Finite Group Databasebeta

← #28620452

Subgroups — #28620452

Order 15104.

1908 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273706C211yesmaximal, characteristic
2C2×#16837558C212yesmaximal, complemented
3C2×#16837557C212yesmaximal, complemented
4#21425104C214yesmaximal, complemented
5#21424404C212yesmaximal, complemented
6#21424403C212yesmaximal, complemented
7#21424006C211yesmaximal, characteristic
8C22×C22⋊Q8×C59C211yesmaximal, characteristic
9C23×Dic118C2211yescharacteristic
10C22×#11745672C2214yescomplemented
11C2×#14273706—28no—
12C2×#14273706—21no—
13C2×#14273706—21no—
14C2×#14273706C2212yes—
15C2×#14273706C2212yes—
16C22×C4⋊Dic59C2211yescharacteristic
17C22×Dic59⋊C4—21no—
18C22×C22⋊Dic59—21no—
19#16837558—28no—
20#16837557—28no—
21C2×#14273651C2214yescomplemented
22C2×#14273650C2214yescomplemented
23C2×C22⋊(C59⋊C8)C2212yes—
24C2×C22⋊(C59⋊C8)C2212yes—
25C2×C22⋊(C59⋊C8)C2212yes—
26#16837164C2214yescomplemented
27C23×Q8×C59C2211yescharacteristic
28C24×C4×C59C2211yescharacteristic
29C2×C22⋊Q8×C59—21no—
30C2×C22⋊Q8×C59—21no—
31C2×C22⋊Q8×C59—28no—
32C2×C22⋊Q8×C59C2212yes—
33C2×C22⋊Q8×C59C2212yes—
34C22×C4⋊C4×C59—21no—
35C22×C4⋊C4×C59C2211yescharacteristic
36C22×C22⋊C4×C59—21no—
37C22×Dic118—21no—
38C22×Dic118—24no—
39C22×Dic118—24no—
40C22×Dic118—42no—
41C22×Dic118C2312yes—
42C23×Dic59—21no—
43C23×Dic59—21no—
44C2×#11745672—28no—
45C2×#11745672—28no—
46C2×#11745672—44no—
47C22×C59⋊C8C2314yes—
48#14273706—28no—
49#14273706—28no—
50#14273706—24no—
51#14273706—24no—
52#14273706—24no—
53#14273706D418yes—
54C2×C4⋊Dic59—21no—
55C2×C4⋊Dic59—24no—
56C2×C4⋊Dic59C2312yes—
57C2×Dic59⋊C4—28no—
58C2×Dic59⋊C4—22no—
59C2×Dic59⋊C4—22no—
60C2×C22⋊Dic59—28no—
61C2×C22⋊Dic59—22no—
62C2×C22⋊Dic59—22no—
63#14273651—28no—
64#14273650—28no—
65C22⋊(C59⋊C8)—24no—
66C22⋊(C59⋊C8)—24no—
67C22⋊(C59⋊C8)—42no—
68C25×C59D411yescharacteristic
69C22×Q8×C59—24no—
70C22×Q8×C59—21no—
71C22×Q8×C59—24no—
72C22×Q8×C59—42no—
73C22×Q8×C59C2312yes—
74C23×C4×C59—21no—
75C23×C4×C59—21no—
76C23×C4×C59—24no—
77C23×C4×C59—24no—
78C23×C4×C59—42no—
79C23×C4×C59D411yescharacteristic
80C23×C4×C59D411yescharacteristic
81C23×C4×C59D411yescharacteristic
82C23×C4×C59C2311yescharacteristic
83C23×C4×C59C2311yescharacteristic
84C23×C4×C59C2311yescharacteristic
85C22⋊Q8×C59—28no—
86C22⋊Q8×C59—28no—
87C22⋊Q8×C59—24no—
88C22⋊Q8×C59—24no—
89C22⋊Q8×C59—24no—
90C22⋊Q8×C59D418yes—
91C2×C4⋊C4×C59—28no—
92C2×C4⋊C4×C59—22no—
93C2×C4⋊C4×C59—22no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59—24no—
96C2×C4⋊C4×C59C2312yes—
97C2×C22⋊C4×C59—28no—
98C2×C22⋊C4×C59—22no—
99C2×C22⋊C4×C59—22no—
100C24×C59—22no—
101C24×C59—22no—
102C24×C59—22no—
103C24×C59—22no—
104C24×C59—41no—
105C24×C59—41no—
106C24×C59Q1614yescomplemented
107C24×C59C2×D411yescharacteristic
108C24×C59C2×D411yescharacteristic
109C24×C59C2×D411yescharacteristic
110C2×Q8×C59—24no—
111C2×Q8×C59—22no—
112C2×Q8×C59—24no—
113C2×Q8×C59—24no—
114C2×Q8×C59—416no—
115C2×Q8×C59—42no—
116C2×Q8×C59—42no—
