Finite Group Databasebeta

← #28600375

Subgroups — #28600375

Order 15104.

812 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic59C211yesmaximal, characteristic
2#21426580C211yesmaximal, characteristic
3#21424840C211yesmaximal, characteristic, complemented
4#21424835C211yesmaximal, characteristic, complemented
5#21424788C211yesmaximal, characteristic, complemented
6#21424788C211yesmaximal, characteristic, complemented
7#21424690C211yesmaximal, characteristic, complemented
8#21424676C211yesmaximal, characteristic, complemented
9#21424409C211yesmaximal, characteristic
10#21424366C211yesmaximal, characteristic, complemented
11#21424352C211yesmaximal, characteristic, complemented
12#21424297C211yesmaximal, characteristic
13#21424191C211yesmaximal, characteristic
14#21423860C211yesmaximal, characteristic
15C2×C59×#485C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18C22×C4×Dic59C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273637C2211yescharacteristic
25C2×#14273637C2211yescharacteristic
26#16837309C2211yescharacteristic
27#16837301C2211yescharacteristic
28#16837301C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837301C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837288—21no—
33#16837277—21no—
34#16837277—21no—
35#16837277—21no—
36#16837262C2211yescharacteristic
37C4⋊C4×Dic59—21no—
38C4⋊C4×Dic59C2211yescharacteristic
39C4×C4⋊Dic59—21no—
40C4×C4⋊Dic59—21no—
41C22⋊C4×Dic59—21no—
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×C22⋊Q8×C59C2211yescharacteristic
53C2×#117×C59C2211yescharacteristic
54C59×#485C418yescomplemented
55C2×#107×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C2×C22⋊C8×C59C2211yescharacteristic
60C23×Dic59D411yescharacteristic
61C23×Dic59D411yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64Q8×Dic59—41no—
65Q8×Dic59—41no—
66#14273686—41no—
67#14273686—41no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—41no—
84C2×C4×Dic59—41no—
85C2×C4×Dic59D411yescharacteristic
86C2×C4×Dic59D411yescharacteristic
87C2×C4×Dic59C2311yescharacteristic
88C2×C4×Dic59C2311yescharacteristic
89#14273661—41no—
90#14273659—41no—
91#14273651—41no—
92#14273651—41no—
93C22⋊(C59⋊C8)—22no—
94#14273637—41no—
95#14273637—41no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C22×Q8×C59C2311yescharacteristic
106C23×C4×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22⋊Q8×C59—22no—
110C22⋊Q8×C59C2×C414yes—
111#117×C59—22no—
112#117×C59C2×C414yes—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C42×C59—21no—
120#107×C59C2×C414yescomplemented
121#106×C59C2×C414yescomplemented
122#103×C59C2×C414yescomplemented
123#103×C59C2×C414yescomplemented
124C22⋊C8×C59C2×C414yes—
125C24×C59C2×D411yescharacteristic
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59C22×C412yes—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59#4112yes—
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—22no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—41no—
150C22×Dic59—41no—
151C22×Dic59—41no—
152C22×Dic59—41no—
153C22×Dic59—41no—
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156C22×Dic59C2×D411yescharacteristic
157C22×Dic59C2×D411yescharacteristic
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C22×C412yes—
163C2×C8×C59C22×C412yes—
164C4⋊C4×C59—24no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59C22×C412yes—
169C4⋊C4×C59C22×C412yes—
170C4⋊C4×C59C22×C412yes—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—24no—
173C42×C59—24no—
174C22⋊Dic59—42no—
175C22⋊Dic59—42no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—42no—
185C4⋊Dic59—42no—
