Finite Group Databasebeta

← #28520277

Subgroups — #28520277

Order 15104.

3494 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837338C212yesmaximal, complemented
2C2×#16837275C212yesmaximal, complemented
3C2×#16837262C211yesmaximal, characteristic, complemented
4C2×#16837258C212yesmaximal
5#21422341C212yesmaximal, complemented
6#21422339C214yesmaximal, complemented
7#21422319C211yesmaximal, characteristic, complemented
8C23×C22⋊C4×C59C211yesmaximal, characteristic
9C22×C4⋊Dic59C2211yescharacteristic, complemented
10C22×C4⋊Dic59C2211yescharacteristic, complemented
11C22×Dic59⋊C4C2214yescomplemented
12C22×C22⋊Dic59C2212yes—
13C22×C22⋊Dic59C2214yes—
14C22×C22⋊Dic59C2214yes—
15C22×C22⋊Dic59C2212yes—
16#16837338—28no—
17#16837275—28no—
18#16837262—24no—
19#16837258—28no—
20C2×C59⋊#95C2212yescomplemented
21C2×C59⋊#95C2214yescomplemented
22C2×C59⋊#95C2212yescomplemented
23C2×C59⋊#95C2212yescomplemented
24C26×C59C2211yescharacteristic
25C24×C4×C59C2211yescharacteristic
26C24×C4×C59C2211yescharacteristic
27C22×C22⋊C4×C59—24no—
28C22×C22⋊C4×C59—28no—
29C22×C22⋊C4×C59C2212yes—
30C22×C22⋊C4×C59C2212yes—
31C23×Dic59C2312yes—
32C23×Dic59C2312yes—
33C23×Dic59C2314yes—
34C2×C4⋊Dic59—22no—
35C2×C4⋊Dic59—22no—
36C2×C4⋊Dic59—24no—
37C2×C4⋊Dic59—42no—
38C2×Dic59⋊C4—28no—
39C2×Dic59⋊C4—28no—
40C2×Dic59⋊C4—44no—
41C2×C22⋊Dic59—24no—
42C2×C22⋊Dic59—24no—
43C2×C22⋊Dic59—24no—
44C2×C22⋊Dic59—24no—
45C2×C22⋊Dic59—24no—
46C2×C22⋊Dic59—28no—
47C2×C22⋊Dic59—28no—
48C2×C22⋊Dic59—28no—
49C2×C22⋊Dic59—28no—
50C2×C22⋊Dic59—44no—
51C2×C22⋊Dic59—44no—
52C2×C22⋊Dic59D418yes—
53C59⋊#95—24no—
54C59⋊#95—24no—
55C59⋊#95—28no—
56C59⋊#95—42no—
57C25×C59—24no—
58C25×C59—22no—
59C25×C59—24no—
60C25×C59—22no—
61C25×C59—24no—
62C25×C59—24no—
63C25×C59—42no—
64C25×C59—42no—
65C25×C59Q814yes—
66C25×C59C2312yes—
67C25×C59C2311yescharacteristic
68C23×C4×C59—22no—
69C23×C4×C59—24no—
70C23×C4×C59—24no—
71C23×C4×C59—24no—
72C23×C4×C59—24no—
73C23×C4×C59—42no—
74C23×C4×C59—42no—
75C23×C4×C59D414yes—
76C23×C4×C59C2311yescharacteristic
77C23×C4×C59C2312yes—
78C23×C4×C59C2311yescharacteristic
79C2×C22⋊C4×C59—232no—
80C2×C22⋊C4×C59—24no—
81C2×C22⋊C4×C59—24no—
82C2×C22⋊C4×C59—24no—
83C2×C22⋊C4×C59—24no—
84C2×C22⋊C4×C59—24no—
85C2×C22⋊C4×C59D418yes—
86C24×C59—22no—
87C24×C59—22no—
88C24×C59—22no—
89C24×C59—22no—
90C24×C59—28no—
91C24×C59—28no—
92C24×C59—216no—
93C24×C59—216no—
94C24×C59—28no—
95C24×C59—22no—
96C24×C59—22no—
97C24×C59—24no—
98C24×C59—22no—
99C24×C59—22no—
100C24×C59—24no—
101C24×C59—24no—
102C24×C59—24no—
103C24×C59—216no—
104C24×C59—48no—
105C24×C59—416no—
106C24×C59—416no—
107C24×C59—416no—
108C24×C59—416no—
109C24×C59—44no—
110C24×C59—44no—
111C24×C59—48no—
112C24×C59—44no—
113C24×C59—48no—
114C24×C59—44no—
115C24×C59C2×D414yes—
116C24×C59C2×D414yes—
117C24×C59C2×D412yes—
118C24×C59C2×Q814yes—
119C24×C59C2×Q812yes—
120C24×C59#4114yes—
121C24×C59#4112yes—
122C24×C59#4114yes—
123C24×C59C2411yescharacteristic
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—28no—
127C22×C4×C59—28no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—24no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—24no—
135C22×C4×C59—28no—
136C22×C4×C59—44no—
137C22×C4×C59—416no—
138C22×C4×C59—416no—
139C22×C4×C59—44no—
140C22×C4×C59—44no—
