Finite Group Databasebeta

← #28622600

Subgroups — #28622600

Order 15104.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838109C211yesmaximal, characteristic, complemented
2C2×#16837692C211yesmaximal, characteristic, complemented
3C2×C4×#11745670C211yesmaximal, characteristic, complemented
4#21425292C212yesmaximal, complemented
5#21425099C212yesmaximal, complemented
6#21425056C211yesmaximal, characteristic, complemented
7#21424891C211yesmaximal, characteristic, complemented
8#21424414C212yesmaximal, complemented
9#21424288C212yesmaximal, complemented
10#21424098C211yesmaximal, characteristic, complemented
11C59×#2537C211yesmaximal, characteristic
12C22×#11745670C2211yescharacteristic
13#16838109—24no—
14#16838109—24no—
15C2×#14273727C2211yescharacteristic
16C2×#14273718—21no—
17C2×#14273706C2212yes—
18C2×#14273704—22no—
19C2×C4×Dic118C2211yescharacteristic
20C2×D4×Dic59—21no—
21C22×C4⋊Dic59C2212yes—
22#16837692—24no—
23C4×#11745670—24no—
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2212yes—
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2212yes—
29C2×#14273648C2211yescharacteristic
30C2×#14273648C2212yes—
31C2×C22⋊(C59⋊C8)C2212yes—
32C2×C22⋊(C59⋊C8)C2212yes—
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837164C2212yes—
35#16837164C2212yes—
36#16837164C2212yes—
37C2×#121×C59—21no—
38C2×#121×C59C2212yes—
39C2×C4×D4×C59C2211yescharacteristic
40C2×C4×D4×C59C2211yescharacteristic
41C59×#394—22no—
42C59×#384C2211yescharacteristic
43C59×#380—21no—
44C59×#380C2212yes—
45C22×Dic118—21no—
46C22×Dic118C2311yescharacteristic
47C23×Dic59—22no—
48C2×#11745670—22no—
49C2×#11745670—22no—
50C2×#11745670—22no—
51C22×C59⋊C8C2311yescharacteristic
52C22×C59⋊C8C2312yes—
53C22×C59⋊C8C2311yescharacteristic
54#14273727—22no—
55#14273727D414yescomplemented
56#14273718—28no—
57#14273706—24no—
58#14273706—24no—
59#14273704—216no—
60C4×Dic118—22no—
61C4×Dic118D414yescomplemented
62D4×Dic59—28no—
63C2×C4⋊Dic59—22no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—24no—
66C2×C4⋊Dic59—24no—
67C2×C4⋊Dic59C2312yes—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×Dic59⋊C4—21no—
70C2×Dic59⋊C4—22no—
71C2×C22⋊Dic59—22no—
72C2×C4×Dic59—21no—
73#14273658—24no—
74#14273658—22no—
75C4⋊(C59⋊C8)—22no—
76#14273650—22no—
77#14273650—42no—
78#14273648—24no—
79#14273648—22no—
80C22⋊(C59⋊C8)—24no—
81C22⋊(C59⋊C8)—42no—
82C4×C59⋊C8—22no—
83C22×D4×C59—22no—
84C22×D4×C59D411yescharacteristic
85C22×D4×C59C2311yescharacteristic
86C23×C4×C59—21no—
87C23×C4×C59C2312yes—
88#121×C59—24no—
89#121×C59—42no—
90#121×C59—44no—
91C4×D4×C59—22no—
92C4×D4×C59—22no—
93C4×D4×C59—22no—
94C4×D4×C59—22no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59D411yescharacteristic
98C2×C4⋊C4×C59C2312yes—
99C2×C4⋊C4×C59C2311yescharacteristic
100C2×C22⋊C4×C59—22no—
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59—21no—
103C2×C22⋊C4×C59D412yes—
104C2×C42×C59C2311yescharacteristic
105#95×C59—22no—
106C24×C59—21no—
107C24×C59C2×D412yes—
108C2×D4×C59—22no—
109C2×D4×C59—22no—
110C2×D4×C59—21no—
111C2×D4×C59—21no—
