Finite Group Databasebeta

← #28588639

Subgroups — #28588639

Order 15104.

1318 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837881C211yesmaximal, characteristic
2#21426696C212yesmaximal, complemented
3#21424172C212yesmaximal, complemented
4#21424083C212yesmaximal, complemented
5#21424082C212yesmaximal, complemented
6#21424012C212yesmaximal
7C4×#14273641C212yesmaximal, complemented
8C4×#14273642C211yesmaximal, characteristic, complemented
9C4×#99×C59C211yesmaximal, characteristic, complemented
10C22×#11745687C2211yescharacteristic
11#16837881—24no—
12#16837881—24no—
13C2×C4×Dic118C411yescharacteristic, complemented
14C2×#14273692C411yescharacteristic, complemented
15C2×D4×Dic59C2211yescharacteristic
16C2×C4×C22⋊D59C411yescharacteristic, complemented
17C2×C4×C22⋊D59C2211yescharacteristic
18C2×(C22⋊D59)⋊C4C411yescharacteristic, complemented
19C2×(C22⋊D59)⋊C4C2211yescharacteristic
20C2×#14273684C411yescharacteristic, complemented
21C2×#14273683C411yescharacteristic, complemented
22C2×#14273683C2211yescharacteristic
23C22×Dic59⋊C4C411yescharacteristic, complemented
24C22×Dic59⋊C4C2211yescharacteristic
25C2×C4⋊C4×D59C411yescharacteristic, complemented
26C2×#14273642C2212yes—
27C2×#14273641C2214yes—
28#16837286—24no—
29#16837284—24no—
30#16837263C2214yes—
31#16837260C2214yes—
32C4×C22⋊Dic59C2212yes—
33#16837169C2212yes—
34#16837168C2214yes—
35C2×C4×D4×C59C2211yescharacteristic
36C2×#99×C59C2212yes—
37C4×C22⋊C4×C59C2212yes—
38C59×#342C2212yes—
39C2×#11745687—24no—
40C2×#11745687—24no—
41C2×#11745687—24no—
42C22×Dic118C2×C411yescharacteristic
43C22×C22⋊D59C2×C411yescharacteristic
44C22×C22⋊D59C2311yescharacteristic
45C23×Dic59C2×C411yescharacteristic
46C23×Dic59C2×C411yescharacteristic
47C23×Dic59C2311yescharacteristic
48C23×Dic59C2311yescharacteristic
49C22×C4×D59C2×C411yescharacteristic
50C22×C4×D59C2×C411yescharacteristic
51C22×C4×D59C2×C411yescharacteristic
52C4×Dic118—22no—
53C4×Dic118—22no—
54#14273692—24no—
55D4×Dic59—24no—
56C4×C22⋊D59—22no—
57C4×C22⋊D59—22no—
58C4×C22⋊D59—22no—
59C4×C22⋊D59—22no—
60(C22⋊D59)⋊C4—24no—
61(C22⋊D59)⋊C4—24no—
62#14273684—22no—
63#14273684—22no—
64#14273683—24no—
65#14273683—24no—
66C2×C4⋊Dic59C2×C411yescharacteristic
67C2×Dic59⋊C4—22no—
68C2×Dic59⋊C4—22no—
69C2×Dic59⋊C4—24no—
70C2×Dic59⋊C4C2×C411yescharacteristic
71C2×Dic59⋊C4C2×C411yescharacteristic
72C2×Dic59⋊C4C2×C411yescharacteristic
73C2×Dic59⋊C4C2×C411yescharacteristic
74C2×Dic59⋊C4C2×C411yescharacteristic
75C2×Dic59⋊C4C2×C411yescharacteristic
76C2×Dic59⋊C4C2×C411yescharacteristic
77C2×Dic59⋊C4C2×C411yescharacteristic
78C2×Dic59⋊C4D414yes—
79C2×Dic59⋊C4D414yes—
80C4⋊C4×D59—24no—
81C2×C22⋊Dic59C2×C411yescharacteristic
82C2×C22⋊Dic59C2314yes—
83C2×C22⋊Dic59C2311yescharacteristic
84C2×(C2×D59)⋊C4C2×C411yescharacteristic
85C2×(C2×D59)⋊C4C2×C411yescharacteristic
86C2×C4×Dic59—22no—
87C2×C4×Dic59C2×C411yescharacteristic
88C2×C4×Dic59C2×C411yescharacteristic
89C2×C4×Dic59C2×C411yescharacteristic
90C2×C4×Dic59C2×C411yescharacteristic
91C2×C4×Dic59C2×C411yescharacteristic
92#14273642—24no—
93#14273641—28no—
94C59⋊#95—22no—
95C59⋊#95—24no—
96C59⋊#95—28no—
97C22×D4×C59C2311yescharacteristic
98C23×C4×C59C2×C411yescharacteristic
99C23×C4×C59C2311yescharacteristic
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C2×C4⋊C4×C59C2×C411yescharacteristic
103C2×C22⋊C4×C59C2×C411yescharacteristic
