Finite Group Databasebeta

← #28628851

Subgroups — #28628851

Order 15104.

756 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837460C211yesmaximal, characteristic
2#21429894C211yesmaximal, characteristic
3#21426712C211yesmaximal, characteristic
4#21425595C211yesmaximal, characteristic, complemented
5#21425590C211yesmaximal, characteristic, complemented
6#21425475C211yesmaximal, characteristic, complemented
7#21425475C211yesmaximal, characteristic, complemented
8#21424837C211yesmaximal, characteristic, complemented
9#21424822C211yesmaximal, characteristic, complemented
10#21424783C211yesmaximal, characteristic, complemented
11#21424769C211yesmaximal, characteristic, complemented
12#21424404C211yesmaximal, characteristic
13#21424291C211yesmaximal, characteristic
14#21424191C211yesmaximal, characteristic
15C2×C59×#485C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18#16837460—24no—
19C2×#14273659C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×C22⋊(C59⋊C8)C2211yescharacteristic
26C2×#14273637C2211yescharacteristic
27C2×C8×Dic59C2211yescharacteristic
28#16837341—21no—
29#16837332—21no—
30#16837330—21no—
31#16837330—21no—
32#16837309C2211yescharacteristic
33#16837309C2211yescharacteristic
34#16837300—21no—
35#16837300—21no—
36#16837298C2211yescharacteristic
37#16837298C2211yescharacteristic
38#16837298C2211yescharacteristic
39#16837298C2211yescharacteristic
40#16837298C2211yescharacteristic
41#16837298C2211yescharacteristic
42#16837262C2211yescharacteristic
43#16837260—21no—
44#16837260—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×C22⋊Q8×C59C2211yescharacteristic
54C2×#117×C59C2211yescharacteristic
55C59×#485—24no—
56C2×#107×C59C2211yescharacteristic
57C2×#106×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C2×#103×C59C2211yescharacteristic
60C2×C22⋊C8×C59C2211yescharacteristic
61C23×Dic59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273686—22no—
65#14273686—22no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—22no—
68C2×C4⋊Dic59—22no—
69C2×C4⋊Dic59D411yescharacteristic
70C2×C4⋊Dic59D411yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59D411yescharacteristic
83C2×C22⋊Dic59D411yescharacteristic
84C2×C4×Dic59C2311yescharacteristic
85C2×C4×Dic59C2311yescharacteristic
86#14273659—22no—
87#14273659—22no—
88Dic59⋊C8—22no—
89Dic59⋊C8—22no—
90#14273651—22no—
91#14273651—22no—
92C22⋊(C59⋊C8)—22no—
93#14273637—22no—
94C8×Dic59—22no—
95C59⋊#95—21no—
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—
105C59⋊#95—21no—
106C22×Q8×C59C2311yescharacteristic
107C23×C4×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C22⋊Q8×C59—22no—
111C22⋊Q8×C59—22no—
112#117×C59—22no—
113#117×C59—22no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C42×C59—21no—
121#107×C59—22no—
122#106×C59—22no—
123#103×C59—22no—
124#103×C59—22no—
125C22⋊C8×C59D414yes—
126C24×C59C2×D411yescharacteristic
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×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×C59—21no—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D412yes—
143C22×C4×C59#4112yes—
144C22×C4×C59C2411yescharacteristic
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—24no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—81no—
151C22×Dic59—81no—
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59C2×D412yes—
161C2×C8×C59C2×D412yes—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—42no—
169C22⋊C4×C59—42no—
170C22⋊C4×C59—42no—
171C42×C59—42no—
172C22⋊Dic59—22no—
173C22⋊Dic59—22no—
174C22⋊Dic59—81no—
175C4⋊Dic59—24no—
176C4⋊Dic59—22no—
177C4⋊Dic59—22no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59#4112yes—
185C4⋊Dic59#4112yes—
186Dic59⋊C4—81no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189Dic59⋊C4—81no—
190Dic59⋊C4—81no—
191Dic59⋊C4—81no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59#12112yes—
209C23×C59C22×D411yescharacteristic
210Q8×C59—41no—
211Q8×C59—41no—
212Q8×C59—41no—
213Q8×C59—41no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59#12112yes—
233C2×C4×C59C22×D411yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C592−1+411yescharacteristic
238C2×C4×C592−1+411yescharacteristic
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—22no—
244C2×Dic59—22no—
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—41no—
254C2×Dic59—41no—
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—84no—
264C8×C59—22no—
265C8×C59—22no—
266C59⋊C8—41no—
267C59⋊C8—41no—
268#24619—591nomaximal, Sylow
269C22×C59—22no—
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—41no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59C2×#12111yescharacteristic
277C22×C59#52511yescharacteristic
278C22×C59#53611yescharacteristic
279C22×C59#57711yescharacteristic
280C22×C59#57711yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57811yescharacteristic
283C4×C59—22no—
284C4×C59—81no—
285C4×C59—81no—
286C4×C59—81no—
287C4×C59—81no—
288C4×C59#54711yescharacteristic
289C4×C59#54711yescharacteristic
290C4×C59#55211yescharacteristic
291C4×C59#55211yescharacteristic
292Dic59—42no—
293Dic59—42no—
294Dic59—41no—
295Dic59—41no—
296Dic59—41no—
297Dic59—41no—
298Dic59—81no—
299Dic59—81no—