117C2×Q8×C59C2×D414yes—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—24no—
123C22×C4×C59—24no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—41no—
136C22×C4×C59—41no—
137C22×C4×C59—41no—
138C22×C4×C59—416no—
139C22×C4×C59—42no—
140C22×C4×C59—42no—
141C22×C4×C59—42no—
142C22×C4×C59—42no—
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D414yes—
147C22×C4×C59C2×D414yes—
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—24no—
150C22×Dic59—24no—
151C22×Dic59—22no—
152C22×Dic59—22no—
153C22×Dic59—42no—
154C22×Dic59—42no—
155C22×Dic59—42no—
156C22×Dic59—42no—
157C2×Dic118—24no—
158C2×Dic118—24no—
159C2×Dic118—24no—
160C2×Dic118—22no—
161C2×Dic118—416no—
162C2×Dic118—42no—
163C2×Dic118—42no—
164C2×Dic118C2×D414yes—
165C4⋊C4×C59—24no—
166C4⋊C4×C59—28no—
167C4⋊C4×C59—28no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59C2×D414yes—
170C22⋊C4×C59—28no—
171C22⋊C4×C59—28no—
172C22⋊Dic59—28no—
173C22⋊Dic59—28no—
174#11745672—416no—
175C4⋊Dic59—24no—
176C4⋊Dic59—22no—
177C4⋊Dic59C2×D414yes—
178Dic59⋊C4—28no—
179Dic59⋊C4—28no—
180C2×C59⋊C8—24no—
181C2×C59⋊C8—24no—
182C2×C59⋊C8—24no—
183C23×C59—21no—
184C23×C59—21no—
185C23×C59—21no—
186C23×C59—21no—
187C23×C59—21no—
188C23×C59—21no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—28no—
198C23×C59—28no—
199C23×C59—44no—
200C23×C59—44no—
201C23×C59—44no—
202C23×C59—44no—
203C23×C59—44no—
204C23×C59#12012yes—
205C23×C59#12012yes—
206C23×C59C2×Q1612yes—
207C23×C59C2×Q1612yes—
208C23×C59C2×Q1612yes—
209C23×C59#13712yes—
210C23×C59#13712yes—
211C23×C59C22×D411yescharacteristic
212Q8×C59—28no—
213Q8×C59—48no—
214Q8×C59—44no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—28no—
220C2×C4×C59—28no—
221C2×C4×C59—21no—
222C2×C4×C59—24no—
223C2×C4×C59—24no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—44no—
227C2×C4×C59—44no—
228C2×C4×C59—44no—
229C2×C4×C59—48no—
230C2×C4×C59—48no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—44no—
236C2×C4×C59#12012yes—
237C2×C4×C59#12012yes—
238C2×C4×C59C22×D412yes—
239C2×C4×C59C22×D412yes—
240C2×C4×C59H(5,𝔽2)12yes—
241C2×Dic59—24no—
242C2×Dic59—24no—
243C2×Dic59—48no—
244C2×Dic59—48no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248C2×Dic59—42no—
249Dic118—28no—
250Dic118—48no—
251Dic118—44no—
252C59⋊C8—44no—
253#23921—591nomaximal, Sylow
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—22no—
259C22×C59—22no—
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—22no—
269C22×C59—416no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59—41no—
273C22×C59—41no—
274C22×C59#45114yescomplemented
275C22×C59#45114yescomplemented
276C22×C59C2×#12011yescharacteristic
277C22×C59C2×#12011yescharacteristic
278C22×C59#53411yescharacteristic
279C22×C59C22×Q1611yescharacteristic
280C22×C59C2×#13711yescharacteristic
281C22×C59C2×#13711yescharacteristic
282C22×C59#57511yescharacteristic
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—41no—
286C4×C59—44no—
287C4×C59—44no—
288C4×C59D4214yes—