141C22×C4×C59C2×D412yes—
142C22×C4×C59C2×D414yes—
143C22×C4×C59C2×D412yes—
144C22×C4×C59C2×D414yes—
145C22×Dic59—24no—
146C22×Dic59—24no—
147C22×Dic59—24no—
148C22×Dic59—24no—
149C22×Dic59—24no—
150C22×Dic59—24no—
151C22×Dic59—22no—
152C22×Dic59—22no—
153C22×Dic59—22no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—24no—
158C22×Dic59—24no—
159C22×Dic59—24no—
160C22×Dic59—24no—
161C22×Dic59C2×D418yes—
162C22⋊C4×C59—216no—
163C22⋊C4×C59—216no—
164C22⋊Dic59—216no—
165C22⋊Dic59—416no—
166C22⋊Dic59—416no—
167C22⋊Dic59—48no—
168C4⋊Dic59—44no—
169C4⋊Dic59—44no—
170Dic59⋊C4—416no—
171C23×C59—22no—
172C23×C59—24no—
173C23×C59—22no—
174C23×C59—24no—
175C23×C59—22no—
176C23×C59—22no—
177C23×C59—24no—
178C23×C59—24no—
179C23×C59—28no—
180C23×C59—28no—
181C23×C59—28no—
182C23×C59—28no—
183C23×C59—232no—
184C23×C59—24no—
185C23×C59—28no—
186C23×C59—24no—
187C23×C59—24no—
188C23×C59—24no—
189C23×C59—24no—
190C23×C59—24no—
191C23×C59—28no—
192C23×C59—28no—
193C23×C59—28no—
194C23×C59—28no—
195C23×C59—28no—
196C23×C59—464no—
197C23×C59—416no—
198C23×C59—416no—
199C23×C59—464no—
200C23×C59—416no—
201C23×C59—416no—
202C23×C59—416no—
203C23×C59—42no—
204C23×C59—44no—
205C23×C59—42no—
206C23×C59—42no—
207C23×C59—42no—
208C23×C59—44no—
209C23×C59—44no—
210C23×C59—44no—
211C23×C59—44no—
212C23×C59—44no—
213C23×C59—416no—
214C23×C59C22⋊Q818yes—
215C23×C59C22⋊Q818yes—
216C23×C59C22⋊Q814yes—
217C23×C59#12314yes—
218C23×C59#12318yes—
219C23×C59#12714yes—
220C23×C59C22×D412yes—
221C23×C59C22×D411yescharacteristic
222C23×C59C22×D412yes—
223C23×C59C22×Q811yescharacteristic
224C23×C59C2×#4112yes—
225C23×C59C2×#4111yescharacteristic
226C23×C59C2×#4112yes—
227C23×C59H(5,𝔽2)12yes—
228C23×C59H(5,𝔽2)12yes—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—28no—
232C2×C4×C59—22no—
233C2×C4×C59—24no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—28no—
237C2×C4×C59—416no—
238C2×C4×C59—416no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59—44no—
242C2×C4×C59—44no—
243C2×C4×C59#12014yes—
244C2×Dic59—28no—
245C2×Dic59—28no—
246C2×Dic59—28no—
247C2×Dic59—44no—
248C2×Dic59—44no—
249C2×Dic59—44no—
250C2×Dic59—44no—
251C2×Dic59—42no—
252C2×Dic59—42no—
253C2×Dic59—42no—
254C2×Dic59—42no—
255C2×Dic59—42no—
256C2×Dic59—42no—
257C2×Dic59—44no—
258C2×Dic59—44no—
259C2×Dic59—44no—
260C2×Dic59—44no—
261#14734—591nomaximal, Sylow
262C22×C59—24no—
263C22×C59—24no—
264C22×C59—24no—
265C22×C59—216no—
266C22×C59—24no—
267C22×C59—216no—
268C22×C59—24no—
269C22×C59—24no—
270C22×C59—28no—
271C22×C59—216no—
272C22×C59—464no—
273C22×C59—44no—
274C22×C59—44no—
275C22×C59—44no—
276C22×C59—48no—
277C22×C59—48no—
278C22×C59—44no—
279C22×C59—48no—
280C22×C59—44no—
281C22×C59—44no—
282C22×C59#39918yescomplemented
283C22×C59C2×#12011yescharacteristic
284C22×C59C2×C22⋊Q812yes—
285C22×C59C2×C22⋊Q811yescharacteristic
286C22×C59C2×C22⋊Q812yes—
287C22×C59C2×#12311yescharacteristic
288C22×C59C2×#12312yes—
289C22×C59#52512yes—
290C22×C59#52511yescharacteristic
291C22×C59C2×#12711yescharacteristic
292C22×C59#53411yescharacteristic
293C22×C59#53511yescharacteristic
294C22×C59#53812yes—
295C22×C59#54312yes—
296C22×C59D4214yes—
297C22×C59#54614yes—
298C22×C59#54614yes—
299C22×C59#55114yes—
300C4×C59—24no—
301C4×C59—42no—
302C4×C59—44no—