186C4⋊Dic59—42no—
187C4⋊Dic59—42no—
188C4⋊Dic59—42no—
189C4⋊Dic59—42no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—41no—
195C4×Dic59—41no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59C4×D412yes—
212C23×C59C22×D411yescharacteristic
213Q8×C59—22no—
214Q8×C59—22no—
215Q8×C59—41no—
216Q8×C59—41no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59C4×D412yes—
236C2×C4×C59C23×C411yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59H(5,𝔽2)11yescharacteristic
239C2×C4×C59H(5,𝔽2)11yescharacteristic
240C2×C4×C592−1+411yescharacteristic
241C2×C4×C592−1+411yescharacteristic
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—44no—
254C2×Dic59—44no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59#13712yes—
268C8×C59—22no—
269C8×C59—22no—
270C59⋊C8—41no—
271C59⋊C8—41no—
272#22083—591nomaximal, Sylow
273C22×C59—22no—
274C22×C59—22no—
275C22×C59—22no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59—41no—
280C22×C59C2×C4×D411yescharacteristic
281C22×C59#53411yescharacteristic
282C22×C59#53611yescharacteristic
283C22×C59C2×#13711yescharacteristic
284C22×C59#57511yescharacteristic
285C22×C59#57511yescharacteristic
286C22×C59#57511yescharacteristic
287C4×C59—22no—
288C4×C59—42no—
289C4×C59—42no—
290C4×C59—42no—
291C4×C59—42no—
292C4×C59#52011yescharacteristic
293C4×C59#52011yescharacteristic
294C4×C59#53911yescharacteristic
295C4×C59#53911yescharacteristic
296Dic59—22no—
297Dic59—41no—
298Dic59—41no—
299Dic59—41no—
300Dic59—81no—
301Dic59—81no—
302Dic59—81no—
303Dic59—81no—
304C2×#485D5911yescharacteristic
305C4×C22⋊Q8—591no—
306#2296—591no—
307#1944—591no—
308#1943—591no—
309#1935—591no—
310#1935—591no—
311#1921—591no—
312#1919—591no—
313#1873—591no—
314#1865—591no—
315#1863—591no—
316#1854—591no—
317#1825—591no—
318#1749—591no—
319C2×C59—42no—
320C2×C59#232211yescharacteristic
321C2×C59#294311yescharacteristic
322C2×C59C4×#13711yescharacteristic
323C2×C59#319611yescharacteristic
324C2×C59#319711yescharacteristic
325C2×C59#323711yescharacteristic
326C2×C59#324011yescharacteristic
327C2×C22⋊Q8C2×D5911yescharacteristic
328C2×C4×Q8—591no—
329C2×#117—591no—
330C2×#117C2×D5911yescharacteristic
331C22×C42—591no—
332#485Dic5918yescomplemented
333C2×#107—591no—
334C2×#107C2×D5911yescharacteristic
335C2×#106—591no—
336C2×#106C2×D5911yescharacteristic
337C2×#103—591no—
338C2×#103—591no—
339C2×#103C2×D5911yescharacteristic
340C2×#103C2×D5911yescharacteristic
341C2×C22⋊C8—591no—
342C2×C22⋊C8C2×D5911yescharacteristic
343C2×#97—591no—
344C2×#97—591no—
345#391—591no—
346#389—591no—
347#389—591no—
348#389—591no—
349#389—591no—
350#389—591no—
351#387—1181no—
352#385—1181no—
353#385—1181no—
354#385—1181no—
355#380—591no—
356C4×C4⋊C4—591no—
357C4×C4⋊C4—1181no—
358C4×C4⋊C4—1181no—
359C4×C4⋊C4—1181no—
360C4×C22⋊C4—1181no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#336—591no—
370#336—591no—
371C59#2208311yescharacteristic, complemented, Sylow
372C22×Q8C22×D5911yescharacteristic
373C23×C4—591no—
374C23×C4—591no—
375C23×C4C22×D5911yescharacteristic
376C22×C8—591no—
377C22×C8—591no—
378C22×C8C22×D5911yescharacteristic
379C22×C8C22×D5911yescharacteristic
380C22⋊Q8—22no—
381C22⋊Q8C2×Dic5914yes—
382C4×Q8—2361no—
383C4×Q8—2361no—
384#117—22no—
385#117—2361no—
386#117—2361no—
387#117C2×Dic5914yes—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
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—1181no—
405C2×C22⋊C4—1181no—
406C2×C42—21no—
407C2×C42—591no—
408C2×C42—591no—
409C2×C42—591no—
410C2×C42—591no—
411C2×C42—1181no—
412C2×C42—1181no—
413C2×C42—1181no—
414C2×C42—2361no—
415C2×C42—2361no—
