Finite Group Databasebeta

← #28625711

Subgroups — #28625711

Order 15104.

1580 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273706C211yesmaximal, characteristic
2C2×#16837641C212yesmaximal, complemented
3C2×#16837557C211yesmaximal, characteristic, complemented
4C22×C22⋊(C59⋊C8)C211yesmaximal, characteristic
5#21425299C212yesmaximal, complemented
6#21425104C212yesmaximal, complemented
7#21424517C212yesmaximal, complemented
8#21424404C211yesmaximal, characteristic, complemented
9#21424283C211yesmaximal, characteristic, complemented
10#21424030C211yesmaximal, characteristic, complemented
11C59×#2848C211yesmaximal, characteristic
12C23×Dic118C2211yescharacteristic
13C22×#11745672C2212yescomplemented
14C23×C59⋊C8C2212yes—
15C2×#14273706—28no—
16C2×#14273706—21no—
17C2×#14273706—21no—
18C2×#14273706C2212yes—
19C2×#14273706C2212yes—
20C22×C4⋊Dic59C2211yescharacteristic
21C22×Dic59⋊C4—21no—
22C22×C22⋊Dic59—21no—
23#16837641—28no—
24#16837557—24no—
25C2×#14273660C2212yescomplemented
26C2×#14273651C2212yescomplemented
27C2×#14273650C2212yescomplemented
28C2×#14273650C2212yescomplemented
29C2×#14273650C2212yescomplemented
30C2×C22⋊(C59⋊C8)—24no—
31C2×C22⋊(C59⋊C8)C2211yescharacteristic
32C2×C22⋊(C59⋊C8)C2212yes—
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34#16837164C2212yescomplemented
35#16837164C2212yescomplemented
36C24×C4×C59C2211yescharacteristic
37C2×C22⋊Q8×C59C2211yescharacteristic
38C2×C22⋊Q8×C59C2211yescharacteristic
39C2×C22⋊Q8×C59C2211yescharacteristic
40C59×#385—21no—
41C59×#385—21no—
42C59×#385—21no—
43C59×#380C2211yescharacteristic
44C59×#380C2211yescharacteristic
45C59×#380C2211yescharacteristic
46C59×#379—21no—
47C22×Dic118—21no—
48C22×Dic118—24no—
49C22×Dic118—22no—
50C22×Dic118—22no—
51C22×Dic118—42no—
52C22×Dic118C2312yes—
53C23×Dic59—21no—
54C23×Dic59—21no—
55C2×#11745672—24no—
56C2×#11745672—24no—
57C2×#11745672—42no—
58C22×C59⋊C8—24no—
59C22×C59⋊C8—24no—
60C22×C59⋊C8—42no—
61C22×C59⋊C8C2312yes—
62C22×C59⋊C8C2312yes—
63#14273706—28no—
64#14273706—28no—
65#14273706—24no—
66#14273706—24no—
67#14273706—24no—
68#14273706D418yes—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—24no—
71C2×C4⋊Dic59C2312yes—
72C2×Dic59⋊C4—28no—
73C2×Dic59⋊C4—22no—
74C2×Dic59⋊C4—22no—
75C2×C22⋊Dic59—28no—
76C2×C22⋊Dic59—22no—
77C2×C22⋊Dic59—22no—
78#14273660—24no—
79#14273651—24no—
80#14273650—24no—
81#14273650—24no—
82#14273650—42no—
83C22⋊(C59⋊C8)—22no—
84C22⋊(C59⋊C8)—24no—
85C22⋊(C59⋊C8)D414yes—
86C25×C59D411yescharacteristic
87C22×Q8×C59C2311yescharacteristic
88C23×C4×C59—24no—
89C23×C4×C59—22no—
90C23×C4×C59—22no—
91C23×C4×C59—42no—
92C23×C4×C59D411yescharacteristic
93C23×C4×C59D411yescharacteristic
94C23×C4×C59D411yescharacteristic
95C23×C4×C59C2311yescharacteristic
96C23×C4×C59C2311yescharacteristic
97C23×C4×C59C2311yescharacteristic
98C22⋊Q8×C59—24no—
99C22⋊Q8×C59—22no—
100C22⋊Q8×C59—22no—
101C22⋊Q8×C59—42no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117C24×C59—22no—
118C24×C59—22no—
119C24×C59—22no—
120C24×C59—22no—
121C24×C59—41no—
122C24×C59—41no—
123C24×C59Q1614yescomplemented
124C24×C59C2×D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C24×C59C2×D411yescharacteristic
127C2×Q8×C59—22no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59—22no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—21no—
143C22×C4×C59—22no—
