Finite Group Databasebeta

← #28610713

Subgroups — #28610713

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837682C211yesmaximal, characteristic
2#21427185C211yesmaximal, characteristic, complemented
3#21425068C211yesmaximal, characteristic, complemented
4#21424862C211yesmaximal, characteristic, complemented
5#21424831C211yesmaximal, characteristic, complemented
6#21424826C211yesmaximal, characteristic, complemented
7#21424685C211yesmaximal, characteristic, complemented
8#21424664C211yesmaximal, characteristic, complemented
9#21424357C211yesmaximal, characteristic
10#21424343C211yesmaximal, characteristic
11#21423971C211yesmaximal, characteristic, complemented
12#21423921C211yesmaximal, characteristic
13#21423921C211yesmaximal, characteristic
14C4×#14273648C211yesmaximal, characteristic
15C59×#2349C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17#16837682C418yescomplemented
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837306—21no—
31#16837306—21no—
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837298—21no—
35C4⋊C4×Dic59—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38C4×C4⋊Dic59C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#385—21no—
55C59×#385—21no—
56C59×#382C2211yescharacteristic
57C59×#380—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C2×#95×C59—21no—
60C22×Dic118—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273727—22no—
66#14273727C2×C414yescomplemented
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77#14273660—41no—
78#14273660C2×C414yescomplemented
79#14273660C2×C414yescomplemented
80#14273658—41no—
81C4⋊(C59⋊C8)C2×C414yescomplemented
82#14273648—22no—
83#14273648C2×C414yes—
84#14273648C2×C414yes—
85#14273648Q814yescomplemented
86#14273638—22no—
87#14273638—41no—
88C4×C59⋊C8—41no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C22×D4×C59C2311yescharacteristic
95C23×C4×C59—21no—
96C23×C4×C59—21no—
97C23×C4×C59—21no—
98C4×D4×C59—22no—
99C4×D4×C59—22no—
100C4×D4×C59—22no—
101C4×D4×C59—41no—
102C4×D4×C59—41no—
103C4×D4×C59C2×C414yes—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C42×C59C2311yescharacteristic
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117#95×C59—21no—
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121#95×C59D411yescharacteristic
122#95×C59D411yescharacteristic
123C24×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59C22×C412yes—
128C2×D4×C59C2×Q812yes—
129C2×D4×C59#4112yes—
130C22×C4×C59—21no—
131C22×C4×C59—24no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—42no—
135C22×C4×C59—42no—
136C22×C4×C59—42no—
137C22×C4×C59—42no—
138C22×C4×C59—81no—
139C22×C4×C59—81no—
140C22×C4×C59—81no—
141C22×C4×C59—81no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C2×Dic118—24no—
154C2×Dic118—81no—
155C2×Dic118—81no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—41no—
163C4⋊C4×C59—81no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59C22×C412yes—
166C22⋊C4×C59—24no—
167C22⋊C4×C59—42no—
168C22⋊C4×C59—42no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59C22×C412yes—
174C42×C59#4112yes—
175C4⋊Dic59—24no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C412yes—
182C4⋊Dic59C22×C412yes—
183C4⋊Dic59C2×Q812yes—
184Dic59⋊C4—81no—
185Dic59⋊C4—81no—
186C4×Dic59—42no—
187C4×Dic59—81no—
188C4×Dic59—81no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C22×C412yes—
199C2×C59⋊C8C22×C412yes—
200C2×C59⋊C8C2×Q812yes—
201C23×C59—22no—
202C23×C59—22no—
203C23×C59—22no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59C22×D411yescharacteristic
209D4×C59—22no—
210D4×C59—22no—
211D4×C59—22no—
212D4×C59C4×Q814yes—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
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×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—84no—
238C2×C4×C59—84no—
239C2×C4×C59—84no—
240C2×C4×C59C4×D412yes—
241C2×C4×C59C4×D412yes—
242C2×C4×C59#13712yes—
243C2×C4×C59C23×C411yescharacteristic
244C2×C4×C59C22×Q811yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59H(5,𝔽2)11yescharacteristic
248C2×C4×C592−1+411yescharacteristic
249C2×Dic59—22no—
250C2×Dic59—22no—
251C2×Dic59—22no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261Dic118—84no—
262C59⋊C8—22no—
263C59⋊C8—22no—
264C59⋊C8—22no—
265C59⋊C8—41no—
266#22979—591nomaximal, Sylow
267C22×C59—24no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59C2×C4×D411yescharacteristic
275C22×C59#54611yescharacteristic
276C22×C59D4×Q811yescharacteristic
277C22×C59C2×#13711yescharacteristic
278C22×C59#57511yescharacteristic
279C22×C59#57611yescharacteristic
280C22×C59#57611yescharacteristic
281C4×C59—22no—
282C4×C59—22no—
283C4×C59—22no—
284C4×C59—42no—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59—41no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59C2×C4×Q811yescharacteristic
291C4×C59#52011yescharacteristic
292C4×C59#55211yescharacteristic
293C4×C59#55411yescharacteristic
294Dic59—42no—
295Dic59—42no—
296Dic59—42no—
297Dic59—81no—
298C2×#474—591no—
299#2369—591no—
300#2349D5911yescharacteristic
301#1993—591no—
302#1949—591no—
303#1943—591no—
304#1942—591no—
305#1921—591no—
306#1918—591no—
307#1864—591no—
308#1862—591no—