104C2×C22⋊C4×C59C2314yes—
105C2×C22⋊C4×C59C2311yescharacteristic
106C2×C42×C59—22no—
107C2×C42×C59C2×C411yescharacteristic
108#99×C59—24no—
109#95×C59—22no—
110C24×C59C22×C411yescharacteristic
111C24×C59C2411yescharacteristic
112C23×D59C22×C411yescharacteristic
113C2×D4×C59—22no—
114C2×D4×C59—22no—
115C2×D4×C59#4114yes—
116C22×C4×C59—21no—
117C22×C4×C59—21no—
118C22×C4×C59—24no—
119C22×C4×C59—81no—
120C22×C4×C59—81no—
121C22×C4×C59C22×C411yescharacteristic
122C22×C4×C59C22×C411yescharacteristic
123C22×C4×C59C22×C411yescharacteristic
124C22×C4×C59C22×C411yescharacteristic
125C22×C4×C59C22×C411yescharacteristic
126C22×C4×C59C2×D412yes—
127C22×C4×C59C2×D412yes—
128C2×C22⋊D59—22no—
129C2×C22⋊D59—22no—
130C2×C22⋊D59—22no—
131C2×C22⋊D59—22no—
132C2×C22⋊D59—22no—
133C2×C22⋊D59#4114yes—
134C22×Dic59—22no—
135C22×Dic59—22no—
136C22×Dic59—22no—
137C22×Dic59—22no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—24no—
141C22×Dic59—81no—
142C22×Dic59—81no—
143C22×Dic59C22×C411yescharacteristic
144C22×Dic59C22×C411yescharacteristic
145C22×Dic59C22×C411yescharacteristic
146C22×Dic59C22×C411yescharacteristic
147C22×Dic59C22×C411yescharacteristic
148C22×Dic59C22×C411yescharacteristic
149C22×Dic59C22×C411yescharacteristic
150C22×Dic59C2×D414yes—
151C22×Dic59C2×D414yes—
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—
163C4⋊C4×C59—22no—
164C22⋊C4×C59—22no—
165C22⋊C4×C59—22no—
166C22⋊C4×C59—24no—
167C22⋊C4×C59—24no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59—82no—
171C22⋊Dic59—22no—
172C22⋊Dic59—22no—
173C22⋊Dic59—24no—
174C22⋊Dic59—24no—
175(C2×D59)⋊C4—21no—
176(C2×D59)⋊C4—21no—
177(C2×D59)⋊C4—22no—
178C4⋊Dic59—22no—
179Dic59⋊C4—21no—
180Dic59⋊C4—21no—
181Dic59⋊C4—22no—
182Dic59⋊C4—22no—
183Dic59⋊C4—22no—
184Dic59⋊C4—84no—
185Dic59⋊C4C22⋊C414yes—
186Dic59⋊C4C22⋊C412yes—
187Dic59⋊C4C22⋊C412yes—
188Dic59⋊C4C22⋊C414yes—
189Dic59⋊C4C22⋊C412yes—
190Dic59⋊C4C22⋊C412yes—
191C4×Dic59—22no—
192C4×Dic59—22no—
193C4×Dic59—22no—
194C4×Dic59—22no—
195C4×Dic59—22no—
196C4×Dic59—82no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59#11712yes—
204C23×C59C23×C411yescharacteristic
205C23×C59C22×D412yes—
206C23×C59C2×#4112yes—
207C23×C59C2×#4112yes—
208C22×D59—21no—
209C22×D59—21no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59#11712yes—
213D4×C59—28no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—44no—
225C2×C4×C59—44no—
226C2×C4×C59—44no—
227C2×C4×C59—81no—
228C2×C4×C59—81no—
229C2×C4×C59C2×C22⋊C412yes—
230C2×C4×C59C2×C22⋊C411yescharacteristic
231C2×C4×C59C2×C22⋊C411yescharacteristic
232C2×C4×C59#11712yes—
233C22⋊D59—28no—
234C22⋊D59—28no—
235C2×Dic59—28no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—28no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—44no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—44no—
251C2×Dic59—44no—
252C2×Dic59—44no—
253C2×Dic59—82no—
254C2×Dic59C2×C22⋊C412yes—
255C2×Dic59C2×C22⋊C412yes—
256C2×Dic59C2×C22⋊C412yes—
257C2×Dic59C2×C22⋊C412yes—
258C2×Dic59#11712yes—
259C4×D59—28no—
260C4×D59—82no—
261C4×D59—82no—
262Dic118—28no—
263#20725—591nomaximal, Sylow
264C22×C59—22no—
265C22×C59—21no—
266C22×C59—22no—
267C22×C59—21no—
268C22×C59—22no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59C22×C22⋊C411yescharacteristic
276C22×C59C2×#11711yescharacteristic
277C22×C59C2×#11711yescharacteristic
278C22×C59#52514yes—
279C2×D59—22no—