144C22×C4×C59—21no—
145C22×C4×C59—22no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—41no—
151C22×C4×C59—41no—
152C22×C4×C59—41no—
153C22×C4×C59—48no—
154C22×C4×C59—48no—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D414yes—
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D412yes—
160C22×C4×C59#4112yes—
161C22×C4×C59C2411yescharacteristic
162C22×Dic59—24no—
163C22×Dic59—24no—
164C22×Dic59—22no—
165C22×Dic59—22no—
166C22×Dic59—42no—
167C22×Dic59—42no—
168C22×Dic59—42no—
169C22×Dic59—42no—
170C2×Dic118—22no—
171C2×Dic118—22no—
172C2×Dic118—22no—
173C2×Dic118—22no—
174C2×Dic118—24no—
175C2×Dic118—22no—
176C2×Dic118—48no—
177C2×Dic118—48no—
178C2×Dic118—42no—
179C2×Dic118—42no—
180C2×Dic118C2×D414yes—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—22no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—22no—
185C4⋊C4×C59—42no—
186C4⋊C4×C59—42no—
187C4⋊C4×C59—42no—
188C22⋊C4×C59—42no—
189C22⋊C4×C59—42no—
190C22⋊C4×C59—42no—
191C22⋊C4×C59—42no—
192C22⋊C4×C59—42no—
193C22⋊C4×C59—42no—
194C22⋊Dic59—28no—
195C22⋊Dic59—28no—
196#11745672—48no—
197C4⋊Dic59—24no—
198C4⋊Dic59—22no—
199C4⋊Dic59C2×D414yes—
200Dic59⋊C4—28no—
201Dic59⋊C4—28no—
202C2×C59⋊C8—22no—
203C2×C59⋊C8—22no—
204C2×C59⋊C8—22no—
205C2×C59⋊C8—22no—
206C2×C59⋊C8—22no—
207C2×C59⋊C8—48no—
208C2×C59⋊C8C2×D414yes—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—28no—
224C23×C59—28no—
225C23×C59—44no—
226C23×C59—44no—
227C23×C59—44no—
228C23×C59—44no—
229C23×C59—44no—
230C23×C59#12012yes—
231C23×C59#12112yes—
232C23×C59C2×Q1612yes—
233C23×C59C2×Q1612yes—
234C23×C59C2×Q1612yes—
235C23×C59#13512yes—
236C23×C59#13712yes—
237C23×C59C22×D411yescharacteristic
238Q8×C59—42no—
239Q8×C59—42no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—24no—
245C2×C4×C59—24no—
246C2×C4×C59—24no—
247C2×C4×C59—24no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—44no—
252C2×C4×C59—44no—
253C2×C4×C59—44no—
254C2×C4×C59—42no—
255C2×C4×C59—42no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—41no—
267C2×C4×C59—41no—
268C2×C4×C59#12012yes—
269C2×C4×C59#12112yes—
270C2×C4×C59C22×D411yescharacteristic
271C2×C4×C59C22×D412yes—
272C2×C4×C59C2×#4111yescharacteristic
273C2×C4×C59H(5,𝔽2)12yes—
274C2×Dic59—24no—
275C2×Dic59—24no—
276C2×Dic59—48no—
277C2×Dic59—48no—
278C2×Dic59—42no—
279C2×Dic59—42no—
280C2×Dic59—42no—
281C2×Dic59—42no—
282Dic118—24no—
283Dic118—24no—
284Dic118—48no—
285Dic118—42no—
286Dic118—42no—
287C59⋊C8—24no—
288C59⋊C8—42no—
289#24380—591nomaximal, Sylow
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—22no—
295C22×C59—22no—
296C22×C59—22no—
297C22×C59—22no—
298C22×C59—22no—
299C22×C59—22no—
300C22×C59—22no—
301C22×C59—22no—
302C22×C59—22no—
303C22×C59—22no—
304C22×C59—22no—
305C22×C59—416no—
306C22×C59—41no—
307C22×C59—41no—
308C22×C59—41no—
309C22×C59—41no—
310C22×C59#45114yescomplemented
311C22×C59#46714yescomplemented
312C22×C59C2×#12011yescharacteristic
313C22×C59C2×#12111yescharacteristic
314C22×C59#53411yescharacteristic
315C22×C59C22×Q1611yescharacteristic
316C22×C59C2×#13511yescharacteristic
317C22×C59C2×#13711yescharacteristic
318C22×C59#57511yescharacteristic
319C4×C59—22no—
320C4×C59—22no—
321C4×C59—41no—
322C4×C59—81no—
323C4×C59—81no—
324C4×C59—81no—
325C4×C59—81no—
326C4×C59D4212yes—
327C4×C59#54612yes—
328Dic59—44no—
329Dic59—44no—
