Finite Group Databasebeta

← #28596754

Subgroups — #28596754

Order 15104.

886 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837760C211yesmaximal, characteristic
2#21426667C211yesmaximal, characteristic, complemented
3#21426643C211yesmaximal, characteristic
4#21425067C211yesmaximal, characteristic, complemented
5#21424911C211yesmaximal, characteristic
6#21424796C211yesmaximal, characteristic
7#21424786C211yesmaximal, characteristic, complemented
8#21424785C211yesmaximal, characteristic, complemented
9#21424228C211yesmaximal, characteristic, complemented
10#21423915C211yesmaximal, characteristic, complemented
11#21423915C211yesmaximal, characteristic, complemented
12#102×Dic59C211yesmaximal, characteristic
13#102×Dic59C211yesmaximal, characteristic
14#97×Dic59C211yesmaximal, characteristic, complemented
15C59×#1936C211yesmaximal, characteristic
16C2×#14273726C2211yescharacteristic
17C2×#14273714C2211yescharacteristic
18C2×#14273692—21no—
19C2×#14273692C2211yescharacteristic
20C2×D4×Dic59C2211yescharacteristic
21#16837760C418yescomplemented
22C2×#14273652C2211yescharacteristic
23C2×#14273652C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30C2×#14273637C2211yescharacteristic
31C2×C8×Dic59C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837306—21no—
35#16837300C2211yescharacteristic
36#16837297—21no—
37#16837290—21no—
38#16837290—21no—
39#16837271C2211yescharacteristic
40#16837267C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43#16837253—21no—
44C22⋊C4×Dic59—21no—
45C22⋊C4×Dic59—21no—
46C42×Dic59C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#102×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C59×#390C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118—21no—
61C22×Dic118C2311yescharacteristic
62C23×Dic59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273726—22no—
66#14273726C2×C414yescomplemented
67#14273714—22no—
68#14273714C2×C414yes—
69#14273692—22no—
70#14273692—22no—
71#14273692—44no—
72D4×Dic59—22no—
73D4×Dic59—22no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59D411yescharacteristic
89C2×C4×Dic59D411yescharacteristic
90C2×C4×Dic59D411yescharacteristic
91C2×C4×Dic59D411yescharacteristic
92C2×C4×Dic59C2311yescharacteristic
93C2×C4×Dic59C2311yescharacteristic
94#14273652—22no—
95#14273652C2×C414yescomplemented
96#14273650—22no—
97#14273650C2×C414yescomplemented
98#14273648—22no—
99#14273648C2×C414yes—
100#14273638—22no—
101#14273637—22no—
102#14273637C2×C414yescomplemented
103C8×Dic59—22no—
104C8×Dic59—22no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C22×D4×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C42×C59C2311yescharacteristic
122#102×C59—22no—
123#102×C59C2×C414yes—
124#97×C59—22no—
125#95×C59—21no—
126C24×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59C22×C412yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—42no—
144C22×Dic59—42no—
145C22×Dic59—42no—
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—24no—
156C2×Dic118—42no—
157C2×Dic118—42no—
158C2×Dic118C22×C412yes—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59C22×C412yes—
164C2×C8×C59#4112yes—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59C22×C412yes—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—42no—
174C42×C59—21no—
175C42×C59—21no—
176C22⋊Dic59—24no—
177C22⋊Dic59—24no—
178C22⋊Dic59—42no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59C22×C412yes—
185Dic59⋊C4—24no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190C4×Dic59—22no—
191C4×Dic59—22no—
192C4×Dic59—41no—
193C4×Dic59—41no—
194C4×Dic59—42no—
195C4×Dic59—42no—
196C4×Dic59—42no—
197C4×Dic59C22×C412yes—
198C4×Dic59#4112yes—
199C4×Dic59#4112yes—
200C4×Dic59#4112yes—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8C22×C412yes—
