Finite Group Databasebeta

← #28638779

Subgroups — #28638779

Order 15104.

824 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837761C211yesmaximal, characteristic
2C2×#16837621C211yesmaximal, characteristic
3#21429311C211yesmaximal, characteristic, complemented
4#21425514C211yesmaximal, characteristic, complemented
5#21425509C211yesmaximal, characteristic
6#21425418C211yesmaximal, characteristic, complemented
7#21425413C211yesmaximal, characteristic
8#21425044C211yesmaximal, characteristic, complemented
9#21424820C211yesmaximal, characteristic, complemented
10#21424437C211yesmaximal, characteristic, complemented
11#21424430C211yesmaximal, characteristic
12#21424111C211yesmaximal, characteristic, complemented
13#21424106C211yesmaximal, characteristic
14#21423929C211yesmaximal, characteristic, complemented
15C59×#2367C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C22×#11745670C2211yescharacteristic
18C2×#14273727C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837761—24no—
21#16837621—24no—
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×#14273638C2211yescharacteristic
34#16837342—21no—
35#16837342—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837307—21no—
39#16837307—21no—
40#16837268C2211yescharacteristic
41#16837268C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#124×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×C4×D4×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#388—21no—
56C59×#388—21no—
57C59×#385—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C22⋊C4×C59—21no—
61C22×Dic118—21no—
62C22×Dic118C2311yescharacteristic
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672—41no—
66C2×#11745670—22no—
67C2×#11745670—22no—
68C2×#11745670—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273727—22no—
74#14273727—22no—
75C4×Dic118—22no—
76C4×Dic118—22no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×C4×Dic59—21no—
83C4⋊(C59⋊C8)—22no—
84C4⋊(C59⋊C8)—41no—
85C4⋊(C59⋊C8)—41no—
86#14273651—22no—
87#14273651—22no—
88#14273651—22no—
89#14273650—22no—
90#14273650—22no—
91#14273648—22no—
92#14273648—22no—
93#14273648—22no—
94#14273638—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59C2311yescharacteristic
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59—21no—
106#124×C59—22no—
107#124×C59D414yes—
108C4×Q8×C59—22no—
109C4×Q8×C59—41no—
110C4×D4×C59—22no—
111C4×D4×C59—41no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59D411yescharacteristic
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C42×C59—21no—
123C2×C42×C59—21no—
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126C2×C42×C59C2311yescharacteristic
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59C2×D412yes—
139C2×D4×C59—21no—
140C2×D4×C59—21no—
141C2×D4×C59—21no—
142C2×D4×C59—21no—
143C2×D4×C59—21no—
144C2×D4×C59C2×D412yes—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—42no—
149C22×C4×C59—42no—
150C22×C4×C59—81no—
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2411yescharacteristic
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—42no—
168C2×Dic118—42no—
169C2×Dic118—42no—
170C2×Dic118#4112yes—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—41no—
175C4⋊C4×C59—41no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59#4112yes—
178C22⋊C4×C59—24no—
179C22⋊C4×C59—24no—
180C22⋊C4×C59—42no—
181C22⋊C4×C59—42no—
182C22⋊C4×C59—81no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—42no—
188C42×C59—81no—
189C42×C59C2×D412yes—
190C42×C59#4112yes—
191#11745672—44no—
192#11745670—44no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—42no—
200Dic59⋊C4—42no—
201C4×Dic59—42no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C23×C59—22no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59C22×D411yescharacteristic
222Q8×C59—22no—
223Q8×C59—22no—
224Q8×C59—41no—
225Q8×C59—41no—
226D4×C59—22no—
227D4×C59—22no—
228D4×C59—41no—
229D4×C59—41no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—22no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—84no—
256C2×C4×C59#12112yes—
257C2×C4×C59#12112yes—
258C2×C4×C59C22×D411yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59H(5,𝔽2)11yescharacteristic
263C2×C4×C592−1+411yescharacteristic
264C2×Dic59—22no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276Dic118—22no—
277Dic118—22no—
278Dic118—41no—
279Dic118—41no—
280Dic118—84no—
281C59⋊C8—41no—
282C59⋊C8—41no—
283C59⋊C8—41no—
284C59⋊C8—41no—
285#25337—591nomaximal, Sylow
286C22×C59—42no—
287C22×C59—42no—
288C22×C59—42no—
289C22×C59—81no—
290C22×C59—81no—
291C22×C59—81no—
292C22×C59—81no—
293C22×C59C2×#12111yescharacteristic
294C22×C59#54611yescharacteristic
295C22×C59#54811yescharacteristic
296C22×C59#57511yescharacteristic
297C22×C59#57511yescharacteristic
298C22×C59#57711yescharacteristic
299C22×C59#57811yescharacteristic
300C4×C59—22no—
301C4×C59—22no—
302C4×C59—42no—
303C4×C59—41no—
304C4×C59—41no—
305C4×C59—42no—
306C4×C59—41no—
307C4×C59—41no—
308C4×C59—81no—
309C4×C59#52911yescharacteristic
310C4×C59#54211yescharacteristic
311C4×C59#54611yescharacteristic
312C4×C59#54711yescharacteristic
313Dic59—42no—
314Dic59—81no—
315Dic59—81no—
316Dic59—81no—
317C2×#489—591no—
318C2×#464—591no—
319#2568—591no—
