Finite Group Databasebeta

← #28608121

Subgroups — #28608121

Order 15104.

866 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
1C2×#16837500C211yesmaximal, characteristic
2#21427043C211yesmaximal, characteristic
3#21424792C211yesmaximal, characteristic
4#21424778C211yesmaximal, characteristic
5#21424685C211yesmaximal, characteristic, complemented
6#21424664C211yesmaximal, characteristic, complemented
7#21424451C211yesmaximal, characteristic
8#21424451C211yesmaximal, characteristic
9#21424229C211yesmaximal, characteristic, complemented
10#21424103C211yesmaximal, characteristic
11#21424101C211yesmaximal, characteristic, complemented
12#21423971C211yesmaximal, characteristic, complemented
13#21423926C211yesmaximal, characteristic, complemented
14C4×#14273651C211yesmaximal, characteristic, complemented
15C59×#2351C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273685—21no—
19#16837500C418yescomplemented
20C2×#14273660C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837294—21no—
34#16837294—21no—
35#16837272C2211yescharacteristic
36#16837259—21no—
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×Q8×C59C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C59×#389—21no—
53C59×#389—21no—
54C59×#389C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#382—21no—
57C59×#382C2211yescharacteristic
58C4×C4⋊C4×C59—21no—
59C4×C4⋊C4×C59C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C2×#11745671—22no—
63C2×#11745671—22no—
64C2×#11745671C2×C414yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C4×D236—22no—
70C4×D236C2×C414yes—
71#14273685—44no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84#14273660C2×C414yes—
85C4⋊(C59⋊C8)—22no—
86#14273651—22no—
87#14273651—22no—
88#14273651C2×C414yes—
89#14273647—41no—
90#14273647—41no—
91#14273647C2×C414yes—
92#14273638—41no—
93#14273638C2×C414yes—
94C4×C59⋊C8—41no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×Q8×C59C2311yescharacteristic
98C4×Q8×C59—22no—
99C4×Q8×C59—22no—
100C4×Q8×C59—22no—
101C4×Q8×C59—41no—
102C4×Q8×C59—41no—
103C4×Q8×C59C2×C414yes—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59D411yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C42×C59—21no—
115C2×C42×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C23×D59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59C22×C412yes—
131C2×Q8×C59#4112yes—
132C2×Q8×C59#4112yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236C22×C412yes—
153C2×C4×D59—24no—
154C2×C4×D59—42no—
155C2×C4×D59—42no—
156C4⋊C4×C59—24no—
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—41no—
164C4⋊C4×C59—42no—
165C4⋊C4×C59—42no—
166C4⋊C4×C59—42no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59C22×C412yes—
169C42×C59—24no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—42no—
175C42×C59—42no—
176C42×C59C22×C412yes—
177C42×C59#4112yes—
178#11745671—28no—
179(C2×D59)⋊C4—24no—
180(C2×D59)⋊C4—42no—
181(C2×D59)⋊C4—42no—
182(C2×D59)⋊C4—81no—
183(C2×D59)⋊C4—81no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—42no—
190C4⋊Dic59—81no—
191C4⋊Dic59C22×C412yes—
192C4×Dic59—42no—
193C4×Dic59—42no—
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⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8C22×C412yes—
205C2×C59⋊C8C22×C412yes—
206C23×C59C22×D411yescharacteristic
207C22×D59—22no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—41no—
214Q8×C59—22no—
215Q8×C59—22no—
216Q8×C59—22no—
217Q8×C59#11714yescomplemented
218C2×C4×C59—22no—
219C2×C4×C59—21no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
