q8×C59—21no—
115C2×Q8×C59—21no—
116C2×Q8×C59C22×C412yes—
117C22×C4×C59—21no—
118C22×C4×C59—21no—
119C22×C4×C59—22no—
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2411yescharacteristic
123C22×Dic59—21no—
124C22×Dic59—21no—
125C22×Dic59—22no—
126C22×Dic59C2×D411yescharacteristic
127C22×Dic59C2×D411yescharacteristic
128C22×Dic59C2×D411yescharacteristic
129C2×C4×D59—21no—
130C2×C4×D59—21no—
131C2×C4×D59—21no—
132C2×C4×D59—21no—
133C2×C4×D59—41no—
134C2×C4×D59—41no—
135C2×C4×D59C22⋊C414yes—
136C2×C4×D59C22⋊C414yes—
137C2×C4×D59SD1614yes—
138C2×C4×D59Q1614yes—
139C2×C4×D59C22×C412yes—
140C2×C4×D59C2×D412yes—
141C2×Dic118—21no—
142C2×Dic118—21no—
143C2×Dic118—21no—
144C2×Dic118—21no—
145C2×Dic118—21no—
146C2×Dic118—24no—
147C2×Dic118—42no—
148C2×Dic118—42no—
149C2×Dic118C22×C412yes—
150C2×C8×C59—22no—
151C2×C8×C59—22no—
152C2×C8×C59—22no—
153C4⋊C4×C59—24no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—42no—
158C4⋊C4×C59—42no—
159C4⋊C4×C59C22×C412yes—
160C42×C59—24no—
161C42×C59C22×C412yes—
162C42×C59C2×D412yes—
163(C2×D59)⋊C4—24no—
164(C2×D59)⋊C4—24no—
165(C2×D59)⋊C4—44no—
166C4⋊Dic59—21no—
167C4⋊Dic59—24no—
168C4⋊Dic59—22no—
169C4⋊Dic59—22no—
170C4⋊Dic59—42no—
171C4⋊Dic59—42no—
172C4⋊Dic59C22×C412yes—
173Dic59⋊C4—24no—
174Dic59⋊C4—24no—
175Dic59⋊C4—24no—
176Dic59⋊C4—24no—
177Dic59⋊C4—24no—
178C4×Dic59—24no—
179C4×Dic59—44no—
180C4×Dic59C22⋊C414yes—
181C4×Dic59C22⋊C414yes—
182C4×Dic59C22×C412yes—
183C4×Dic59C2×D412yes—
184C2×C59⋊C8—22no—
185C2×C59⋊C8—22no—
186C2×C59⋊C8—22no—
187C8×D59—48no—
188C23×C59C22×D411yescharacteristic
189C22×D59—21no—
190C22×D59—21no—
191C22×D59—21no—
192C22×D59—21no—
193C22×D59C2×C22⋊C412yes—
194C22×D59C2×SD1612yes—
195C22×D59C2×Q1612yes—
196Q8×C59—22no—
197Q8×C59—22no—
198Q8×C59—41no—
199Q8×C59—41no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—22no—
203C2×C4×C59—22no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—42no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59C2×C22⋊C412yes—
212C2×C4×C59C2×C22⋊C412yes—
213C2×C4×C59C2×SD1612yes—
214C2×C4×C59C2×Q1612yes—
215C2×C4×C59C23×C411yescharacteristic
216C2×C4×C59C22×D411yescharacteristic
217C2×C4×C59H(5,𝔽2)11yescharacteristic
218C2×C4×C59H(5,𝔽2)11yescharacteristic
219C2×C4×C592−1+411yescharacteristic
220C2×C4×C592−1+411yescharacteristic
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—
237C2×Dic59C2×SD1612yes—
238C2×Dic59C2×Q1612yes—
239C4×D59—22no—
240C4×D59—22no—
241C4×D59—22no—
242C4×D59—22no—
243C4×D59—44no—
244C4×D59#103116yescomplemented
245Dic118—22no—
246Dic118—22no—
247Dic118—48no—
248Dic118—41no—
249Dic118—41no—
250C8×C59—42no—
251C59⋊C8—42no—
252#21885—591nomaximal, Sylow
253C22×C59C22×C22⋊C411yescharacteristic
254C22×C59#53711yescharacteristic
255C22×C59#53711yescharacteristic
256C22×C59C22×SD1611yescharacteristic
257C22×C59C22×Q1611yescharacteristic
258C22×C59#57511yescharacteristic
259C22×C59#57511yescharacteristic
260C2×D59—22no—
261C2×D59—22no—
262C2×D59—22no—
263C2×D59—22no—
264C2×D59—22no—
265C2×D59—22no—
266C2×D59C2×#10314yes—
267C4×C59—22no—
268C4×C59—42no—
269C4×C59—42no—