416#107—2361no—
417#107C2×Dic5914yescomplemented
418#106—2361no—
419#106C2×Dic5914yescomplemented
420#103—2361no—
421#103—2361no—
422#103C2×Dic5914yescomplemented
423#103C2×Dic5914yescomplemented
424C22⋊C8—1182no—
425C22⋊C8C2×Dic5914yes—
426#97—2361no—
427#97—2361no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437C24D4×D5911yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C22×Dic5912yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—591no—
453C22×C4—591no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—1182no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—2361no—
462C22×C4—2361no—
463C22×C4—2361no—
464C22×C4—2361no—
465C22×C4—2361no—
466C22×C4D4×D5911yescharacteristic
467C22×C4#1174568712yes—
468C22×C4C22×Dic5912yes—
469C22×C4C23×D5911yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8C22×Dic5912yes—
481C2×C8C22×Dic5912yes—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C4⋊C4—2362no—
497C4⋊C4—2362no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4C22×Dic5912yes—
501C4⋊C4C22×Dic5912yes—
502C4⋊C4C22×Dic5912yes—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—2362no—
506C22⋊C4—2362no—
507C42—24no—
508C42—1181no—
509C42—1181no—
510C42—1181no—
511C42—1181no—
512C42—2361no—
513C42—2361no—
514C42—2362no—
515C42—2362no—
516C42—2362no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23D4×Dic5912yes—
524C23C2×D4×D5911yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—592no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—2364no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2364no—
573C2×C4D4×Dic5912yes—
574C2×C4C23×Dic5911yescharacteristic
575C2×C4C2×#1174568711yescharacteristic
576C2×C4#1427377311yescharacteristic
577C2×C4#1427377311yescharacteristic
578C2×C4#1427377411yescharacteristic
579C2×C4#1427377411yescharacteristic
580C8—22no—
581C8—22no—
582C8—2361no—
583C8—2361no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×D4×Dic5911yescharacteristic
592C22#1683796311yescharacteristic
593C22#1683797411yescharacteristic
594C22#137×D5911yescharacteristic
595C22#1683819911yescharacteristic
596C22#1683819911yescharacteristic
597C22#1683820111yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—1182no—
604C4—4721no—
605C4—4721no—
606C4—4721no—
607C4—4721no—
608C4—2361no—
609C4—2361no—
610C4—2361no—
611C4#1683788511yescharacteristic
612C4#1683788511yescharacteristic
613C4#1683799611yescharacteristic
614C4#1683799611yescharacteristic
615C2—42no—
616C2#2142676811yescharacteristic
617C2#2143289711yescharacteristic
618C2#137×Dic5911yescharacteristic
619C2#2143493011yescharacteristic
620C2#2143494011yescharacteristic
621C2#2143530711yescharacteristic
622C2#2143534011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#485C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterC2×#485D59
hypocenterC59#22083
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
3. #21424840C2semidirect
4. #21424835C2semidirect
5. #21424788C2semidirect
6. #21424788C2semidirect
7. #21424690C2semidirect
8. #21424676C2semidirect
10. #21424366C2semidirect
11. #21424352C2semidirect
54. C59×#485C4semidirect
120. #107×C59C2×C4semidirect
121. #106×C59C2×C4semidirect
122. #103×C59C2×C4semidirect
123. #103×C59C2×C4semidirect
332. #485Dic59semidirect
371. C59#22083semidirect
417. #107C2×Dic59semidirect
419. #106C2×Dic59semidirect
422. #103C2×Dic59semidirect
423. #103C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic592
#214265802
#214248402
#214248352
#214247882
#214246902
#214246762
#214244092
#214243662
#214243522
#214242972
#214241912
#214238602
C2×C59×#4852
#2208359