206C2×C59⋊C8#4112yes—
207C23×C59—22no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59C22×D411yescharacteristic
215D4×C59—22no—
216D4×C59—22no—
217D4×C59—41no—
218D4×C59—41no—
219C2×C4×C59—22no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59C23×C411yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—22no—
247C2×Dic59—22no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—84no—
259C2×Dic59C4×D412yes—
260C2×Dic59C4×D412yes—
261C2×Dic59#12412yes—
262C2×Dic59#12412yes—
263C2×Dic59AGL(1,ℤ/8)12yes—
264C2×Dic59#13712yes—
265Dic118—22no—
266Dic118—22no—
267Dic118—48no—
268Dic118—41no—
269Dic118—41no—
270C8×C59—22no—
271C8×C59—22no—
272C59⋊C8—22no—
273C59⋊C8—22no—
274C4×#489—591nomaximal, Sylow
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59—81no—
279C22×C59—81no—
280C22×C59—81no—
281C22×C59—81no—
282C22×C59C2×C4×D411yescharacteristic
283C22×C59C2×#12411yescharacteristic
284C22×C59#53211yescharacteristic
285C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C59C2×#13711yescharacteristic
287C22×C59#57511yescharacteristic
288C22×C59#57511yescharacteristic
289C4×C59—41no—
290C4×C59—41no—
291C4×C59—42no—
292C4×C59—81no—
293C4×C59—81no—
294C4×C59C4×#4111yescharacteristic
295C4×C59C4×#4111yescharacteristic
296C4×C59#52911yescharacteristic
297C4×C59#52911yescharacteristic
298Dic59—22no—
299Dic59—22no—
300Dic59—22no—
301Dic59—42no—
302Dic59—42no—
303Dic59—81no—
304Dic59—81no—
305Dic59#48914yescomplemented
306C2×#489—591no—
307C4×C4⋊Q8—591no—
308C4×#124—591no—
309#1993—591no—
310#1959—591no—
311#1936—591no—
312#1936D5911yescharacteristic
313#1935—591no—
314#1935—591no—
315#1836—591no—
316C4×#103—591no—
317C4×#103—591no—
318C4×#102—591no—
319C4×#102—591no—
320C4×#97—591no—
321C2×C59—82no—
322C2×C59C4×#12411yescharacteristic
323C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
324C2×C59C4×#13711yescharacteristic
325C2×C59C2×#48911yescharacteristic
326C2×C59#313511yescharacteristic
327C2×C59#317111yescharacteristic
328C2×C59#317211yescharacteristic
329C2×C4⋊Q8—591no—
330C2×#124—591no—
331C2×C4×Q8—591no—
332C2×C4×Q8—1181no—
333C2×C4×D4—591no—
334#489—598no—
335C2×#103—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#102—591no—
340C2×#102—591no—
341C2×#102C2×D5911yescharacteristic
342C2×#102C2×D5911yescharacteristic
343C2×#97—591no—
344C2×#97—591no—
345C2×#97—591no—
346C2×#97C2×D5911yescharacteristic
347C2×C4×C8—591no—
348C2×C4×C8—591no—
349#391—591no—
350#391—1181no—
351#390C2×D5911yescharacteristic
352#389—591no—
353#389—1181no—
354#389C2×D5911yescharacteristic
355#388—1181no—
356#388—1181no—
357#384—591no—
358#382—591no—
359C4×C4⋊C4—591no—
360C4×C4⋊C4—591no—
361C4×C4⋊C4—1181no—
362C4×C22⋊C4—1181no—
363C4×C22⋊C4—1181no—
364C43—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340C2×D5911yescharacteristic
372#340C2×D5911yescharacteristic
373C59C4×#48911yescharacteristic, complemented, Sylow
374C22×Q8—591no—
375C22×Q8—1181no—
376C22×D4C22×D5911yescharacteristic
377C23×C4—1181no—
378C22×C8—591no—
379C22×C8—591no—
380C22×C8C22×D5911yescharacteristic
381C22×C8C22×D5911yescharacteristic
382C4⋊Q8—594no—
383C4⋊Q8—1182no—
384#124—594no—
385#124—1182no—
386C4×Q8—1182no—
387C4×Q8—1182no—
388C4×Q8—2364no—
389C4×D4—1182no—
390C4×D4—1182no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1181no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C42—591no—
409C2×C42—591no—
410C2×C42—591no—
411C2×C42—591no—
412C2×C42—591no—
413C2×C42—591no—
414C2×C42—1181no—
415C2×C42—1181no—
416C2×C42—1181no—
417C2×C42C22×D5911yescharacteristic
418#103—594no—
419#103—594no—
420#103—1182no—
421#103—1182no—
422#102—22no—
423#102—594no—
424#102—1182no—
425#102C4×D5914yes—
426#97—22no—
427#97—594no—
428#97—1182no—