303Dic59—48no—
304Dic59—82no—
305Dic59—82no—
306C23×C22⋊C4D5911yescharacteristic
307C2×#399—592no—
308C2×#385—592no—
309C2×#380—591no—
310C2×#379—592no—
311#1439—594no—
312#1439—592no—
313C22⋊#95—591no—
314C2×C59—28no—
315C2×C59—48no—
316C2×C59C2×#39911yescharacteristic
317C2×C59#240511yescharacteristic
318C2×C59#241814yes—
319C2×C59#243311yescharacteristic
320C2×C59#243714yes—
321C2×C59#248211yescharacteristic
322C2×C59#253911yescharacteristic
323C2×C59#262611yescharacteristic
324C2×C59#284811yescharacteristic
325C26C2×D5911yescharacteristic
326C24×C4C2×D5911yescharacteristic
327C24×C4C2×D5911yescharacteristic
328C22×C4⋊C4—591no—
329C22×C4⋊C4—594no—
330C22×C4⋊C4—591no—
331C22×C22⋊C4—24no—
332C22×C22⋊C4—28no—
333C22×C22⋊C4—592no—
334C22×C22⋊C4—592no—
335C22×C22⋊C4—594no—
336C22×C22⋊C4—594no—
337C22×C22⋊C4C2×D5912yes—
338C22×C22⋊C4C2×D5912yes—
339#399—1188no—
340#385—1188no—
341#380—1184no—
342#379—1188no—
343C2×#95—592no—
344C2×#95—592no—
345C2×#95—594no—
346C2×#95—592no—
347C59#1473411yescharacteristic, complemented, Sylow
348C25—24no—
349C25—22no—
350C25—24no—
351C25—22no—
352C25—24no—
353C25—24no—
354C25—42no—
355C25—42no—
356C25Dic11814yes—
357C25C22×D5912yes—
358C25C22×D5911yescharacteristic
359C23×C4—24no—
360C23×C4—22no—
361C23×C4—24no—
362C23×C4—24no—
363C23×C4—24no—
364C23×C4—42no—
365C23×C4—42no—
366C23×C4—594no—
367C23×C4—592no—
368C23×C4—592no—
369C23×C4C22⋊D5914yes—
370C23×C4C22×D5912yes—
371C23×C4C22×D5911yescharacteristic
372C23×C4C22×D5911yescharacteristic
373C2×C4⋊C4—1188no—
374C2×C4⋊C4—1182no—
375C2×C4⋊C4—1182no—
376C2×C4⋊C4—1188no—
377C2×C4⋊C4—1184no—
378C2×C4⋊C4—2364no—
379C2×C4⋊C4—2362no—
380C2×C22⋊C4—232no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—598no—
387C2×C22⋊C4—1184no—
388C2×C22⋊C4—1184no—
389C2×C22⋊C4—1188no—
390C2×C22⋊C4—1188no—
391C2×C22⋊C4—1188no—
392C2×C22⋊C4—1184no—
393C2×C22⋊C4—1184no—
394C2×C22⋊C4—1184no—
395C2×C22⋊C4—1188no—
396C2×C22⋊C4—2364no—
397C2×C22⋊C4—2364no—
398C2×C22⋊C4C22⋊D5918yes—
399#95—1188no—
400#95—1184no—
401#95—1184no—
402#95—2362no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—28no—
408C24—216no—
409C24—28no—
410C24—216no—
411C24—28no—
412C24—24no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—22no—
417C24—22no—
418C24—24no—
419C24—24no—
420C24—216no—
421C24—48no—
422C24—416no—
423C24—416no—
424C24—416no—
425C24—416no—
426C24—44no—
427C24—44no—
428C24—48no—
429C24—44no—
430C24—48no—
431C24—44no—
432C24C2×Dic11814yes—
433C24C2×Dic11812yes—
434C24#1174568514yes—
435C24#1174568512yes—
436C24D4×D5914yes—
437C24#1174568714yes—
438C24C2×C22⋊D5914yes—
439C24C2×C22⋊D5912yes—
440C24C23×D5911yescharacteristic
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—28no—
453C22×C4—44no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—598no—
459C22×C4—1184no—
460C22×C4—1184no—
461C22×C4—1184no—
462C22×C4—1184no—
463C22×C4—1184no—
464C22×C4—1184no—
465C22×C4—1182no—
466C22×C4—1182no—
467C22×C4—1182no—
468C22×C4—1182no—
469C22×C4—1182no—
470C22×C4—1182no—
471C22×C4—1184no—
472C22×C4—1184no—
473C22×C4—1184no—
474C22×C4—1184no—
475C22×C4C2×C22⋊D5912yes—
476C22×C4C2×C22⋊D5914yes—
477C22×C4C2×C22⋊D5914yes—
478C22×C4C2×C22⋊D5912yes—
479C4⋊C4—23616no—
480C4⋊C4—2364no—
481C4⋊C4—2364no—
482C22⋊C4—216no—
483C22⋊C4—216no—
