Finite Group Databasebeta

← #28640745

Subgroups — #28640745

Order 15104.

790 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837692C211yesmaximal, characteristic
2C2×#16837533C211yesmaximal, characteristic, complemented
3#21429081C211yesmaximal, characteristic, complemented
4#21425598C211yesmaximal, characteristic, complemented
5#21425597C211yesmaximal, characteristic, complemented
6#21425484C211yesmaximal, characteristic
7#21425470C211yesmaximal, characteristic
8#21425068C211yesmaximal, characteristic, complemented
9#21424862C211yesmaximal, characteristic, complemented
10#21424831C211yesmaximal, characteristic, complemented
11#21424821C211yesmaximal, characteristic, complemented
12#21424796C211yesmaximal, characteristic
13#21424782C211yesmaximal, characteristic
14#21423921C211yesmaximal, characteristic
15C59×#2580C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273725—21no—
18#16837692—24no—
19#16837533—24no—
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273638C2211yescharacteristic
32C2×#14273638C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837332—21no—
35#16837332—21no—
36#16837307—21no—
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837300C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C59×#399—21no—
53C59×#393—21no—
54C59×#390C2211yescharacteristic
55C59×#390C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#385—21no—
59C59×#385—21no—
60C59×#382C2211yescharacteristic
61C22×Dic118—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273727—22no—
67#14273727—22no—
68#14273725—44no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×C4×Dic59—21no—
79#14273660—22no—
80#14273658—22no—
81#14273658—22no—
82#14273658—22no—
83C4⋊(C59⋊C8)—22no—
84C4⋊(C59⋊C8)—22no—
85C4⋊(C59⋊C8)D414yes—
86#14273648—22no—
87#14273648—22no—
88#14273648—22no—
89#14273648Q814yescomplemented
90#14273638—22no—
91#14273638—22no—
92C4×C59⋊C8—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C22×Q8×C59D411yescharacteristic
99C22×D4×C59C2311yescharacteristic
100C23×C4×C59—21no—
101C4×Q8×C59—22no—
102C4×Q8×C59—22no—
103C4×D4×C59—22no—
104C4×D4×C59—22no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125C24×C59—21no—
126C2×Q8×C59—22no—
127C2×Q8×C59—22no—
128C2×Q8×C59—41no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59C2×Q812yes—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—42no—
139C22×C4×C59—81no—
140C22×C4×C59—81no—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2411yescharacteristic
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C2×Dic118—42no—
153C2×Dic118—42no—
154C2×Dic118—42no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59—81no—
166C22⋊C4×C59—42no—
167C22⋊C4×C59—42no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—81no—
170C22⋊C4×C59—81no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59C2×D412yes—
176C42×C59#4112yes—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—42no—
184C4⋊Dic59C2×Q812yes—
185C4⋊Dic59#4112yes—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190C4×Dic59—42no—
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⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8C2×D412yes—
201C2×C59⋊C8C2×D412yes—
202C2×C59⋊C8C2×Q812yes—
203C23×C59—22no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59C22×D411yescharacteristic
211Q8×C59—44no—
212D4×C59—22no—
213D4×C59—22no—
214D4×C59—41no—
215D4×C59—41no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
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—21no—
230C2×C4×C59—21no—
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—84no—
241C2×C4×C59#12112yes—
242C2×C4×C59#12112yes—
243C2×C4×C59AGL(1,ℤ/8)12yes—
244C2×C4×C59C22×D411yescharacteristic
245C2×C4×C59C22×Q811yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C592−1+411yescharacteristic
249C2×C4×C592−1+411yescharacteristic
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—
261C2×Dic59—41no—
262Dic118—84no—
263C59⋊C8—22no—
264C59⋊C8—22no—
265C59⋊C8—22no—
266C59⋊C8—41no—
267#25485—591nomaximal, Sylow
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—81no—
272C22×C59—81no—
273C22×C59—81no—
274C22×C59—81no—
275C22×C59C2×#12111yescharacteristic
276C22×C59#54811yescharacteristic
277C22×C59D4×Q811yescharacteristic
278C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C59#57511yescharacteristic
280C22×C59#57711yescharacteristic
281C22×C59#57711yescharacteristic
282C4×C59—22no—
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59—41no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59—81no—
291C4×C59#53311yescharacteristic
292C4×C59#54111yescharacteristic
293C4×C59#54711yescharacteristic
294C4×C59D4×Q811yescharacteristic
295Dic59—42no—
296Dic59—42no—
297Dic59—81no—
298Dic59—81no—
299C2×#476—591no—
300C2×#446—591no—
301#2580D5911yescharacteristic
302#2546—591no—
303#2088—591no—
304#2088—591no—
305#2067—591no—
306#2065—591no—
307#1993—591no—
308#1949—591no—
309#1943—591no—
