Finite Group Databasebeta

← #28611780

Subgroups — #28611780

Order 15104.

826 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837691C211yesmaximal, characteristic
2#21427182C211yesmaximal, characteristic, complemented
3#21427041C211yesmaximal, characteristic
4#21425055C211yesmaximal, characteristic, complemented
5#21424858C211yesmaximal, characteristic, complemented
6#21424778C211yesmaximal, characteristic
7#21424777C211yesmaximal, characteristic
8#21424724C211yesmaximal, characteristic
9#21424721C211yesmaximal, characteristic
10#21424679C211yesmaximal, characteristic, complemented
11#106×Dic59C211yesmaximal, characteristic, complemented
12C4×#14273658C211yesmaximal, characteristic, complemented
13#21423957C211yesmaximal, characteristic, complemented
14#21423910C211yesmaximal, characteristic
15C59×#1920C211yesmaximal, characteristic, complemented
16C2×#14273726C2211yescharacteristic
17C2×#14273689C2211yescharacteristic
18C2×#14273684—21no—
19#16837691C418yescomplemented
20C2×#14273659C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×C4×C59⋊C8C2211yescharacteristic
29C2×C8×Dic59C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31#16837307—21no—
32#16837306—21no—
33#16837304C2211yescharacteristic
34#16837303C2211yescharacteristic
35#16837298C2211yescharacteristic
36#16837297—21no—
37#16837293—21no—
38#16837291—21no—
39#16837271C2211yescharacteristic
40#16837261—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4×Dic59⋊C4C2211yescharacteristic
45C4×C4⋊Dic59C2211yescharacteristic
46#16837253—21no—
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
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—
69#14273684—44no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4D411yescharacteristic
78C2×Dic59⋊C4D411yescharacteristic
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59C2311yescharacteristic
89C2×C4×Dic59C2311yescharacteristic
90#14273659—22no—
91#14273658—22no—
92#14273658C2×C414yes—
93Dic59⋊C8C2×C414yes—
94#14273649—22no—
95#14273649C2×C414yes—
96#14273647—22no—
97#14273647C2×C414yes—
98C4×C59⋊C8—22no—
99C8×Dic59—22no—
100C8×Dic59—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117#106×C59C2×C414yes—
118#95×C59—21no—
119#95×C59—21no—
120C23×D59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2411yescharacteristic
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59C2×D411yescharacteristic
132C22×Dic59C2×D411yescharacteristic
133C22×Dic59C2×D411yescharacteristic
134C22×Dic59C2×D411yescharacteristic
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236—21no—
140C2×D236C22×C412yes—
141C2×C4×D59—24no—
142C2×C4×D59—42no—
143C2×C4×D59—42no—
144C2×Dic118—24no—
145C2×Dic118—81no—
146C2×Dic118—81no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59C22×C412yes—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—81no—
160C4⋊C4×C59C22×C412yes—
161C4⋊C4×C59C22×C412yes—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—42no—
165(C2×D59)⋊C4—24no—
166(C2×D59)⋊C4—42no—
167(C2×D59)⋊C4—42no—
168(C2×D59)⋊C4—81no—
169(C2×D59)⋊C4—81no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59C22×C412yes—
175C4⋊Dic59#4112yes—
176Dic59⋊C4—24no—
177Dic59⋊C4—22no—
178Dic59⋊C4—42no—
179Dic59⋊C4—81no—
180Dic59⋊C4—81no—
181Dic59⋊C4SD1614yes—
182C4×Dic59—21no—
183C4×Dic59—21no—
184C4×Dic59—42no—
185C4×Dic59—42no—
186C4×Dic59—81no—
187C4×Dic59—81no—
188C4×Dic59C22×C412yes—
189C4×Dic59#4112yes—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8C22×C412yes—
195C2×C59⋊C8#4112yes—
196C23×C59C22×D411yescharacteristic
197C22×D59—22no—
198C22×D59—41no—
199C22×D59—41no—
200C22×D59—41no—
201C22×D59—41no—
202C22×D59—41no—
203C22×D59—41no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—22no—