112C2×D4×C59—21no—
113C2×D4×C59—21no—
114C2×D4×C59—21no—
115C2×D4×C59—44no—
116C2×D4×C59—44no—
117C2×D4×C59—82no—
118C2×D4×C59SD1614yescomplemented
119C2×D4×C59#4112yes—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59—22no—
128C22×C4×C59—42no—
129C22×C4×C59—42no—
130C22×C4×C59—42no—
131C22×C4×C59C2×D412yes—
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2411yescharacteristic
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—22no—
138C22×Dic59—28no—
139C22×Dic59—82no—
140C22×Dic59—82no—
141C2×Dic118—21no—
142C2×Dic118—21no—
143C2×Dic118—21no—
144C2×Dic118—21no—
145C2×Dic118—21no—
146C2×Dic118—24no—
147C2×Dic118—81no—
148C2×Dic118—81no—
149C2×Dic118C2×D412yes—
150C4⋊C4×C59—22no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—22no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—42no—
156C4⋊C4×C59—81no—
157C22⋊C4×C59—24no—
158C22⋊C4×C59—44no—
159C22⋊C4×C59—44no—
160C22⋊C4×C59—42no—
161C42×C59C2×D412yes—
162C42×C59C2×D412yes—
163C22⋊Dic59—28no—
164#11745670—28no—
165C4⋊Dic59—28no—
166C4⋊Dic59—24no—
167C4⋊Dic59—21no—
168C4⋊Dic59—22no—
169C4⋊Dic59C2×D412yes—
170C4⋊Dic59C2×D414yes—
171Dic59⋊C4—28no—
172Dic59⋊C4—24no—
173C4×Dic59—24no—
174C2×C59⋊C8—21no—
175C2×C59⋊C8—21no—
176C2×C59⋊C8—21no—
177C2×C59⋊C8—22no—
178C2×C59⋊C8—22no—
179C2×C59⋊C8—21no—
180C2×C59⋊C8—22no—
181C2×C59⋊C8—21no—
182C2×C59⋊C8#4112yes—
183C23×C59—22no—
184C23×C59—22no—
185C23×C59—22no—
186C23×C59—22no—
187C23×C59—22no—
188C23×C59—22no—
189C23×C59—22no—
190C23×C59—41no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59C2×SD1614yes—
197C23×C59C22×D411yescharacteristic
198D4×C59—22no—
199D4×C59—22no—
200D4×C59—28no—
201D4×C59—41no—
202D4×C59—41no—
203D4×C59—88no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—22no—
207C2×C4×C59—22no—
208C2×C4×C59—22no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59—42no—
212C2×C4×C59—42no—
213C2×C4×C59—41no—
214C2×C4×C59—42no—
215C2×C4×C59—41no—
216C2×C4×C59—42no—
217C2×C4×C59—41no—
218C2×C4×C59—42no—
219C2×C4×C59—84no—
220C2×C4×C59#12012yes—
221C2×C4×C59#12012yes—
222C2×C4×C59C2×SD1612yes—
223C2×C4×C59#13712yes—
224C2×C4×C59C22×D411yescharacteristic
225C2×C4×C59C22×D411yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59H(5,𝔽2)11yescharacteristic
229C2×C4×C592−1+411yescharacteristic
230C2×Dic59—22no—
231C2×Dic59—22no—
232C2×Dic59—24no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—42no—
236C2×Dic59—41no—
237C2×Dic59—42no—
238C2×Dic59—41no—
239C2×Dic59—88no—
240Dic118—22no—
241Dic118—22no—
242Dic118—41no—
243Dic118—41no—
244Dic118—84no—
245C59⋊C8—22no—
246C59⋊C8—41no—
247C59⋊C8—42no—
248#24118—591nomaximal, Sylow
249C22×C59—24no—
250C22×C59—24no—
251C22×C59—24no—
252C22×C59—42no—
253C22×C59—42no—
254C22×C59—42no—
255C22×C59—42no—
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59C2×#12011yescharacteristic
264C22×C59#54611yescharacteristic
265C22×C59#54711yescharacteristic
266C22×C59C22×SD1611yescharacteristic
267C22×C59C2×#13711yescharacteristic
268C22×C59#57611yescharacteristic
269C22×C59#57711yescharacteristic
270C4×C59—22no—
271C4×C59—42no—