289Dic59—44no—
290Dic59—44no—
291C22×C22⋊Q8—591no—
292C22×C22⋊Q8D5911yescharacteristic
293C2×#451—592no—
294C2×#451—592no—
295#2001—594no—
296#1872—592no—
297#1872—592no—
298#1779—591no—
299C2×C59—24no—
300C2×C59—24no—
301C2×C59—42no—
302C2×C59#240311yescharacteristic
303C2×C59C2×#45111yescharacteristic
304C2×C59C2×#45111yescharacteristic
305C2×C59#300711yescharacteristic
306C2×C59#300711yescharacteristic
307C2×C59#318911yescharacteristic
308C2×C59#319511yescharacteristic
309C23×Q8—591no—
310C23×Q8C2×D5911yescharacteristic
311C24×C4C2×D5911yescharacteristic
312C22×Q16—594no—
313C2×C22⋊Q8—21no—
314C2×C22⋊Q8—21no—
315C2×C22⋊Q8—28no—
316C2×C22⋊Q8—592no—
317C2×C22⋊Q8—592no—
318C2×C22⋊Q8—1188no—
319C2×C22⋊Q8—1181no—
320C2×C22⋊Q8—1181no—
321C2×C22⋊Q8C2×D5912yes—
322C2×C22⋊Q8C2×D5912yes—
323C22×C4⋊C4—21no—
324C22×C4⋊C4—591no—
325C22×C4⋊C4—1181no—
326C22×C4⋊C4C2×D5911yescharacteristic
327C22×C22⋊C4—21no—
328C22×C22⋊C4—1181no—
329#451—1188no—
330#451—1188no—
331C2×#103—594no—
332C2×#103—594no—
333C2×C22⋊C8—592no—
334C2×C22⋊C8—592no—
335C2×C22⋊C8—592no—
336#340—594no—
337C59#2392111yescharacteristic, complemented, Sylow
338C25C22⋊D5911yescharacteristic
339C22×Q8—24no—
340C22×Q8—21no—
341C22×Q8—24no—
342C22×Q8—42no—
343C22×Q8—592no—
344C22×Q8—1184no—
345C22×Q8—1181no—
346C22×Q8—1184no—
347C22×Q8—2362no—
348C22×Q8C22×D5912yes—
349C23×C4—21no—
350C23×C4—21no—
351C23×C4—24no—
352C23×C4—24no—
353C23×C4—42no—
354C23×C4—1181no—
355C23×C4—1181no—
356C23×C4C22⋊D5911yescharacteristic
357C23×C4C22⋊D5911yescharacteristic
358C23×C4C22⋊D5911yescharacteristic
359C23×C4C22×D5911yescharacteristic
360C23×C4C22×D5911yescharacteristic
361C23×C4C22×D5911yescharacteristic
362C2×Q16—1188no—
363C2×Q16—1188no—
364C2×Q16—2364no—
365C22×C8—594no—
366C22⋊Q8—28no—
367C22⋊Q8—28no—
368C22⋊Q8—24no—
369C22⋊Q8—24no—
370C22⋊Q8—24no—
371C22⋊Q8—598no—
372C22⋊Q8—1184no—
373C22⋊Q8—1184no—
374C22⋊Q8—1188no—
375C22⋊Q8—1188no—
376C22⋊Q8—1184no—
377C22⋊Q8C22⋊D5918yes—
378C2×C4⋊C4—28no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—592no—
384C2×C4⋊C4—1184no—
385C2×C4⋊C4—1188no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1182no—
388C2×C4⋊C4—1182no—
389C2×C4⋊C4C22×D5912yes—
390C2×C22⋊C4—28no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—1188no—
394C2×C22⋊C4—1182no—
395C2×C22⋊C4—1182no—
396#103—1188no—
397#103—1188no—
398C22⋊C8—1184no—
399C22⋊C8—1184no—
400C22⋊C8—2362no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—41no—
406C24—41no—
407C24#1174567214yescomplemented
408C24C2×C22⋊D5911yescharacteristic
409C24C2×C22⋊D5911yescharacteristic
410C24C2×C22⋊D5911yescharacteristic
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—416no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—594no—
419C2×Q8—1184no—
420C2×Q8—1182no—
421C2×Q8—1184no—
422C2×Q8—1184no—
423C2×Q8—23616no—
424C2×Q8—2362no—
425C2×Q8—2362no—
426C2×Q8C2×C22⋊D5914yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—416no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—1184no—
