Finite Group Databasebeta

← #28623040

Subgroups — #28623040

Order 15104.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837952C211yesmaximal, characteristic
2C2×#16837687C211yesmaximal, characteristic, complemented
3C2×#16837602C211yesmaximal, characteristic, complemented
4#21425285C212yesmaximal, complemented
5#21425097C212yesmaximal, complemented
6#21425048C211yesmaximal, characteristic, complemented
7#21425023C211yesmaximal, characteristic, complemented
8#21424433C212yesmaximal, complemented
9#21424356C212yesmaximal, complemented
10#21424076C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C59C211yesmaximal, characteristic, complemented
12C22×#11745685—21no—
13C22×#11745671C2212yes—
14#16837952—24no—
15#16837952—24no—
16C2×#14273727C2211yescharacteristic
17C2×#14273724C2211yescharacteristic
18C2×#14273715—21no—
19C2×#14273701—21no—
20C2×#14273701C2212yes—
21C2×C4×D236—21no—
22C2×C4×D236C2212yes—
23#16837687—24no—
24#16837602—24no—
25C2×#14273658C2212yes—
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273651C2212yes—
29C2×#14273647C2212yes—
30C2×#14273647C2212yes—
31C2×#14273647C2212yes—
32C2×C22⋊(C59⋊C8)C2212yes—
33#16837164C2212yes—
34#16837164C2212yes—
35#16837157C2212yes—
36C2×C22⋊Q8×C59C2211yescharacteristic
37C2×C4×Q8×C59C2211yescharacteristic
38C22×C42×C59C2211yescharacteristic
39C59×#391C2211yescharacteristic
40C59×#389C2211yescharacteristic
41C59×#387—21no—
42C59×#385—21no—
43C59×#380C2211yescharacteristic
44C4×C4⋊C4×C59—21no—
45C4×C4⋊C4×C59C2211yescharacteristic
46C4×C22⋊C4×C59—21no—
47C2×#11745685—28no—
48C2×#11745685—44no—
49C2×#11745685—44no—
50C22×Dic118—21no—
51C22×D236—21no—
52C22×D236—21no—
53C22×D236C2312yes—
54C22×C22⋊D59—22no—
55C22×C4×D59—22no—
56C2×#11745671—24no—
57C2×#11745671—24no—
58C2×#11745671—42no—
59C22×C59⋊C8C2312yes—
60C22×C59⋊C8C2312yes—
61#14273727—22no—
62#14273727D414yes—
63#14273724—22no—
64#14273724D414yes—
65#14273715—28no—
66#14273701—28no—
67#14273701—24no—
68#14273701—24no—
69C4×D236—28no—
70C4×D236—24no—
71C4×D236—24no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2312yes—
74C2×(C2×D59)⋊C4—22no—
75C2×(C2×D59)⋊C4—22no—
76#14273658—24no—
77C4⋊(C59⋊C8)—22no—
78C4⋊(C59⋊C8)—22no—
79#14273651—24no—
80#14273647—24no—
81#14273647—24no—
82#14273647—42no—
83C22⋊(C59⋊C8)—42no—
84C22×Q8×C59C2311yescharacteristic
85C23×C4×C59D411yescharacteristic
86C23×C4×C59D411yescharacteristic
87C23×C4×C59C2311yescharacteristic
88C22⋊Q8×C59—22no—
89C22⋊Q8×C59—22no—
90C4×Q8×C59—22no—
91C4×Q8×C59—22no—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59—21no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59C2311yescharacteristic
96C2×C4⋊C4×C59C2311yescharacteristic
97C2×C4⋊C4×C59C2311yescharacteristic
98C2×C22⋊C4×C59—21no—
99C2×C22⋊C4×C59—21no—
100C2×C42×C59—22no—
101C2×C42×C59—22no—
102C2×C42×C59—21no—
103C2×C42×C59—21no—
104C2×C42×C59D411yescharacteristic
105C2×C42×C59D411yescharacteristic
106C2×C42×C59C2311yescharacteristic
107C2×C42×C59C2311yescharacteristic
108#95×C59—21no—
109#95×C59—21no—
110#95×C59—21no—
111#95×C59—21no—
112#95×C59—21no—
113C24×C59C2×D411yescharacteristic