272C4×C59—81no—
273C4×C59—82no—
274C4×C59#44814yes—
275C4×C59#52511yescharacteristic
276C4×C59#53711yescharacteristic
277C4×C59#54811yescharacteristic
278C4×C59#55011yescharacteristic
279Dic59—42no—
280Dic59—44no—
281Dic59—42no—
282C2×#554—591no—
283C2×#476—591no—
284C2×C4×SD16—591no—
285#2537D5911yescharacteristic
286#2035—592no—
287#2000—592no—
288#1990—591no—
289#1954—591no—
290#1874—592no—
291#1853—592no—
292#1803—591no—
293C2×C59—44no—
294C2×C59—82no—
295C2×C59#240711yescharacteristic
296C2×C59C2×#44811yescharacteristic
297C2×C59#301211yescharacteristic
298C2×C59#329511yescharacteristic
299C2×C59#333111yescharacteristic
300C2×C59#333511yescharacteristic
301C2×C59#335711yescharacteristic
302C22×SD16—591no—
303#554—1184no—
304#554—1184no—
305C2×C4⋊Q8—591no—
306C2×#125—1181no—
307C2×C22⋊Q8—592no—
308C2×C22⋊Q8—1182no—
309C2×#121—21no—
310C2×#121C2×D5912yes—
311C2×C4×Q8—591no—
312C2×C4×D4—1181no—
313C2×C4×D4C2×D5911yescharacteristic
314C2×C4×D4C2×D5911yescharacteristic
315C22×C4⋊C4—592no—
316#476—1184no—
317C4×SD16—1184no—
318C2×#106—591no—
319C2×#106—592no—
320C2×C4⋊C8—591no—
321C2×#103—591no—
322C2×#103—592no—
323C2×#102—591no—
324C2×#102—592no—
325C2×C22⋊C8—592no—
326C2×C22⋊C8—592no—
327C2×C4×C8—591no—
328#394—22no—
329#384C2×D5911yescharacteristic
330#380—21no—
331#380C2×D5912yes—
332#340—592no—
333#340—592no—
334#340—592no—
335C59#2411811yescharacteristic, complemented, Sylow
336C22×Q8—591no—
337C22×Q8—1181no—
338C22×D4—22no—
339C22×D4C22⋊D5911yescharacteristic
340C22×D4C22×D5911yescharacteristic
341C23×C4—21no—
342C23×C4—1182no—
343C23×C4C22×D5912yes—
344C2×SD16—1182no—
345C2×SD16—1182no—
346C2×SD16—1182no—
347C22×C8—591no—
348C22×C8—591no—
349C22×C8—592no—
350C4⋊Q8—594no—
351C4⋊Q8—1182no—
352#125—1188no—
353C22⋊Q8—11816no—
354C22⋊Q8—1184no—
355C22⋊Q8—1184no—
356#121—24no—
357#121—42no—
358#121—44no—
359C4×Q8—594no—
360C4×Q8—1182no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—1188no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—592no—
370C2×C4⋊C4—1184no—
371C2×C4⋊C4—1184no—
372C2×C4⋊C4—1182no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1182no—
376C2×C4⋊C4C22⋊D5911yescharacteristic
377C2×C4⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5912yes—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—1182no—
383C2×C22⋊C4C22⋊D5912yes—
384C2×C42—1181no—
385C2×C42C22×D5911yescharacteristic
386#106—1182no—
387#106—1184no—
388C4⋊C8—1182no—
389#103—1182no—
390#103—2362no—
391#102—1182no—
392#102—1184no—
393C22⋊C8—1184no—
394C22⋊C8—2362no—
395C4×C8—1182no—
396#95—22no—
397C24—21no—
398C24C2×C22⋊D5912yes—
399C2×Q8—592no—
400C2×Q8—1184no—
401C2×Q8—1181no—
402C2×Q8—1181no—
403C2×Q8—1181no—
404C2×Q8—1181no—
405C2×Q8—1181no—
406C2×Q8—4721no—
407C2×Q8—4721no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—82no—
418C2×D4#1174567014yescomplemented
419C2×D4#1174568712yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—1188no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1182no—
435C22×C4—4722no—
436C22×C4—4722no—
437C22×C4C2×C22⋊D5912yes—
