Finite Group Databasebeta

← #28554600

Subgroups — #28554600

Order 15104.

1210 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837881C211yesmaximal, characteristic
2#21433125C211yesmaximal, characteristic, complemented
3#21422439C212yesmaximal, complemented
4#21422432C212yesmaximal, complemented
5#21422428C212yesmaximal, complemented
6#21422425C212yesmaximal, complemented
7#21422412C212yesmaximal, complemented
8C2×#16837127C212yesmaximal, complemented
9C59×#2973C211yesmaximal, characteristic, complemented
10C22×#11745687C411yescharacteristic
11#16837881—24no—
12#16837881—24no—
13C2×C4×Dic118C2211yescharacteristic
14C2×#14273692C411yescharacteristic
15C2×D4×Dic59C411yescharacteristic
16C2×C4×C22⋊D59C2211yescharacteristic
17C2×C4×C22⋊D59C2211yescharacteristic
18C2×(C22⋊D59)⋊C4C411yescharacteristic
19C2×(C22⋊D59)⋊C4C411yescharacteristic
20C2×#14273684C2211yescharacteristic
21C2×#14273683C411yescharacteristic
22C2×#14273683C411yescharacteristic
23C22×Dic59⋊C4C2211yescharacteristic
24C22×Dic59⋊C4C2211yescharacteristic
25C2×C4⋊C4×D59C411yescharacteristic
26#16837474—24no—
27#16837459C2212yes—
28#16837366C2212yes—
29C2×C22⋊(C59⋊C8)C2212yes—
30#16837127C2214yes—
31#16837127C2214yes—
32#16837127C2214yes—
33#16837125C2214yes—
34C2×C4×D4×C59C2211yescharacteristic
35C59×#435—24no—
36C59×#432C2212yes—
37C59×#407C2212yes—
38C2×C22⋊C8×C59C2212yes—
39C2×#11745687—24no—
40C2×#11745687—24no—
41C2×#11745687—24no—
42C22×Dic118C2×C411yescharacteristic
43C22×C22⋊D59C2×C411yescharacteristic
44C22×C22⋊D59C2×C411yescharacteristic
45C23×Dic59C2×C411yescharacteristic
46C23×Dic59C2×C411yescharacteristic
47C23×Dic59C2×C411yescharacteristic
48C23×Dic59C2×C411yescharacteristic
49C22×C4×D59C2×C411yescharacteristic
50C22×C4×D59C2×C411yescharacteristic
51C22×C4×D59C2×C411yescharacteristic
52C2×#11745664—22no—
53C22×C59⋊C8—22no—
54C4×Dic118—22no—
55C4×Dic118—22no—
56#14273692—24no—
57D4×Dic59—24no—
58C4×C22⋊D59—22no—
59C4×C22⋊D59—22no—
60C4×C22⋊D59—22no—
61C4×C22⋊D59—22no—
62(C22⋊D59)⋊C4—24no—
63(C22⋊D59)⋊C4—24no—
64#14273684—22no—
65#14273684—22no—
66#14273683—24no—
67#14273683—24no—
68C2×C4⋊Dic59C2×C411yescharacteristic
69C2×Dic59⋊C4—22no—
70C2×Dic59⋊C4—22no—
71C2×Dic59⋊C4—22no—
72C2×Dic59⋊C4—22no—
73C2×Dic59⋊C4C2×C411yescharacteristic
74C2×Dic59⋊C4C2×C411yescharacteristic
75C2×Dic59⋊C4C2×C411yescharacteristic
76C2×Dic59⋊C4C2×C411yescharacteristic
77C2×Dic59⋊C4C2×C411yescharacteristic
78C2×Dic59⋊C4C2311yescharacteristic
79C2×Dic59⋊C4C2311yescharacteristic
80C2×Dic59⋊C4C2311yescharacteristic
81C4⋊C4×D59—24no—
82C2×C22⋊Dic59C2×C411yescharacteristic