330C22×C22⋊Q8—591no—
331C2×#467—592no—
332C2×#451—591no—
333C22×C22⋊C8—591no—
334#2848D5911yescharacteristic
335#2036—592no—
336#2001—592no—
337#1892—592no—
338#1872—591no—
339#1852—591no—
340#1787—591no—
341C2×C59—24no—
342C2×C59—24no—
343C2×C59—42no—
344C2×C59#241011yescharacteristic
345C2×C59C2×#45111yescharacteristic
346C2×C59#300211yescharacteristic
347C2×C59C2×#46711yescharacteristic
348C2×C59#305611yescharacteristic
349C2×C59#318911yescharacteristic
350C2×C59#319011yescharacteristic
351C23×Q8—591no—
352C24×C4C2×D5911yescharacteristic
353C22×Q16—592no—
354C23×C8—592no—
355C2×C22⋊Q8—592no—
356C2×C22⋊Q8—592no—
357C2×C22⋊Q8—1188no—
358C2×C22⋊Q8—1181no—
359C2×C22⋊Q8—1181no—
360C2×C22⋊Q8C2×D5911yescharacteristic
361C2×C22⋊Q8C2×D5911yescharacteristic
362C2×C22⋊Q8C2×D5911yescharacteristic
363C22×C4⋊C4—591no—
364C22×C4⋊C4—1181no—
365C22×C22⋊C4—1181no—
366#467—1188no—
367#451—1184no—
368C2×#107—592no—
369C2×#103—592no—
370C2×#103—592no—
371C2×#103—592no—
372C2×#103—592no—
373C2×C22⋊C8—591no—
374C2×C22⋊C8—591no—
375C2×C22⋊C8—592no—
376C2×C22⋊C8—1184no—
377#385—21no—
378#385—21no—
379#385—21no—
380#380C2×D5911yescharacteristic
381#380C2×D5911yescharacteristic
382#380C2×D5911yescharacteristic
383#379—21no—
384#340—592no—
385#340—592no—
386C59#2438011yescharacteristic, complemented, Sylow
387C25C22⋊D5911yescharacteristic
388C22×Q8—592no—
389C22×Q8—1184no—
390C22×Q8—1181no—
391C22×Q8—1182no—
392C22×Q8—1182no—
393C22×Q8—2362no—
394C22×Q8C22×D5911yescharacteristic
395C23×C4—24no—
396C23×C4—22no—
397C23×C4—22no—
398C23×C4—42no—
399C23×C4—1181no—
400C23×C4—1181no—
401C23×C4C22⋊D5911yescharacteristic
402C23×C4C22⋊D5911yescharacteristic
403C23×C4C22⋊D5911yescharacteristic
404C23×C4C22×D5911yescharacteristic
405C23×C4C22×D5911yescharacteristic
406C23×C4C22×D5911yescharacteristic
407C2×Q16—1184no—
408C2×Q16—1184no—
409C2×Q16—2362no—
410C22×C8—592no—
411C22×C8—592no—
412C22×C8—1184no—
413C22×C8—1184no—
414C22×C8—2362no—
415C22⋊Q8—22no—
416C22⋊Q8—24no—
417C22⋊Q8—22no—
418C22⋊Q8—42no—
419C22⋊Q8—598no—
420C22⋊Q8—1184no—
421C22⋊Q8—1184no—
422C22⋊Q8—1188no—
423C22⋊Q8—1188no—
424C22⋊Q8—1184no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—592no—
429C2×C4⋊C4—1184no—
430C2×C4⋊C4—1188no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1182no—
433C2×C4⋊C4—1182no—
434C2×C4⋊C4C22×D5911yescharacteristic
435C2×C4⋊C4C22×D5911yescharacteristic
436C2×C4⋊C4C22×D5911yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1188no—
444C2×C22⋊C4—1182no—
445C2×C22⋊C4—1182no—
446#107—1184no—
447#103—1184no—
448#103—1184no—
449#103—1184no—
450#103—2362no—
451C22⋊C8—594no—
452C22⋊C8—1184no—
453C22⋊C8—1182no—
454#95—21no—
455#95—21no—
456#95—21no—
457C24—22no—
458C24—22no—
459C24—22no—
460C24—22no—
461C24—41no—
462C24—41no—
463C24#1174567214yescomplemented
464C24C2×C22⋊D5911yescharacteristic
465C24C2×C22⋊D5911yescharacteristic
466C24C2×C22⋊D5911yescharacteristic
467C2×Q8—22no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—594no—
472C2×Q8—1184no—
473C2×Q8—1182no—
474C2×Q8—1182no—
475C2×Q8—1182no—
476C2×Q8—1182no—
477C2×Q8—1182no—
478C2×Q8—2368no—
479C2×Q8—2368no—
480C2×Q8—2362no—
481C2×Q8—2362no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—1184no—
507C22×C4—1184no—
508C22×C4—1182no—
509C22×C4—1182no—
510C22×C4—2362no—
511C22×C4—2362no—
512C22×C4—2362no—
513C22×C4—2362no—
