Finite Group Databasebeta

← #28608111

Subgroups — #28608111

Order 15104.

942 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837512C211yesmaximal, characteristic
2#21427007C211yesmaximal, characteristic, complemented
3#21424796C211yesmaximal, characteristic, complemented
4#21424782C211yesmaximal, characteristic, complemented
5#21424681C211yesmaximal, characteristic, complemented
6#21424674C211yesmaximal, characteristic, complemented
7#21424447C211yesmaximal, characteristic
8#21424447C211yesmaximal, characteristic
9#21424231C211yesmaximal, characteristic, complemented
10#21424106C211yesmaximal, characteristic, complemented
11#21424093C211yesmaximal, characteristic, complemented
12#21423971C211yesmaximal, characteristic, complemented
13#21423924C211yesmaximal, characteristic, complemented
14C4×#14273647C211yesmaximal, characteristic, complemented
15C59×#2354C211yesmaximal, characteristic, complemented
16C22×#11745669C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20#16837512C418yescomplemented
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×C4×C59⋊C8C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837273—21no—
35#16837273—21no—
36#16837268C2211yescharacteristic
37#16837261—21no—
38#95×D59—21no—
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×#117×C59—21no—
51C59×#390C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#388—21no—
54C59×#388—21no—
55C59×#384C2211yescharacteristic
56C59×#380—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64C2×#11745669—22no—
65C2×#11745669—22no—
66C2×#11745669C2×C414yes—
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C4×D236—22no—
72C4×D236—22no—
73C4×D236—22no—
74C4×D236—41no—
75C4×D236—41no—
76C4×D236C2×C414yes—
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85#14273658C2×C414yescomplemented
86C4⋊(C59⋊C8)—22no—
87#14273648—41no—
88#14273648—41no—
89#14273648C2×C414yescomplemented
90#14273647—22no—
91#14273647—22no—
92#14273647C2×C414yes—
93#14273638—41no—
94#14273638C2×C414yescomplemented
95C4×C59⋊C8—41no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C22×D4×C59C2311yescharacteristic
101C23×C4×C59—21no—
102C4×D4×C59—22no—
103C4×D4×C59C2×C414yescomplemented
104#117×C59—44no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59—21no—
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59—21no—
127C23×D59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59C22×C412yes—
134C22×C4×C59—21no—
135C22×C4×C59—24no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—42no—
139C22×C4×C59—42no—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2411yescharacteristic
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236C22×C412yes—
154C2×D236#4112yes—
155C2×D236#4112yes—
156C2×C4×D59—24no—
157C2×C4×D59—42no—
158C2×C4×D59—42no—
159C2×C4×D59—42no—
160C2×C4×D59—42no—
161C2×C4×D59—81no—
162C2×C4×D59—81no—
163C2×C4×D59—81no—
164C2×C4×D59—81no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—41no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59—81no—
174C4⋊C4×C59C22×C412yes—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—81no—
179C22⋊C4×C59—81no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—42no—
185C42×C59—42no—
186C42×C59C22×C412yes—
187C42×C59#4112yes—
188#11745669—28no—
189(C2×D59)⋊C4—24no—
190(C2×D59)⋊C4—42no—
191(C2×D59)⋊C4—42no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59C22×C412yes—
198Dic59⋊C4—81no—
199Dic59⋊C4—81no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8C22×C412yes—
211C2×C59⋊C8C22×C412yes—
212C23×C59—22no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59C22×D411yescharacteristic
220C22×D59—22no—
221C22×D59—22no—
222C22×D59—22no—
223C22×D59—41no—
224C22×D59—41no—
225C22×D59—41no—
226C22×D59—41no—
227D4×C59—22no—
228D4×C59—22no—
229D4×C59—41no—
230D4×C59—41no—
