Finite Group Databasebeta

← #28610215

Subgroups — #28610215

Order 15104.

902 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837718C211yesmaximal, characteristic
2#21426963C211yesmaximal, characteristic, complemented
3#21424836C211yesmaximal, characteristic, complemented
4#21424782C211yesmaximal, characteristic, complemented
5#21424713C211yesmaximal, characteristic, complemented
6#21424674C211yesmaximal, characteristic, complemented
7#21424671C211yesmaximal, characteristic, complemented
8#21424535C211yesmaximal, characteristic
9#21424357C211yesmaximal, characteristic, complemented
10#21424295C211yesmaximal, characteristic, complemented
11C4×#14273658C211yesmaximal, characteristic, complemented
12#21423921C211yesmaximal, characteristic, complemented
13#21423880C211yesmaximal, characteristic, complemented
14#21423817C211yesmaximal, characteristic, complemented
15C59×#2363C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273683—21no—
18C22×C4⋊Dic59C2211yescharacteristic
19#16837718C418yescomplemented
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837287—21no—
34#16837287—21no—
35#16837263—21no—
36#16837262C2211yescharacteristic
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×#121×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×#117×C59C2211yescharacteristic
52C59×#390—21no—
53C59×#390C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#386—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C4×C22⋊C4×C59C2211yescharacteristic
60C23×Dic59—21no—
61C2×#11745664—22no—
62C2×#11745664—22no—
63C2×#11745664—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273683—44no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—22no—
71C2×C4⋊Dic59C2×C414yescomplemented
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×C22⋊Dic59—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81#14273660—41no—
82#14273658—22no—
83#14273658C2×C414yescomplemented
84#14273658C2×C414yescomplemented
85#14273648—22no—
86#14273648—41no—
87#14273648C2×C414yes—
88#14273648C2×C414yes—
89C22⋊(C59⋊C8)C2×C414yescomplemented
90#14273638—41no—
91C4×C59⋊C8—22no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×D4×C59—21no—
98C22×D4×C59C2311yescharacteristic
99C23×C4×C59—21no—
100C23×C4×C59C2311yescharacteristic
101#121×C59—22no—
102#121×C59C2×C414yes—
103C4×D4×C59—22no—
104C4×D4×C59—41no—
105#117×C59—22no—
106#117×C59—41no—
107C2×C4⋊C4×C59—21no—
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×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C42×C59—21no—
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59—21no—
127C24×C59C2×D411yescharacteristic
128C2×D4×C59—24no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—81no—
135C2×D4×C59—81no—
136C2×D4×C59C22×C412yes—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—42no—
145C22×C4×C59—42no—
146C22×C4×C59—81no—
147C22×C4×C59C22×C412yes—
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59#4112yes—
154C22×C4×C59C2411yescharacteristic
155C22×Dic59—24no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—42no—
161C22×Dic59—42no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—81no—
167C4⋊C4×C59C22×C412yes—
168C4⋊C4×C59#4112yes—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—41no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—42no—
175C22⋊C4×C59—81no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—42no—
181C42×C59—81no—
182C22⋊Dic59—42no—
183C4⋊Dic59—24no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59C22×C412yes—
191C4⋊Dic59C22×C412yes—
192Dic59⋊C4—42no—
193Dic59⋊C4—42no—
194C4×Dic59—42no—
195C4×Dic59—42no—
196#11745664—44no—
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⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8C22×C412yes—
207C2×C59⋊C8C22×C412yes—
208C2×C59⋊C8#4112yes—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—22no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59C4×D412yes—
223C23×C59C22×D411yescharacteristic
224D4×C59—22no—
225D4×C59—22no—
226D4×C59—41no—
227D4×C59—41no—
228D4×C59—84no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—22no—
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—21no—
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—84no—
256C2×C4×C59C4×D412yes—
257C2×C4×C59AGL(1,ℤ/8)12yes—
258C2×C4×C59C23×C411yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59H(5,𝔽2)11yescharacteristic
263C2×C4×C592−1+411yescharacteristic
264C2×Dic59—22no—
265C2×Dic59—22no—
266C2×Dic59—48no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C59⋊C8—22no—
278C59⋊C8—22no—
279C59⋊C8—22no—
280C59⋊C8—41no—
281#22943—591nomaximal, Sylow
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—22no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—42no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59—81no—
296C22×C59C2×C4×D411yescharacteristic
297C22×C59#54611yescharacteristic
298C22×C59#54711yescharacteristic
299C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C59#57511yescharacteristic
301C22×C59#57711yescharacteristic
302C22×C59#57711yescharacteristic
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—41no—
306C4×C59—41no—
307C4×C59—42no—
308C4×C59—41no—
309C4×C59—42no—
310C4×C59—81no—
311C4×C59C4×#4111yescharacteristic
312C4×C59#52011yescharacteristic
313C4×C59#55111yescharacteristic
314C4×C59#55211yescharacteristic
315Dic59—42no—
316Dic59—42no—
317Dic59—81no—
318Dic59—81no—
319C2×#481—591no—
320#2363D5911yescharacteristic
321#2344—591no—
322#1944—591no—
323#1934—591no—
324#1925—591no—
325#1919—591no—
326#1919—591no—
327#1895—591no—
328#1864—591no—
329#1854—591no—
330C4×#106—591no—
331#1762—591no—
332#1753—591no—
333#1736—591no—
