Finite Group Databasebeta

← #28591752

Subgroups — #28591752

Order 15104.

922 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×#16837504C211yesmaximal, characteristic
2#21426560C211yesmaximal, characteristic
3C42×Dic118C211yesmaximal, characteristic, complemented
4#21424724C211yesmaximal, characteristic
5#21424717C211yesmaximal, characteristic
6#21424689C211yesmaximal, characteristic, complemented
7#21424668C211yesmaximal, characteristic, complemented
8#21424450C211yesmaximal, characteristic
9#21424450C211yesmaximal, characteristic
10#21424367C211yesmaximal, characteristic, complemented
11#21424353C211yesmaximal, characteristic, complemented
12#21423993C211yesmaximal, characteristic, complemented
13#21423927C211yesmaximal, characteristic, complemented
14#21423922C211yesmaximal, characteristic
15C59×#1745C211yesmaximal, characteristic, complemented
16C22×#11745660C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20C2×C4×D236C2211yescharacteristic
21C2×#14273684—21no—
22#16837504C418yescomplemented
23C2×#14273661C2211yescharacteristic
24C2×#14273661C2211yescharacteristic
25C2×#14273659C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28C2×#14273652C2211yescharacteristic
29C2×#14273649C2211yescharacteristic
30C2×#14273649C2211yescharacteristic
31C2×#14273649C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837304C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837293—21no—
37#16837293—21no—
38C4×Dic59⋊C4—21no—
39C4×Dic59⋊C4—21no—
40C4×Dic59⋊C4—21no—
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45C4×(C2×D59)⋊C4—21no—
46C42×Dic59—21no—
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C2×#97×C59C2211yescharacteristic
54C2×#97×C59C2211yescharacteristic
55C43×C59C2211yescharacteristic
56C59×#336C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C2×#11745660—22no—
64C2×#11745660—22no—
65C2×#11745660C2×C414yes—
66C4×Dic118—22no—
67C4×Dic118—22no—
68C4×Dic118—22no—
69C4×Dic118—22no—
70C4×Dic118—22no—
71C4×Dic118C2×C414yes—
72C4×D236—22no—
73C4×D236C2×C414yes—
74#14273684—44no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4—21no—
88C2×(C2×D59)⋊C4—21no—
89C2×(C2×D59)⋊C4—21no—
90C2×(C2×D59)⋊C4—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59—21no—
95C2×C4×Dic59—21no—
96C2×C4×Dic59—21no—
97C2×C4×Dic59—21no—
98#14273661—22no—
99#14273661C2×C414yes—
100#14273659—22no—
101#14273652—22no—
102#14273652—22no—
103#14273652C2×C414yes—
104#14273649—22no—
105#14273649—22no—
106#14273649C2×C414yes—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C22×C8×C59C2311yescharacteristic
117C2×C42×C59D411yescharacteristic
118C2×C42×C59D411yescharacteristic
119C2×C42×C59D411yescharacteristic
120C2×C42×C59D411yescharacteristic
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#97×C59—22no—
125#97×C59C2×C414yes—
126C23×D59—21no—
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2411yescharacteristic
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236C22×C412yes—
147C2×C4×D59—24no—
148C2×C4×D59—42no—
149C2×C4×D59—42no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118C22×C412yes—
154C2×Dic118#4112yes—
155C2×Dic118#4112yes—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59C22×C412yes—
167C2×C8×C59C22×C412yes—
168C42×C59—22no—
169C42×C59—22no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—41no—
175C42×C59—41no—
176C42×C59C22×C412yes—
177C42×C59#4112yes—
178(C2×D59)⋊C4—24no—
179(C2×D59)⋊C4—42no—
180(C2×D59)⋊C4—42no—
181(C2×D59)⋊C4—42no—
182(C2×D59)⋊C4—42no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59C22×C412yes—
192C4⋊Dic59C22×C412yes—
193C4⋊Dic59#4112yes—
194C4⋊Dic59#4112yes—
195Dic59⋊C4—24no—
196Dic59⋊C4—24no—
197Dic59⋊C4—24no—
198Dic59⋊C4—42no—
199Dic59⋊C4—42no—
200C4×Dic59—24no—
201C4×Dic59—24no—
202C4×Dic59—24no—
203C4×Dic59—42no—
204C4×Dic59—42no—
205C4×Dic59—42no—
206C4×Dic59—42no—
207#11745660—28no—
208C23×C59C22×D411yescharacteristic
