Finite Group Databasebeta

← #28600894

Subgroups — #28600894

Order 15104.

740 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×#16837691C211yesmaximal, characteristic
2#21426843C211yesmaximal, characteristic, complemented
3#21426735C211yesmaximal, characteristic
4#21425046C211yesmaximal, characteristic, complemented
5#21424865C211yesmaximal, characteristic, complemented
6#21424836C211yesmaximal, characteristic, complemented
7#21424792C211yesmaximal, characteristic
8#21424779C211yesmaximal, characteristic
9#21424722C211yesmaximal, characteristic
10#21424714C211yesmaximal, characteristic
11#21424684C211yesmaximal, characteristic, complemented
12#21424676C211yesmaximal, characteristic, complemented
13#21424105C211yesmaximal, characteristic
14#21423954C211yesmaximal, characteristic, complemented
15C59×#1941C211yesmaximal, characteristic, complemented
16C2×#14273726C2211yescharacteristic
17C2×#14273689C2211yescharacteristic
18#16837691C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273649C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
28#16837307—21no—
29#16837306—21no—
30#16837304C2211yescharacteristic
31#16837303C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837294—21no—
35#16837292—21no—
36#16837284—21no—
37#16837283—21no—
38#16837267C2211yescharacteristic
39#16837264—21no—
40#16837259—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44#16837248C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118—21no—
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273726—22no—
66#14273726C2×C414yes—
67#14273689—22no—
68#14273689C2×C414yes—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59C2311yescharacteristic
84C2×C4×Dic59C2311yescharacteristic
85#14273660—41no—
86#14273658C2×C414yes—
87Dic59⋊C8—22no—
88Dic59⋊C8C2×C414yes—
89#14273649—41no—
90#14273649C2×C414yes—
91#14273647—41no—
92#14273647C2×C414yes—
93#14273638—41no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95D411yescharacteristic
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C59⋊#95D411yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C42×C59C2311yescharacteristic
114#106×C59—41no—
115#106×C59C2×C414yes—
116C4×C8×C59—41no—
117#95×C59—21no—
118#95×C59—21no—
119C23×D59—21no—
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×Dic59C2×D411yescharacteristic
131C22×Dic59C2×D411yescharacteristic
132C22×Dic59C2×D411yescharacteristic
133C22×Dic59C2×D411yescharacteristic
134C2×D236—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236C22×C412yes—
140C2×C4×D59—24no—
141C2×C4×D59—81no—
142C2×C4×D59—81no—
143C2×Dic118—24no—
144C2×Dic118—81no—
145C2×Dic118—81no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59C22×C412yes—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—81no—
159C4⋊C4×C59—81no—
160C4⋊C4×C59C22×C412yes—
161C4⋊C4×C59C22×C412yes—
162C42×C59—21no—
163C42×C59—21no—
164(C2×D59)⋊C4—24no—
165(C2×D59)⋊C4—81no—
166(C2×D59)⋊C4—81no—
167(C2×D59)⋊C4—81no—
168(C2×D59)⋊C4—81no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—81no—
175C4⋊Dic59C22×C412yes—
176Dic59⋊C4—24no—
177Dic59⋊C4—81no—
178C4×Dic59—21no—
179C4×Dic59—21no—
180C4×Dic59—81no—
181C4×Dic59—81no—
182C4×Dic59—81no—
183C4×Dic59C22×C412yes—
184C4×Dic59#4112yes—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8C22×C412yes—
191C23×C59C22×D411yescharacteristic
192C22×D59—22no—
193C22×D59—41no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C2×C4×C59—22no—
200C2×C4×C59—22no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
