Finite Group Databasebeta

← #28624057

Subgroups — #28624057

Order 15104.

1490 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837604C211yesmaximal, characteristic, complemented
2C2×#16837541C212yesmaximal, complemented
3C2×C4×#11745671C211yesmaximal, characteristic, complemented
4#21429009C211yesmaximal, characteristic
5#21425289C212yesmaximal, complemented
6#21425097C212yesmaximal, complemented
7#21424899C211yesmaximal, characteristic, complemented
8#21424891C211yesmaximal, characteristic, complemented
9#21424101C211yesmaximal, characteristic, complemented
10#21424031C212yesmaximal, complemented
11C2×D4×Q8×C59C211yesmaximal, characteristic, complemented
12C22×#11745671C2212yes—
13C22×#11745671C2211yescharacteristic
14C2×#14273701C2212yes—
15C2×#14273698—21no—
16C2×C4×D236C2211yescharacteristic
17#16837604—24no—
18#16837541—28no—
19C4×#11745671—24no—
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273651C2212yes—
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2212yes—
25C2×#14273651C2212yes—
26C2×#14273651C2212yes—
27C2×#14273647C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2212yes—
29C2×C22⋊(C59⋊C8)C2212yes—
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837318—22no—
32#16837272C2211yescharacteristic
33#16837262C2212yes—
34#16837259—21no—
35#16837164C2212yes—
36C23×Q8×C59C2212yes—
37D4×Q8×C59—24no—
38D4×Q8×C59—24no—
39C2×C4⋊Q8×C59—21no—
40C2×C4⋊Q8×C59C2211yescharacteristic
41C2×C22⋊Q8×C59—22no—
42C2×C22⋊Q8×C59C2212yes—
43C2×C4×Q8×C59C2211yescharacteristic
44C2×C4×D4×C59—21no—
45C2×C4×D4×C59C2211yescharacteristic
46C22×D236C2311yescharacteristic
47C22×C22⋊D59—22no—
48C22×C4×D59—21no—
49C2×#11745671—22no—
50C2×#11745671—22no—
51C2×#11745671—24no—
52C2×#11745671—24no—
53C2×#11745671—42no—
54C2×#11745671D414yescomplemented
55C22×C59⋊C8C2311yescharacteristic
56C22×C59⋊C8C2312yes—
57C22×C59⋊C8C2311yescharacteristic
58#14273701—24no—
59#14273701—42no—
60#14273698—44no—
61C4×D236—22no—
62C4×D236—22no—
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59C2312yes—
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C22⋊Dic59—22no—
68C2×(C2×D59)⋊C4—22no—
69C2×(C2×D59)⋊C4—21no—
70#14273658—22no—
71C4⋊(C59⋊C8)—22no—
72#14273651—22no—
73#14273651—24no—
74#14273651—24no—
75#14273651—24no—
76#14273651—42no—
77#14273647—22no—
78C22⋊(C59⋊C8)—24no—
79C22⋊(C59⋊C8)—42no—
80C4×C59⋊C8—22no—
81C59⋊#95—22no—
82C22×Q8×C59—24no—
83C22×Q8×C59—24no—
84C22×Q8×C59—22no—
85C22×Q8×C59—24no—
86C22×Q8×C59—24no—
87C22×Q8×C59—42no—
88C22×Q8×C59—42no—
89C22×Q8×C59C2312yes—
90C22×Q8×C59C2311yescharacteristic
91C22×D4×C59D411yescharacteristic
92C23×C4×C59—22no—
93C23×C4×C59C2312yes—
94C4⋊Q8×C59—28no—
95C4⋊Q8×C59—22no—
96C4⋊Q8×C59D414yescomplemented
97C22⋊Q8×C59—216no—
98C22⋊Q8×C59—24no—
99C22⋊Q8×C59—24no—
100C4×Q8×C59—22no—
101C4×Q8×C59D414yescomplemented
102C4×D4×C59—28no—
103C4×D4×C59—22no—
104C4×D4×C59—22no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—22no—
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—22no—
112C2×C22⋊C4×C59D412yes—
113C2×C42×C59—21no—
114C2×C42×C59C2311yescharacteristic
115C24×C59C2×D412yes—
116C23×D59—21no—
117C2×Q8×C59—28no—
118C2×Q8×C59—21no—
119C2×Q8×C59—22no—
120C2×Q8×C59—21no—
121C2×Q8×C59—21no—
122C2×Q8×C59—22no—
123C2×Q8×C59—22no—
124C2×Q8×C59—22no—
125C2×Q8×C59—22no—
126C2×Q8×C59—44no—
127C2×Q8×C59—44no—
128C2×Q8×C59—48no—
129C2×Q8×C59—48no—
130C2×Q8×C59—48no—
131C2×Q8×C59—48no—