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—41no—
245C2×C4×C59C4×D412yes—
246C2×C4×C59C4×D412yes—
247C2×C4×C59AGL(1,ℤ/8)12yes—
248C2×C4×C59C23×C411yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59H(5,𝔽2)11yescharacteristic
253C2×C4×C592−1+411yescharacteristic
254C2×Dic59—22no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263D236—22no—
264D236—22no—
265D236—41no—
266D236—41no—
267C4×D59—48no—
268C59⋊C8—22no—
269C59⋊C8—22no—
270C59⋊C8—41no—
271C59⋊C8—41no—
272#22781—591nomaximal, Sylow
273C22×C59C2×C4×D411yescharacteristic
274C22×C59#54611yescharacteristic
275C22×C59#54811yescharacteristic
276C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C59#57511yescharacteristic
278C22×C59#57711yescharacteristic
279C22×C59#57811yescharacteristic
280C2×D59—42no—
281C2×D59—42no—
282C2×D59—42no—
283C2×D59—81no—
284C2×D59—81no—
285C2×D59—81no—
286C2×D59—81no—
287C4×C59—24no—
288C4×C59—22no—
289C4×C59—22no—
290C4×C59—22no—
291C4×C59—42no—
292C4×C59—41no—
293C4×C59—41no—
294C4×C59—41no—
295C4×C59—81no—
296C4×C59—81no—
297C4×C59C2×#11711yescharacteristic
298C4×C59#52011yescharacteristic
299C4×C59#55211yescharacteristic
300C4×C59#55311yescharacteristic
301Dic59—42no—
302Dic59—81no—
303Dic59—81no—
304C2×#440—591no—
305#2354—591no—
306#2351D5911yescharacteristic, complemented
307#1936—591no—
308#1934—591no—
309#1921—591no—
310#1918—591no—
311#1880—591no—
312#1880—591no—
313#1836—591no—
314#1804—591no—
315#1803—591no—
316#1771—591no—
317#1763—591no—
318C4×#103—591no—
319C2×C59#235211yescharacteristic
320C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
321C2×C59#294711yescharacteristic
322C2×C59#330811yescharacteristic
323C2×C59#332111yescharacteristic
324C2×C59#336811yescharacteristic
325C2×C59#337711yescharacteristic
326D59—82no—
327C22×SD16—591no—
328C2×C4×Q8C2×D5911yescharacteristic
329C2×C4×Q8C2×D5911yescharacteristic
330C2×C4×Q8C2×D5911yescharacteristic
331C2×C4×D4—591no—
332C2×#117—1181no—
333#440—598no—
334C2×#107—591no—
335C2×C4⋊C8—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#102—591no—
340C2×#102—591no—
341C2×#102—591no—
342C2×#97—591no—
343C2×#97—591no—
344C2×C4×C8—591no—
345#390—591no—
346#390—591no—
347#389—21no—
348#389—21no—
349#389C2×D5911yescharacteristic
350#389C2×D5911yescharacteristic
351#388—1181no—
352#388—1181no—
353#384—591no—
354#382—21no—
355#382C2×D5911yescharacteristic
356#380—1181no—
357C4×C4⋊C4—21no—
358C4×C4⋊C4—591no—
359C4×C4⋊C4—591no—
360C4×C4⋊C4C2×D5911yescharacteristic
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#2278111yescharacteristic, complemented, Sylow
372C22×Q8C22×D5911yescharacteristic
373C22×D4—591no—
374C23×C4—1181no—
375C2×SD16—594no—
376C2×SD16—1182no—
377C2×SD16—1182no—
378C22×C8—591no—
379C22×C8—591no—
380C22×C8—591no—
381C22×C8—591no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8—22no—
385C4×Q8—41no—
386C4×Q8—41no—
387C4×Q8C4×D5914yes—
388C4×D4—594no—
389C4×D4—1182no—
390#117—2364no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4C22⋊D5911yescharacteristic
402C2×C4⋊C4C22⋊D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C22⋊C4—1181no—
409C2×C22⋊C4—1181no—
410C2×C22⋊C4—1181no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—1181no—
415C2×C42—1181no—
416C2×C42C22×D5911yescharacteristic
417C2×C42C22×D5911yescharacteristic
418C2×C42C22×D5911yescharacteristic
419#107—594no—
420C4⋊C8—1182no—
421#103—594no—
422#103—1182no—
423#103—1182no—
424#102—594no—
425#102—2361no—
426#102—2361no—
427#97—594no—
428#97—2361no—
429C4×C8—2361no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1181no—
