Finite Group Databasebeta

← #28610142

Subgroups — #28610142

Order 15104.

958 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×#16837715C211yesmaximal, characteristic
2#21427112C211yesmaximal, characteristic
3#21426965C211yesmaximal, characteristic, complemented
4#21424833C211yesmaximal, characteristic, complemented
5#21424794C211yesmaximal, characteristic, complemented
6#21424721C211yesmaximal, characteristic, complemented
7#21424691C211yesmaximal, characteristic, complemented
8#21424686C211yesmaximal, characteristic, complemented
9#21424524C211yesmaximal, characteristic
10#21424358C211yesmaximal, characteristic, complemented
11#21424289C211yesmaximal, characteristic, complemented
12#21423992C211yesmaximal, characteristic, complemented
13#21423920C211yesmaximal, characteristic, complemented
14#21423872C211yesmaximal, characteristic, complemented
15C2×C59×#340C211yesmaximal, characteristic, complemented
16C2×#14273701C2211yescharacteristic
17C2×#14273689C2211yescharacteristic
18C2×C4×C22⋊D59—21no—
19C2×#14273686—21no—
20#16837715C418yescomplemented
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×#14273637C2211yescharacteristic
31C2×#14273637C2211yescharacteristic
32#16837303C2211yescharacteristic
33#16837301C2211yescharacteristic
34#16837301C2211yescharacteristic
35#16837298C2211yescharacteristic
36#16837292—21no—
37#16837288—21no—
38#16837288—21no—
39#16837281—21no—
40#16837273—21no—
41#16837265—21no—
42#16837262C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4⋊C4×Dic59C2211yescharacteristic
45C4⋊C4×Dic59C2211yescharacteristic
46C4×C22⋊Dic59C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C23×C8×C59C2211yescharacteristic
54C22×C4⋊C4×C59C2211yescharacteristic
55C22×C4⋊C4×C59C2211yescharacteristic
56C59×#340C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C22⋊D59—21no—
62C22×C4×D59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273701—22no—
66#14273701C2×C414yes—
67#14273689—22no—
68#14273689—41no—
69C4×C22⋊D59—44no—
70#14273686—44no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59D411yescharacteristic
79C2×C22⋊Dic59D411yescharacteristic
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59C2311yescharacteristic
88C2×C4×Dic59C2311yescharacteristic
89#14273660C2×C414yescomplemented
90#14273660C2×C414yescomplemented
91#14273658—22no—
92#14273658—41no—
93#14273649—22no—
94#14273649—41no—
95#14273647C2×C414yescomplemented
96#14273647C2×C414yescomplemented
97C22⋊(C59⋊C8)C2×C414yescomplemented
98#14273637—22no—
99#14273637—41no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C23×C4×C59—21no—
110C23×C4×C59—21no—
111C23×C4×C59C2311yescharacteristic
112C22×C8×C59—22no—
113C22×C8×C59—22no—
114C22×C8×C59—41no—
115C22×C8×C59C2311yescharacteristic
116C22×C8×C59C2311yescharacteristic
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59—22no—
119C2×C4⋊C4×C59—22no—
120C2×C4⋊C4×C59—21no—
121C2×C4⋊C4×C59—41no—
122C2×C4⋊C4×C59C2×C414yescomplemented
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C4⋊C4×C59C2311yescharacteristic
126C2×C4⋊C4×C59C2311yescharacteristic
127C24×C59C2×D411yescharacteristic
128C23×D59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—24no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—42no—
139C22×C4×C59—42no—
140C22×C4×C59—42no—
141C22×C4×C59—42no—
142C22×C4×C59—81no—
143C22×C4×C59C22×C412yes—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59#4112yes—
146C22×C4×C59C2411yescharacteristic
147C2×C22⋊D59—24no—
148C2×C22⋊D59—42no—
149C2×C22⋊D59—42no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156C22×Dic59C2×D411yescharacteristic
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236—21no—
162C2×D236C22×C412yes—
163C2×C4×D59—42no—
164C2×C4×D59—42no—
165C2×C4×D59—81no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59—44no—
172C2×C8×C59#4112yes—
173C4⋊C4×C59—24no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—42no—
180C4⋊C4×C59C22×C412yes—
181C4⋊C4×C59C22×C412yes—
182C4⋊C4×C59#4112yes—
183C22⋊Dic59—22no—
184C22⋊Dic59—22no—
185C22⋊Dic59—42no—
186(C2×D59)⋊C4—24no—
187(C2×D59)⋊C4—42no—
188(C2×D59)⋊C4—42no—
189(C2×D59)⋊C4—81no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—42no—
196C4⋊Dic59—42no—
197C4⋊Dic59C22×C412yes—
198Dic59⋊C4—42no—