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—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—48no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59C4×D412yes—
257C2×C4×C59C4×D412yes—
258C2×C4×C59AGL(1,ℤ/8)12yes—
259C2×C4×C59C23×C411yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59H(5,𝔽2)11yescharacteristic
264C2×C4×C592−1+411yescharacteristic
265C2×Dic59—22no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274D236—22no—
275D236—22no—
276D236—22no—
277D236#11714yescomplemented
278C4×D59—84no—
279C4×D59—84no—
280C4×D59—84no—
281C59⋊C8—22no—
282C59⋊C8—22no—
283C59⋊C8—41no—
284C59⋊C8—41no—
285#22780—591nomaximal, Sylow
286C22×C59—42no—
287C22×C59—42no—
288C22×C59—42no—
289C22×C59—81no—
290C22×C59—81no—
291C22×C59—81no—
292C22×C59—81no—
293C22×C59C2×C4×D411yescharacteristic
294C22×C59#54611yescharacteristic
295C22×C59#54811yescharacteristic
296C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C59#57511yescharacteristic
298C22×C59#57611yescharacteristic
299C22×C59#57711yescharacteristic
300C2×D59—24no—
301C2×D59—42no—
302C2×D59—42no—
303C2×D59—42no—
304C2×D59—42no—
305C2×D59—42no—
306C2×D59—42no—
307C4×C59—22no—
308C4×C59—22no—
309C4×C59—22no—
310C4×C59—42no—
311C4×C59—41no—
312C4×C59—41no—
313C4×C59—41no—
314C4×C59—81no—
315C4×C59—81no—
316C4×C59C2×#11711yescharacteristic
317C4×C59#52011yescharacteristic
318C4×C59#55211yescharacteristic
319C4×C59#55311yescharacteristic
320Dic59—42no—
321Dic59—81no—
322Dic59—81no—
323C2×#442—591no—
324#2354D5911yescharacteristic, complemented
325#2349—591no—
326#1936—591no—
327#1934—591no—
328#1920—591no—
329#1919—591no—
330#1879—591no—
331#1879—591no—
332#1836—591no—
333#1804—591no—
334#1802—591no—
335#1771—591no—
336#1762—591no—
337C4×#102—591no—
338C2×C59—82no—
339C2×C59#235211yescharacteristic
340C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
341C2×C59#294611yescharacteristic
342C2×C59#330611yescharacteristic
343C2×C59#331911yescharacteristic
344C2×C59#336611yescharacteristic
345C2×C59#337611yescharacteristic
346D59—44no—
347C22×D8—591no—
348C2×C4×D4—591no—
349C2×C4×D4—591no—
350C2×C4×D4—591no—
351C2×C4×D4C2×D5911yescharacteristic
352C2×#117—21no—
353#442—598no—
354C2×#106—591no—
355C2×C4⋊C8—591no—
356C2×#102—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×#102—591no—
362C2×#97—591no—
363C2×#97—591no—
364C2×C4×C8—591no—
365#390C2×D5911yescharacteristic
366#390C2×D5911yescharacteristic
367#389—591no—
368#389—591no—
369#388—21no—
370#388—21no—
371#385—1181no—
372#385—1181no—
373#384C2×D5911yescharacteristic
374#382—591no—
375#380—21no—
376#380—1181no—
377C4×C4⋊C4C2×D5911yescharacteristic
378C4×C4⋊C4C2×D5911yescharacteristic
379C4×C4⋊C4C2×D5911yescharacteristic
380C2×#95—1181no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#336—591no—
390#336—591no—
391C59#2278011yescharacteristic, complemented, Sylow
392C22×D4—591no—
393C22×D4C22×D5911yescharacteristic
394C23×C4—21no—
395C23×C4—1181no—
396C23×C4—1181no—
397C23×C4—1181no—
398C2×D8—594no—
399C2×D8—1182no—
400C2×D8—1182no—
401C22×C8—591no—
402C22×C8—591no—
403C22×C8—591no—
404C22×C8—591no—
405C4×D4—22no—
406C4×D4—594no—
407C4×D4—1182no—
408C4×D4—1182no—
409C4×D4—1182no—
410C4×D4—2361no—
411C4×D4—2361no—
412C4×D4C4×D5914yescomplemented
413#117—44no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4C22⋊D5911yescharacteristic
422C2×C4⋊C4C22⋊D5911yescharacteristic
423C2×C4⋊C4C22×D5911yescharacteristic
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C4⋊C4C22×D5911yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4—1181no—
433C2×C22⋊C4—1181no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D5911yescharacteristic
437C2×C42C22×D5911yescharacteristic
438C2×C42C22×D5911yescharacteristic
439#106—594no—
440C4⋊C8—1182no—
441#102—594no—
442#102—594no—
443#102—1182no—
444#102—1182no—
445#102—2361no—
446#102—2361no—
447#97—594no—
448#97—2361no—
449C4×C8—2361no—
450#95—21no—
