Finite Group Databasebeta

← #28596797

Subgroups — #28596797

Order 15104.

826 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837762C211yesmaximal, characteristic
2#21426842C211yesmaximal, characteristic, complemented
3#21426804C211yesmaximal, characteristic
4#21425056C211yesmaximal, characteristic, complemented
5#21424905C211yesmaximal, characteristic
6#21424791C211yesmaximal, characteristic
7#21424784C211yesmaximal, characteristic, complemented
8#21424777C211yesmaximal, characteristic
9#21424770C211yesmaximal, characteristic, complemented
10#21423925C211yesmaximal, characteristic, complemented
11#21423920C211yesmaximal, characteristic
12#21423915C211yesmaximal, characteristic, complemented
13#21423910C211yesmaximal, characteristic
14#21423784C211yesmaximal, characteristic, complemented
15C59×#1836C211yesmaximal, characteristic, complemented
16C2×#14273727C2211yescharacteristic
17C2×#14273715C2211yescharacteristic
18C2×#14273692C2211yescharacteristic
19C2×#14273689C2211yescharacteristic
20#16837762C418yescomplemented
21C2×#14273652C2211yescharacteristic
22C2×#14273652C2211yescharacteristic
23C2×#14273650C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30C2×#14273637C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33C2×C8×Dic59C2211yescharacteristic
34#16837307C2211yescharacteristic
35#16837306—21no—
36#16837306—21no—
37#16837303C2211yescharacteristic
38#16837298C2211yescharacteristic
39#16837298C2211yescharacteristic
40#16837292—21no—
41#16837292—21no—
42#16837287—21no—
43#16837287—21no—
44#16837273—21no—
45#16837273—21no—
46C4⋊C4×Dic59C2211yescharacteristic
47C4⋊C4×Dic59C2211yescharacteristic
48#16837248C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#97×C59C2211yescharacteristic
54C59×#384C2211yescharacteristic
55C59×#382C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118—21no—
61C22×Dic118C2311yescharacteristic
62C22×D236C2311yescharacteristic
63C22×C4×D59—21no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273727—22no—
67#14273727C2×C414yescomplemented
68#14273715—22no—
69#14273715C2×C414yes—
70#14273692—22no—
71#14273692—22no—
72#14273689—22no—
73#14273689—22no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59C2311yescharacteristic
89C2×C4×Dic59C2311yescharacteristic
90#14273652C2×C414yescomplemented
91#14273652C2×C414yescomplemented
92#14273650—22no—
93#14273650—22no—
94#14273649C2×C414yes—
95#14273649C2×C414yes—
96#14273647—22no—
97#14273647—22no—
98#14273637—22no—
99#14273637—22no—
100C4×C59⋊C8—22no—
101C8×Dic59—22no—
102C8×Dic59—22no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95D411yescharacteristic
110C59⋊#95D411yescharacteristic
111C59⋊#95D411yescharacteristic
112C59⋊#95D411yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#97×C59C2×C414yescomplemented
123#95×C59—21no—
124#95×C59—21no—
125C23×D59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2411yescharacteristic
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59C2×D411yescharacteristic
137C22×Dic59C2×D411yescharacteristic
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C2×D236—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236C22×C412yes—
146C2×C4×D59—42no—
147C2×C4×D59—81no—
148C2×C4×D59—81no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—24no—
155C2×Dic118—81no—
156C2×Dic118—81no—
157C2×Dic118C22×C412yes—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C22×C412yes—
163C2×C8×C59C22×C412yes—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—81no—
173C4⋊C4×C59—81no—
174C42×C59C22×C412yes—
175C42×C59#4112yes—
176(C2×D59)⋊C4—24no—
177(C2×D59)⋊C4—24no—
178(C2×D59)⋊C4—42no—
179(C2×D59)⋊C4—81no—
180(C2×D59)⋊C4—81no—
181C4⋊Dic59—24no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59C22×C412yes—
185C4⋊Dic59C22×C412yes—
186Dic59⋊C4—42no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—42no—