114C23×D59—22no—
115C2×Q8×C59—21no—
116C2×Q8×C59—21no—
117C2×Q8×C59—21no—
118C2×Q8×C59—21no—
119C2×Q8×C59—21no—
120C2×Q8×C59#4112yescomplemented
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—41no—
135C22×C4×C59—41no—
136C22×C4×C59SD1614yescomplemented
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2411yescharacteristic
143C2×C22⋊D59—28no—
144C2×C22⋊D59—44no—
145C2×C22⋊D59—44no—
146C22×Dic59—22no—
147#11745685—432no—
148C2×D236—22no—
149C2×D236—22no—
150C2×D236—22no—
151C2×D236—22no—
152C2×D236—24no—
153C2×D236—24no—
154C2×D236—22no—
155C2×D236—42no—
156C2×D236—42no—
157C2×D236—42no—
158C2×D236—42no—
159C2×D236C2×D414yes—
160C2×C4×D59—28no—
161C2×C4×D59—44no—
162C2×C4×D59—44no—
163C2×Dic118—24no—
164C2×Dic118—42no—
165C2×Dic118—42no—
166C4⋊C4×C59—24no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59—42no—
174C4⋊C4×C59#4112yescomplemented
175C22⋊C4×C59—24no—
176C22⋊C4×C59—42no—
177C42×C59—22no—
178C42×C59—22no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—42no—
182C42×C59—42no—
183C42×C59C2×D412yes—
184C42×C59C2×D412yes—
185#11745671—48no—
186(C2×D59)⋊C4—28no—
187(C2×D59)⋊C4—28no—
188C4⋊Dic59—24no—
189C4⋊Dic59—22no—
190C4⋊Dic59C2×D414yes—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C2×C59⋊C8—22no—
194C2×C59⋊C8—22no—
195C2×C59⋊C8—22no—
196C2×C59⋊C8—22no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59C2×SD1612yes—
204C23×C59C22×D411yescharacteristic
205C22×D59—24no—
206C22×D59—42no—
207C22×D59—42no—
208C22×D59—42no—
209C22×D59—42no—
210C22×D59—42no—
211C22×D59—42no—
212Q8×C59—22no—
213Q8×C59—22no—
214Q8×C59—41no—
215Q8×C59—41no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—28no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—44no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59#12012yes—
245C2×C4×C59#12012yes—
246C2×C4×C59C2×SD1612yes—
247C2×C4×C59C2×SD1612yes—
248C2×C4×C59AGL(1,ℤ/8)12yes—
249C2×C4×C59C22×D411yescharacteristic
250C2×C4×C59C22×D411yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59H(5,𝔽2)11yescharacteristic
254C2×C4×C592−1+411yescharacteristic
255C22⋊D59—416no—
256C2×Dic59—24no—
257C2×Dic59—42no—
258C2×Dic59—42no—
259D236—24no—
260D236—24no—
261D236—48no—
262D236—42no—
263D236—42no—
264D236—48no—
265C4×D59—416no—
266Dic118—48no—
267C59⋊C8—42no—
268C59⋊C8—42no—
269#24154—591nomaximal, Sylow
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—22no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59C2×#12011yescharacteristic
278C22×C59#52511yescharacteristic
279C22×C59#53711yescharacteristic
280C22×C59C22×SD1611yescharacteristic
281C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C59#57511yescharacteristic
283C22×C59#57511yescharacteristic
284C2×D59—44no—
285C2×D59—44no—
286C2×D59—44no—
287C2×D59—82no—
288C2×D59—82no—
289C2×D59—82no—
290C2×D59—82no—
291C4×C59—22no—
292C4×C59—22no—
293C4×C59—42no—
294C4×C59—41no—
295C4×C59—42no—
296C4×C59—41no—
297C4×C59—81no—
298C4×C59—81no—
299C4×C59#45014yes—
300C4×C59#54611yescharacteristic
301C4×C59#54711yescharacteristic
302C4×C59#54811yescharacteristic
303C4×C59#55011yescharacteristic
304Dic59—44no—
305C2×#532—591no—