199C4×Dic59—21no—
200C4×Dic59—21no—
201C4×Dic59—21no—
202C4×Dic59—21no—
203C4×Dic59—42no—
204C4×Dic59—42no—
205C4×Dic59—42no—
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—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59C4×D412yes—
217C23×C59AGL(1,ℤ/8)12yes—
218C23×C59AGL(1,ℤ/8)12yes—
219C23×C59C22×D411yescharacteristic
220C22×D59—22no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225C22×D59—41no—
226C22×D59—41no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—48no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—84no—
247C2×C4×C59C4×D412yes—
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
254C22⋊D59—48no—
255C2×Dic59—22no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—21no—
267C2×Dic59—21no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276D236—22no—
277D236—22no—
278D236—41no—
279D236—41no—
280C4×D59—84no—
281C8×C59—22no—
282C8×C59—41no—
283C59⋊C8—22no—
284C59⋊C8—22no—
285#22938—591nomaximal, Sylow
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59#44214yes—
293C22×C59C2×C4×D411yescharacteristic
294C22×C59#54611yescharacteristic
295C22×C59#54711yescharacteristic
296C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C59#57711yescharacteristic
299C22×C59#57811yescharacteristic
300C2×D59—42no—
301C2×D59—42no—
302C2×D59—42no—
303C2×D59—81no—
304C2×D59—81no—
305C2×D59—81no—
306C2×D59—81no—
307C4×C59—22no—
308C4×C59—42no—
309C4×C59—42no—
310C4×C59—42no—
311C4×C59—81no—
312C4×C59C4×#4111yescharacteristic
313C4×C59#52011yescharacteristic
314C4×C59#55111yescharacteristic
315C4×C59#55211yescharacteristic
316Dic59—41no—
317Dic59—41no—
318Dic59—42no—
319Dic59—41no—
320Dic59—41no—
321Dic59—81no—
322Dic59—81no—
323C2×#480—591no—
324#2361—591no—
325#2344—591no—
326#1943—591no—
327#1936—591no—
328#1926—591no—
329#1921—591no—
330#1921—591no—
331#1893—591no—
332#1864—591no—
333#1853—591no—
334#1776—591no—
335#1762—591no—
336#1752—591no—
337C2×#340D5911yescharacteristic, complemented
338C2×C59—24no—
339C2×C59#235911yescharacteristic
340C2×C59C2×#44211yescharacteristic
341C2×C59#294711yescharacteristic
342C2×C59#330811yescharacteristic
343C2×C59#331011yescharacteristic
344C2×C59#334311yescharacteristic
345C2×C59#334511yescharacteristic
346D59—82no—
347C23×C8C2×D5911yescharacteristic
348C2×#121—591no—
349C2×C4×D4—591no—
350C2×C4×D4—1181no—
351C2×#117—1181no—
352C22×C4⋊C4C2×D5911yescharacteristic
353C22×C4⋊C4C2×D5911yescharacteristic
354#480—598no—
355C2×#107—591no—
356C2×#107—591no—
357C2×#106—591no—
358C2×#106—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×#102—591no—
362C2×#102—591no—
363C2×C22⋊C8—591no—
364C2×#97—591no—
365C2×#97—591no—
366#390—591no—
367#389—591no—
368#389—591no—
369#389—591no—
370#388—1181no—
371#387—1181no—
372#387—1181no—
373#386—1181no—
374#385—1181no—
375#381—1181no—
376#380—591no—
377C4×C4⋊C4—591no—
378C4×C4⋊C4—591no—
379C4×C4⋊C4—591no—
380C4×C22⋊C4—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340C2×D5911yescharacteristic
386#340C2×D5911yescharacteristic
387#340C2×D5911yescharacteristic
388#340C2×D5911yescharacteristic
389#336—591no—
390#336—591no—
391C59#2293811yescharacteristic, complemented, Sylow
392C22×D4—591no—
393C22×D4—1181no—
394C23×C4—21no—
395C23×C4—21no—
396C23×C4—1181no—
397C23×C4C22×D5911yescharacteristic
398C22×C8—22no—
399C22×C8—22no—
400C22×C8—41no—
401C22×C8—591no—
402C22×C8—591no—
403C22×C8C22×D5911yescharacteristic
404C22×C8C22×D5911yescharacteristic
405#121—594no—
406#121—1182no—
407C4×D4—1182no—
408C4×D4—2361no—
409C4×D4—2364no—
410#117—2364no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—41no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4C4×D5914yescomplemented
423C2×C4⋊C4C22×D5911yescharacteristic
424C2×C4⋊C4C22×D5911yescharacteristic
425C2×C4⋊C4C22×D5911yescharacteristic
426C2×C4⋊C4C22×D5911yescharacteristic
427C2×C22⋊C4—591no—
428C2×C22⋊C4—591no—
429C2×C22⋊C4—1181no—
430C2×C22⋊C4—1181no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4—1181no—
433C2×C22⋊C4—1181no—
434C2×C42—591no—
435C2×C42—591no—
436C2×C42—1181no—
437C2×C42—1181no—
438C2×C42—1181no—
439#107—594no—
440#107—594no—
441#106—1182no—
442#106—2361no—
443#102—594no—
444#102—594no—
445#102—1182no—
446#102—2361no—
447C22⋊C8—594no—
448#97—1182no—
449#97—2361no—