83C2×C22⋊Dic59C2×C411yescharacteristic
84C2×(C2×D59)⋊C4C2×C411yescharacteristic
85C2×(C2×D59)⋊C4C2311yescharacteristic
86C2×C4×Dic59C2×C411yescharacteristic
87C2×C4×Dic59C2×C411yescharacteristic
88C2×C4×Dic59C2×C411yescharacteristic
89C2×C4×Dic59C2×C411yescharacteristic
90C2×C4×Dic59C2×C411yescharacteristic
91C4⋊(C59⋊C8)—22no—
92C22⋊(C59⋊C8)—22no—
93C22⋊(C59⋊C8)—22no—
94C22⋊(C59⋊C8)C2314yes—
95#14273638—22no—
96C22×D4×C59C2×C411yescharacteristic
97C23×C4×C59C2311yescharacteristic
98C23×C4×C59C2311yescharacteristic
99C2×M16×C59—22no—
100C22×C8×C59—22no—
101C4×D4×C59D414yes—
102C4×D4×C59D414yes—
103C2×C4⋊C4×C59C2×C411yescharacteristic
104C2×C22⋊C4×C59C2×C411yescharacteristic
105C2×C22⋊C4×C59C2×C411yescharacteristic
106C2×C42×C59C2311yescharacteristic
107C4⋊C8×C59—22no—
108C22⋊C8×C59—22no—
109C22⋊C8×C59—22no—
110C22⋊C8×C59C2314yes—
111#97×C59—22no—
112C24×C59C22×C411yescharacteristic
113C24×C59C22×C411yescharacteristic
114C23×D59C22×C411yescharacteristic
115C2×D4×C59—22no—
116C2×D4×C59—22no—
117C2×D4×C59C22⋊C414yes—
118C22×C4×C59—81no—
119C22×C4×C59—81no—
120C22×C4×C59C22×C411yescharacteristic
121C22×C4×C59C22×C411yescharacteristic
122C22×C4×C59C22×C411yescharacteristic
123C22×C4×C59C22×C411yescharacteristic
124C22×C4×C59C2×D412yes—
125C22×C4×C59C2×D412yes—
126C22×C4×C59C2×D412yes—
127C22×C4×C59C2×D412yes—
128C22×C4×C59C2411yescharacteristic
129C2×C22⋊D59—22no—
130C2×C22⋊D59—22no—
131C2×C22⋊D59—22no—
132C2×C22⋊D59—22no—
133C2×C22⋊D59—22no—
134C2×C22⋊D59M1614yescomplemented
135C22×Dic59—22no—
136C22×Dic59—22no—
137C22×Dic59—22no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—22no—
141C22×Dic59—22no—
142C22×Dic59—81no—
143C22×Dic59—81no—
144C22×Dic59M1614yescomplemented
145C22×Dic59C22×C411yescharacteristic
146C22×Dic59C22×C411yescharacteristic
147C22×Dic59C22×C411yescharacteristic
148C22×Dic59C22×C411yescharacteristic
149C22×Dic59C22×C411yescharacteristic
150C22×Dic59C22×C411yescharacteristic
151C22×Dic59C22×C411yescharacteristic
152#11745687—232no—
153C2×C4×D59—22no—
154C2×C4×D59—22no—
155C2×C4×D59—22no—
156C2×C4×D59—22no—
157C2×C4×D59—22no—
158C2×C4×D59—81no—
159C2×C4×D59—81no—
160C2×Dic118—22no—
161C2×Dic118—22no—
162C2×Dic118—22no—
163M16×C59—82no—
164C2×C8×C59—24no—
165C2×C8×C59—82no—
166C4⋊C4×C59C22⋊C414yes—
167C22⋊C4×C59C22⋊C414yes—
168C22⋊C4×C59C22⋊C414yes—
169C42×C59C2×D412yes—
170C42×C59C2×D412yes—
171C22⋊Dic59—22no—
172C22⋊Dic59—22no—
173(C2×D59)⋊C4—22no—
174(C2×D59)⋊C4—21no—
175(C2×D59)⋊C4—21no—
176C4⋊Dic59—22no—
177Dic59⋊C4—22no—
178Dic59⋊C4—22no—
179Dic59⋊C4—22no—
180Dic59⋊C4—22no—
181Dic59⋊C4—22no—
182Dic59⋊C4—21no—