132C2×Q8×C59C2×D414yes—
133C2×Q8×C59C2×D412yes—
134C2×Q8×C59C2×D412yes—
135C2×Q8×C59#4112yes—
136C2×D4×C59—22no—
137C2×D4×C59—22no—
138C2×D4×C59SD1614yescomplemented
139C22×C4×C59—21no—
140C22×C4×C59—28no—
141C22×C4×C59—21no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—22no—
148C22×C4×C59—22no—
149C22×C4×C59—44no—
150C22×C4×C59—44no—
151C22×C4×C59C2×D412yes—
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2411yescharacteristic
155C2×C22⋊D59—44no—
156C2×C22⋊D59—44no—
157C2×C22⋊D59—82no—
158C22×Dic59—21no—
159C22×Dic59—22no—
160C2×D236—21no—
161C2×D236—21no—
162C2×D236—21no—
163C2×D236—21no—
164C2×D236—21no—
165C2×D236C2×D412yes—
166C2×C4×D59—42no—
167C2×C4×D59—42no—
168C2×C4×D59—42no—
169C4⋊C4×C59—28no—
170C4⋊C4×C59—24no—
171C4⋊C4×C59—24no—
172C4⋊C4×C59—24no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59C2×D412yes—
176C22⋊C4×C59—28no—
177C22⋊C4×C59—24no—
178C42×C59—24no—
179C42×C59C2×D412yes—
180C42×C59C2×D412yes—
181C22⋊Dic59—44no—
182#11745671—28no—
183#11745671—48no—
184(C2×D59)⋊C4—44no—
185(C2×D59)⋊C4—42no—
186C4⋊Dic59—21no—
187C4⋊Dic59—22no—
188C4⋊Dic59—21no—
189C4⋊Dic59—22no—
190C4⋊Dic59—42no—
191C4⋊Dic59—81no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—22no—
196C2×C59⋊C8—22no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—22no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8C2×D412yes—
201C23×C59—22no—
202C23×C59—22no—
203C23×C59—22no—
204C23×C59—22no—
205C23×C59—22no—
206C23×C59—22no—
207C23×C59C2×SD1614yes—
208C23×C59C22×D411yescharacteristic
209C22×D59—22no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—41no—
214C22×D59—41no—
215C22×D59—41no—
216Q8×C59—22no—
217Q8×C59—22no—
218Q8×C59—22no—
219Q8×C59—24no—
220Q8×C59—24no—
221Q8×C59—416no—
222Q8×C59—42no—
223Q8×C59—42no—
224Q8×C59#12114yescomplemented
225D4×C59—28no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—22no—
232C2×C4×C59—24no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—416no—
237C2×C4×C59—42no—
238C2×C4×C59—42no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59#12012yes—
246C2×C4×C59#12012yes—
247C2×C4×C59C2×SD1612yes—
248C2×C4×C59#13712yes—
249C2×C4×C59C22×D411yescharacteristic
250C2×C4×C59C22×D411yescharacteristic
251C2×C4×C59C22×D411yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59H(5,𝔽2)11yescharacteristic
254C2×C4×C59H(5,𝔽2)11yescharacteristic
255C22⋊D59—88no—
256C2×Dic59—41no—
257C2×Dic59—42no—
258C2×Dic59—42no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—42no—
262D236—22no—
263D236—22no—
264D236—41no—
265D236—41no—
266C4×D59—84no—
267C59⋊C8—22no—
268C59⋊C8—41no—
269C59⋊C8—42no—
270#24239—591nomaximal, Sylow
271C22×C59—24no—
272C22×C59—24no—
273C22×C59—24no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59C2×#12011yescharacteristic
279C22×C59D4211yescharacteristic
280C22×C59#54611yescharacteristic
281C22×C59C22×SD1611yescharacteristic
282C22×C59C2×#13711yescharacteristic
283C22×C59#57711yescharacteristic
284C22×C59#57811yescharacteristic
285C2×D59—42no—
286C2×D59—42no—
287C2×D59—42no—
288C2×D59—81no—
289C2×D59—81no—
290C2×D59—81no—
291C2×D59—81no—
292C4×C59—24no—
293C4×C59—22no—
294C4×C59—42no—
295C4×C59—44no—
296C4×C59—42no—
297C4×C59#44814yescomplemented
298C4×C59C2×#12111yescharacteristic
299C4×C59#53511yescharacteristic
300C4×C59D4211yescharacteristic
301C4×C59#54611yescharacteristic
302Dic59—81no—
303Dic59—82no—
304C2×D4×Q8D5911yescharacteristic, complemented
