Finite Group Databasebeta

← #28556534

Subgroups — #28556534

Order 15104.

1010 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433125C211yesmaximal, characteristic, complemented
2#21427036C211yesmaximal, characteristic, complemented
3#21422652C212yesmaximal, complemented
4#21422647C211yesmaximal, characteristic, complemented
5#21422643C211yesmaximal, characteristic, complemented
6#21422639C212yesmaximal, complemented
7#21422608C211yesmaximal, characteristic, complemented
8#21422604C211yesmaximal, characteristic, complemented
9#21422600C212yesmaximal, complemented
10#21422587C212yesmaximal, complemented
11C59×#3506C211yesmaximal, characteristic
12#16837474—22no—
13#16837474—22no—
14#16837459C2211yescharacteristic
15#16837459C2211yescharacteristic
16#16837366C2212yes—
17C2×C22⋊(C59⋊C8)C2212yes—
18#16837295—21no—
19#16837288—21no—
20#16837282—21no—
21#16837270C2211yescharacteristic
22#16837265C2211yescharacteristic
23#16837262C2211yescharacteristic
24C22⋊C4×Dic59—21no—
25C4×C22⋊Dic59C2211yescharacteristic
26#16837248C2211yescharacteristic
27C2×C59⋊#95C2211yescharacteristic
28#16837139C2212yes—
29#16837139C2212yes—
30#16837139C2212yes—
31#16837139C2212yes—
32#16837136C2212yes—
33#16837136C2212yes—
34#16837136C2212yes—
35#16837136C2212yes—
36C59×#559—22no—
37C59×#556C412yescomplemented
38C59×#555C412yescomplemented
39C59×#553—22no—
40C59×#550C412yescomplemented
41C59×#547C412yescomplemented
42C59×#538—21no—
43C59×#538—21no—
44C59×#535—21no—
45C59×#533C2211yescharacteristic
46C59×#532C2211yescharacteristic
47C59×#525C2212yes—
48C59×#525C2212yes—
49C59×#520—21no—
50C2×C4×D4×C59C2211yescharacteristic
51C23×Dic59—21no—
52C2×#11745664—22no—
53C22×C59⋊C8—22no—
54C2×C4⋊Dic59—21no—
55C2×C22⋊Dic59—21no—
56C2×C22⋊Dic59—21no—
57C2×C22⋊Dic59—21no—
58C2×C4×Dic59—21no—
59C2×C4×Dic59—21no—
60C4⋊(C59⋊C8)—21no—
61C4⋊(C59⋊C8)—21no—
62C22⋊(C59⋊C8)—22no—
63C22⋊(C59⋊C8)—22no—
64C22⋊(C59⋊C8)C2312yes—
65C22⋊(C59⋊C8)C2312yes—
66#14273638—21no—
67#14273638—21no—
68C59⋊#95—21no—
69C59⋊#95—21no—
70C59⋊#95—21no—
71C59⋊#95—21no—
72C59⋊#95—21no—
73C59⋊#95C2311yescharacteristic
74C59⋊#95C2311yescharacteristic
75C59⋊#95C2311yescharacteristic
76C59⋊#95C2311yescharacteristic
77C2×#41×C59—21no—
78C2×#41×C59C2×C412yes—
79C2×#41×C59C2×C412yescomplemented
80C22×D4×C59D411yescharacteristic
81C23×C4×C59C2311yescharacteristic
82C23×C4×C59C2311yescharacteristic
83C4⋊Q8×C59C2×C411yescharacteristic, complemented
84C4⋊Q8×C59C2×C411yescharacteristic, complemented
85C4⋊Q8×C59C2×C411yescharacteristic, complemented
86#127×C59—22no—
87#127×C59C2×C411yescharacteristic, complemented
88#126×C59—22no—
89#126×C59—22no—
90#125×C59—21no—
91#124×C59—22no—
92#124×C59—22no—
93#124×C59—22no—
94#124×C59—21no—
95#123×C59—22no—
96#123×C59—22no—
97#123×C59—22no—
98#123×C59—22no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59C2×C412yes—
101C22⋊Q8×C59C2×C412yescomplemented
102C22⋊Q8×C59C2312yes—
103#121×C59—22no—
104#121×C59—22no—
105#121×C59—21no—
106#121×C59—22no—
107#121×C59—22no—
108#121×C59C2×C412yes—
109#121×C59C2×C412yescomplemented
110#121×C59C2312yes—
111#120×C59—22no—
112#120×C59—22no—
113C4×Q8×C59—22no—
114C4×Q8×C59—21no—
115C4×Q8×C59C2×C412yescomplemented
116C4×D4×C59—22no—
117C4×D4×C59—22no—
118C4×D4×C59—22no—
119C4×D4×C59—22no—
120C4×D4×C59—22no—
121C4×D4×C59—21no—
122C4×D4×C59C2×C412yes—
123C4×D4×C59C2×C412yes—
124C4×D4×C59C2×C412yescomplemented
125C4×D4×C59D412yes—
126C4×D4×C59D412yes—
127#117×C59—21no—
128#117×C59—22no—
129#117×C59C2×C412yes—
130#117×C59C2×C412yescomplemented
131C2×C4⋊C4×C59—22no—
132C2×C4⋊C4×C59D411yescharacteristic
133C2×C22⋊C4×C59D411yescharacteristic
134C2×C22⋊C4×C59D411yescharacteristic
135C2×C42×C59C2311yescharacteristic
136C24×C59C2×D411yescharacteristic