306C2×#475—591no—
307C2×#460—591no—
308C4×C22⋊Q8D5911yescharacteristic, complemented
309#2034—592no—
310#2000—592no—
311#1988—591no—
312#1982—591no—
313#1877—592no—
314#1864—592no—
315#1797—591no—
316C2×C59—42no—
317C2×C59#240711yescharacteristic
318C2×C59C2×#45011yescharacteristic
319C2×C59#300511yescharacteristic
320C2×C59#310311yescharacteristic
321C2×C59#311511yescharacteristic
322C2×C59#324611yescharacteristic
323C2×C59#325111yescharacteristic
324D59—84no—
325C22×#41—1181no—
326C22×SD16—592no—
327#532—1184no—
328#532—1184no—
329C2×C4⋊Q8—591no—
330C2×#127—591no—
331C2×#124—1181no—
332C2×C22⋊Q8C2×D5911yescharacteristic
333C2×#121—592no—
334C2×#121—1181no—
335C2×C4×Q8C2×D5911yescharacteristic
336C2×C4×D4—592no—
337C2×C4×D4—1181no—
338C22×C42C2×D5911yescharacteristic
339#475—1184no—
340#460—1184no—
341C2×#106—592no—
342C2×C4⋊C8—591no—
343C2×C4⋊C8—591no—
344C2×#103—592no—
345C2×#102—592no—
346C2×#102—592no—
347C2×#102—592no—
348C2×C22⋊C8—592no—
349#391C2×D5911yescharacteristic
350#389C2×D5911yescharacteristic
351#387—21no—
352#385—21no—
353#380C2×D5911yescharacteristic
354C4×C4⋊C4—21no—
355C4×C4⋊C4C2×D5911yescharacteristic
356C4×C22⋊C4—21no—
357#340—592no—
358#340—592no—
359#336—592no—
360C59#2415411yescharacteristic, complemented, Sylow
361C2×#41—1188no—
362C2×#41—2364no—
363C2×#41—2364no—
364C22×Q8—1181no—
365C22×Q8C22×D5911yescharacteristic
366C22×D4—592no—
367C22×D4—1182no—
368C22×D4—1181no—
369C22×D4—1181no—
370C23×C4—1182no—
371C23×C4C22⋊D5911yescharacteristic
372C23×C4C22⋊D5911yescharacteristic
373C23×C4C22×D5911yescharacteristic
374C2×SD16—1184no—
375C2×SD16—1184no—
376C2×SD16—2362no—
377C22×C8—592no—
378C22×C8—592no—
379C4⋊Q8—594no—
380C4⋊Q8—1182no—
381#127—594no—
382#127—1182no—
383#124—1188no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386#121—1184no—
387#121—1184no—
388#121—1188no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×D4—1184no—
392C4×D4—1184no—
393C4×D4—1188no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—592no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1182no—
405C2×C22⋊C4—1182no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42C22⋊D5911yescharacteristic
411C2×C42C22⋊D5911yescharacteristic
412C2×C42C22×D5911yescharacteristic
413C2×C42C22×D5911yescharacteristic
414#106—1184no—
415C4⋊C8—1182no—
416C4⋊C8—1182no—
417#103—1184no—
418#102—1184no—
419#102—1184no—
420#102—2362no—
421C22⋊C8—2362no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427C24—1182no—
428C24C2×C22⋊D5911yescharacteristic
429#41—23632no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—1184no—
436C2×Q8—2362no—
437C2×Q8—2362no—
438C2×Q8#1174568512yescomplemented
439C2×D4—594no—
440C2×D4—1188no—
441C2×D4—1184no—
442C2×D4—1184no—
443C2×D4—1182no—
444C2×D4—1182no—
445C2×D4—1182no—
446C2×D4—1182no—
447C2×D4—1182no—
448C2×D4—2362no—
449C2×D4—2362no—
450C2×D4—2364no—
451C2×D4—2364no—
452C2×D4—2362no—
453C2×D4—2362no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—1188no—
470C22×C4—1182no—
471C22×C4—2364no—
472C22×C4—2364no—
473C22×C4#1174567114yescomplemented
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C23×D5911yescharacteristic
