Finite Group Databasebeta

← #28606319

Subgroups — #28606319

Order 15104.

778 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×#16837439C211yesmaximal, characteristic
2#21426988C211yesmaximal, characteristic
3#21426977C211yesmaximal, characteristic
4#21424823C211yesmaximal, characteristic, complemented
5#21424821C211yesmaximal, characteristic, complemented
6#21424689C211yesmaximal, characteristic, complemented
7#21424687C211yesmaximal, characteristic, complemented
8#21424669C211yesmaximal, characteristic, complemented
9#21424189C211yesmaximal, characteristic
10#21424189C211yesmaximal, characteristic
11#21424120C211yesmaximal, characteristic
12C4×#14273658C211yesmaximal, characteristic, complemented
13#21423984C211yesmaximal, characteristic, complemented
14#21423866C211yesmaximal, characteristic
15C59×#1918C211yesmaximal, characteristic, complemented
16C22×C8×D59C2211yescharacteristic
17C2×#14273685C2211yescharacteristic
18C2×#14273685C2211yescharacteristic
19#16837439C418yescomplemented
20C2×#14273661C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×(C2×D59)⋊C8C2211yescharacteristic
26C2×C4×C59⋊C8C2211yescharacteristic
27C2×C8×Dic59C2211yescharacteristic
28#16837301C2211yescharacteristic
29#16837300C2211yescharacteristic
30#16837298C2211yescharacteristic
31#16837294—21no—
32#16837293—21no—
33#16837292—21no—
34#16837291—21no—
35#16837269—21no—
36#16837266—21no—
37#16837259C2211yescharacteristic
38#16837259C2211yescharacteristic
39C4⋊C4×Dic59—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×(C2×D59)⋊C4C2211yescharacteristic
44#16837248—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×C4×D59C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C2×C8×D59—22no—
64C2×C8×D59—22no—
65C2×C8×D59C2×C414yes—
66#14273685—22no—
67#14273685—22no—
68#14273685C2×C414yes—
69#14273685C2×C414yes—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4D411yescharacteristic
82C2×(C2×D59)⋊C4D411yescharacteristic
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59C2311yescharacteristic
89#14273661C2×C414yes—
90#14273658—22no—
91#14273658—22no—
92#14273658C2×C414yes—
93#14273658C2×C414yes—
94(C2×D59)⋊C8—22no—
95C4×C59⋊C8—22no—
96C8×Dic59—22no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95D411yescharacteristic
106C59⋊#95D411yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C42×C59—21no—
115C2×C42×C59C2311yescharacteristic
116#107×C59C2×C414yes—
117#95×C59—21no—
118#95×C59—21no—
119C23×D59C2×D411yescharacteristic
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2411yescharacteristic
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59C2×D411yescharacteristic
132C22×Dic59C2×D411yescharacteristic
133C22×Dic59C2×D411yescharacteristic
134C2×C4×D59—21no—
135C2×C4×D59—21no—
136C2×C4×D59—21no—
137C2×C4×D59—21no—
138C2×C4×D59C22×C412yes—
139C2×C4×D59#4112yes—
140C2×C8×C59—21no—
141C2×C8×C59—21no—
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59C22×C412yes—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—42no—
153C4⋊C4×C59C22×C412yes—
154C4⋊C4×C59C22×C412yes—
155C42×C59—21no—
156C42×C59—21no—
157C42×C59—42no—
158(C2×D59)⋊C4—24no—
159(C2×D59)⋊C4—24no—
160(C2×D59)⋊C4—22no—
161(C2×D59)⋊C4—22no—
162(C2×D59)⋊C4—81no—
163(C2×D59)⋊C4—81no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—42no—
171C4⋊Dic59C22×C412yes—
172C4⋊Dic59C22×C412yes—
173Dic59⋊C4—81no—
174Dic59⋊C4—81no—
175C4×Dic59—21no—
176C4×Dic59—21no—
177C4×Dic59—24no—
178C4×Dic59—24no—
179C4×Dic59—42no—
180C4×Dic59—81no—
181C4×Dic59—81no—
182C2×C59⋊C8—21no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8C22×C412yes—
186C2×C59⋊C8#4112yes—
187C2×C59⋊C8#4112yes—
188C8×D59—28no—
189C23×C59C22×D411yescharacteristic
