Finite Group Databasebeta

← #28627411

Subgroups — #28627411

Order 15104.

1188 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837553C211yesmaximal, characteristic
2C2×#16837459C211yesmaximal, characteristic, complemented
3#21432435C211yesmaximal, characteristic, complemented
4#21425484C212yesmaximal, complemented
5#21425413C212yesmaximal, complemented
6#21424976C212yesmaximal, complemented
7#21424796C212yesmaximal, complemented
8#21424403C211yesmaximal, characteristic, complemented
9#21424284C211yesmaximal, characteristic, complemented
10#21424027C211yesmaximal, characteristic
11C2×C59×#390C211yesmaximal, characteristic
12C22×#11745670C2212yes—
13C2×#14273706C2211yescharacteristic
14#16837553—24no—
15#16837459—24no—
16C2×#14273660C2212yescomplemented
17C2×C4⋊(C59⋊C8)C2212yescomplemented
18C2×#14273650C2212yescomplemented
19C2×#14273648C2212yes—
20C2×#14273648C2212yes—
21C2×#14273648C2212yes—
22C2×C22⋊(C59⋊C8)C2211yescharacteristic
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273638C2212yescomplemented
25#16837308C2211yescharacteristic, complemented
26#16837308C2211yescharacteristic, complemented
27#16837300C2211yescharacteristic, complemented
28#16837300C2211yescharacteristic, complemented
29#16837286—21no—
30#16837286—21no—
31#16837275—21no—
32#16837265—21no—
33#16837262C2211yescharacteristic
34#16837164C2212yescomplemented
35#16837164C2212yescomplemented
36C23×D4×C59C2211yescharacteristic
37C22×C4⋊C4×C59C2211yescharacteristic
38C22×C22⋊C4×C59—21no—
39C22×C22⋊C4×C59—21no—
40C22×C42×C59C2211yescharacteristic
41C59×#390—21no—
42C59×#390—21no—
43C59×#390C2212yes—
44C59×#390C2212yes—
45C22×Dic118C2311yescharacteristic
46C2×#11745670—24no—
47C2×#11745670—24no—
48C2×#11745670—42no—
49C22×C59⋊C8C2312yes—
50C22×C59⋊C8C2312yes—
51#14273706—22no—
52#14273706—22no—
53C2×C4⋊Dic59C2311yescharacteristic
54C2×C4⋊Dic59C2311yescharacteristic
55C2×C4⋊Dic59C2311yescharacteristic
56C2×Dic59⋊C4—21no—
57C2×Dic59⋊C4—21no—
58C2×Dic59⋊C4—21no—
59C2×C22⋊Dic59—21no—
60C2×C22⋊Dic59—21no—
61#14273660—42no—
62C4⋊(C59⋊C8)—24no—
63#14273650—42no—
64#14273648—24no—
65#14273648—24no—
66#14273648—42no—
67C22⋊(C59⋊C8)—22no—
68C22⋊(C59⋊C8)D414yes—
69#14273638—24no—
70C59⋊#95—21no—
71C59⋊#95—21no—
72C59⋊#95—21no—
73C59⋊#95—21no—
74C59⋊#95—21no—
75C59⋊#95—21no—
76C59⋊#95—21no—
77C25×C59—21no—
78C22×D4×C59—24no—
79C22×D4×C59—21no—
80C22×D4×C59—22no—
81C22×D4×C59—22no—
82C22×D4×C59—42no—
83C22×D4×C59C2312yes—
84C23×C4×C59—21no—
85C23×C4×C59D411yescharacteristic
86C23×C4×C59D411yescharacteristic
87C23×C4×C59C2311yescharacteristic
88C2×C4⋊C4×C59—21no—
89C2×C4⋊C4×C59—42no—
90C2×C4⋊C4×C59C2312yes—
91C2×C22⋊C4×C59—22no—
92C2×C22⋊C4×C59—22no—
93C2×C22⋊C4×C59—22no—
94C2×C22⋊C4×C59—22no—
95C2×C22⋊C4×C59—44no—
96C2×C22⋊C4×C59—44no—
97C2×C42×C59—22no—
98C2×C42×C59—22no—
99C2×C42×C59D411yescharacteristic
100C2×C42×C59D411yescharacteristic
101C2×C42×C59C2311yescharacteristic
102C2×C42×C59C2311yescharacteristic
103C24×C59—24no—
104C24×C59—22no—
105C24×C59—42no—
106C24×C59—42no—
107C24×C59—42no—
108C24×C59—42no—
109C24×C59—42no—
110C24×C59—42no—
111C24×C59C2×D411yescharacteristic
112C2×D4×C59—24no—
113C2×D4×C59—22no—
114C2×D4×C59—22no—
115C2×D4×C59—22no—
116C2×D4×C59—22no—
117C2×D4×C59—22no—
118C2×D4×C59—48no—
119C2×D4×C59—48no—
120C2×D4×C59—42no—
121C2×D4×C59—42no—
122C2×D4×C59#4114yes—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—41no—
132C22×C4×C59—42no—
133C22×C4×C59—42no—
134C22×C4×C59—42no—
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D412yes—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59#4112yes—
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118—22no—
150C2×Dic118—22no—
151C4⋊C4×C59—22no—
152C4⋊C4×C59—22no—
153C4⋊C4×C59—42no—
154C22⋊C4×C59—44no—
155C22⋊C4×C59—44no—
156C22⋊C4×C59—44no—
157C22⋊C4×C59—44no—
158C42×C59—22no—