209C22×D59—22no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—41no—
214C22×D59—41no—
215C22×D59—41no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
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—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59C4×D412yes—
231C2×C4×C59C4×D412yes—
232C2×C4×C59AGL(1,ℤ/8)12yes—
233C2×C4×C59#13712yes—
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—22no—
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—22no—
246C2×Dic59—22no—
247C2×Dic59—41no—
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—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261D236—22no—
262D236—22no—
263D236—41no—
264D236—41no—
265C4×D59—48no—
266Dic118—22no—
267Dic118—22no—
268Dic118—22no—
269Dic118#11714yescomplemented
270C8×C59—22no—
271C8×C59—22no—
272C8×C59—41no—
273C8×C59—41no—
274#21124—591nomaximal, Sylow
275C22×C59C2×C4×D411yescharacteristic
276C22×C59#52511yescharacteristic
277C22×C59#52511yescharacteristic
278C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C59C2×#13711yescharacteristic
280C22×C59#57511yescharacteristic
281C22×C59#57511yescharacteristic
282C2×D59—42no—
283C2×D59—42no—
284C2×D59—42no—
285C2×D59—81no—
286C2×D59—81no—
287C2×D59—81no—
288C2×D59—81no—
289C4×C59—22no—
290C4×C59—22no—
291C4×C59—22no—
292C4×C59—22no—
293C4×C59—41no—
294C4×C59—41no—
295C4×C59C2×#11711yescharacteristic
296C4×C59C4×#4111yescharacteristic
297C4×C59#52911yescharacteristic
298C4×C59#52911yescharacteristic
299Dic59—24no—
300Dic59—42no—
301Dic59—42no—
302Dic59—42no—
303Dic59—42no—
304Dic59—81no—
305Dic59—81no—
306C2×#440—591no—
307#2293—591no—
308C42×Q8—591no—
309#1926—591no—
310#1925—591no—
311#1921—591no—
312#1918—591no—
313#1880—591no—
314#1880—591no—
315#1865—591no—
316#1863—591no—
317#1776—591no—
318#1763—591no—
319#1762—591no—
320#1745D5911yescharacteristic, complemented
321C2×C59#229011yescharacteristic
322C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
323C2×C59C4×#13711yescharacteristic
324C2×C59#311111yescharacteristic
325C2×C59#311211yescharacteristic
326C2×C59#311511yescharacteristic
327C2×C59#311711yescharacteristic
328D59—82no—
329C22×SD16—591no—
330C2×C4×Q8—591no—
331C2×C4×Q8—591no—
332C2×C4×Q8—591no—
333C2×C4×D4—591no—
334C2×#117—1181no—
335#440—598no—
336C2×#107—591no—
337C2×#107—591no—
338C2×#106—591no—
339C2×#103—591no—
340C2×#103—591no—
341C2×#103—591no—
342C2×#102—591no—
343C2×#102—591no—
344C2×#102—591no—
345C2×#97C2×D5911yescharacteristic
346C2×#97C2×D5911yescharacteristic
347#390—591no—
348#390—591no—
349#389—591no—
350#389—591no—
351#388—1181no—
352#388—1181no—
353C4×C4⋊C4—591no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357C4×C4⋊C4—1181no—
358C4×C4⋊C4—1181no—
359C4×C4⋊C4—1181no—
360C4×C22⋊C4—1181no—
361C43—1181no—
362C43C2×D5911yescharacteristic
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#336C2×D5911yescharacteristic
370#336C2×D5911yescharacteristic
371#336C2×D5911yescharacteristic
372#336C2×D5911yescharacteristic
373C59#2112411yescharacteristic, complemented, Sylow
374C22×Q8—591no—
375C22×D4—591no—
376C23×C4—1181no—
377C2×SD16—594no—
378C2×SD16—1182no—
379C2×SD16—1182no—
380C22×C8C22×D5911yescharacteristic
381C22×C8C22×D5911yescharacteristic
382C22×C8C22×D5911yescharacteristic
383C22×C8C22×D5911yescharacteristic
384C4×Q8—594no—
385C4×Q8—1182no—
386C4×Q8—1182no—
387C4×Q8—1182no—
388C4×Q8—1182no—
389C4×Q8—1182no—
390C4×D4—594no—
391C4×D4—1182no—
392#117—2364no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C22⋊C4—1181no—
405C2×C22⋊C4—1181no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C22⋊C4—1181no—
409C2×C42—1181no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42—1181no—
413C2×C42—1181no—
414C2×C42—1181no—
415C2×C42—1181no—
416C2×C42D23611yescharacteristic
417C2×C42D23611yescharacteristic
418C2×C42D23611yescharacteristic
419C2×C42D23611yescharacteristic
420C2×C42C22×D5911yescharacteristic