190C22×D59—21no—
191C22×D59—21no—
192C22×D59—21no—
193C22×D59—21no—
194C22×D59—21no—
195C22×D59—21no—
196C22×D59C4×D412yes—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—22no—
200C2×C4×C59—22no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
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#13512yes—
215C2×C4×C59C23×C411yescharacteristic
216C2×C4×C59C2×#4111yescharacteristic
217C2×C4×C59C2×#4111yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
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—21no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—41no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59C4×D412yes—
241C2×Dic59#13512yes—
242C4×D59—22no—
243C4×D59—22no—
244C4×D59—22no—
245C4×D59—22no—
246C8×C59—22no—
247C8×C59—41no—
248C59⋊C8—41no—
249C59⋊C8#11714yes—
250#22610—591nomaximal, Sylow
251C22×C59C2×C4×D411yescharacteristic
252C22×C59#54711yescharacteristic
253C22×C59#54711yescharacteristic
254C22×C59C2×#13511yescharacteristic
255C22×C59C2×#13511yescharacteristic
256C22×C59#57711yescharacteristic
257C22×C59#57711yescharacteristic
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—41no—
267C4×C59—42no—
268C4×C59—42no—
269C4×C59—81no—
270C4×C59—81no—
271C4×C59C2×#11711yescharacteristic
272C4×C59#51911yescharacteristic
273C4×C59#55211yescharacteristic
274C4×C59#55211yescharacteristic
275Dic59—22no—
276Dic59—22no—
277Dic59—41no—
278Dic59—42no—
279Dic59—42no—
280Dic59—81no—
281Dic59—81no—
282C2×#427—591no—
283#2347—591no—
284#2346—591no—
285#1941—591no—
286#1941—591no—
287#1921—591no—
288#1921—591no—
289#1918—591no—
290#1918D5911yescharacteristic, complemented
291#1825—591no—
292#1825—591no—
293#1807—591no—
294C4×#106—591no—
295C4×#106—591no—
296C8×C22⋊C4—591no—
297C2×C59#234411yescharacteristic
298C2×C59C4×#13511yescharacteristic
299C2×C59#295211yescharacteristic
300C2×C59#333711yescharacteristic
301C2×C59#333711yescharacteristic
302C2×C59#334011yescharacteristic
303C2×C59#334011yescharacteristic
304D59—42no—
305C23×C8—591no—
306C2×#117—591no—
307C2×#117—591no—
308#427—598no—
309C2×#107—591no—
310C2×#107C2×D5911yescharacteristic
311C2×#106—591no—
312C2×#106—591no—
313C2×#106—591no—
314C2×#106—591no—
315C2×C22⋊C8—591no—
316C2×C4×C8—591no—
317C2×C4×C8—591no—
318#389—591no—
319#389—591no—
320#389—591no—
321#389C2×D5911yescharacteristic
322#388—1181no—
323#388—1181no—
324#388—1181no—
325#388—1181no—
326#383—1181no—
327#382—1181no—
328#380—591no—
329#380—591no—
330C4×C4⋊C4—591no—
331C4×C4⋊C4—591no—
332C4×C4⋊C4—591no—
333C4×C4⋊C4—1181no—
334C4×C4⋊C4C2×D5911yescharacteristic
335C4×C22⋊C4—591no—
336#376—1181no—
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#336—591no—
348#336C2×D5911yescharacteristic
349C59#2261011yescharacteristic, complemented, Sylow
350C23×C4—591no—
351C22×C8—591no—
352C22×C8—594no—
353C22×C8—591no—
354C22×C8—1182no—
355C22×C8—1182no—
356C22×C8C22×D5911yescharacteristic
357C22×C8C22×D5911yescharacteristic
358#117—594no—
359#117—594no—
360#117—1182no—
361#117—1182no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—591no—
364C2×C4⋊C4—591no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—1181no—
368C2×C4⋊C4—1181no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4C22×D5911yescharacteristic
371C2×C4⋊C4C22×D5911yescharacteristic
372C2×C4⋊C4C22×D5911yescharacteristic
373C2×C4⋊C4C22×D5911yescharacteristic
374C2×C22⋊C4—591no—
375C2×C22⋊C4—591no—
376C2×C22⋊C4—1181no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C42—21no—
381C2×C42—591no—
382C2×C42—1181no—
383C2×C42—1181no—
384C2×C42—1181no—
385C2×C42—1181no—
386C2×C42—1181no—
387C2×C42C22×D5911yescharacteristic
388#107—594no—
389#107C4×D5914yes—
390#106—594no—
391#106—594no—