159C42×C59—22no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—21no—
164C22⋊Dic59—42no—
165C22⋊Dic59—42no—
166#11745670—48no—
167C4⋊Dic59—21no—
168C4⋊Dic59—22no—
169C4⋊Dic59—21no—
170C4⋊Dic59—22no—
171Dic59⋊C4—42no—
172Dic59⋊C4—42no—
173Dic59⋊C4—42no—
174C2×C59⋊C8—22no—
175C2×C59⋊C8—22no—
176C2×C59⋊C8—22no—
177C2×C59⋊C8—22no—
178C2×C59⋊C8—22no—
179C2×C59⋊C8C2×D414yes—
180C23×C59—21no—
181C23×C59—21no—
182C23×C59—21no—
183C23×C59—21no—
184C23×C59—24no—
185C23×C59—48no—
186C23×C59—48no—
187C23×C59—48no—
188C23×C59—42no—
189C23×C59—42no—
190C23×C59—42no—
191C23×C59—42no—
192C23×C59—42no—
193C23×C59—42no—
194C23×C59—84no—
195C23×C59—84no—
196C23×C59—84no—
197C23×C59—84no—
198C23×C59#12112yes—
199C23×C59AGL(1,ℤ/8)12yes—
200C23×C59#13712yes—
201C23×C59C22×D411yescharacteristic
202D4×C59—24no—
203D4×C59—24no—
204D4×C59—48no—
205D4×C59—42no—
206D4×C59—42no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—24no—
212C2×C4×C59—24no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—44no—
223C2×C4×C59—44no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—84no—
228C2×C4×C59#12112yes—
229C2×C4×C59AGL(1,ℤ/8)12yes—
230C2×C4×C59C22×D411yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4112yes—
233C2×C4×C59H(5,𝔽2)12yes—
234C2×Dic59—41no—
235C2×Dic59—41no—
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—
246Dic118—42no—
247Dic118—42no—
248C59⋊C8—24no—
249C59⋊C8—42no—
250#24507—591nomaximal, Sylow
251C22×C59—22no—
252C22×C59—22no—
253C22×C59—22no—
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—44no—
258C22×C59—44no—
259C22×C59—44no—
260C22×C59—816no—
261C22×C59—82no—
262C22×C59—82no—
263C22×C59—82no—
264C22×C59—82no—
265C22×C59#46814yes—
266C22×C59C2×#12111yescharacteristic
267C22×C59#52511yescharacteristic
268C22×C59#53511yescharacteristic
269C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C59C2×#13711yescharacteristic
272C22×C59#57511yescharacteristic
273C4×C59—22no—
274C4×C59—22no—
275C4×C59—41no—
276C4×C59—41no—
277C4×C59—41no—
278C4×C59—82no—
279C4×C59#54612yes—
280C4×C59#55112yes—
281Dic59—81no—
282Dic59—81no—
283Dic59—81no—
284Dic59—81no—
285C2×#450—591no—
286C2×#432—591no—
287#2855—591no—
288C2×#390D5911yescharacteristic
289#2067—592no—
290#2055—592no—
291#1973—592no—
292#1936—592no—
293#1872—591no—
294#1852—591no—
295#1786—591no—
296C2×C59—24no—
297C2×C59—84no—
298C2×C59#241111yescharacteristic
299C2×C59C2×#46811yescharacteristic
300C2×C59#305511yescharacteristic
301C2×C59#311011yescharacteristic
302C2×C59#311411yescharacteristic
303C2×C59#321511yescharacteristic
304C2×C59#321711yescharacteristic
305C23×D4C2×D5911yescharacteristic
306C22×SD16—592no—
307C2×C22⋊Q8—591no—
308C22×C4⋊C4C2×D5911yescharacteristic
309C22×C22⋊C4—21no—
310C22×C22⋊C4—21no—
311C22×C42C2×D5911yescharacteristic
312#450—1184no—
313#432—1184no—
314C2×#107—592no—
315C2×C4⋊C8—592no—
316C2×#103—592no—
317C2×#102—592no—
318C2×#102—592no—
319C2×#102—592no—
320C2×C22⋊C8—591no—
321C2×C22⋊C8—591no—
322C2×#97—592no—
323#391—591no—
324#391—591no—
325#390—21no—
326#390—21no—
327#390C2×D5912yes—
328#390C2×D5912yes—
329#389—591no—
330#389—591no—
331#387—1181no—
332#387—1181no—
333#385—1181no—
334#381—1181no—
335#380—591no—
336#340—592no—
337#340—592no—
338C59#2450711yescharacteristic, complemented, Sylow
339C25—21no—
340C22×Q8—591no—
341C22×D4—24no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—21no—
345C22×D4—42no—
346C22×D4C22×D5912yes—
347C23×C4—21no—
348C23×C4C22⋊D5911yescharacteristic
349C23×C4C22⋊D5911yescharacteristic
350C23×C4C22×D5911yescharacteristic
351C2×SD16—1184no—
352C2×SD16—1184no—
353C2×SD16—2362no—
354C22×C8—592no—
355C22×C8—592no—
356C22⋊Q8—1182no—
357C22⋊Q8—1182no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—591no—
361C2×C4⋊C4—591no—
362C2×C4⋊C4—591no—
363C2×C4⋊C4—1181no—
364C2×C4⋊C4—1181no—
365C2×C4⋊C4—1181no—