334C2×C59—42no—
335C2×C59—82no—
336C2×C59#235911yescharacteristic
337C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
338C2×C59#294911yescharacteristic
339C2×C59#332311yescharacteristic
340C2×C59#333211yescharacteristic
341C2×C59#334511yescharacteristic
342C2×C59#334811yescharacteristic
343C22×M16—591no—
344C2×#121C2×D5911yescharacteristic
345C2×C4×D4C2×D5911yescharacteristic
346C2×#117—1181no—
347C2×#117C2×D5911yescharacteristic
348C22×C4⋊C4—591no—
349#481—598no—
350C2×#107—591no—
351C2×#106—591no—
352C2×#106—591no—
353C2×#106—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×#102—591no—
358C2×C22⋊C8—591no—
359C2×#97—591no—
360C2×C4×C8—591no—
361#390—21no—
362#390C2×D5911yescharacteristic
363#389—591no—
364#389—591no—
365#389C2×D5911yescharacteristic
366#388—21no—
367#387—1181no—
368#387—1181no—
369#386—21no—
370#381—1181no—
371#380—591no—
372C4×C4⋊C4—591no—
373C4×C4⋊C4—591no—
374C4×C4⋊C4C2×D5911yescharacteristic
375C4×C22⋊C4—21no—
376C4×C22⋊C4C2×D5911yescharacteristic
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#336—591no—
386#336—591no—
387C59#2294311yescharacteristic, complemented, Sylow
388C22×D4—21no—
389C22×D4C22×D5911yescharacteristic
390C23×C4—21no—
391C23×C4—1181no—
392C23×C4C22×D5911yescharacteristic
393C2×M16—1182no—
394C2×M16—1182no—
395C2×M16—2361no—
396C22×C8—591no—
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400#121—22no—
401#121C4×D5914yes—
402C4×D4—22no—
403C4×D4—41no—
404#117—22no—
405#117—41no—
406#117—2364no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—594no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—1182no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1181no—
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—1181no—
426C2×C22⋊C4C22⋊D5911yescharacteristic
427C2×C22⋊C4C22⋊D5911yescharacteristic
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—1181no—
431C2×C42—1181no—
432C2×C42C22×D5911yescharacteristic
433C2×C42C22×D5911yescharacteristic
434#107—2361no—
435#106—594no—
436#106—594no—
437#106—1182no—
438#102—594no—
439#102—594no—
440#102—1182no—
441#102—2361no—
442C22⋊C8—594no—
443#97—2361no—
444C4×C8—1182no—
445#95—21no—
446#95—21no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95C22⋊D5911yescharacteristic
453#95C22⋊D5911yescharacteristic
454C24—21no—
455C24C2×C22⋊D5911yescharacteristic
456C2×D4—24no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—81no—
463C2×D4—81no—
464C2×D4C2×C4×D5912yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—81no—
475C22×C4—1184no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—2362no—
481C22×C4—2362no—
482C22×C4C2×C4×D5912yes—
483C22×C4#1174568512yes—
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C23×D5911yescharacteristic
490M16—2364no—
491C2×C8—592no—
492C2×C8—592no—
493C2×C8—592no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—81no—
508C4⋊C4—592no—
509C4⋊C4—592no—
510C4⋊C4—1184no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—2362no—
518C4⋊C4—2362no—
519C4⋊C4C2×C4×D5912yes—
520C4⋊C4#1174568712yes—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—41no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—2362no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—42no—
534C42—81no—
535C42—2362no—
536C42—2362no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23C4×C22⋊D5912yes—
551C23C22×C22⋊D5911yescharacteristic
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—1182no—
585C2×C4—1182no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2368no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4C4×C22⋊D5912yes—
598C2×C4#1427375212yes—
599C2×C4C22×C4×D5911yescharacteristic
600C2×C4#41×D5911yescharacteristic
601C2×C4C2×#1174568711yescharacteristic
602C2×C4C2×#1174568511yescharacteristic
603C2×C4#1427377211yescharacteristic
604C2×C4#1427377411yescharacteristic
605C8—1182no—
606C8—1182no—
607C8—1182no—
608C8—2361no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×C4×C22⋊D5911yescharacteristic
624C22#1683804611yescharacteristic
625C22#1683805111yescharacteristic
626C22C2×#1427375211yescharacteristic
627C22#1683820011yescharacteristic
628C22#1683821311yescharacteristic
629C22#1683821311yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—81no—
638C4—4721no—
639C4—4721no—
640C4—2362no—
641C4—2362no—
642C4C4×#1174568711yescharacteristic
643C4#1683788111yescharacteristic
644C4#1683808111yescharacteristic
645C4#1683809211yescharacteristic
646C2—42no—
647C2—82no—
648C2#2142708211yescharacteristic
649C2C4×#1427375211yescharacteristic
650C2#2143295311yescharacteristic
651C2#2143631211yescharacteristic
652C2#2143642311yescharacteristic
653C2#2143656011yescharacteristic
654C2#2143661111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2363C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
cosocleC22×C4×C59C24
hypercenter#2363D59
hypocenterC59#22943
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
2. #21426963C2semidirect
3. #21424836C2semidirect
4. #21424782C2semidirect
5. #21424713C2semidirect
6. #21424674C2semidirect
7. #21424671C2semidirect
9. #21424357C2semidirect
10. #21424295C2semidirect
11. C4×#14273658C2semidirect
12. #21423921C2semidirect
13. #21423880C2semidirect
14. #21423817C2semidirect
19. #16837718C4semidirect
71. C2×C4⋊Dic59C2×C4semidirect
83. #14273658C2×C4semidirect
84. #14273658C2×C4semidirect
89. C22⋊(C59⋊C8)C2×C4semidirect
387. C59#22943semidirect

Maximal subgroups

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

subgroupindex
C2×#168377182
#214269632
#214248362
#214247822
#214247132
#214246742
#214246712
#214245352
#214243572
#214242952
C4×#142736582
#214239212
#214238802
#214238172
C59×#23632
#2294359