429#97—1182no—
430C4×C8—1182no—
431C4×C8—1182no—
432#95—21no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440C24—21no—
441C2×Q8—592no—
442C2×Q8—1184no—
443C2×Q8—1181no—
444C2×Q8—1181no—
445C2×Q8—1181no—
446C2×Q8—1181no—
447C2×Q8—1181no—
448C2×Q8—2362no—
449C2×Q8—2362no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4C2×C4×D5912yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—591no—
460C22×C4—591no—
461C22×C4—591no—
462C22×C4—591no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—1181no—
466C22×C4—1181no—
467C22×C4—2362no—
468C22×C4—2362no—
469C22×C4—2362no—
470C22×C4D4×D5911yescharacteristic
471C22×C4D4×D5911yescharacteristic
472C22×C4C23×D5911yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—592no—
478C2×C8—592no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8C2×C4×D5912yes—
484C2×C8#1174568712yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—592no—
492C4⋊C4—1184no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C4⋊C4C2×C4×D5912yes—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—1184no—
506C22⋊C4—1184no—
507C22⋊C4—2362no—
508C42—21no—
509C42—21no—
510C42—592no—
511C42—592no—
512C42—592no—
513C42—592no—
514C42—1182no—
515C42—1182no—
516C42—2361no—
517C42—2361no—
518C42—2362no—
519C42—2362no—
520C42—2362no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C2×D4×D5911yescharacteristic
529Q8—1182no—
530Q8—1182no—
531Q8—2368no—
532Q8—2361no—
533Q8—2361no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—592no—
554C2×C4—592no—
555C2×C4—592no—
556C2×C4—592no—
557C2×C4—592no—
558C2×C4—592no—
559C2×C4—1181no—
560C2×C4—1181no—
561C2×C4—1181no—
562C2×C4—1181no—
563C2×C4—1181no—
564C2×C4—1181no—
565C2×C4—1182no—
566C2×C4—1182no—
567C2×C4—4724no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4C22×C4×D5911yescharacteristic
579C2×C4#41×D5911yescharacteristic
580C2×C4#41×D5911yescharacteristic
581C2×C4#41×D5911yescharacteristic
582C2×C4#41×D5911yescharacteristic
583C2×C4C2×#1174568711yescharacteristic
584C8—22no—
585C8—22no—
586C8—1182no—
587C8—1182no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D5911yescharacteristic
596C22#124×D5911yescharacteristic
597C22#1683794711yescharacteristic
598C22AGL(1,ℤ/8)×D5911yescharacteristic
599C22#137×D5911yescharacteristic
600C22#1683819811yescharacteristic
601C22#1683819811yescharacteristic
602C4—41no—
603C4—41no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—594no—
608C4—1182no—
609C4—1182no—
610C4—1182no—
611C4—4721no—
612C4—4721no—
613C4—2362no—
614C4—2362no—
615C4C4×#1174568711yescharacteristic
616C4#1683787011yescharacteristic
617C4#1683792911yescharacteristic
618C4#1683793511yescharacteristic
619C2—82no—
620C2#2142663711yescharacteristic
621C2#2143290711yescharacteristic
622C2#2143291811yescharacteristic
623C2#489×D5911yescharacteristic
624C2#2143444011yescharacteristic
625C2#2143468611yescharacteristic
626C2#2143469511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1936C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1936D59
hypocenterC59C4×#489
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #21426667C2semidirect
4. #21425067C2semidirect
7. #21424786C2semidirect
8. #21424785C2semidirect
9. #21424228C2semidirect
10. #21423915C2semidirect
11. #21423915C2semidirect
14. #97×Dic59C2semidirect
21. #16837760C4semidirect
66. #14273726C2×C4semidirect
95. #14273652C2×C4semidirect
97. #14273650C2×C4semidirect
102. #14273637C2×C4semidirect
305. Dic59#489semidirect
373. C59C4×#489semidirect

Maximal subgroups

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

subgroupindex
C2×#168377602
#214266672
#214266432
#214250672
#214249112
#214247962
#214247862
#214247852
#214242282
#214239152
#102×Dic592
#97×Dic592
C59×#19362
C4×#48959