192C4×Dic59—81no—
193C4×Dic59—81no—
194C4×Dic59#4112yes—
195C4×Dic59#4112yes—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8#4112yes—
201C2×C59⋊C8#4112yes—
202C23×C59C22×D411yescharacteristic
203C22×D59—22no—
204C22×D59—41no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
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×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59C4×D412yes—
227C2×C4×C59C4×D412yes—
228C2×C4×C59C23×C411yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—22no—
239C2×Dic59—22no—
240C2×Dic59—22no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59#12412yes—
254C2×Dic59#12412yes—
255D236—22no—
256D236—22no—
257D236—41no—
258D236—41no—
259C4×D59—84no—
260Dic118—22no—
261Dic118—22no—
262Dic118—41no—
263Dic118—41no—
264Dic118—84no—
265C8×C59—22no—
266C8×C59—22no—
267C59⋊C8—22no—
268C59⋊C8—22no—
269#21654—591nomaximal, Sylow
270C22×C59C2×C4×D411yescharacteristic
271C22×C59C2×#12411yescharacteristic
272C22×C59#53211yescharacteristic
273C22×C59#57611yescharacteristic
274C22×C59#57711yescharacteristic
275C22×C59#57711yescharacteristic
276C22×C59#57811yescharacteristic
277C2×D59—42no—
278C2×D59—42no—
279C2×D59—42no—
280C2×D59—81no—
281C2×D59—81no—
282C2×D59—81no—
283C2×D59—81no—
284C4×C59—22no—
285C4×C59—22no—
286C4×C59—81no—
287C4×C59—81no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59C4×#4111yescharacteristic
291C4×C59C4×#4111yescharacteristic
292C4×C59#52911yescharacteristic
293C4×C59#52911yescharacteristic
294Dic59—22no—
295Dic59—22no—
296Dic59—41no—
297Dic59—41no—
298Dic59—42no—
299Dic59—42no—
300Dic59—42no—
301C2×#489—591no—
302#2331—591no—
303#2326—591no—
304#1990—591no—
305#1958—591no—
306#1936—591no—
307#1935—591no—
308#1934—591no—
309#1933—591no—
310#1836D5911yescharacteristic, complemented
311#1763—591no—
312#1762—591no—
313C4×#103—591no—
314C4×#102—591no—
315#1726—591no—
316C2×C59C4×#12411yescharacteristic
317C2×C59#294611yescharacteristic
318C2×C59#294711yescharacteristic
319C2×C59#313611yescharacteristic
320C2×C59#313711yescharacteristic
321C2×C59#317311yescharacteristic
322C2×C59#317511yescharacteristic
323D59—82no—
324C2×C4⋊Q8—591no—
325C2×#124—591no—
326C2×C4×Q8—591no—
327C2×C4×D4—591no—
328#489—598no—
329C2×#103—591no—
330C2×#103—591no—
331C2×#103—591no—
332C2×#103—591no—
333C2×#102—591no—
334C2×#102—591no—
335C2×#102—591no—
336C2×#102—591no—
337C2×#97—591no—
338C2×#97—591no—
339C2×#97C2×D5911yescharacteristic
340C2×C4×C8—591no—
341C2×C4×C8—591no—
342C2×C4×C8—591no—
343#391—591no—
344#391—1181no—
345#391—1181no—
346#390—591no—
347#389—591no—
348#389—591no—
349#388—1181no—
350#388—1181no—
351#387—1181no—
352#387—1181no—
353#385—1181no—
354#385—1181no—
355#384C2×D5911yescharacteristic
356#382C2×D5911yescharacteristic
357C4×C4⋊C4—591no—
358C4×C4⋊C4—591no—
359#376—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340C2×D5911yescharacteristic
365#340C2×D5911yescharacteristic
366#340C2×D5911yescharacteristic
367#340C2×D5911yescharacteristic
368C59#2165411yescharacteristic, complemented, Sylow
369C22×Q8—591no—
370C22×Q8—1181no—
371C22×D4—591no—
372C23×C4—1181no—
373C22×C8—591no—
374C22×C8—591no—
375C22×C8C22×D5911yescharacteristic
376C22×C8C22×D5911yescharacteristic
377C4⋊Q8—594no—
378C4⋊Q8—1182no—
379#124—594no—
380#124—1182no—
381C4×Q8—1182no—
382C4×Q8—1182no—
383C4×D4—1182no—
384C4×D4—1182no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4C22×D5911yescharacteristic
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C22⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C42—591no—
403C2×C42—591no—
404C2×C42—1181no—
405C2×C42—1181no—
406C2×C42—1181no—
407C2×C42C22×D5911yescharacteristic
408#103—594no—
409#103—594no—
410#103—1182no—
411#103—1182no—
412#102—594no—
413#102—594no—
414#102—1182no—
415#102—1182no—
416#97—1182no—
417#97—1182no—
418#97C4×D5914yescomplemented