480SD16—2368no—
481C2×C8—1182no—
482C2×C8—1182no—
483C2×C8—1182no—
484C2×C8—1182no—
485C2×C8—1182no—
486C2×C8—1182no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—594no—
496C4⋊C4—1184no—
497C4⋊C4—1182no—
498C4⋊C4#1174568512yescomplemented
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—1188no—
502C22⋊C4—1188no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—42no—
509C42D4×D5912yes—
510C42D4×D5912yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—1184no—
518C23—2362no—
519C23—2362no—
520C23—2362no—
521C23—2362no—
522C23—2362no—
523C23—2362no—
524C23C2×#1174567112yes—
525C23C22×C22⋊D5911yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—2368no—
531D4—1184no—
532D4—1184no—
533D4—2368no—
534D4—23616no—
535D4—2362no—
536D4—2368no—
537D4—2362no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—1184no—
567C2×C4—23616no—
568C2×C4—2362no—
569C2×C4—2362no—
570C2×C4#1427369512yes—
571C2×C4#1427369512yes—
572C2×C4C2×#1174567112yes—
573C2×C4C2×#1174567112yes—
574C2×C4#1427375212yes—
575C2×C4C2×D4×D5911yescharacteristic
576C2×C4C2×D4×D5911yescharacteristic
577C2×C4C2×#1174568511yescharacteristic
578C2×C4C2×#1174568511yescharacteristic
579C2×C4#1427377311yescharacteristic
580C2×C4#1427377411yescharacteristic
581C8—2362no—
582C8—2362no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—4722no—
591C22—4722no—
592C22—4722no—
593C22—4722no—
594C22—2364no—
595C22—2364no—
596C22—2364no—
597C22C2×#1427369511yescharacteristic
598C22#1683791311yescharacteristic
599C22#1683798011yescharacteristic
600C22C22×#1174567111yescharacteristic
601C22C2×#1427375211yescharacteristic
602C22#1683820011yescharacteristic
603C22#1683820011yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—2364no—
613C4#1683755214yes—
614C4#1683804811yescharacteristic
615C4#1683805711yescharacteristic
616C4#1683806111yescharacteristic
617C4#1683807311yescharacteristic
618C2—42no—
619C2—4724no—
620C2#2142744111yescharacteristic
621C2C2×#1683755211yescharacteristic
622C2#2143333611yescharacteristic
623C2#2143412711yescharacteristic
624C2#2143424711yescharacteristic
625C2#2143540811yescharacteristic
626C2#2143545811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC4×C22⋊Q8×C59C2
socleC23×C59C22×D4
prime radicalC2×#16837952C2
hypercenterC4×C22⋊Q8D59
hypocenterC59#24154
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16837687C2semidirect
3. C2×#16837602C2semidirect
4. #21425285C2semidirect
5. #21425097C2semidirect
6. #21425048C2semidirect
7. #21425023C2semidirect
8. #21424433C2semidirect
9. #21424356C2semidirect
10. #21424076C2semidirect
11. C4×C22⋊Q8×C59C2semidirect
120. C2×Q8×C59#41semidirect
136. C22×C4×C59SD16semidirect
174. C4⋊C4×C59#41semidirect
308. C4×C22⋊Q8D59semidirect
360. C59#24154semidirect
438. C2×Q8#11745685semidirect
473. C22×C4#11745671semidirect
498. C4⋊C4#11745685semidirect

Maximal subgroups

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

subgroupindex
C2×#168379522
C2×#168376872
C2×#168376022
#214252852
#214250972
#214250482
#214250232
#214244332
#214243562
#214240762
C4×C22⋊Q8×C592
#2415459