435#95—1181no—
436#95C22⋊D5911yescharacteristic
437#95C22⋊D5911yescharacteristic
438C24—1181no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D5912yes—
443C2×Q8#1174568712yes—
444C2×Q8#1174568912yes—
445C2×D4—592no—
446C2×D4—1181no—
447C2×D4—1181no—
448C2×D4—1181no—
449C2×D4—1181no—
450C2×D4—1181no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—1181no—
456C22×C4—1184no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—2362no—
460C22×C4—2362no—
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C2×C22⋊D5911yescharacteristic
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C23×D5911yescharacteristic
468SD16—1188no—
469C2×C8—592no—
470C2×C8—592no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—592no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—4721no—
500C4⋊C4—2362no—
501C4⋊C4C2×C4×D5912yes—
502C22⋊C4—1184no—
503C22⋊C4—4721no—
504C22⋊C4—4721no—
505C22⋊C4—2362no—
506C22⋊C4—2362no—
507C42—24no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—42no—
513C42—42no—
514C42—2362no—
515C42—2362no—
516C42C2×C4×D5912yes—
517C42#1174568512yes—
518C23—1182no—
519C23—2361no—
520C23—2361no—
521C23—2361no—
522C23—2361no—
523C23—2361no—
524C23—2361no—
525C23C22×C22⋊D5911yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8#1427368514yescomplemented
530D4—1182no—
531D4—1182no—
532D4—2361no—
533D4—2361no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
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—1182no—
562C2×C4—2368no—
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×C4C4×C22⋊D5912yes—
572C2×C4C4×C22⋊D5912yes—
573C2×C4#1427375212yes—
574C2×C4C22×C4×D5911yescharacteristic
575C2×C4C2×#1174568711yescharacteristic
576C2×C4C2×#1174568511yescharacteristic
577C2×C4C2×#1174568911yescharacteristic
578C2×C4#1427377211yescharacteristic
579C2×C4#1427377511yescharacteristic
580C8—1182no—
581C8—1182no—
582C8—2361no—
583C8—2361no—
584C22—4721no—
585C22—4721no—
586C22—4721no—
587C22—4721no—
588C22—2362no—
589C22—2362no—
590C22—2362no—
591C22C2×C4×C22⋊D5911yescharacteristic
592C22#1683804611yescharacteristic
593C22#1683806311yescharacteristic
594C22C2×#1427375211yescharacteristic
595C22#1683820011yescharacteristic
596C22#1683821011yescharacteristic
597C22#1683821711yescharacteristic
598C4—22no—
599C4—24no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—4721no—
609C4—4721no—
610C4—2362no—
611C4C2×#1427368511yescharacteristic
612C4#1683788311yescharacteristic
613C4#1683808711yescharacteristic
614C4#1683810011yescharacteristic
615C2—4722no—
616C2#2142702111yescharacteristic
617C2C4×#1427375211yescharacteristic
618C2#2143293911yescharacteristic
619C2#2143608711yescharacteristic
620C2#2143628211yescharacteristic
621C2#2143682511yescharacteristic
622C2#2143693811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2351C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#2351D59
hypocenterC59#22781
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
5. #21424685C2semidirect
6. #21424664C2semidirect
9. #21424229C2semidirect
11. #21424101C2semidirect
12. #21423971C2semidirect
13. #21423926C2semidirect
14. C4×#14273651C2semidirect
15. C59×#2351C2semidirect
19. #16837500C4semidirect
217. Q8×C59#117semidirect
306. #2351D59semidirect
371. C59#22781semidirect
529. Q8#14273685semidirect

Maximal subgroups

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

subgroupindex
C2×#168375002
#214270432
#214247922
#214247782
#214246852
#214246642
#214244512
#214242292
#214241032
#214241012
#214239712
#214239262
C4×#142736512
C59×#23512
#2278159