419C4×C8—1182no—
420C4×C8—1182no—
421C4×C8—1182no—
422#95—21no—
423#95—21no—
424#95—591no—
425#95—591no—
426#95—591no—
427#95—591no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434C24—1181no—
435C2×Q8—592no—
436C2×Q8—1184no—
437C2×Q8—1181no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C2×Q8—1181no—
442C2×Q8—4721no—
443C2×Q8—4721no—
444C2×D4—592no—
445C2×D4—1181no—
446C2×D4—1181no—
447C2×D4—1181no—
448C2×D4—1181no—
449C2×D4—1181no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—591no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—4721no—
462C22×C4—4721no—
463C22×C4—2362no—
464C22×C4D4×D5911yescharacteristic
465C22×C4D4×D5911yescharacteristic
466C22×C4C23×D5911yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—592no—
472C2×C8—592no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8C2×C4×D5912yes—
478C2×C8C2×C4×D5912yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—592no—
490C4⋊C4—592no—
491C4⋊C4—1184no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—4721no—
495C4⋊C4—4721no—
496C4⋊C4—2362no—
497C22⋊C4—1184no—
498C22⋊C4—1184no—
499C22⋊C4—4721no—
500C22⋊C4—4721no—
501C22⋊C4—2362no—
502C42—592no—
503C42—592no—
504C42—1181no—
505C42—1181no—
506C42—4721no—
507C42—4721no—
508C42—2362no—
509C42C2×C4×D5912yes—
510C42#1174568912yes—
511C23—1182no—
512C23—2361no—
513C23—2361no—
514C23—2361no—
515C23—2361no—
516C23—2361no—
517C23—2361no—
518C23C2×D4×D5911yescharacteristic
519Q8—1182no—
520Q8—1182no—
521Q8—4724no—
522Q8—2361no—
523Q8—2361no—
524D4—1182no—
525D4—1182no—
526D4—2361no—
527D4—2361no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—592no—
545C2×C4—592no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1182no—
557C2×C4—1182no—
558C2×C4—1182no—
559C2×C4—4724no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4#1427368912yes—
567C2×C4#1427368912yes—
568C2×C4C22×C4×D5911yescharacteristic
569C2×C4#41×D5911yescharacteristic
570C2×C4#41×D5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C2×C4#41×D5911yescharacteristic
573C2×C4C2×#1174568911yescharacteristic
574C8—22no—
575C8—22no—
576C8—1182no—
577C8—1182no—
578C22—4721no—
579C22—4721no—
580C22—4721no—
581C22—4721no—
582C22—2362no—
583C22—2362no—
584C22—2362no—
585C22C2×#1427368911yescharacteristic
586C22#124×D5911yescharacteristic
587C22#1683795111yescharacteristic
588C22#1683820611yescharacteristic
589C22#1683821111yescharacteristic
590C22#1683821211yescharacteristic
591C22#1683821811yescharacteristic
592C4—22no—
593C4—22no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—1182no—
599C4—1182no—
600C4—2361no—
601C4—2362no—
602C4—2362no—
603C4—2361no—
604C4—2362no—
605C4#1683787011yescharacteristic
606C4#1683787011yescharacteristic
607C4#1683793711yescharacteristic
608C4#1683793711yescharacteristic
609C2—4722no—
610C2#2142663511yescharacteristic
611C2#2143292611yescharacteristic
612C2#2143293711yescharacteristic
613C2#2143445111yescharacteristic
614C2#2143446211yescharacteristic
615C2#2143471211yescharacteristic
616C2#2143473411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1836C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1836D59
hypocenterC59#21654
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
2. #21426842C2semidirect
4. #21425056C2semidirect
7. #21424784C2semidirect
9. #21424770C2semidirect
10. #21423925C2semidirect
12. #21423915C2semidirect
14. #21423784C2semidirect
15. C59×#1836C2semidirect
20. #16837762C4semidirect
67. #14273727C2×C4semidirect
90. #14273652C2×C4semidirect
91. #14273652C2×C4semidirect
122. #97×C59C2×C4semidirect
310. #1836D59semidirect
368. C59#21654semidirect
418. #97C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377622
#214268422
#214268042
#214250562
#214249052
#214247912
#214247842
#214247772
#214247702
#214239252
#214239202
#214239152
#214239102
#214237842
C59×#18362
#2165459