Finite Group Databasebeta

← #28591743

Subgroups — #28591743

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×#16837505C211yesmaximal, characteristic
2#21426554C211yesmaximal, characteristic
3C4×Q8×Dic59C211yesmaximal, characteristic, complemented
4#21424722C211yesmaximal, characteristic
5#21424717C211yesmaximal, characteristic
6#21424691C211yesmaximal, characteristic, complemented
7#21424664C211yesmaximal, characteristic, complemented
8#21424451C211yesmaximal, characteristic
9#21424449C211yesmaximal, characteristic
10#21424352C211yesmaximal, characteristic, complemented
11#21423996C211yesmaximal, characteristic, complemented
12#21423928C211yesmaximal, characteristic, complemented
13#21423920C211yesmaximal, characteristic
14#21423843C211yesmaximal, characteristic, complemented
15C59×#1865C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C2×Q8×Dic59C2211yescharacteristic
19C2×#14273689C2211yescharacteristic
20C2×#14273685—21no—
21#16837505C418yescomplemented
22C2×#14273661C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30#16837304C2211yescharacteristic
31#16837303C2211yescharacteristic
32#16837301C2211yescharacteristic
33#16837294—21no—
34#16837292—21no—
35C4⋊C4×Dic59—21no—
36C4⋊C4×Dic59—21no—
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4×C4⋊Dic59—21no—
41C4×C4⋊Dic59C2211yescharacteristic
42#16837249—21no—
43C42×Dic59—21no—
44C42×Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837157C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×Q8×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×#103×C59C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C2×#11745671—22no—
63C2×#11745671—22no—
64C2×#11745671C2×C414yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67Q8×Dic59—22no—
68Q8×Dic59—22no—
69Q8×Dic59—22no—
70Q8×Dic59C2×C414yes—
71#14273689—22no—
72#14273689C2×C414yes—
73#14273685—44no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
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×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91C2×C4×Dic59D411yescharacteristic
92C2×C4×Dic59D411yescharacteristic
93C2×C4×Dic59D411yescharacteristic
94C2×C4×Dic59D411yescharacteristic
95C2×C4×Dic59C2311yescharacteristic
96C2×C4×Dic59C2311yescharacteristic
97#14273661—22no—
98#14273660C2×C414yes—
99#14273651—22no—
100#14273649—22no—
101#14273649C2×C414yes—
102#14273647—22no—
103#14273637—22no—
104#14273637C2×C414yes—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C22×Q8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C4×Q8×C59—22no—
114C4×Q8×C59—22no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C42×C59—21no—
121C2×C42×C59C2311yescharacteristic
122#106×C59—22no—
123#103×C59—22no—
124#103×C59C2×C414yes—
125#95×C59—21no—
126C23×D59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59C22×C412yes—
131C2×Q8×C59#4112yes—
132C2×Q8×C59#4112yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2411yescharacteristic
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59C2×D411yescharacteristic
144C22×Dic59C2×D411yescharacteristic
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236C22×C412yes—
153C2×C4×D59—24no—
154C2×C4×D59—42no—
155C2×C4×D59—42no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59C22×C412yes—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59C22×C412yes—
169C4⋊C4×C59#4112yes—
170C42×C59—24no—
171C42×C59—21no—
172C42×C59—21no—
173#11745671—28no—
174(C2×D59)⋊C4—24no—
175(C2×D59)⋊C4—42no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—42no—
179C4⋊Dic59—24no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—24no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—42no—