484C22⋊C4—11816no—
485C22⋊C4—23616no—
486C22⋊C4—23616no—
487C22⋊C4—2368no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—232no—
501C23—24no—
502C23—28no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—464no—
514C23—416no—
515C23—416no—
516C23—464no—
517C23—416no—
518C23—416no—
519C23—416no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—416no—
531C23#1427370418yes—
532C23#1427370618yes—
533C23#1427370614yes—
534C23#1427370814yes—
535C23#1427370918yes—
536C23#1427372314yes—
537C23C2×D4×D5912yes—
538C23C22×C22⋊D5911yescharacteristic
539C23C22×C22⋊D5912yes—
540C23C22×Dic11811yescharacteristic
541C23C2×#1174568712yes—
542C23C2×#1174568511yescharacteristic
543C23C2×#1174568512yes—
544C23#1427377212yes—
545C23#1427377212yes—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—416no—
555C2×C4—416no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—1188no—
561C2×C4—1188no—
562C2×C4—1188no—
563C2×C4—2362no—
564C2×C4—2362no—
565C2×C4—2362no—
566C2×C4—2362no—
567C2×C4—2362no—
568C2×C4—2362no—
569C2×C4—2364no—
570C2×C4—2364no—
571C2×C4—2364no—
572C2×C4—2364no—
573C2×C4—2364no—
574C2×C4—2364no—
575C2×C4—2364no—
576C2×C4—2364no—
577C2×C4#1427369614yes—
578C22—24no—
579C22—24no—
580C22—216no—
581C22—24no—
582C22—216no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—216no—
587C22—28no—
588C22—464no—
589C22—44no—
590C22—44no—
591C22—48no—
592C22—48no—
593C22—48no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22#1683733818yescomplemented
599C22C2×#1427369611yescharacteristic
600C22C2×#1427370412yes—
601C22C2×#1427370612yes—
602C22C2×#1427370611yescharacteristic
603C22C2×#1427370811yescharacteristic
604C22C2×#1427370912yes—
605C22#1683791311yescharacteristic
606C22#1683791312yes—
607C22C2×#1427372311yescharacteristic
608C22#1683796111yescharacteristic
609C22#1683796911yescharacteristic
610C22#1683799012yes—
611C22#1683802812yes—
612C22D4×C22⋊D5914yes—
613C22#1683804214yes—
614C22#1683804614yes—
615C22#1683807814yes—
616C4—24no—
617C4—44no—
618C4—42no—
619C4—4722no—
620C4—4722no—
621C4—2368no—
622C2—28no—
623C2—48no—
624C2C2×#1683733811yescharacteristic
625C2#2142741911yescharacteristic
626C2#2142755814yes—
627C2#2142772911yescharacteristic
628C2#2142775714yes—
629C2#2142837011yescharacteristic
630C2#2142902711yescharacteristic
631C2#2142990111yescharacteristic
632C2#2143238311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D59
commutator, socle, cosocleC24×C59C24
FittingC23×C22⋊C4×C59C2
prime radicalC26×C59C22
hypercenterC23×C22⋊C4D59
hypocenterC59#14734
LCSG ▷ C24×C59 ▷ C59
UCSC1 ◁ C24 ◁ C23×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16837338C2semidirect
2. C2×#16837275C2semidirect
3. C2×#16837262C2semidirect
5. #21422341C2semidirect
6. #21422339C2semidirect
7. #21422319C2semidirect
9. C22×C4⋊Dic59C22semidirect
10. C22×C4⋊Dic59C22semidirect
11. C22×Dic59⋊C4C22semidirect
20. C2×C59⋊#95C22semidirect
21. C2×C59⋊#95C22semidirect
22. C2×C59⋊#95C22semidirect
23. C2×C59⋊#95C22semidirect
282. C22×C59#399semidirect
347. C59#14734semidirect
598. C22#16837338semidirect

Maximal subgroups

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

subgroupindex
C2×#168373382
C2×#168372752
C2×#168372622
C2×#168372582
#214223412
#214223392
#214223192
C23×C22⋊C4×C592
#1473459