183Dic59⋊C4—21no—
184Dic59⋊C4—21no—
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188C4×Dic59—22no—
189C4×Dic59—22no—
190C4×Dic59—22no—
191C4×Dic59—22no—
192C4×Dic59—22no—
193#11745664—82no—
194C2×C59⋊C8—24no—
195C2×C59⋊C8—82no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59C2×C22⋊C412yes—
203C23×C59C2×C22⋊C412yes—
204C23×C59C2×M1612yes—
205C23×C59C2×M1612yes—
206C23×C59C23×C411yescharacteristic
207C22×D59—21no—
208C22×D59—21no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59C2×M1612yes—
212D4×C59—28no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59C2×C22⋊C412yes—
224C2×C4×C59C2×C22⋊C412yes—
225C2×C4×C59C2×C22⋊C412yes—
226C2×C4×C59C2×C22⋊C412yes—
227C2×C4×C59C22×D411yescharacteristic
228C2×C4×C59C22×D411yescharacteristic
229C22⋊D59—28no—
230C22⋊D59—28no—
231C2×Dic59—28no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—28no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—82no—
248C2×Dic59—82no—
249C2×Dic59C2×M1612yes—
250C2×Dic59C2×M1612yes—
251C2×Dic59C2×M1612yes—
252C4×D59—28no—
253C4×D59—82no—
254C4×D59—82no—
255Dic118—28no—
256C8×C59—44no—
257C59⋊C8—44no—
258#17368—591nomaximal, Sylow
259C22×C59—21no—
260C22×C59—22no—
261C22×C59—21no—
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59#40714yes—
271C22×C59C22×C22⋊C411yescharacteristic
272C22×C59C22×M1611yescharacteristic
273C22×C59C22×M1611yescharacteristic
274C2×D59—22no—
275C2×D59—42no—
276C2×D59—41no—
277C2×D59—41no—
278C2×D59—42no—
279C4×C59—22no—
280C4×C59—21no—
281C4×C59—21no—
282C4×C59—22no—
283C4×C59—22no—
284C4×C59—22no—
285Dic59—22no—
286Dic59—22no—
287Dic59—22no—
288Dic59—41no—
289Dic59—41no—
290Dic59—41no—
291Dic59—41no—
292C2×#520—591no—
293#2973—591no—
294#2973D5911yescharacteristic, complemented
295#1467—592no—
296#1465—592no—
297#1464—592no—
298#1463—592no—
299#1459—592no—
300C2×#324—592no—
301C2×C59—21no—
302C2×C59—21no—
303C2×C59—24no—
304C2×C59—42no—
305C2×C59C2×#40711yescharacteristic
306C2×C59#288311yescharacteristic
307C2×C59#288911yescharacteristic
308D59—42no—
309C22×#41—591no—
310#520—1184no—
311#520—1184no—
312C2×C4×Q8—591no—
313C2×C4×Q8—591no—
314C2×C4×D4—591no—
315C2×C4×D4—591no—
316C2×C4×D4—591no—
317C2×C4×D4—591no—
318C2×C4×D4—591no—
319C2×C4×D4C2×D5911yescharacteristic
320C2×#117—591no—
321C2×#117—591no—
322C2×#117—591no—
323C22×C4⋊C4—591no—
324C22×C4⋊C4—591no—
325C22×C4⋊C4—591no—
326#435—24no—
327#435—1184no—
328#432—592no—
329#432C2×D5912yes—
330#407—592no—
331#407C2×D5912yes—
332C2×C22⋊C8—592no—
333C2×C22⋊C8C2×D5912yes—
334#324—594no—
335#324—594no—
336#324—594no—
337#323—594no—
338C59#1736811yescharacteristic, complemented, Sylow