270C4×C59—82no—
271C4×C59C2×#10314yes—
272C4×C59#52011yescharacteristic
273C4×C59#52011yescharacteristic
274C4×C59#53511yescharacteristic
275C4×C59#53611yescharacteristic
276Dic59—22no—
277Dic59—22no—
278Dic59—42no—
279Dic59—42no—
280Dic59—82no—
281Dic59C2×#10314yes—
282C2×#533—591no—
283C22×C4⋊C8—591no—
284#2285—591no—
285#1987—592no—
286#1981—592no—
287#1892—592no—
288#1892—592no—
289#1803—591no—
290#1803—591no—
291#1803—591no—
292#1803D5911yescharacteristic, complemented
293C2×C59#231111yescharacteristic
294C2×C59C22×#10311yescharacteristic
295C2×C59#289611yescharacteristic
296C2×C59#324611yescharacteristic
297C2×C59#324611yescharacteristic
298C2×C59#324911yescharacteristic
299C2×C59#324911yescharacteristic
300D59—24no—
301C23×C8—592no—
302#533—598no—
303#533—1184no—
304C2×C4⋊Q8—591no—
305C2×C4⋊Q8—591no—
306C2×#125—1181no—
307C2×C22⋊Q8—591no—
308C2×C22⋊Q8—591no—
309C2×C22⋊Q8—1181no—
310C2×C4×Q8—591no—
311C2×C4×Q8—1181no—
312C2×C4×Q8C2×D5911yescharacteristic
313C2×#117—1181no—
314C2×#117—1181no—
315C22×C42—591no—
316C2×C4⋊C8—592no—
317C2×C4⋊C8—591no—
318C2×C4⋊C8—1184no—
319C2×C4⋊C8C2×D5911yescharacteristic
320C2×#103—592no—
321C2×#103—592no—
322C2×#103—592no—
323C2×#103C2×D5912yes—
324#384—591no—
325#384C2×D5911yescharacteristic
326#382—1181no—
327#380—592no—
328#380—1181no—
329C4×C4⋊C4—592no—
330C4×C4⋊C4—1181no—
331#340—592no—
332#340—592no—
333#340—592no—
334#340C2×D5912yes—
335C59#2188511yescharacteristic, complemented, Sylow
336C22×Q8—591no—
337C22×Q8—1181no—
338C22×Q8C22×D5911yescharacteristic
339C23×C4—591no—
340C23×C4—591no—
341C23×C4—591no—
342C22×C8—592no—
343C22×C8—1184no—
344C22×C8—2362no—
345C22×C8—2362no—
346C22×C8C22×D5912yes—
347C4⋊Q8—594no—
348C4⋊Q8—594no—
349C4⋊Q8—1182no—
350C4⋊Q8—1182no—
351#125—1188no—
352C22⋊Q8—594no—
353C22⋊Q8—594no—
354C22⋊Q8—1182no—
355C22⋊Q8—1188no—
356C22⋊Q8—1182no—
357C4×Q8—22no—
358C4×Q8—594no—
359C4×Q8—1188no—
360C4×Q8—1182no—
361C4×Q8C4×D5914yes—
362#117—1188no—
363#117—2364no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—592no—
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—1181no—
377C2×C4⋊C4C22×D5912yes—
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C22⋊C4—1182no—
382C2×C42—21no—
383C2×C42—591no—
384C2×C42—591no—
385C2×C42—591no—
386C2×C42—594no—
387C2×C42—594no—
388C2×C42—1181no—
389C2×C42—1182no—
390C2×C42C22×D5911yescharacteristic
391C4⋊C8—22no—
392C4⋊C8—1184no—
393C4⋊C8—1182no—
394#103—42no—
395#103—2362no—
396#103—2362no—
397#103—2362no—
398#95—1182no—
399#95—1182no—
400C24—591no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—592no—
407C2×Q8—1184no—
408C2×Q8—1181no—
409C2×Q8—1181no—
410C2×Q8—1181no—
411C2×Q8—1181no—
412C2×Q8—1181no—
413C2×Q8—2362no—
414C2×Q8—2362no—
415C2×Q8C2×C4×D5912yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—591no—
420C22×C4—594no—
421C22×C4—594no—
422C22×C4—594no—
423C22×C4—591no—
424C22×C4—591no—
425C22×C4—592no—
426C22×C4—592no—
427C22×C4—594no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1182no—
435C22×C4—2361no—
436C22×C4—2361no—
437C22×C4D4×D5911yescharacteristic