366C2×C4⋊C4C22×D5912yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4—1181no—
374C2×C22⋊C4—1181no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42C22⋊D5911yescharacteristic
378C2×C42C22⋊D5911yescharacteristic
379C2×C42C22×D5911yescharacteristic
380C2×C42C22×D5911yescharacteristic
381#107—2362no—
382C4⋊C8—1184no—
383#103—2362no—
384#102—1184no—
385#102—1184no—
386#102—2362no—
387C22⋊C8—594no—
388C22⋊C8—1182no—
389#97—1184no—
390#95—1181no—
391#95—1181no—
392#95—1181no—
393#95—1181no—
394#95—1181no—
395#95—1181no—
396#95—1181no—
397C24—24no—
398C24—22no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24C2×C22⋊D5911yescharacteristic
406C2×Q8—1181no—
407C2×Q8—1181no—
408C2×Q8—1182no—
409C2×Q8—1182no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4#1174568514yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4D4×D5912yes—
438C22×C4#1174568712yes—
439C22×C4C2×C22⋊D5911yescharacteristic
440C22×C4C2×C22⋊D5911yescharacteristic
441C22×C4C2×C22⋊D5911yescharacteristic
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C23×D5911yescharacteristic
445SD16—2368no—
446C2×C8—594no—
447C2×C8—1182no—
448C2×C8—1182no—
449C2×C8—1182no—
450C2×C8—1182no—
451C2×C8—1182no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1182no—
458C4⋊C4—1182no—
459C4⋊C4—2362no—
460C4⋊C4—2362no—
461C4⋊C4—2362no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—2362no—
467C22⋊C4—2362no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—24no—
479C23—48no—
480C23—48no—
481C23—48no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23#1427369912yes—
493C23#1427375312yes—
494C23#1427375712yes—
495C23C22×C22⋊D5911yescharacteristic
496Q8—2362no—
497Q8—2362no—
498D4—24no—
499D4—24no—
500D4—48no—
501D4—42no—
502D4—42no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—84no—
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—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4#1427369912yes—
537C2×C4#1427375312yes—
538C2×C4C2×D4×D5911yescharacteristic
539C2×C4C2×#1174568711yescharacteristic
540C2×C4C2×#1174568512yes—
541C2×C4#1427377212yes—
542C8—1184no—
543C8—2362no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—816no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22#1683764614yes—
559C22C2×#1427369911yescharacteristic
560C22#1683791311yescharacteristic
561C22#1683796911yescharacteristic
562C22C2×#1427375311yescharacteristic
563C22C2×#1427375311yescharacteristic
564C22C2×#1427375711yescharacteristic
565C22#1683820011yescharacteristic
566C4—22no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—82no—
572C4—4721no—
573C4—4721no—
574C4—4721no—
575C4—4721no—
576C4#1683804212yes—
577C4#1683807812yes—
578C2—24no—
579C2—84no—
580C2#2142748111yescharacteristic
581C2C2×#1683764611yescharacteristic
582C2#2143371111yescharacteristic
583C2#2143419511yescharacteristic
584C2#2143423411yescharacteristic
585C2#2143509911yescharacteristic
586C2#2143511411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#390C2
socleC23×C59C22×D4
prime radicalC23×D4×C59C22
hypercenterC2×#390D59
hypocenterC59#24507
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#16837459C2semidirect
3. #21432435C2semidirect
4. #21425484C2semidirect
5. #21425413C2semidirect
6. #21424976C2semidirect
7. #21424796C2semidirect
8. #21424403C2semidirect
9. #21424284C2semidirect
16. C2×#14273660C22semidirect
17. C2×C4⋊(C59⋊C8)C22semidirect
18. C2×#14273650C22semidirect
24. C2×#14273638C22semidirect
25. #16837308C22semidirect
26. #16837308C22semidirect
27. #16837300C22semidirect
28. #16837300C22semidirect
34. #16837164C22semidirect
35. #16837164C22semidirect
338. C59#24507semidirect

Maximal subgroups

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

subgroupindex
C2×#168375532
C2×#168374592
#214324352
#214254842
#214254132
#214249762
#214247962
#214244032
#214242842
#214240272
C2×C59×#3902
#2450759