305C2×#461—591no—
306C2×#448—592no—
307C2×C4×SD16—591no—
308#2537—591no—
309#2035—592no—
310#2000—592no—
311#1956—591no—
312#1954—591no—
313#1803—591no—
314#1787—592no—
315C2×C59—44no—
316C2×C59#240811yescharacteristic
317C2×C59C2×#44811yescharacteristic
318C2×C59#301311yescharacteristic
319C2×C59#327611yescharacteristic
320C2×C59D4×SD1611yescharacteristic
321C2×C59#329811yescharacteristic
322C2×C59#332911yescharacteristic
323D59—82no—
324C23×Q8C2×D5912yes—
325C22×SD16—591no—
326C22×SD16—592no—
327D4×Q8—24no—
328D4×Q8—24no—
329C2×C4⋊Q8—21no—
330C2×C4⋊Q8C2×D5911yescharacteristic
331C2×C22⋊Q8—22no—
332C2×C22⋊Q8C2×D5912yes—
333C2×#121—592no—
334C2×#121—1181no—
335C2×C4×Q8C2×D5911yescharacteristic
336C2×C4×D4—21no—
337C2×C4×D4—591no—
338C2×C4×D4C2×D5911yescharacteristic
339#461—1184no—
340#448—1188no—
341C4×SD16—1184no—
342C2×#106—591no—
343C2×C4⋊C8—591no—
344C2×#103—592no—
345C2×#103—591no—
346C2×#103—592no—
347C2×#103—592no—
348C2×#103—592no—
349C2×#102—591no—
350C2×C22⋊C8—592no—
351C2×C22⋊C8—592no—
352C2×C4×C8—591no—
353#394—1182no—
354#384—591no—
355#380—592no—
356#380—1181no—
357#340—592no—
358C59#2423911yescharacteristic, complemented, Sylow
359C22×Q8—24no—
360C22×Q8—24no—
361C22×Q8—22no—
362C22×Q8—24no—
363C22×Q8—24no—
364C22×Q8—42no—
365C22×Q8—42no—
366C22×Q8C22×D5912yes—
367C22×Q8C22×D5911yescharacteristic
368C22×D4—591no—
369C22×D4—1182no—
370C22×D4C22⋊D5911yescharacteristic
371C23×C4—22no—
372C23×C4—1181no—
373C23×C4C22×D5912yes—
374C2×SD16—594no—
375C2×SD16—1182no—
376C2×SD16—1182no—
377C2×SD16—1184no—
378C2×SD16—1184no—
379C2×SD16—2362no—
380C22×C8—591no—
381C22×C8—592no—
382C22×C8—591no—
383C4⋊Q8—28no—
384C4⋊Q8—22no—
385C4⋊Q8C22⋊D5914yescomplemented
386C22⋊Q8—216no—
387C22⋊Q8—24no—
388C22⋊Q8—24no—
389#121—1184no—
390#121—2362no—
391#121—2364no—
392C4×Q8—22no—
393C4×Q8C22⋊D5914yescomplemented
394C4×D4—28no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—1182no—
398C4×D4—1182no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—592no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4C22⋊D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C22⋊C4—22no—
410C2×C22⋊C4—1182no—
411C2×C22⋊C4—1181no—
412C2×C22⋊C4—1182no—
413C2×C22⋊C4C22⋊D5912yes—
414C2×C42—21no—
415C2×C42C22×D5911yescharacteristic
416#106—1182no—
417C4⋊C8—1182no—
418#103—1184no—
419#103—1184no—
420#103—1182no—
421#103—1184no—
422#103—2362no—
423#102—1182no—
424C22⋊C8—1184no—
425C22⋊C8—2362no—
426C4×C8—1182no—
427#95—1182no—
428C24—1181no—
429C24C2×C22⋊D5912yes—
430C2×Q8—28no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—48no—
442C2×Q8—48no—
443C2×Q8—48no—
444C2×Q8—48no—
445C2×Q8D4×D5912yes—
446C2×Q8#1174568712yes—
447C2×Q8C2×C22⋊D5914yes—
448C2×Q8C2×C22⋊D5912yes—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—592no—
452C2×D4—1181no—
453C2×D4—1181no—
454C2×D4—1181no—
455C2×D4—1181no—
456C2×D4—1181no—
457C2×D4—4722no—
458C2×D4—2364no—
459C2×D4—2364no—
460C2×D4#1174567114yescomplemented
461C22×C4—21no—
462C22×C4—28no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—1181no—
474C22×C4—1182no—
475C22×C4—2362no—
476C22×C4—2362no—
477C22×C4—2362no—
478C22×C4C2×C22⋊D5912yes—
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C2×C22⋊D5911yescharacteristic
481C22×C4C23×D5911yescharacteristic
482SD16—1188no—
483SD16—2368no—
484C2×C8—592no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1182no—
488C2×C8—1181no—