186C4⋊Dic59C22×C412yes—
187C4⋊Dic59#4112yes—
188C4×Dic59—24no—
189C4×Dic59—22no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—24no—
193C4×Dic59—22no—
194C4×Dic59—41no—
195C4×Dic59—41no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C4×Dic59—42no—
200C4×Dic59C22×C412yes—
201C4×Dic59#4112yes—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8C22×C412yes—
208C23×C59C22×D411yescharacteristic
209C22×D59—22no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—41no—
214C22×D59—41no—
215C22×D59—41no—
216Q8×C59—22no—
217Q8×C59—22no—
218Q8×C59—22no—
219Q8×C59#11714yescomplemented
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59AGL(1,ℤ/8)12yes—
238C2×C4×C59C23×C411yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—22no—
251C2×Dic59—21no—
252C2×Dic59—22no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59C4×D412yes—
263C2×Dic59C4×D412yes—
264C2×Dic59#13712yes—
265D236—22no—
266D236—22no—
267D236—41no—
268D236—41no—
269C4×D59—48no—
270C8×C59—22no—
271C8×C59—41no—
272C59⋊C8—22no—
273C59⋊C8—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—24no—
291C4×C59—41no—
292C4×C59—42no—
293C4×C59—42no—
294C4×C59—81no—
295C4×C59C2×#11711yescharacteristic
296C4×C59C4×#4111yescharacteristic
297C4×C59#52911yescharacteristic
298C4×C59#52911yescharacteristic
299Dic59—22no—
300Dic59—22no—
301Dic59—22no—
302Dic59—41no—
303Dic59—42no—
304Dic59—42no—
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#1865D5911yescharacteristic, complemented
316#1863—591no—
317#1776—591no—
318#1763—591no—
319#1762—591no—
320#1745—591no—
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×Q8C2×D5911yescharacteristic
333C2×C4×D4—591no—
334C2×#117—1181no—
335#440—598no—
336C2×#107—591no—
337C2×#107—591no—
338C2×#106C2×D5911yescharacteristic
339C2×#103—591no—
340C2×#103C2×D5911yescharacteristic
341C2×#103C2×D5911yescharacteristic
342C2×#102—591no—
343C2×#102—591no—
344C2×#102—591no—
345C2×#97—591no—
346C2×#97—591no—
347#390—591no—
348#390—591no—
349#389—591no—
350#389C2×D5911yescharacteristic
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—591no—
362C43—1181no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340C2×D5911yescharacteristic
369#336—591no—
370#336—591no—
371#336—591no—
372#336C2×D5911yescharacteristic
373C59#2112411yescharacteristic, complemented, Sylow
374C22×Q8C22×D5911yescharacteristic
375C22×D4—591no—
376C23×C4—1181no—
377C2×SD16—594no—
378C2×SD16—1182no—
379C2×SD16—1182no—
380C22×C8—591no—
381C22×C8—591no—
382C22×C8C22×D5911yescharacteristic
383C22×C8C22×D5911yescharacteristic
384C4×Q8—22no—
385C4×Q8—22no—
386C4×Q8—594no—
387C4×Q8—1182no—
388C4×Q8—1182no—
389C4×Q8—1182no—
390C4×D4—594no—
391C4×D4—1182no—
392#117—2364no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C22⋊C4—1181no—
405C2×C22⋊C4—1181no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C22⋊C4—1181no—
409C2×C42—21no—
410C2×C42—591no—
411C2×C42—591no—
412C2×C42—591no—
413C2×C42—591no—
414C2×C42—591no—
415C2×C42—591no—
416C2×C42—1181no—
417C2×C42—1181no—
418C2×C42—1181no—
419C2×C42—1181no—
420C2×C42—1181no—
421C2×C42—1181no—
422C2×C42C22×D5911yescharacteristic
423#107—594no—
424#107—1182no—
425#106—22no—
426#103—22no—
427#103—1182no—
428#103C4×D5914yes—
429#102—594no—
430#102—1182no—
431#102—1182no—
432#97—594no—
433#97—1182no—
434#95—21no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440C24—1181no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D5912yes—
445C2×Q8#1174568712yes—
446C2×Q8#1174568912yes—
447C2×D4—592no—
448C2×D4—1181no—
449C2×D4—1181no—
450C2×D4—1181no—
451C2×D4—1181no—