438C22×C4C2×C22⋊D5911yescharacteristic
439C22×C4C2×C22⋊D5911yescharacteristic
440C22×C4C23×D5911yescharacteristic
441SD16—1188no—
442C2×C8—592no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1182no—
447C2×C8—1182no—
448C2×C8—1181no—
449C2×C8—1182no—
450C2×C8—1181no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—81no—
458C4⋊C4—592no—
459C4⋊C4—594no—
460C4⋊C4—1188no—
461C4⋊C4—1188no—
462C4⋊C4—1184no—
463C4⋊C4—1184no—
464C4⋊C4—1181no—
465C4⋊C4—1182no—
466C22⋊C4—24no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—1188no—
471C42—1184no—
472C42D4×D5912yes—
473C42D4×D5912yes—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C2×#1174567014yes—
488C23C22×C22⋊D5911yescharacteristic
489Q8—1182no—
490Q8—1182no—
491Q8—4724no—
492Q8—2361no—
493Q8—2361no—
494D4—22no—
495D4—28no—
496D4—22no—
497D4—41no—
498D4—41no—
499D4—88no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—84no—
516C2×C4—1182no—
517C2×C4—1182no—
518C2×C4—1184no—
519C2×C4—4728no—
520C2×C4—2361no—
521C2×C4—2362no—
522C2×C4—2361no—
523C2×C4—2362no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4#1427369512yes—
527C2×C4#1427369512yes—
528C2×C4C2×#1174567012yes—
529C2×C4#1427375712yes—
530C2×C4C2×D4×D5911yescharacteristic
531C2×C4C2×D4×D5911yescharacteristic
532C2×C4#41×D5911yescharacteristic
533C2×C4C2×#1174568711yescharacteristic
534C2×C4#1427377211yescharacteristic
535C2×C4#1427377411yescharacteristic
536C8—1182no—
537C8—2361no—
538C8—2362no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#1427369511yescharacteristic
554C22#1683804611yescharacteristic
555C22#1683805111yescharacteristic
556C22C22×#1174567011yescharacteristic
557C22C2×#1427375711yescharacteristic
558C22#1683820511yescharacteristic
559C22#1683821011yescharacteristic
560C4—22no—
561C4—42no—
562C4—82no—
563C4—81no—
564C4—2362no—
565C4—2364no—
566C4—2362no—
567C4#1683754014yes—
568C4#1683791211yescharacteristic
569C4#1683798111yescharacteristic
570C4#1683805811yescharacteristic
571C4#1683807011yescharacteristic
572C2—44no—
573C2—82no—
574C2#2142743711yescharacteristic
575C2C2×#1683754011yescharacteristic
576C2#2143339711yescharacteristic
577C2#2143592011yescharacteristic
578C2#2143641211yescharacteristic
579C2#2143645311yescharacteristic
580C2#2143671611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
Fitting, prime radicalC59×#2537C2
socleC23×C59C22×D4
hypercenter#2537D59
hypocenterC59#24118
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#16838109C2semidirect
2. C2×#16837692C2semidirect
3. C2×C4×#11745670C2semidirect
4. #21425292C2semidirect
5. #21425099C2semidirect
6. #21425056C2semidirect
7. #21424891C2semidirect
8. #21424414C2semidirect
9. #21424288C2semidirect
10. #21424098C2semidirect
55. #14273727D4semidirect
61. C4×Dic118D4semidirect
118. C2×D4×C59SD16semidirect
335. C59#24118semidirect
418. C2×D4#11745670semidirect

Maximal subgroups

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

subgroupindex
C2×#168381092
C2×#168376922
C2×C4×#117456702
#214252922
#214250992
#214250562
#214248912
#214244142
#214242882
#214240982
C59×#25372
#2411859