339C2×#41—1184no—
340C2×#41—1184no—
341C2×#41—1184no—
342C22×Q8—591no—
343C22×D4—591no—
344C22×D4—591no—
345C22×D4C4×D5911yescharacteristic
346C23×C4—591no—
347C23×C4—591no—
348C23×C4—591no—
349C23×C4—591no—
350C23×C4—591no—
351C23×C4—591no—
352C23×C4—591no—
353C23×C4C22×D5911yescharacteristic
354C23×C4C22×D5911yescharacteristic
355C2×M16—22no—
356C2×M16—1182no—
357C22×C8—22no—
358C22×C8—1182no—
359C4×Q8—1184no—
360C4×Q8—1182no—
361C4×Q8—1182no—
362C4×D4—1184no—
363C4×D4—1182no—
364C4×D4—1182no—
365C4×D4—1182no—
366C4×D4—1182no—
367C4×D4—1184no—
368C4×D4—1184no—
369C4×D4D23614yes—
370C4×D4C22⋊D5914yes—
371#117—1184no—
372#117—1184no—
373#117—1182no—
374#117—1182no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—1182no—
385C2×C4⋊C4—1182no—
386C2×C4⋊C4—1182no—
387C2×C4⋊C4—1182no—
388C2×C4⋊C4—1184no—
389C2×C4⋊C4C4×D5911yescharacteristic
390C2×C22⋊C4—591no—
391C2×C22⋊C4—591no—
392C2×C22⋊C4—591no—
393C2×C22⋊C4—591no—
394C2×C22⋊C4C4×D5911yescharacteristic
395C2×C22⋊C4C4×D5911yescharacteristic
396C2×C42—591no—
397C2×C42—591no—
398C2×C42—591no—
399C2×C42—591no—
400C2×C42—591no—
401C2×C42C22×D5911yescharacteristic
402C4⋊C8—22no—
403C4⋊C8—1182no—
404C22⋊C8—22no—
405C22⋊C8—22no—
406C22⋊C8—594no—
407C22⋊C8—1182no—
408C22⋊C8—1182no—
409C22⋊C8C22×D5914yes—
410#97—22no—
411#97—1182no—
412C24—591no—
413C24C2×C4×D5911yescharacteristic
414C24C2×C4×D5911yescharacteristic
415#41—11832no—
416C2×Q8—1182no—
417C2×Q8—1182no—
418C2×Q8—1182no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—594no—
422C2×D4—1182no—
423C2×D4—1182no—
424C2×D4—1182no—
425C2×D4—1182no—
426C2×D4—1182no—
427C2×D4(C2×D59)⋊C414yes—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—594no—
431C22×C4—591no—
432C22×C4—591no—
433C22×C4—591no—
434C22×C4—591no—
435C22×C4—591no—
436C22×C4—591no—
437C22×C4—591no—
438C22×C4—1182no—
439C22×C4—1182no—
440C22×C4—1182no—
441C22×C4—1182no—
442C22×C4—1182no—
443C22×C4—1182no—
444C22×C4—1182no—
445C22×C4—1182no—
446C22×C4—1182no—
447C22×C4—1182no—
448C22×C4—1182no—
449C22×C4—1182no—
450C22×C4—4721no—
451C22×C4—4721no—
452C22×C4—4721no—
453C22×C4—4721no—
454C22×C4C2×C4×D5911yescharacteristic
455C22×C4C2×C4×D5911yescharacteristic
456C22×C4C2×C4×D5911yescharacteristic
457C22×C4C2×C4×D5911yescharacteristic
458C22×C4C2×D23612yes—
459C22×C4C2×D23612yes—
460C22×C4C2×C22⋊D5912yes—
461C22×C4C2×C22⋊D5912yes—
462C22×C4C23×D5911yescharacteristic
463M16—82no—
464M16—4722no—
465C2×C8—24no—
466C2×C8—82no—
467C2×C8—1184no—
468C2×C8—4722no—
469C4⋊C4—1182no—
470C4⋊C4—1182no—
471C4⋊C4—1182no—
472C4⋊C4—1182no—
473C4⋊C4—1182no—