300C2×#485D5911yescharacteristic
301C2×#433—591no—
302#2625—591no—
303#2316—591no—
304#2088—591no—
305#2087—591no—
306#2066—591no—
307#2066—591no—
308#1944—591no—
309#1941—591no—
310#1935—591no—
311#1933—591no—
312#1872—591no—
313#1853—591no—
314#1825—591no—
315C2×C59—42no—
316C2×C59#241311yescharacteristic
317C2×C59#306011yescharacteristic
318C2×C59#306311yescharacteristic
319C2×C59#312111yescharacteristic
320C2×C59#312311yescharacteristic
321C2×C59#324211yescharacteristic
322C2×C59#324411yescharacteristic
323C2×C22⋊Q8C2×D5911yescharacteristic
324C2×#117—591no—
325C2×#117C2×D5911yescharacteristic
326C22×C4⋊C4—591no—
327#485—24no—
328#433—1184no—
329C2×#107C2×D5911yescharacteristic
330C2×#106—591no—
331C2×#106—591no—
332C2×#106C2×D5911yescharacteristic
333C2×C4⋊C8—591no—
334C2×C4⋊C8—591no—
335C2×#103—591no—
336C2×#103—591no—
337C2×#103C2×D5911yescharacteristic
338C2×#103C2×D5911yescharacteristic
339C2×C22⋊C8—591no—
340C2×C22⋊C8C2×D5911yescharacteristic
341C2×#97—591no—
342C2×C4×C8—591no—
343#400—1181no—
344#398—1181no—
345#397—1181no—
346#397—1181no—
347#391—591no—
348#391—591no—
349#389—591no—
350#389—591no—
351#389—591no—
352#389—591no—
353#389—591no—
354#389—591no—
355#389—1181no—
356#389—1181no—
357#380—591no—
358#380—1181no—
359#380—1181no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368C59#2461911yescharacteristic, complemented, Sylow
369C22×Q8C22×D5911yescharacteristic
370C23×C4—1181no—
371C23×C4C22×D5911yescharacteristic
372C22×C8—591no—
373C22×C8—591no—
374C22×C8C22×D5911yescharacteristic
375C22×C8C22×D5911yescharacteristic
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378#117—22no—
379#117—22no—
380#117—1182no—
381#117—1182no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1182no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—591no—
404C2×C22⋊C4—591no—
405C2×C22⋊C4—1181no—
406C2×C42—21no—
407C2×C42—591no—
408C2×C42—591no—
409#107—22no—
410#106—22no—
411#106—1182no—
412#106—1182no—
413C4⋊C8—1182no—
414C4⋊C8—1182no—
415#103—22no—
416#103—22no—
417#103—1182no—
418#103—1182no—
419C22⋊C8—1182no—
420C22⋊C8C22⋊D5914yes—
421#97—1182no—
422C4×C8—1182no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434C24D4×D5911yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—591no—
450C22×C4—591no—
451C22×C4—591no—
452C22×C4—591no—
453C22×C4—1184no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—4721no—
459C22×C4—4721no—
460C22×C4D4×D5911yescharacteristic
461C22×C4#1174568712yes—
462C22×C4C2×C22⋊D5912yes—
463C22×C4C23×D5911yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8C2×C22⋊D5912yes—
475C2×C8C2×C22⋊D5912yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—592no—
484C4⋊C4—592no—
485C4⋊C4—1184no—
486C4⋊C4—1182no—
487C4⋊C4—1182no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—4721no—
495C4⋊C4—4721no—
496C4⋊C4—4721no—
497C4⋊C4—4721no—
498C4⋊C4—4721no—
499C4⋊C4—4721no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—1182no—
503C22⋊C4—1182no—
504C22⋊C4—4721no—
505C42—42no—
506C42—1181no—
507C42—1181no—
508C42—1181no—
509C42—1181no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23#1427369812yes—
517C23C2×D4×D5911yescharacteristic
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1182no—
545C2×C4—1182no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—4724no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4#1427369812yes—
566C2×C4C22×C22⋊D5911yescharacteristic
567C2×C4#41×D5911yescharacteristic
568C2×C4#41×D5911yescharacteristic
569C2×C4C2×#1174568711yescharacteristic
570C2×C4#1427377411yescharacteristic
571C2×C4#1427377411yescharacteristic
572C8—22no—
573C8—22no—
574C8—2361no—
575C8—2361no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1427369811yescharacteristic
584C22#1683791011yescharacteristic
585C22#1683797411yescharacteristic
586C22#1683820911yescharacteristic
587C22#1683820911yescharacteristic
588C22#1683821111yescharacteristic
589C22#1683821811yescharacteristic
590C4—22no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—4721no—
596C4—4721no—
597C4—2361no—
598C4—2361no—
599C4—2361no—
600C4—2361no—
601C4—2362no—
602C4—2362no—
603C4#1683805111yescharacteristic
604C4#1683805111yescharacteristic
605C4#1683808911yescharacteristic
606C4#1683808911yescharacteristic
607C2—42no—
608C2#2142749911yescharacteristic
609C2#2143375411yescharacteristic
610C2#2143378111yescharacteristic
611C2#2143432811yescharacteristic
612C2#2143434411yescharacteristic
613C2#2143537011yescharacteristic
614C2#2143540011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#485C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenterC2×#485D59
hypocenterC59#24619
LCSG ▷ C22×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
4. #21425595C2semidirect
5. #21425590C2semidirect
6. #21425475C2semidirect
7. #21425475C2semidirect
8. #21424837C2semidirect
9. #21424822C2semidirect
10. #21424783C2semidirect
11. #21424769C2semidirect
368. C59#24619semidirect

Maximal subgroups

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

subgroupindex
C2×#168374602
#214298942
#214267122
#214255952
#214255902
#214254752
#214248372
#214248222
#214247832
#214247692
#214244042
#214242912
#214241912
C2×C59×#4852
#2461959