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423#107—594no—
424#107—1182no—
425#106—1182no—
426#103—594no—
427#103—1182no—
428#103—1182no—
429#102—594no—
430#102—1182no—
431#102—1182no—
432#97—22no—
433#97C4×D5914yes—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440C24—1181no—
441C2×Q8—592no—
442C2×Q8—592no—
443C2×Q8—592no—
444C2×Q8—1181no—
445C2×Q8—1181no—
446C2×Q8—1181no—
447C2×D4—592no—
448C2×D4—1181no—
449C2×D4—1181no—
450C2×D4—1181no—
451C2×D4—1181no—
452C2×D4—1181no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1184no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4C2×D23611yescharacteristic
464C22×C4C2×D23611yescharacteristic
465C22×C4C2×D23611yescharacteristic
466C22×C4C2×D23611yescharacteristic
467C22×C4C2×D23611yescharacteristic
468C22×C4C2×D23611yescharacteristic
469C22×C4C23×D5911yescharacteristic
470SD16—1188no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8C2×C4×D5912yes—
482C2×C8C2×C4×D5912yes—
483C4⋊C4—592no—
484C4⋊C4—592no—
485C4⋊C4—592no—
486C4⋊C4—592no—
487C4⋊C4—1184no—
488C4⋊C4—1184no—
489C4⋊C4—1184no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C22⋊C4—1184no—
501C22⋊C4—2362no—
502C22⋊C4—2362no—
503C22⋊C4—2362no—
504C22⋊C4—2362no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—41no—
512C42—41no—
513C42—1184no—
514C42—1184no—
515C42—1184no—
516C42—2362no—
517C42—2362no—
518C42—2362no—
519C42—2362no—
520C42C2×C4×D5912yes—
521C42#1174568512yes—
522C23—1182no—
523C23—2361no—
524C23—2361no—
525C23—2361no—
526C23—2361no—
527C23—2361no—
528C23—2361no—
529C23C22×D23611yescharacteristic
530Q8—594no—
531Q8—1182no—
532Q8—1182no—
533Q8—1182no—
534D4—1182no—
535D4—1182no—
536D4—2361no—
537D4—2361no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—1182no—
553C2×C4—1182no—
554C2×C4—1182no—
555C2×C4—1182no—
556C2×C4—1182no—
557C2×C4—1182no—
558C2×C4—1182no—
559C2×C4—2368no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4C4×D23612yes—
575C2×C4C4×D23612yes—
576C2×C4#1427375512yes—
577C2×C4#1427376012yes—
578C2×C4C22×C4×D5911yescharacteristic
579C2×C4#41×D5911yescharacteristic
580C2×C4#41×D5911yescharacteristic
581C2×C4#41×D5911yescharacteristic
582C2×C4#41×D5911yescharacteristic
583C2×C4C2×#1174568511yescharacteristic
584C8—22no—
585C8—22no—
586C8—41no—
587C8—41no—
588C22—4721no—
589C22—4721no—
590C22—4721no—
591C22—4721no—
592C22—2362no—
593C22—2362no—
594C22—2362no—
595C22C2×C4×D23611yescharacteristic
596C22#1683791411yescharacteristic
597C22#1683791411yescharacteristic
598C22C2×#1427375511yescharacteristic
599C22C2×#1427376011yescharacteristic
600C22#1683820211yescharacteristic
601C22#1683820211yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—1184no—
609C4—4721no—
610C4—4721no—
611C4—2362no—
612C4—2362no—
613C4—2362no—
614C4—2362no—
615C4#117×D5911yescharacteristic
616C4#1683787011yescharacteristic
617C4#1683793011yescharacteristic
618C4#1683793011yescharacteristic
619C2—4722no—
620C2#2142654011yescharacteristic
621C2C4×#1427375511yescharacteristic
622C2C4×#1427376011yescharacteristic
623C2#2143420711yescharacteristic
624C2#2143421911yescharacteristic
625C2#2143424811yescharacteristic
626C2#2143426911yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1745C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1745D59
hypocenterC59#21124
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
3. C42×Dic118C2semidirect
6. #21424689C2semidirect
7. #21424668C2semidirect
10. #21424367C2semidirect
11. #21424353C2semidirect
12. #21423993C2semidirect
13. #21423927C2semidirect
15. C59×#1745C2semidirect
22. #16837504C4semidirect
269. Dic118#117semidirect
320. #1745D59semidirect
373. C59#21124semidirect

Maximal subgroups

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

subgroupindex
C2×#168375042
#214265602
C42×Dic1182
#214247242
#214247172
#214246892
#214246682
#214244502
#214243672
#214243532
#214239932
#214239272
#214239222
C59×#17452
#2112459