451#95—21no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456#95C22⋊D5911yescharacteristic
457#95C22⋊D5911yescharacteristic
458C24—21no—
459C24—1181no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—592no—
466C2×D4—592no—
467C2×D4—592no—
468C2×D4—1181no—
469C2×D4—1181no—
470C2×D4—1181no—
471C2×D4C2×C4×D5912yes—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—1181no—
479C22×C4—1184no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—4721no—
483C22×C4—4721no—
484C22×C4—4721no—
485C22×C4—4721no—
486C22×C4—2362no—
487C22×C4—2362no—
488C22×C4—2362no—
489C22×C4—2362no—
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C23×D5911yescharacteristic
497D8—1188no—
498C2×C8—592no—
499C2×C8—592no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—41no—
517C4⋊C4—42no—
518C4⋊C4—81no—
519C4⋊C4—592no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—4721no—
526C4⋊C4—4721no—
527C4⋊C4C2×C4×D5912yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—81no—
532C22⋊C4—81no—
533C22⋊C4—1184no—
534C22⋊C4—2362no—
535C22⋊C4—2362no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—42no—
542C42C2×C4×D5912yes—
543C42#1174568512yes—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—1182no—
552C23—1182no—
553C23—1182no—
554C23—2361no—
555C23—2361no—
556C23—2361no—
557C23—2361no—
558C23C22×C22⋊D5911yescharacteristic
559D4—22no—
560D4—22no—
561D4—41no—
562D4—41no—
563D4—594no—
564D4—1182no—
565D4—1182no—
566D4—1182no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—48no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—1182no—
593C2×C4—4724no—
594C2×C4—4724no—
595C2×C4—4724no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4C4×C22⋊D5912yes—
605C2×C4C4×C22⋊D5912yes—
606C2×C4#1427375312yes—
607C2×C4C22×C4×D5911yescharacteristic
608C2×C4#41×D5911yescharacteristic
609C2×C4#41×D5911yescharacteristic
610C2×C4C2×#1174568511yescharacteristic
611C2×C4#1427377311yescharacteristic
612C2×C4#1427377411yescharacteristic
613C8—1182no—
614C8—1182no—
615C8—2361no—
616C8—2361no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—1184no—
625C22—2362no—
626C22—2362no—
627C22—2362no—
628C22—2362no—
629C22—2362no—
630C22—2362no—
631C22C2×C4×C22⋊D5911yescharacteristic
632C22#1683804011yescharacteristic
633C22#1683805911yescharacteristic
634C22C2×#1427375311yescharacteristic
635C22#1683820011yescharacteristic
636C22#1683820511yescharacteristic
637C22#1683821011yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—81no—
646C4—81no—
647C4—4721no—
648C4—4721no—
649C4—2362no—
650C4#117×D5911yescharacteristic
651C4#1683788211yescharacteristic
652C4#1683809111yescharacteristic
653C4#1683809611yescharacteristic
654C2—82no—
655C2—2364no—
656C2#2142702611yescharacteristic
657C2C4×#1427375311yescharacteristic
658C2#2143292411yescharacteristic
659C2#2143605111yescharacteristic
660C2#2143624611yescharacteristic
661C2#2143679911yescharacteristic
662C2#2143691711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2354C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
cosocleC22×C4×C59C24
hypercenter#2354D59
hypocenterC59#22780
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #21427007C2semidirect
3. #21424796C2semidirect
4. #21424782C2semidirect
5. #21424681C2semidirect
6. #21424674C2semidirect
9. #21424231C2semidirect
10. #21424106C2semidirect
11. #21424093C2semidirect
12. #21423971C2semidirect
13. #21423924C2semidirect
14. C4×#14273647C2semidirect
15. C59×#2354C2semidirect
20. #16837512C4semidirect
85. #14273658C2×C4semidirect
89. #14273648C2×C4semidirect
94. #14273638C2×C4semidirect
103. C4×D4×C59C2×C4semidirect
277. D236#117semidirect
324. #2354D59semidirect
391. C59#22780semidirect
412. C4×D4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168375122
#214270072
#214247962
#214247822
#214246812
#214246742
#214244472
#214242312
#214241062
#214240932
#214239712
#214239242
C4×#142736472
C59×#23542
#2278059