514C22×C4D4×D5914yes—
515C22×C4D4×D5912yes—
516C22×C4#1174568712yes—
517C22×C4C2×C22⋊D5911yescharacteristic
518C22×C4C2×C22⋊D5911yescharacteristic
519C22×C4C2×C22⋊D5911yescharacteristic
520C22×C4C23×D5911yescharacteristic
521Q16—2368no—
522C2×C8—594no—
523C2×C8—1182no—
524C2×C8—1182no—
525C2×C8—1182no—
526C2×C8—1182no—
527C2×C8—1182no—
528C2×C8—2368no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—594no—
537C4⋊C4—1188no—
538C4⋊C4—1184no—
539C4⋊C4—1188no—
540C4⋊C4—1182no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—1188no—
548C22⋊C4—1188no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—28no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—28no—
564C23—21no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23#1427369512yes—
571C23#1427369912yes—
572C23C2×#1174567212yes—
573C23C2×#1174567212yes—
574C23C2×#1174567212yes—
575C23#1427374712yes—
576C23#1427375712yes—
577C23C22×C22⋊D5911yescharacteristic
578Q8—42no—
579Q8—42no—
580Q8—1184no—
581Q8—1184no—
582Q8—2368no—
583Q8—2362no—
584Q8—2362no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—1184no—
614C2×C4—1184no—
615C2×C4—2368no—
616C2×C4—2368no—
617C2×C4—2362no—
618C2×C4—2362no—
619C2×C4—2362no—
620C2×C4—2362no—
621C2×C4#1427369512yes—
622C2×C4#1427369912yes—
623C2×C4C2×D4×D5911yescharacteristic
624C2×C4C2×D4×D5912yes—
625C2×C4C2×#1174568711yescharacteristic
626C2×C4#1427377212yes—
627C8—1184no—
628C8—2362no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—416no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22#1683755714yescomplemented
650C22#1683764114yescomplemented
651C22C2×#1427369511yescharacteristic
652C22C2×#1427369911yescharacteristic
653C22#1683796111yescharacteristic
654C22C22×#1174567211yescharacteristic
655C22C2×#1427374711yescharacteristic
656C22C2×#1427375711yescharacteristic
657C22#1683820011yescharacteristic
658C4—22no—
659C4—22no—
660C4—41no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—2364no—
666C4—2364no—
667C4#1683803412yes—
668C4#1683803912yes—
669C2—24no—
670C2—24no—
671C2—42no—
672C2#2142747111yescharacteristic
673C2C2×#1683755711yescharacteristic
674C2#2143331211yescharacteristic
675C2C2×#1683764111yescharacteristic
676C2#2143372811yescharacteristic
677C2#2143486811yescharacteristic
678C2#2143487311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2848C2
socleC23×C59C22×D4
prime radicalC25×C59D4
hypercenter#2848D59
hypocenterC59#24380
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
2. C2×#16837641C2semidirect
3. C2×#16837557C2semidirect
5. #21425299C2semidirect
6. #21425104C2semidirect
7. #21424517C2semidirect
8. #21424404C2semidirect
9. #21424283C2semidirect
10. #21424030C2semidirect
13. C22×#11745672C22semidirect
25. C2×#14273660C22semidirect
26. C2×#14273651C22semidirect
27. C2×#14273650C22semidirect
28. C2×#14273650C22semidirect
29. C2×#14273650C22semidirect
34. #16837164C22semidirect
35. #16837164C22semidirect
123. C24×C59Q16semidirect
310. C22×C59#451semidirect
311. C22×C59#467semidirect
386. C59#24380semidirect
463. C24#11745672semidirect
649. C22#16837557semidirect
650. C22#16837641semidirect

Maximal subgroups

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

subgroupindex
C22×#142737062
C2×#168376412
C2×#168375572
C22×C22⋊(C59⋊C8)2
#214252992
#214251042
#214245172
#214244042
#214242832
#214240302
C59×#28482
#2438059