310#1941—591no—
311#1936—591no—
312#1934—591no—
313#1762—591no—
314C2×C59—82no—
315C2×C59#242911yescharacteristic
316C2×C59#305511yescharacteristic
317C2×C59#305711yescharacteristic
318C2×C59#337111yescharacteristic
319C2×C59#337311yescharacteristic
320C2×C59#338411yescharacteristic
321C2×C59#338811yescharacteristic
322C2×C4⋊Q8—591no—
323C2×C4⋊Q8—1181no—
324C2×C4×Q8C2×D5911yescharacteristic
325C2×C4×D4C2×D5911yescharacteristic
326#476—1184no—
327#446—1184no—
328C2×#107—591no—
329C2×#106—591no—
330C2×#106—591no—
331C2×#106—591no—
332C2×C4⋊C8—591no—
333C2×C4⋊C8—591no—
334C2×C4⋊C8—591no—
335C2×#102—591no—
336C2×#102—591no—
337C2×#102—591no—
338C2×#102—591no—
339C2×#97—591no—
340C2×#97—591no—
341C2×C4×C8—591no—
342#399—21no—
343#398—1181no—
344#398—1181no—
345#393—21no—
346#391—1181no—
347#390C2×D5911yescharacteristic
348#390C2×D5911yescharacteristic
349#389—591no—
350#389—591no—
351#389—591no—
352#389—591no—
353#389C2×D5911yescharacteristic
354#389C2×D5911yescharacteristic
355#385—21no—
356#385—21no—
357#382C2×D5911yescharacteristic
358C4×C4⋊C4—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367C59#2548511yescharacteristic, complemented, Sylow
368C22×Q8—1181no—
369C22×Q8C22⋊D5911yescharacteristic
370C22×D4C22×D5911yescharacteristic
371C23×C4—21no—
372C22×C8—591no—
373C22×C8—591no—
374C22×C8—591no—
375C22×C8—591no—
376C4⋊Q8—1182no—
377C4⋊Q8—1182no—
378C4⋊Q8—2364no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×D4—22no—
382C4×D4—22no—
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⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4C22⋊D5911yescharacteristic
395C2×C4⋊C4C22⋊D5911yescharacteristic
396C2×C4⋊C4C22⋊D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C42—1181no—
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#107—1182no—
410#106—1182no—
411#106—1182no—
412#106—1182no—
413C4⋊C8—594no—
414C4⋊C8—1182no—
415C4⋊C8—1182no—
416#102—594no—
417#102—1182no—
418#102—1182no—
419#102—1182no—
420#97—1182no—
421#97—1182no—
422C4×C8—1182no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432C24—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—41no—
436C2×Q8—2362no—
437C2×Q8—2362no—
438C2×Q8—2362no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4Q8×D5912yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
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—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—592no—
486C4⋊C4—592no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C4⋊C4—2362no—
497C4⋊C4—2362no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—2362no—
508C42D4×D5912yes—
509C42#1174568712yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C22×C22⋊D5911yescharacteristic
518Q8—44no—
519Q8—4724no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
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—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
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—41no—
548C2×C4—84no—
549C2×C4—1182no—
550C2×C4—1182no—
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×C4—2361no—
560C2×C4—2361no—
561C2×C4#1427369912yes—
562C2×C4#1427369912yes—
563C2×C4#1427375212yes—
564C2×C4C2×D4×D5911yescharacteristic
565C2×C4C2×Q8×D5911yescharacteristic
566C2×C4#41×D5911yescharacteristic
567C2×C4C2×#1174568711yescharacteristic
568C2×C4#1427377411yescharacteristic
569C2×C4#1427377511yescharacteristic
570C8—1182no—
571C8—1182no—
572C8—1182no—
573C8—2361no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#1427369911yescharacteristic
582C22#1683805911yescharacteristic
583C22Q8×C22⋊D5911yescharacteristic
584C22C2×#1427375211yescharacteristic
585C22#1683820011yescharacteristic
586C22#1683821311yescharacteristic
587C22#1683821311yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—4721no—
598C4—4721no—
599C4—2362no—
600C4—2362no—
601C4#1683795411yescharacteristic
602C4#1683801211yescharacteristic
603C4#1683805411yescharacteristic
604C4#1683806511yescharacteristic
605C2—82no—
606C2#2142769711yescharacteristic
607C2#2143371711yescharacteristic
608C2#2143373211yescharacteristic
609C2#2143685811yescharacteristic
610C2#2143688411yescharacteristic
611C2#2143699511yescharacteristic
612C2#2143704111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2580C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2580D59
hypocenterC59#25485
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. C2×#16837533C2semidirect
3. #21429081C2semidirect
4. #21425598C2semidirect
5. #21425597C2semidirect
8. #21425068C2semidirect
9. #21424862C2semidirect
10. #21424831C2semidirect
11. #21424821C2semidirect
89. #14273648Q8semidirect
367. C59#25485semidirect

Maximal subgroups

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

subgroupindex
C2×#168376922
C2×#168375332
#214290812
#214255982
#214255972
#214254842
#214254702
#214250682
#214248622
#214248312
#214248212
#214247962
#214247822
#214239212
C59×#25802
#2548559