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—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59C23×C411yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×C4×C59H(5,𝔽2)11yescharacteristic
220C2×C4×C592−1+411yescharacteristic
221C2×C4×C592−1+411yescharacteristic
222C2×C4×C592−1+411yescharacteristic
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—22no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
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×Dic59C4×D412yes—
243C2×Dic59C4×D412yes—
244D236—22no—
245D236—22no—
246D236—41no—
247D236—41no—
248C4×D59—84no—
249Dic118—84no—
250C8×C59—22no—
251C8×C59—41no—
252C59⋊C8—22no—
253C59⋊C8—41no—
254#22123—591nomaximal, Sylow
255C22×C59C2×C4×D411yescharacteristic
256C22×C59#53611yescharacteristic
257C22×C59#53711yescharacteristic
258C22×C59#57611yescharacteristic
259C22×C59#57711yescharacteristic
260C22×C59#57711yescharacteristic
261C22×C59#57711yescharacteristic
262C2×D59—42no—
263C2×D59—42no—
264C2×D59—42no—
265C2×D59—81no—
266C2×D59—81no—
267C2×D59—81no—
268C2×D59—81no—
269C4×C59—42no—
270C4×C59—42no—
271C4×C59—41no—
272C4×C59—41no—
273C4×C59—81no—
274C4×C59—81no—
275C4×C59#51911yescharacteristic
276C4×C59#52011yescharacteristic
277C4×C59#54111yescharacteristic
278C4×C59#54211yescharacteristic
279Dic59—22no—
280Dic59—22no—
281Dic59—42no—
282Dic59—41no—
283Dic59—41no—
284Dic59—81no—
285Dic59—81no—
286C2×#476—591no—
287#2331—591no—
288#2319—591no—
289#1988—591no—
290#1949—591no—
291#1944—591no—
292#1941D5911yescharacteristic, complemented
293#1936—591no—
294#1934—591no—
295#1926—591no—
296#1925—591no—
297#1920—591no—
298#1919—591no—
299#1804—591no—
300#1768—591no—
301C2×C59#232311yescharacteristic
302C2×C59#294611yescharacteristic
303C2×C59#294911yescharacteristic
304C2×C59#324111yescharacteristic
305C2×C59#324411yescharacteristic
306C2×C59#325711yescharacteristic
307C2×C59#326311yescharacteristic
308D59—82no—
309C2×C4⋊Q8—591no—
310C2×C4×D4—591no—
311#476—598no—
312C2×#107—591no—
313C2×#106—591no—
314C2×#106C2×D5911yescharacteristic
315C2×#106C2×D5911yescharacteristic
316C2×C4⋊C8—591no—
317C2×C4⋊C8—591no—
318C2×#102—591no—
319C2×#102—591no—
320C2×#102—591no—
321C2×#102—591no—
322C2×#97—591no—
323C2×C4×C8C2×D5911yescharacteristic
324#391—1181no—
325#391—1181no—
326#390—591no—
327#390—591no—
328#389—591no—
329#389—591no—
330#389C2×D5911yescharacteristic
331#389C2×D5911yescharacteristic
332#388—1181no—
333#388—1181no—
334#387—1181no—
335#387—1181no—
336#382—591no—
337#381—1181no—
338#380—1181no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4—591no—
342#376—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340C2×D5911yescharacteristic
350#340C2×D5911yescharacteristic
351#336—591no—
352#336—591no—
353C59#2212311yescharacteristic, complemented, Sylow
354C22×Q8—1181no—
355C22×D4—591no—
356C23×C4—1181no—
357C22×C8—591no—
358C22×C8—591no—
359C22×C8C22×D5911yescharacteristic
360C22×C8C22×D5911yescharacteristic
361C4⋊Q8—594no—
362C4⋊Q8—1182no—
363C4×D4—594no—
364C4×D4—1182no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4C22×D5911yescharacteristic
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C4⋊C4C22×D5911yescharacteristic
376C2×C4⋊C4C22×D5911yescharacteristic
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C22⋊C4—1181no—
382C2×C42—591no—
383C2×C42—591no—
384C2×C42—1181no—
385C2×C42—1181no—
386C2×C42—1181no—
387C2×C42C22×D5911yescharacteristic
388#107—2361no—
389#106—41no—
390#106—594no—
391#106C4×D5914yes—
392C4⋊C8—594no—
393C4⋊C8—1182no—