438C22×C4D4×D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—1182no—
444C2×C8—1182no—
445C2×C8—1182no—
446C2×C8—2368no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—592no—
454C4⋊C4—1184no—
455C4⋊C4—1184no—
456C4⋊C4—1184no—
457C4⋊C4—1184no—
458C4⋊C4—1184no—
459C4⋊C4—1184no—
460C4⋊C4—1181no—
461C4⋊C4—1182no—
462C4⋊C4—1182no—
463C4⋊C4—2362no—
464C4⋊C4—2362no—
465C4⋊C4C2×C4×D5912yes—
466C22⋊C4—1184no—
467C22⋊C4—1184no—
468C22⋊C4—2364no—
469C42—24no—
470C42—594no—
471C42—594no—
472C42—592no—
473C42—592no—
474C42—1184no—
475C42—2364no—
476C42C2×C4×D5912yes—
477C42D4×D5912yes—
478C23—592no—
479C23—592no—
480C23—592no—
481C23—1181no—
482C23—1181no—
483C23—1181no—
484C23—1181no—
485C23C2×D4×D5911yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490Q8—1182no—
491Q8—1182no—
492Q8—2368no—
493Q8—2361no—
494Q8—2361no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—592no—
507C2×C4—592no—
508C2×C4—592no—
509C2×C4—5916no—
510C2×C4—592no—
511C2×C4—592no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4—1182no—
517C2×C4—1182no—
518C2×C4—1182no—
519C2×C4—1182no—
520C2×C4—1182no—
521C2×C4—1182no—
522C2×C4—2364no—
523C2×C4—2362no—
524C2×C4—2362no—
525C2×C4—2362no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4C22⋊C4×D5912yes—
531C2×C4C22⋊C4×D5912yes—
532C2×C4SD16×D5912yes—
533C2×C4Q16×D5912yes—
534C2×C4C22×C4×D5911yescharacteristic
535C2×C4C2×D4×D5911yescharacteristic
536C2×C4#1427377211yescharacteristic
537C2×C4#1427377311yescharacteristic
538C2×C4#1427377411yescharacteristic
539C2×C4#1427377511yescharacteristic
540C8—42no—
541C8—2362no—
542C22—594no—
543C22—1182no—
544C22—1182no—
545C22—1182no—
546C22—1182no—
547C22—1182no—
548C22—1182no—
549C22C2×C22⋊C4×D5911yescharacteristic
550C22#1683798111yescharacteristic
551C22#1683798211yescharacteristic
552C22C2×SD16×D5911yescharacteristic
553C22C2×Q16×D5911yescharacteristic
554C22#1683819811yescharacteristic
555C22#1683820111yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—594no—
561C4—1182no—
562C4—1182no—
563C4—4722no—
564C4—2362no—
565C4—2362no—
566C4#103×D5914yes—
567C4#1683788211yescharacteristic
568C4#1683788311yescharacteristic
569C4#1683796811yescharacteristic
570C4#1683797511yescharacteristic
571C2—1184no—
572C2#2142668511yescharacteristic
573C2C2×#103×D5911yescharacteristic
574C2#2143261211yescharacteristic
575C2#2143541011yescharacteristic
576C2#2143541111yescharacteristic
577C2#2143543911yescharacteristic
578C2#2143544011yescharacteristic
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#21885
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. #21425042C2semidirect
5. #21425017C2semidirect
8. #21424102C2semidirect
9. #21424101C2semidirect
10. #21424098C2semidirect
11. C59×#1803C2semidirect
14. #16837956C4semidirect
244. C4×D59#103semidirect
292. #1803D59semidirect
335. C59#21885semidirect

Maximal subgroups

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

subgroupindex
C2×#168379562
C2×C4⋊C8×D592
#214265012
#214250422
#214250172
#214245182
#214245162
#214241022
#214241012
#214240982
C59×#18032
#2188559