450#95—591no—
451#95—591no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459C24—1181no—
460C24D4×D5911yescharacteristic
461C2×D4—592no—
462C2×D4—1184no—
463C2×D4—1181no—
464C2×D4—1181no—
465C2×D4—1181no—
466C2×D4—1181no—
467C2×D4—1181no—
468C2×D4—2362no—
469C2×D4—2362no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—81no—
484C22×C4—591no—
485C22×C4—591no—
486C22×C4—591no—
487C22×C4—591no—
488C22×C4—1181no—
489C22×C4—1181no—
490C22×C4—1181no—
491C22×C4—4721no—
492C22×C4—2362no—
493C22×C4—2362no—
494C22×C4C2×C4×D5912yes—
495C22×C4D4×D5911yescharacteristic
496C22×C4#1174568712yes—
497C22×C4C23×D5911yescharacteristic
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—44no—
504C2×C8—592no—
505C2×C8—592no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8#1174568512yes—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—592no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—2362no—
525C4⋊C4—2362no—
526C4⋊C4—2362no—
527C4⋊C4C2×C4×D5912yes—
528C4⋊C4C2×C4×D5912yes—
529C4⋊C4#1174568512yes—
530C22⋊C4—1182no—
531C22⋊C4—1182no—
532C22⋊C4—1184no—
533C22⋊C4—4721no—
534C22⋊C4—2362no—
535C22⋊C4—2362no—
536C22⋊C4—2362no—
537C42—1181no—
538C42—1181no—
539C42—1181no—
540C42—1181no—
541C42—2362no—
542C42—2362no—
543C42—2362no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—1182no—
549C23—2361no—
550C23—2361no—
551C23—2361no—
552C23—2361no—
553C23—2361no—
554C23—2361no—
555C23(C22⋊D59)⋊C412yes—
556C23#1427375112yes—
557C23#1427375412yes—
558C23C2×D4×D5911yescharacteristic
559D4—1182no—
560D4—1182no—
561D4—2368no—
562D4—2361no—
563D4—2361no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—48no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—1181no—
585C2×C4—1181no—
586C2×C4—1181no—
587C2×C4—1181no—
588C2×C4—1181no—
589C2×C4—1181no—
590C2×C4—1181no—
591C2×C4—1181no—
592C2×C4—1181no—
593C2×C4—1181no—
594C2×C4—1181no—
595C2×C4—1181no—
596C2×C4—1182no—
597C2×C4—4724no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4(C22⋊D59)⋊C412yes—
607C2×C4C22×C4×D5911yescharacteristic
608C2×C4C2×#1174568711yescharacteristic
609C2×C4C2×#1174568511yescharacteristic
610C2×C4C2×#1174568511yescharacteristic
611C2×C4#1427377211yescharacteristic
612C2×C4#1427377511yescharacteristic
613C8—22no—
614C8—41no—
615C8—1182no—
616C8—1182no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—4721no—
624C22—4721no—
625C22—4721no—
626C22—4721no—
627C22—2362no—
628C22—2362no—
629C22—2362no—
630C22#1683751114yes—
631C22C2×(C22⋊D59)⋊C411yescharacteristic
632C22#1683804211yescharacteristic
633C22#1683805311yescharacteristic
634C22C2×#1427375111yescharacteristic
635C22C2×#1427375411yescharacteristic
636C22#1683820811yescharacteristic
637C22#1683821511yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—4721no—
644C4—4721no—
645C4—2362no—
646C4—2361no—
647C4—2361no—
648C4—2361no—
649C4—2361no—
650C4C4×#1174568511yescharacteristic
651C4#1683788311yescharacteristic
652C4#1683807811yescharacteristic
653C4#1683808711yescharacteristic
654C2—24no—
655C2—4722no—
656C2#2142708411yescharacteristic
657C2C2×#1683751111yescharacteristic
658C2#2143293311yescharacteristic
659C2#2143608311yescharacteristic
660C2#2143611311yescharacteristic
661C2#2143653211yescharacteristic
662C2#2143656211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#340C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenterC2×#340D59
hypocenterC59#22938
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #21426965C2semidirect
4. #21424833C2semidirect
5. #21424794C2semidirect
6. #21424721C2semidirect
7. #21424691C2semidirect
8. #21424686C2semidirect
10. #21424358C2semidirect
11. #21424289C2semidirect
12. #21423992C2semidirect
13. #21423920C2semidirect
14. #21423872C2semidirect
15. C2×C59×#340C2semidirect
20. #16837715C4semidirect
89. #14273660C2×C4semidirect
90. #14273660C2×C4semidirect
95. #14273647C2×C4semidirect
96. #14273647C2×C4semidirect
97. C22⋊(C59⋊C8)C2×C4semidirect
122. C2×C4⋊C4×C59C2×C4semidirect
337. C2×#340D59semidirect
391. C59#22938semidirect
422. C2×C4⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377152
#214271122
#214269652
#214248332
#214247942
#214247212
#214246912
#214246862
#214245242
#214243582
#214242892
#214239922
#214239202
#214238722
C2×C59×#3402
#2293859