474C4⋊C4—1182no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4(C2×D59)⋊C414yes—
482C22⋊C4—1182no—
483C22⋊C4—1182no—
484C22⋊C4—1182no—
485C22⋊C4—1181no—
486C22⋊C4—1181no—
487C22⋊C4(C2×D59)⋊C414yes—
488C22⋊C4(C2×D59)⋊C414yes—
489C42—1182no—
490C42—1182no—
491C42—1182no—
492C42—1182no—
493C42—1182no—
494C42C2×D23612yes—
495C42C2×C22⋊D5912yes—
496C23—21no—
497C23—21no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—592no—
503C23—1181no—
504C23—1181no—
505C23—2361no—
506C23—2361no—
507C23C2×(C2×D59)⋊C412yes—
508C23C2×(C2×D59)⋊C412yes—
509C23M16×D5912yes—
510C23M16×D5912yes—
511C23C22×C4×D5911yescharacteristic
512Q8—1188no—
513D4—28no—
514D4—1188no—
515D4—1188no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—592no—
527C2×C4—592no—
528C2×C4—592no—
529C2×C4—1188no—
530C2×C4—1188no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1188no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—4722no—
543C2×C4—4722no—
544C2×C4—4722no—
545C2×C4—4722no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4C2×(C2×D59)⋊C412yes—
551C2×C4C2×(C2×D59)⋊C412yes—
552C2×C4C2×(C2×D59)⋊C412yes—
553C2×C4C2×(C2×D59)⋊C412yes—
554C2×C4C22×C22⋊D5911yescharacteristic
555C2×C4C22×D23611yescharacteristic
556C8—44no—
557C8—2364no—
558C22—21no—
559C22—22no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—1182no—
570C22—2362no—
571C22—2362no—
572C22—2361no—
573C22—2361no—
574C22#1683736514yes—
575C22C22×(C2×D59)⋊C411yescharacteristic
576C22C2×M16×D5911yescharacteristic
577C22C2×M16×D5911yescharacteristic
578C4—22no—
579C4—22no—
580C4—21no—
581C4—21no—
582C4—22no—
583C4—22no—
584C4—1182no—
585C4—1182no—
586C4—1182no—
587C4—2361no—
588C4—2361no—
589C4—2361no—
590C4—2361no—
591C2—21no—
592C2—21no—
593C2—24no—
594C2—42no—
595C2—2362no—
596C2C2×#1683736511yescharacteristic
597C2#2143255611yescharacteristic
598C2#2143257811yescharacteristic
subgroupG/H
centerC22C22×(C2×D59)⋊C4
commutatorC23×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2973C2
socleC22×C59C22×C22⋊C4
prime radicalC22×#11745687C4
cosocleC22×C4×C59C24
hypercenter#2973D59
hypocenterC59#17368
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21433125C2semidirect
3. #21422439C2semidirect
4. #21422432C2semidirect
5. #21422428C2semidirect
6. #21422425C2semidirect
7. #21422412C2semidirect
8. C2×#16837127C2semidirect
9. C59×#2973C2semidirect
134. C2×C22⋊D59M16semidirect
144. C22×Dic59M16semidirect
294. #2973D59semidirect
338. C59#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#168378812
#214331252
#214224392
#214224322
#214224282
#214224252
#214224122
C2×#168371272
C59×#29732
#1736859