489C2×C8—1182no—
490C2×C8—1181no—
491C2×C8—1182no—
492C2×C8—1181no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—1182no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1182no—
503C4⋊C4—4721no—
504C4⋊C4—2362no—
505C4⋊C4C2×C22⋊D5912yes—
506C22⋊C4—28no—
507C22⋊C4—24no—
508C22⋊C4—2364no—
509C22⋊C4—2364no—
510C22⋊C4—2362no—
511C42—24no—
512C42C2×C22⋊D5912yes—
513C42C2×C22⋊D5912yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—1182no—
521C23—2361no—
522C23—2361no—
523C23—2361no—
524C23—2361no—
525C23—2361no—
526C23—2361no—
527C23C2×#1174567114yes—
528C23C22×C22⋊D5911yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—24no—
533Q8—24no—
534Q8—416no—
535Q8—42no—
536Q8—42no—
537Q8#1427369814yescomplemented
538D4—28no—
539D4—1182no—
540D4—1182no—
541D4—4728no—
542D4—2361no—
543D4—2361no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—416no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—4724no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2362no—
567C2×C4—2361no—
568C2×C4—2362no—
569C2×C4—2362no—
570C2×C4#1427369612yes—
571C2×C4#1427369612yes—
572C2×C4C2×#1174567112yes—
573C2×C4#1427375812yes—
574C2×C4C2×D4×D5911yescharacteristic
575C2×C4C22×C22⋊D5911yescharacteristic
576C2×C4C22×C22⋊D5911yescharacteristic
577C2×C4C2×#1174568711yescharacteristic
578C2×C4#1427377211yescharacteristic
579C2×C4#1427377211yescharacteristic
580C8—1182no—
581C8—2361no—
582C8—2362no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—4721no—
591C22—4721no—
592C22—4721no—
593C22—4721no—
594C22—2362no—
595C22—2362no—
596C22—2362no—
597C22C2×#1427369611yescharacteristic
598C22D4×C22⋊D5911yescharacteristic
599C22#1683804611yescharacteristic
600C22C22×#1174567111yescharacteristic
601C22C2×#1427375811yescharacteristic
602C22#1683821011yescharacteristic
603C22#1683821711yescharacteristic
604C4—22no—
605C4—24no—
606C4—42no—
607C4—42no—
608C4—44no—
609C4—4722no—
610C4—4721no—
611C4#1683754114yescomplemented
612C4C2×#1427369811yescharacteristic
613C4#1683796911yescharacteristic
614C4D4×C22⋊D5911yescharacteristic
615C4#1683804611yescharacteristic
616C2—44no—
617C2—4722no—
618C2#2142744611yescharacteristic
619C2C2×#1683754111yescharacteristic
620C2#2143341111yescharacteristic
621C2#2143574211yescharacteristic
622C2D4×#1174567111yescharacteristic
623C2#2143596111yescharacteristic
624C2#2143639011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×D4×Q8×C59C2
socleC23×C59C22×D4
prime radical#21429009C2
hypercenterC2×D4×Q8D59
hypocenterC59#24239
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16837604C2semidirect
2. C2×#16837541C2semidirect
3. C2×C4×#11745671C2semidirect
5. #21425289C2semidirect
6. #21425097C2semidirect
7. #21424899C2semidirect
8. #21424891C2semidirect
9. #21424101C2semidirect
10. #21424031C2semidirect
11. C2×D4×Q8×C59C2semidirect
54. C2×#11745671D4semidirect
96. C4⋊Q8×C59D4semidirect
101. C4×Q8×C59D4semidirect
138. C2×D4×C59SD16semidirect
224. Q8×C59#121semidirect
297. C4×C59#448semidirect
304. C2×D4×Q8D59semidirect
358. C59#24239semidirect
385. C4⋊Q8C22⋊D59semidirect
393. C4×Q8C22⋊D59semidirect
460. C2×D4#11745671semidirect
537. Q8#14273698semidirect
611. C4#16837541semidirect

Maximal subgroups

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

subgroupindex
C2×#168376042
C2×#168375412
C2×C4×#117456712
#214290092
#214252892
#214250972
#214248992
#214248912
#214241012
#214240312
C2×D4×Q8×C592
#2423959