392#106—1182no—
393#106—1182no—
394C22⋊C8—1182no—
395C4×C8—1182no—
396C4×C8—1182no—
397#95—21no—
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—
409C24—591no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—591no—
415C22×C4—591no—
416C22×C4—592no—
417C22×C4—592no—
418C22×C4—591no—
419C22×C4—1181no—
420C22×C4—1181no—
421C22×C4—1181no—
422C22×C4—1181no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4—1181no—
427C22×C4D4×D5911yescharacteristic
428C22×C4D4×D5911yescharacteristic
429C22×C4C23×D5911yescharacteristic
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—592no—
436C2×C8—592no—
437C2×C8—592no—
438C2×C8—1188no—
439C2×C8—1181no—
440C2×C8—1181no—
441C2×C8—1181no—
442C2×C8C2×C4×D5912yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—592no—
451C4⋊C4—592no—
452C4⋊C4—1181no—
453C4⋊C4—1181no—
454C4⋊C4—1181no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—4721no—
459C4⋊C4—4721no—
460C4⋊C4—2362no—
461C4⋊C4C2×C4×D5912yes—
462C4⋊C4C2×C4×D5912yes—
463C22⋊C4—1182no—
464C22⋊C4—1184no—
465C22⋊C4—1184no—
466C22⋊C4—1182no—
467C22⋊C4—4721no—
468C22⋊C4—4721no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—1184no—
473C42—1184no—
474C42—1181no—
475C42—1181no—
476C42—4721no—
477C42—4721no—
478C42—2362no—
479C23—592no—
480C23—1181no—
481C23—1181no—
482C23—1181no—
483C23—1181no—
484C23—1181no—
485C23—1181no—
486C23C2×D4×D5911yescharacteristic
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—592no—
505C2×C4—592no—
506C2×C4—1181no—
507C2×C4—1181no—
508C2×C4—1181no—
509C2×C4—1181no—
510C2×C4—1182no—
511C2×C4—1182no—
512C2×C4—1182no—
513C2×C4—1182no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4—1181no—
517C2×C4—1182no—
518C2×C4—1182no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4#1427374612yes—
530C2×C4C22×C4×D5911yescharacteristic
531C2×C4#41×D5911yescharacteristic
532C2×C4#41×D5911yescharacteristic
533C2×C4#41×D5911yescharacteristic
534C2×C4#1427377411yescharacteristic
535C2×C4#1427377511yescharacteristic
536C8—22no—
537C8—41no—
538C8—594no—
539C8—2361no—
540C22—1182no—
541C22—1182no—
542C22—1182no—
543C22—2361no—
544C22—2361no—
545C22—2361no—
546C22—2361no—
547C22C4×D4×D5911yescharacteristic
548C22#1683805011yescharacteristic
549C22#1683805411yescharacteristic
550C22#135×D5911yescharacteristic
551C22C2×#1427374611yescharacteristic
552C22#1683821111yescharacteristic
553C22#1683821111yescharacteristic
554C4—22no—
555C4—42no—
556C4—42no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—1182no—
561C4—1182no—
562C4—4721no—
563C4—4721no—
564C4—2361no—
565C4—2362no—
566C4—2362no—
567C4#117×D5911yescharacteristic
568C4#1683787811yescharacteristic
569C4#1683808411yescharacteristic
570C4#1683808811yescharacteristic
571C2—2362no—
572C2#2142696211yescharacteristic
573C2C4×#1427374611yescharacteristic
574C2#2143297611yescharacteristic
575C2#2143647211yescharacteristic
576C2#2143647911yescharacteristic
577C2#2143650211yescharacteristic
578C2#2143650611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1918C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1918D59
hypocenterC59#22610
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
4. #21424823C2semidirect
5. #21424821C2semidirect
6. #21424689C2semidirect
7. #21424687C2semidirect
8. #21424669C2semidirect
12. C4×#14273658C2semidirect
13. #21423984C2semidirect
15. C59×#1918C2semidirect
19. #16837439C4semidirect
290. #1918D59semidirect
349. C59#22610semidirect

Maximal subgroups

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

subgroupindex
C2×#168374392
#214269882
#214269772
#214248232
#214248212
#214246892
#214246872
#214246692
#214241892
#214241202
C4×#142736582
#214239842
#214238662
C59×#19182
#2261059