394#102—594no—
395#102—594no—
396#102—2361no—
397#102—2361no—
398#97—2361no—
399C4×C8—41no—
400#95—21no—
401#95—21no—
402#95—591no—
403#95—591no—
404#95—591no—
405#95—591no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413C24—1181no—
414C2×Q8—1184no—
415C2×Q8—4721no—
416C2×Q8—4721no—
417C2×D4—592no—
418C2×D4—1181no—
419C2×D4—1181no—
420C2×D4—1181no—
421C2×D4—1181no—
422C2×D4—1181no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—591no—
428C22×C4—591no—
429C22×C4—591no—
430C22×C4—591no—
431C22×C4—1184no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—4721no—
436C22×C4—4721no—
437C22×C4D4×D5911yescharacteristic
438C22×C4D4×D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—592no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8C2×C4×D5912yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—592no—
461C4⋊C4—1184no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—4721no—
468C4⋊C4—4721no—
469C4⋊C4C2×C4×D5912yes—
470C4⋊C4C2×C4×D5912yes—
471C22⋊C4—1184no—
472C22⋊C4—4721no—
473C22⋊C4—4721no—
474C22⋊C4—4721no—
475C22⋊C4—4721no—
476C42—21no—
477C42—21no—
478C42—592no—
479C42—592no—
480C42—1181no—
481C42—1181no—
482C42—4721no—
483C42—4721no—
484C42—4721no—
485C23—1182no—
486C23—2361no—
487C23—2361no—
488C23—2361no—
489C23—2361no—
490C23—2361no—
491C23—2361no—
492C23C2×D4×D5911yescharacteristic
493Q8—4724no—
494D4—1182no—
495D4—1182no—
496D4—2361no—
497D4—2361no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—592no—
517C2×C4—592no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1182no—
529C2×C4—4724no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4C22×C4×D5911yescharacteristic
539C2×C4#41×D5911yescharacteristic
540C2×C4#1427377211yescharacteristic
541C2×C4#1427377411yescharacteristic
542C2×C4#1427377511yescharacteristic
543C2×C4#1427377511yescharacteristic
544C8—22no—
545C8—41no—
546C8—1182no—
547C8—2361no—
548C22—4721no—
549C22—4721no—
550C22—4721no—
551C22—4721no—
552C22—2362no—
553C22—2362no—
554C22—2362no—
555C22C4×D4×D5911yescharacteristic
556C22#1683797711yescharacteristic
557C22#1683797911yescharacteristic
558C22#1683820411yescharacteristic
559C22#1683820811yescharacteristic
560C22#1683820811yescharacteristic
561C22#1683820911yescharacteristic
562C4—42no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—1182no—
569C4—1182no—
570C4—4721no—
571C4—4721no—
572C4—2361no—
573C4—2362no—
574C4—2361no—
575C4#1683787811yescharacteristic
576C4#1683788311yescharacteristic
577C4#1683801111yescharacteristic
578C4#1683802211yescharacteristic
579C2—4722no—
580C2#2142677311yescharacteristic
581C2#2143292311yescharacteristic
582C2#2143295011yescharacteristic
583C2#2143535211yescharacteristic
584C2#2143539711yescharacteristic
585C2#2143551011yescharacteristic
586C2#2143556011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1941C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1941D59
hypocenterC59#22123
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
2. #21426843C2semidirect
4. #21425046C2semidirect
5. #21424865C2semidirect
6. #21424836C2semidirect
11. #21424684C2semidirect
12. #21424676C2semidirect
14. #21423954C2semidirect
15. C59×#1941C2semidirect
18. #16837691C4semidirect
292. #1941D59semidirect
353. C59#22123semidirect

Maximal subgroups

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

subgroupindex
C2×#168376912
#214268432
#214267352
#214250462
#214248652
#214248362
#214247922
#214247792
#214247222
#214247142
#214246842
#214246762
#214241052
#214239542
C59×#19412
#2212359