453C22×C4—1184no—
454C22×C4—1182no—
455C22×C4—1182no—
456C22×C4—2362no—
457C22×C4—2362no—
458C22×C4—2362no—
459C22×C4—2362no—
460C22×C4D4×D5914yes—
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C2×C22⋊D5914yes—
465C22×C4C23×D5911yescharacteristic
466Q16—23616no—
467C2×C8—1184no—
468C2×C8—1184no—
469C2×C8—1184no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—22no—
474C4⋊C4—594no—
475C4⋊C4—1188no—
476C4⋊C4—1184no—
477C4⋊C4—1188no—
478C4⋊C4—1182no—
479C4⋊C4C2×C22⋊D5914yes—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—1188no—
483C22⋊C4—1188no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—28no—
498C23—28no—
499C23—21no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23#1427369512yes—
506C23#1427369612yes—
507C23C2×#1174567212yes—
508C23C2×#1174567212yes—
509C23C2×#1174567212yes—
510C23#1427375712yes—
511C23#1427375812yes—
512C23C22×C22⋊D5911yescharacteristic
513Q8—28no—
514Q8—48no—
515Q8—44no—
516Q8—1188no—
517Q8—2368no—
518Q8—2364no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—1184no—
541C2×C4—1184no—
542C2×C4—2368no—
543C2×C4—2368no—
544C2×C4—2362no—
545C2×C4—2362no—
546C2×C4—2362no—
547C2×C4—2362no—
548C2×C4#1427369512yes—
549C2×C4#1427369612yes—
550C2×C4C2×D4×D5912yes—
551C2×C4C22×C22⋊D5912yes—
552C2×C4#1427377212yes—
553C8—2364no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—416no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22#1683755714yescomplemented
575C22#1683755814yescomplemented
576C22C2×#1427369511yescharacteristic
577C22C2×#1427369611yescharacteristic
578C22#1683796111yescharacteristic
579C22C22×#1174567211yescharacteristic
580C22C2×#1427375711yescharacteristic
581C22C2×#1427375811yescharacteristic
582C22#1683820011yescharacteristic
583C4—22no—
584C4—22no—
585C4—44no—
586C4—41no—
587C4—44no—
588C4—2364no—
589C4—2364no—
590C4D4×C22⋊D5914yes—
591C2—24no—
592C2—24no—
593C2—42no—
594C2#2142740611yescharacteristic
595C2C2×#1683755711yescharacteristic
596C2C2×#1683755811yescharacteristic
597C2#2143335411yescharacteristic
598C2#2143335611yescharacteristic
599C2#2143486811yescharacteristic
600C2#2143491711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC22×C22⋊Q8×C59C2
socleC23×C59C22×D4
prime radicalC25×C59D4
hypercenterC22×C22⋊Q8D59
hypocenterC59#23921
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16837558C2semidirect
3. C2×#16837557C2semidirect
4. #21425104C2semidirect
5. #21424404C2semidirect
6. #21424403C2semidirect
10. C22×#11745672C22semidirect
21. C2×#14273651C22semidirect
22. C2×#14273650C22semidirect
26. #16837164C22semidirect
106. C24×C59Q16semidirect
274. C22×C59#451semidirect
275. C22×C59#451semidirect
337. C59#23921semidirect
407. C24#11745672semidirect
574. C22#16837557semidirect
575. C22#16837558semidirect

Maximal subgroups

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

subgroupindex
C22×#142737062
C2×#168375582
C2×#168375572
#214251042
#214244042
#214244032
#214240062
C22×C22⋊Q8×C592
#2392159