309#1771—591no—
310#1762—591no—
311#1762—591no—
312C4×#102—591no—
313C2×C59—44no—
314C2×C59#236011yescharacteristic
315C2×C59C4×#13711yescharacteristic
316C2×C59#294811yescharacteristic
317C2×C59#331411yescharacteristic
318C2×C59#331911yescharacteristic
319C2×C59#338511yescharacteristic
320C2×C59#338911yescharacteristic
321C2×C4⋊Q8—591no—
322C2×C4×D4C2×D5911yescharacteristic
323C2×C4×D4C2×D5911yescharacteristic
324C2×C4×D4C2×D5911yescharacteristic
325#474—598no—
326C2×#107—591no—
327C2×#107—591no—
328C2×#107—591no—
329C2×#106—591no—
330C2×C4⋊C8—591no—
331C2×#102—591no—
332C2×#102—591no—
333C2×#102—591no—
334C2×#102—591no—
335C2×#97—591no—
336C2×#97—591no—
337C2×C4×C8—591no—
338#391—1181no—
339#391—1181no—
340#389—591no—
341#389—591no—
342#389—1181no—
343#389C2×D5911yescharacteristic
344#389C2×D5911yescharacteristic
345#385—21no—
346#385—21no—
347#382C2×D5911yescharacteristic
348#380—21no—
349C4×C4⋊C4—591no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—1181no—
353C4×C4⋊C4C2×D5911yescharacteristic
354C2×#95—21no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#336—591no—
364#336—591no—
365C59#2297911yescharacteristic, complemented, Sylow
366C22×Q8—1181no—
367C22×D4C22×D5911yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4—21no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8—591no—
374C22×C8—591no—
375C4⋊Q8—594no—
376C4⋊Q8—1182no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—41no—
381C4×D4—41no—
382C4×D4C4×D5914yes—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4C22⋊D5911yescharacteristic
393C2×C4⋊C4C22⋊D5911yescharacteristic
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—1181no—
401C2×C42—1181no—
402C2×C42—1181no—
403C2×C42C22×D5911yescharacteristic
404C2×C42C22×D5911yescharacteristic
405C2×C42C22×D5911yescharacteristic
406#107—594no—
407#107—594no—
408#107—2361no—
409#106—2361no—
410C4⋊C8—594no—
411#102—594no—
412#102—594no—
413#102—594no—
414#102—1182no—
415#97—1182no—
416#97—2361no—
417C4×C8—2361no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95C22⋊D5911yescharacteristic
428#95C22⋊D5911yescharacteristic
429C24—21no—
430C2×Q8—1184no—
431C2×Q8—4721no—
432C2×Q8—4721no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4C2×C4×D5912yes—
437C2×D4#1174568712yes—
438C2×D4Q8×D5912yes—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4C2×C22⋊D5911yescharacteristic
456C22×C4C2×C22⋊D5911yescharacteristic
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C23×D5911yescharacteristic
462C2×C8—592no—
463C2×C8—592no—
464C2×C8—592no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—592no—
484C4⋊C4—592no—
485C4⋊C4—592no—
486C4⋊C4—1184no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—4721no—
493C4⋊C4—4721no—
494C4⋊C4C2×C4×D5912yes—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—4721no—
503C42—4721no—
504C42—2362no—
505C42C2×C4×D5912yes—
506C42#1174568512yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C22×C22⋊D5911yescharacteristic
515Q8—4724no—
516D4—22no—
517D4—22no—
518D4—22no—
519D4#1427369214yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—1182no—
548C2×C4—1182no—
549C2×C4—1182no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4C4×C22⋊D5912yes—
560C2×C4C4×C22⋊D5912yes—
561C2×C4#1427375712yes—
562C2×C4C22×C4×D5911yescharacteristic
563C2×C4C2×Q8×D5911yescharacteristic
564C2×C4C2×#1174568711yescharacteristic
565C2×C4C2×#1174568511yescharacteristic
566C2×C4#1427377211yescharacteristic
567C2×C4#1427377511yescharacteristic
568C8—1182no—
569C8—1182no—
570C8—1182no—
571C8—2361no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×C4×C22⋊D5911yescharacteristic
580C22#1683804611yescharacteristic
581C22Q8×C22⋊D5911yescharacteristic
582C22C2×#1427375711yescharacteristic
583C22#1683820011yescharacteristic
584C22#1683820511yescharacteristic
585C22#1683820511yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4—4721no—
596C4—2362no—
597C4—2362no—
598C4—2362no—
599C4C2×#1427369211yescharacteristic
600C4#1683788311yescharacteristic
601C4#1683808711yescharacteristic
602C4#1683810711yescharacteristic
603C2—44no—
604C2#2142710111yescharacteristic
605C2C4×#1427375711yescharacteristic
606C2#2143294611yescharacteristic
607C2#2143617711yescharacteristic
608C2#2143625211yescharacteristic
609C2#2143700611yescharacteristic
610C2#2143705011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2349C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#2349D59
hypocenterC59#22979
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #21427185C2semidirect
3. #21425068C2semidirect
4. #21424862C2semidirect
5. #21424831C2semidirect
6. #21424826C2semidirect
7. #21424685C2semidirect
8. #21424664C2semidirect
11. #21423971C2semidirect
17. #16837682C4semidirect
66. #14273727C2×C4semidirect
78. #14273660C2×C4semidirect
79. #14273660C2×C4semidirect
81. C4⋊(C59⋊C8)C2×C4semidirect
85. #14273648Q8semidirect
365. C59#22979semidirect

Maximal subgroups

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

subgroupindex
C2×#168376822
#214271852
#214250682
#214248622
#214248312
#214248262
#214246852
#214246642
#214243572
#214243432
#214239712
#214239212
C4×#142736482
C59×#23492
#2297959