280C2×D59—42no—
281C2×D59—41no—
282C2×D59—41no—
283C2×D59—42no—
284C4×C59—22no—
285C4×C59—21no—
286C4×C59—21no—
287C4×C59—22no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—84no—
291Dic59—22no—
292Dic59—22no—
293Dic59—22no—
294Dic59—22no—
295Dic59—22no—
296Dic59—41no—
297Dic59—41no—
298Dic59—84no—
299C2×#520—591no—
300#2313—592no—
301#1820—592no—
302#1799—592no—
303#1799—592no—
304#1782—592no—
305C4×#99—592no—
306C4×#99—591no—
307C4×#99D5911yescharacteristic, complemented
308C2×C59—24no—
309C2×C59—21no—
310C2×C59—21no—
311C2×C59—42no—
312C2×C59#228011yescharacteristic
313C2×C59#288311yescharacteristic
314C2×C59#288411yescharacteristic
315D59—42no—
316C22×#41—591no—
317#520—1184no—
318#520—1184no—
319C2×C4×Q8—591no—
320C2×C4×Q8—591no—
321C2×C4×D4—591no—
322C2×C4×D4—591no—
323C2×C4×D4—591no—
324C2×C4×D4—591no—
325C2×C4×D4—591no—
326C2×C4×D4C2×D5911yescharacteristic
327C2×#117—591no—
328C2×#117—591no—
329C2×#117—591no—
330C22×C4⋊C4—591no—
331C22×C4⋊C4—591no—
332C22×C4⋊C4—591no—
333C2×#99—592no—
334C2×#99—594no—
335C2×#99C2×D5912yes—
336#387—1184no—
337#387—1184no—
338#381—594no—
339#380—594no—
340C4×C22⋊C4—592no—
341C4×C22⋊C4C2×D5912yes—
342#342—594no—
343#342—592no—
344#342C2×D5912yes—
345C59#2072511yescharacteristic, complemented, Sylow
346C2×#41—1184no—
347C2×#41—1184no—
348C2×#41—1184no—
349C22×Q8—591no—
350C22×D4—591no—
351C22×D4—591no—
352C22×D4C22×D5911yescharacteristic
353C23×C4—591no—
354C23×C4—591no—
355C23×C4—591no—
356C23×C4—591no—
357C23×C4—591no—
358C23×C4—591no—
359C23×C4—591no—
360C23×C4C4×D5911yescharacteristic
361C23×C4C22×D5911yescharacteristic
362C4×Q8—1182no—
363C4×Q8—1182no—
364C4×Q8—1184no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—1184no—
368C4×D4—1184no—
369C4×D4—1182no—
370C4×D4—1182no—
371C4×D4—1184no—
372C4×D4—1182no—
373C4×D4—1182no—
374#117—1182no—
375#117—1182no—
376#117—1184no—
377#117—1184no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—594no—
383C2×C4⋊C4—594no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1184no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1182no—
392C2×C4⋊C4—1184no—
393C2×C4⋊C4C4×D5911yescharacteristic
394C2×C22⋊C4—591no—
395C2×C22⋊C4—591no—
396C2×C22⋊C4—591no—
397C2×C22⋊C4—591no—
398C2×C22⋊C4—594no—
399C2×C22⋊C4C4×D5911yescharacteristic
400C2×C22⋊C4C22×D5914yes—
401C2×C22⋊C4C22×D5911yescharacteristic
402C2×C42—22no—
403C2×C42—591no—
404C2×C42—591no—
405C2×C42—591no—
406C2×C42—591no—
407C2×C42—591no—
408C2×C42—1182no—
409C2×C42C4×D5911yescharacteristic
410#99—24no—
411#99—1188no—
412#99—1184no—
413#95—22no—
414#95—1184no—
415#95—1188no—
416#95—1182no—
417C24—591no—
418C24C2×C4×D5911yescharacteristic
419C24C23×D5911yescharacteristic
420#41—11832no—
421C2×Q8—1182no—
422C2×Q8—1182no—
423C2×Q8—1182no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—594no—
427C2×D4—1182no—
428C2×D4—1182no—
429C2×D4—1182no—
430C2×D4—1182no—
431C2×D4—1182no—
432C2×D4#1174568514yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—591no—
439C22×C4—591no—
440C22×C4—591no—
441C22×C4—594no—
442C22×C4—591no—
443C22×C4—591no—
444C22×C4—594no—
445C22×C4—591no—
446C22×C4—591no—
447C22×C4—1182no—
448C22×C4—1182no—
449C22×C4—1182no—
450C22×C4—1182no—
451C22×C4—1182no—
452C22×C4—1182no—
453C22×C4—1182no—
454C22×C4—1182no—
455C22×C4—1182no—
456C22×C4—1182no—
457C22×C4—1182no—
458C22×C4—1184no—
459C22×C4—4721no—
460C22×C4—4721no—