207C2×C4×C59—22no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
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×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59C2×SD1612yes—
222C2×C4×C59C23×C411yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C59C2×#4111yescharacteristic
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C592−1+411yescharacteristic
227C2×C4×C592−1+411yescharacteristic
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—22no—
233C2×Dic59—21no—
234C2×Dic59—22no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59C4×D412yes—
245C2×Dic59C4×D412yes—
246C2×Dic59C2×SD1612yes—
247C2×Dic59C2×SD1612yes—
248C2×Dic59#13512yes—
249D236—22no—
250D236—22no—
251D236—41no—
252D236—41no—
253C4×D59—48no—
254Dic118—84no—
255C8×C59—22no—
256C8×C59—41no—
257C59⋊C8—22no—
258C59⋊C8—22no—
259#23061—591nomaximal, Sylow
260C22×C59C2×C4×D411yescharacteristic
261C22×C59#54711yescharacteristic
262C22×C59#54811yescharacteristic
263C22×C59C22×SD1611yescharacteristic
264C22×C59C2×#13511yescharacteristic
265C22×C59#57611yescharacteristic
266C22×C59#57711yescharacteristic
267C2×D59—42no—
268C2×D59—42no—
269C2×D59—42no—
270C2×D59—81no—
271C2×D59—81no—
272C2×D59—81no—
273C2×D59—81no—
274C4×C59—22no—
275C4×C59—41no—
276C4×C59—42no—
277C4×C59—42no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59C4×#4111yescharacteristic
281C4×C59#51911yescharacteristic
282C4×C59#55211yescharacteristic
283C4×C59#55611yescharacteristic
284Dic59—22no—
285Dic59—22no—
286Dic59—41no—
287Dic59—42no—
288Dic59—42no—
289Dic59—81no—
290Dic59C4×SD1614yes—
291C2×#476—591no—
292#2369—591no—
293#2354—591no—
294#1990—591no—
295#1948—591no—
296#1934—591no—
297#1934—591no—
298#1926—591no—
299#1926—591no—
300#1920—591no—
301#1920D5911yescharacteristic, complemented
302C4×#106—591no—
303C4×#106—591no—
304C8×C4⋊C4—591no—
305C4×#102—591no—
306C2×C59#236211yescharacteristic
307C2×C59C2×C4×SD1611yescharacteristic
308C2×C59#294611yescharacteristic
309C2×C59#333511yescharacteristic
310C2×C59#334111yescharacteristic
311C2×C59#336011yescharacteristic
312C2×C59#336211yescharacteristic
313D59—82no—
314C2×C4⋊Q8—591no—
315C2×C4×D4—591no—
316C2×#117—1181no—
317#476—598no—
318C2×#106—591no—
319C2×#106—591no—
320C2×#106—591no—
321C2×#106C2×D5911yescharacteristic
322C2×C4⋊C8—591no—
323C2×#102—591no—
324C2×#102—591no—
325C2×#102—591no—
326C2×#102—591no—
327C2×C4×C8—591no—
328C2×C4×C8—591no—
329C2×C4×C8—591no—
330#391—1181no—
331#391—1181no—
332#390—591no—
333#390—591no—
334#389—591no—
335#389—1181no—
336#389C2×D5911yescharacteristic
337#388—1181no—
338#388—1181no—
339#384—591no—
340#380—1181no—
341C4×C4⋊C4—591no—
342C4×C4⋊C4—591no—
343C4×C4⋊C4—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C4⋊C4—1181no—
347C4×C4⋊C4C2×D5911yescharacteristic
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340C2×D5911yescharacteristic
354#340C2×D5911yescharacteristic
355#340C2×D5911yescharacteristic
356#336—591no—
357#336C2×D5911yescharacteristic
358C59#2306111yescharacteristic, complemented, Sylow
359C22×Q8—1181no—
360C22×D4—591no—
361C23×C4—1181no—
362C22×C8—591no—
363C22×C8—591no—
364C22×C8C22×D5911yescharacteristic
365C22×C8C22×D5911yescharacteristic
366C4⋊Q8—594no—
367C4⋊Q8—1182no—
368C4×D4—594no—
369C4×D4—1182no—
370#117—2364no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C22⋊C4—1181no—
386C2×C22⋊C4—1181no—
387C2×C22⋊C4—1181no—
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—1181no—
390C2×C42—21no—
391C2×C42—591no—
392C2×C42—591no—
393C2×C42—1181no—
394C2×C42—1181no—
395C2×C42—1181no—
396C2×C42—1181no—
397C2×C42C22×D5911yescharacteristic
398#106—594no—
399#106—1182no—
400#106—1182no—
401#106C4×D5914yes—
402C4⋊C8—594no—
403#102—594no—
404#102—594no—
405#102—1182no—
406#102—1182no—