320#2367D5911yescharacteristic
321#2074—591no—
322#2073—591no—
323#2056—591no—
324#2055—591no—
325#1987—591no—
326#1940—591no—
327#1878—591no—
328#1877—591no—
329#1805—591no—
330#1804—591no—
331#1763—591no—
332C2×C59—82no—
333C2×C59#242711yescharacteristic
334C2×C59#304011yescharacteristic
335C2×C59#304611yescharacteristic
336C2×C59#331111yescharacteristic
337C2×C59#331311yescharacteristic
338C2×C59#337111yescharacteristic
339C2×C59#337711yescharacteristic
340C22×Q16—591no—
341C22×SD16—591no—
342C2×C4⋊Q8—591no—
343C2×#124C2×D5911yescharacteristic
344C2×C4×Q8—591no—
345C2×C4×Q8C2×D5911yescharacteristic
346C2×C4×D4C2×D5911yescharacteristic
347#489—1184no—
348#464—1184no—
349C2×C4⋊C8—591no—
350C2×C4⋊C8—591no—
351C2×C4⋊C8—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×#97—591no—
361#400—1181no—
362#400—1181no—
363#391—591no—
364#391—591no—
365#391—1181no—
366#391—1181no—
367#391C2×D5911yescharacteristic
368#390C2×D5911yescharacteristic
369#388—21no—
370#388—21no—
371#385—21no—
372#382—591no—
373#382—591no—
374C4×C4⋊C4C2×D5911yescharacteristic
375C4×C4⋊C4C2×D5911yescharacteristic
376C4×C22⋊C4—21no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#336—591no—
384#336—591no—
385C59#2533711yescharacteristic, complemented, Sylow
386C22×Q8—591no—
387C22×Q8—1181no—
388C22×Q8C22×D5911yescharacteristic
389C22×D4C22×D5911yescharacteristic
390C23×C4—21no—
391C2×Q16—1182no—
392C2×Q16—1182no—
393C2×Q16—2361no—
394C2×SD16—1182no—
395C2×SD16—1182no—
396C2×SD16—2361no—
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8—591no—
401C4⋊Q8—1182no—
402C4⋊Q8—1182no—
403#124—22no—
404#124D23614yes—
405C4×Q8—22no—
406C4×Q8—41no—
407C4×Q8—1182no—
408C4×Q8—1182no—
409C4×D4—22no—
410C4×D4—41no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1181no—
417C2×C4⋊C4C22⋊D5911yescharacteristic
418C2×C4⋊C4C22⋊D5911yescharacteristic
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C42—21no—
427C2×C42—21no—
428C2×C42—1181no—
429C2×C42C22×D5911yescharacteristic
430C2×C42C22×D5911yescharacteristic
431C2×C42C22×D5911yescharacteristic
432C4⋊C8—1182no—
433C4⋊C8—2361no—
434C4⋊C8—2361no—
435#103—1182no—
436#103—1182no—
437#103—1182no—
438#103—1182no—
439#103—1182no—
440#102—1182no—
441#102—1182no—
442#102—1182no—
443#97—1182no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95C22⋊D5911yescharacteristic
456#95C22⋊D5911yescharacteristic
457C24—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—592no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C2×Q8—1181no—
467C2×Q8—1181no—
468C2×Q8—1181no—
469C2×Q8—2362no—
470C2×Q8—2362no—
471C2×Q8—2362no—
472C2×Q8C2×D23612yes—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C2×D23612yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C23×D5911yescharacteristic
496Q16—2364no—
497SD16—2364no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—42no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—2362no—
523C4⋊C4—2362no—
524C4⋊C4#1174568712yes—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—81no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—81no—
536C42—2362no—
537C42C2×D23612yes—
538C42#1174568512yes—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C22×C22⋊D5911yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—1182no—
552Q8—1182no—
553Q8—4724no—
554Q8—2361no—
555Q8—2361no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—1182no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4#1427370112yes—
599C2×C4#1427370112yes—
600C2×C4C22×D23611yescharacteristic
601C2×C4#41×D5911yescharacteristic
602C2×C4C2×#1174568711yescharacteristic
603C2×C4C2×#1174568511yescharacteristic
604C2×C4#1427377211yescharacteristic
605C2×C4#1427377411yescharacteristic
606C8—2361no—
607C8—2361no—
608C8—2361no—
609C8—2361no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#1427370111yescharacteristic
618C22#1683804611yescharacteristic
619C22#1683805911yescharacteristic
620C22#1683820011yescharacteristic
621C22#1683820011yescharacteristic
622C22#1683821311yescharacteristic
623C22#1683821711yescharacteristic
624C4—22no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—4721no—
634C4—4721no—
635C4—4721no—
636C4—2362no—
637C4#1683793411yescharacteristic
638C4#1683801711yescharacteristic
639C4#1683804711yescharacteristic
640C4#1683805611yescharacteristic
641C2—82no—
642C2#2142766711yescharacteristic
643C2#2143359011yescharacteristic
644C2#2143364611yescharacteristic
645C2#2143613211yescharacteristic
646C2#2143616211yescharacteristic
647C2#2143685811yescharacteristic
648C2#2143693211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2367C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2367D59
hypocenterC59#25337
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
3. #21429311C2semidirect
4. #21425514C2semidirect
6. #21425418C2semidirect
8. #21425044C2semidirect
9. #21424820C2semidirect
10. #21424437C2semidirect
12. #21424111C2semidirect
14. #21423929C2semidirect
385. C59#25337semidirect

Maximal subgroups

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

subgroupindex
C2×#168377612
C2×#168376212
#214293112
#214255142
#214255092
#214254182
#214254132
#214250442
#214248202
#214244372
#214244302
#214241112
#214241062
#214239292
C59×#23672
#2533759