452C2×D4—1181no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—591no—
458C22×C4—591no—
459C22×C4—591no—
460C22×C4—591no—
461C22×C4—1184no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—2362no—
466C22×C4—2362no—
467C22×C4D4×D5911yescharacteristic
468C22×C4D4×D5911yescharacteristic
469C22×C4C23×D5911yescharacteristic
470SD16—1188no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—592no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8C2×C4×D5912yes—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—592no—
490C4⋊C4—592no—
491C4⋊C4—1184no—
492C4⋊C4—1184no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—2362no—
498C4⋊C4C2×C4×D5912yes—
499C4⋊C4#1174568712yes—
500C22⋊C4—1184no—
501C22⋊C4—2362no—
502C22⋊C4—2362no—
503C22⋊C4—2362no—
504C22⋊C4—2362no—
505C42—24no—
506C42—21no—
507C42—21no—
508C42—592no—
509C42—592no—
510C42—1182no—
511C42—1184no—
512C42—1184no—
513C42—1182no—
514C42—1181no—
515C42—1181no—
516C42—2361no—
517C42—2361no—
518C42—2362no—
519C42—2362no—
520C42—2362no—
521C42—2362no—
522C23—1182no—
523C23—2361no—
524C23—2361no—
525C23—2361no—
526C23—2361no—
527C23—2361no—
528C23—2361no—
529C23C2×D4×D5911yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1427368514yescomplemented
534D4—1182no—
535D4—1182no—
536D4—2361no—
537D4—2361no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—592no—
556C2×C4—592no—
557C2×C4—592no—
558C2×C4—1181no—
559C2×C4—1181no—
560C2×C4—1181no—
561C2×C4—1181no—
562C2×C4—1181no—
563C2×C4—1181no—
564C2×C4—1182no—
565C2×C4—1181no—
566C2×C4—1181no—
567C2×C4—1181no—
568C2×C4—1182no—
569C2×C4—2368no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4#1427375412yes—
578C2×C4C22×C4×D5911yescharacteristic
579C2×C4#41×D5911yescharacteristic
580C2×C4#41×D5911yescharacteristic
581C2×C4C2×#1174568711yescharacteristic
582C2×C4C2×#1174568711yescharacteristic
583C2×C4C2×#1174568911yescharacteristic
584C8—22no—
585C8—41no—
586C8—1182no—
587C8—2361no—
588C22—4721no—
589C22—4721no—
590C22—4721no—
591C22—4721no—
592C22—2362no—
593C22—2362no—
594C22—2362no—
595C22C4×D4×D5911yescharacteristic
596C22#1683790811yescharacteristic
597C22#1683791511yescharacteristic
598C22C2×#1427375411yescharacteristic
599C22#137×D5911yescharacteristic
600C22#1683819911yescharacteristic
601C22#1683820111yescharacteristic
602C4—24no—
603C4—22no—
604C4—42no—
605C4—41no—
606C4—42no—
607C4—81no—
608C4—1182no—
609C4—1182no—
610C4—1182no—
611C4—4721no—
612C4—2362no—
613C4—2361no—
614C4—2362no—
615C4C2×#1427368511yescharacteristic
616C4C4×#1174568711yescharacteristic
617C4#1683792911yescharacteristic
618C4#1683793811yescharacteristic
619C2—4722no—
620C2#2142653211yescharacteristic
621C2C4×#1427375411yescharacteristic
622C2#2143291211yescharacteristic
623C2#2143420611yescharacteristic
624C2#2143421311yescharacteristic
625C2#2143425011yescharacteristic
626C2#2143427211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1865C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1865D59
hypocenterC59#21124
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
3. C4×Q8×Dic59C2semidirect
6. #21424691C2semidirect
7. #21424664C2semidirect
10. #21424352C2semidirect
11. #21423996C2semidirect
12. #21423928C2semidirect
14. #21423843C2semidirect
15. C59×#1865C2semidirect
21. #16837505C4semidirect
219. Q8×C59#117semidirect
315. #1865D59semidirect
373. C59#21124semidirect
533. Q8#14273685semidirect

Maximal subgroups

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

subgroupindex
C2×#168375052
#214265542
C4×Q8×Dic592
#214247222
#214247172
#214246912
#214246642
#214244512
#214244492
#214243522
#214239962
#214239282
#214239202
#214238432
C59×#18652
#2112459