407C4×C8—1182no—
408C4×C8—1182no—
409C4×C8—1182no—
410#95—21no—
411#95—21no—
412#95—591no—
413#95—591no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419C24—1181no—
420C2×Q8—1184no—
421C2×Q8—4721no—
422C2×Q8—4721no—
423C2×D4—592no—
424C2×D4—1181no—
425C2×D4—1181no—
426C2×D4—1181no—
427C2×D4—1181no—
428C2×D4—1181no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—591no—
434C22×C4—591no—
435C22×C4—591no—
436C22×C4—591no—
437C22×C4—1184no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—2362no—
442C22×C4—2362no—
443C22×C4D4×D5911yescharacteristic
444C22×C4D4×D5911yescharacteristic
445C22×C4C23×D5911yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—592no—
452C2×C8—592no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8C2×C4×D5912yes—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—81no—
465C4⋊C4—594no—
466C4⋊C4—592no—
467C4⋊C4—592no—
468C4⋊C4—1182no—
469C4⋊C4—1184no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—4721no—
475C4⋊C4—4721no—
476C4⋊C4—2362no—
477C4⋊C4C2×C4×D5912yes—
478C4⋊C4C2×C4×D5912yes—
479C22⋊C4—1184no—
480C22⋊C4—4721no—
481C22⋊C4—4721no—
482C22⋊C4—2362no—
483C22⋊C4—2362no—
484C42—21no—
485C42—21no—
486C42—42no—
487C42—592no—
488C42—592no—
489C42—1181no—
490C42—1181no—
491C42—4721no—
492C42—4721no—
493C42—2362no—
494C42—2362no—
495C23—1182no—
496C23—2361no—
497C23—2361no—
498C23—2361no—
499C23—2361no—
500C23—2361no—
501C23—2361no—
502C23C2×D4×D5911yescharacteristic
503Q8—4724no—
504D4—1182no—
505D4—1182no—
506D4—2361no—
507D4—2361no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—592no—
526C2×C4—592no—
527C2×C4—592no—
528C2×C4—592no—
529C2×C4—592no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1182no—
538C2×C4—1182no—
539C2×C4—2368no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4SD16×D5912yes—
548C2×C4C22×C4×D5911yescharacteristic
549C2×C4#41×D5911yescharacteristic
550C2×C4#41×D5911yescharacteristic
551C2×C4#41×D5911yescharacteristic
552C2×C4#1427377411yescharacteristic
553C2×C4#1427377511yescharacteristic
554C8—22no—
555C8—41no—
556C8—1182no—
557C8—1182no—
558C22—4721no—
559C22—4721no—
560C22—4721no—
561C22—4721no—
562C22—2362no—
563C22—2362no—
564C22—2362no—
565C22C4×D4×D5911yescharacteristic
566C22#1683805011yescharacteristic
567C22#1683806211yescharacteristic
568C22C2×SD16×D5911yescharacteristic
569C22#135×D5911yescharacteristic
570C22#1683820611yescharacteristic
571C22#1683821111yescharacteristic
572C4—22no—
573C4—42no—
574C4—42no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—594no—
579C4—1182no—
580C4—1182no—
581C4—4721no—
582C4—2362no—
583C4—2362no—
584C4—2361no—
585C4#1683787011yescharacteristic
586C4#1683787811yescharacteristic
587C4#1683808411yescharacteristic
588C4#1683811911yescharacteristic
589C2—4722no—
590C2#2142711811yescharacteristic
591C2C4×SD16×D5911yescharacteristic
592C2#2143292711yescharacteristic
593C2#2143644611yescharacteristic
594C2#2143651311yescharacteristic
595C2#2143674811yescharacteristic
596C2#2143676611yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1920C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1920D59
hypocenterC59#23061
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
2. #21427182C2semidirect
4. #21425055C2semidirect
5. #21424858C2semidirect
10. #21424679C2semidirect
11. #106×Dic59C2semidirect
12. C4×#14273658C2semidirect
13. #21423957C2semidirect
15. C59×#1920C2semidirect
19. #16837691C4semidirect
301. #1920D59semidirect
358. C59#23061semidirect

Maximal subgroups

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

subgroupindex
C2×#168376912
#214271822
#214270412
#214250552
#214248582
#214247782
#214247772
#214247242
#214247212
#214246792
#106×Dic592
C4×#142736582
#214239572
#214239102
C59×#19202
#2306159