461C22×C4—4721no—
462C22×C4—4721no—
463C22×C4C2×C4×D5911yescharacteristic
464C22×C4C2×C4×D5911yescharacteristic
465C22×C4C2×C4×D5911yescharacteristic
466C22×C4C2×C4×D5911yescharacteristic
467C22×C4C2×C4×D5911yescharacteristic
468C22×C4D4×D5912yes—
469C22×C4D4×D5912yes—
470C4⋊C4—22no—
471C4⋊C4—592no—
472C4⋊C4—592no—
473C4⋊C4—594no—
474C4⋊C4—592no—
475C4⋊C4—592no—
476C4⋊C4—594no—
477C4⋊C4—1182no—
478C4⋊C4—1182no—
479C4⋊C4—1182no—
480C4⋊C4—1182no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—4724no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—1182no—
489C22⋊C4—1182no—
490C22⋊C4—1182no—
491C22⋊C4—1181no—
492C22⋊C4—1181no—
493C22⋊C4—1184no—
494C22⋊C4—1184no—
495C42—21no—
496C42—21no—
497C42—82no—
498C42—1182no—
499C42—1182no—
500C42—1182no—
501C42—1182no—
502C42—1182no—
503C42—4722no—
504C23—21no—
505C23—21no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—592no—
511C23—1181no—
512C23—1181no—
513C23—2361no—
514C23—2361no—
515C23#1427368412yes—
516C23C22×C4×D5911yescharacteristic
517C23C2×D4×D5912yes—
518C23#41×D5912yes—
519C23C2×#1174568512yes—
520Q8—1188no—
521D4—28no—
522D4—1188no—
523D4—1188no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—592no—
540C2×C4—592no—
541C2×C4—592no—
542C2×C4—592no—
543C2×C4—592no—
544C2×C4—1188no—
545C2×C4—1188no—
546C2×C4—1188no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—4722no—
556C2×C4—4722no—
557C2×C4—4722no—
558C2×C4—2364no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2364no—
562C2×C4—2364no—
563C2×C4—2364no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4C22⋊C4×D5911yescharacteristic
567C2×C4C22⋊C4×D5911yescharacteristic
568C2×C4C22⋊C4×D5912yes—
569C2×C4#1427368412yes—
570C22—22no—
571C22—21no—
572C22—22no—
573C22—21no—
574C22—22no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—41no—
580C22—41no—
581C22—1182no—
582C22—2362no—
583C22—2362no—
584C22—2361no—
585C22—2361no—
586C22C2×C22⋊C4×D5911yescharacteristic
587C22#117×D5911yescharacteristic
588C22C2×#1427368411yescharacteristic
589C22#1683791114yes—
590C4—22no—
591C4—21no—
592C4—21no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—84no—
597C4—1182no—
598C4—1182no—
599C4—1182no—
600C4—1182no—
601C4—1182no—
602C4—4724no—
603C4—2361no—
604C4—2361no—
605C2—24no—
606C2—21no—
607C2—21no—
608C2—42no—
609C2—2362no—
610C2#2142647711yescharacteristic
611C2#2143255411yescharacteristic
612C2#2143255811yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D59
commutatorC23×C59C23×C4
FrattiniC24C23×D59
FittingC4×#99×C59C2
socleC22×C59C22×C22⋊C4
prime radicalC22×#11745687C22
cosocleC24×C59C24
hypercenterC4×#99D59
hypocenterC59#20725
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #21426696C2semidirect
3. #21424172C2semidirect
4. #21424083C2semidirect
5. #21424082C2semidirect
7. C4×#14273641C2semidirect
8. C4×#14273642C2semidirect
9. C4×#99×C59C2semidirect
13. C2×C4×Dic118C4semidirect
14. C2×#14273692C4semidirect
16. C2×C4×C22⋊D59C4semidirect
18. C2×(C22⋊D59)⋊C4C4semidirect
20. C2×#14273684C4semidirect
21. C2×#14273683C4semidirect
23. C22×Dic59⋊C4C4semidirect
25. C2×C4⋊C4×D59C4semidirect
307. C4×#99D59semidirect
345. C59#20725semidirect

Maximal subgroups

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

subgroupindex
C2×#168378812
#214266962
#214241722
#214240832
#214240822
#214240122
C4×#142736412
C4×#142736422
C4×#99×C592
#2072559