137C24×C59C2×D411yescharacteristic
138#41×C59—42no—
139#41×C59—42no—
140#41×C59—82no—
141#41×C59—82no—
142C2×Q8×C59—21no—
143C2×Q8×C59—21no—
144C2×Q8×C59C22×C412yes—
145C2×D4×C59—22no—
146C2×D4×C59—22no—
147C2×D4×C59—22no—
148C2×D4×C59—22no—
149C2×D4×C59—22no—
150C2×D4×C59—21no—
151C2×D4×C59—21no—
152C2×D4×C59—41no—
153C2×D4×C59—41no—
154C2×D4×C59C22⋊C412yes—
155C2×D4×C59C22⋊C412yes—
156C2×D4×C59C22×C412yes—
157C22×C4×C59—22no—
158C22×C4×C59—22no—
159C22×C4×C59—22no—
160C22×C4×C59—81no—
161C22×C4×C59—81no—
162C22×C4×C59C22×C412yes—
163C22×C4×C59C22×C412yes—
164C22×C4×C59C2×D411yescharacteristic
165C22×C4×C59C2×D411yescharacteristic
166C22×C4×C59C2×D411yescharacteristic
167C22×C4×C59C2×D411yescharacteristic
168C22×C4×C59C2×D412yes—
169C22×C4×C59C2×D412yes—
170C22×C4×C59C2411yescharacteristic
171C22×Dic59—21no—
172C22×Dic59—21no—
173C22×Dic59—21no—
174C22×Dic59—21no—
175C22×Dic59—42no—
176C22×Dic59—42no—
177C22×Dic59—81no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—22no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—22no—
185C4⋊C4×C59—22no—
186C4⋊C4×C59C22⋊C412yes—
187C4⋊C4×C59C22⋊C412yes—
188C4⋊C4×C59C22×C412yes—
189C4⋊C4×C59C22×C412yes—
190C22⋊C4×C59—22no—
191C22⋊C4×C59—22no—
192C22⋊C4×C59—22no—
193C22⋊C4×C59—22no—
194C22⋊C4×C59—22no—
195C22⋊C4×C59—22no—
196C22⋊C4×C59—22no—
197C22⋊C4×C59—22no—
198C22⋊C4×C59—22no—
199C22⋊C4×C59C22⋊C412yes—
200C22⋊C4×C59C22⋊C412yes—
201C22⋊C4×C59C22⋊C412yes—
202C22⋊C4×C59C22⋊C412yes—
203C42×C59—22no—
204C42×C59—22no—
205C42×C59—22no—
206C42×C59C22×C411yescharacteristic
207C42×C59C22×C411yescharacteristic
208C42×C59C2×D411yescharacteristic
209C42×C59C2×D411yescharacteristic
210C22⋊Dic59—42no—
211C22⋊Dic59—42no—
212C22⋊Dic59—81no—
213C4⋊Dic59—81no—
214C4×Dic59—42no—
215C4×Dic59—81no—
216#11745664—82no—
217C2×C59⋊C8—22no—
218C2×C59⋊C8—22no—
219C2×C59⋊C8—82no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—22no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59C2×C22⋊C412yes—
230C23×C59C2×C22⋊C412yes—
231C23×C59C22×D411yescharacteristic
232Q8×C59—42no—
233Q8×C59—42no—
234Q8×C59—81no—
235D4×C59—42no—
236D4×C59—42no—
237D4×C59—42no—
238D4×C59—42no—
239D4×C59—42no—
240D4×C59—42no—
241D4×C59—81no—
242D4×C59—82no—
243D4×C59—82no—
244D4×C59—82no—
245D4×C59—82no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—22no—
249C2×C4×C59—22no—
250C2×C4×C59—22no—
251C2×C4×C59—42no—
252C2×C4×C59—42no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—42no—
258C2×C4×C59—42no—
259C2×C4×C59—44no—
260C2×C4×C59—82no—
261C2×C4×C59—82no—
262C2×C4×C59C2×C22⋊C411yescharacteristic
263C2×C4×C59C2×C22⋊C412yes—
264C2×C4×C59C2×C22⋊C412yes—
265C2×C4×C59C2×C22⋊C411yescharacteristic
266C2×C4×C59C2×C22⋊C411yescharacteristic
267C2×C4×C59C2×C22⋊C411yescharacteristic
268C2×C4×C59C23×C411yescharacteristic
269C2×C4×C59C22×D411yescharacteristic
270C2×Dic59—22no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—84no—
283C59⋊C8—42no—
284C59⋊C8—42no—
285#17629—591nomaximal, Sylow
286C22×C59—21no—
287C22×C59—21no—
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—42no—
295C22×C59—42no—
296C22×C59—42no—
297C22×C59—42no—
298C22×C59—42no—
299C22×C59—81no—
300C22×C59—81no—
301C22×C59C22×C22⋊C411yescharacteristic
302C22×C59#57511yescharacteristic
303C22×C59#57511yescharacteristic
304C4×C59—21no—
305C4×C59—21no—
306C4×C59—22no—
307C4×C59—22no—
308C4×C59—21no—
309C4×C59—21no—
310C4×C59—21no—
311C4×C59—21no—
312C4×C59—42no—
313C4×C59—42no—
314C4×C59—42no—
315Dic59—42no—
316Dic59—81no—
317Dic59—81no—
318Dic59—81no—
319#3506D5911yescharacteristic
320#2973—591no—
321#2353—591no—
322#1516—592no—
323#1515—591no—
324#1514—591no—
325#1513—592no—
326#1506—591no—
327#1505—591no—
328#1504—592no—
329#1501—592no—
330C2×C59—21no—
331C2×C59—21no—
332C2×C59—42no—
333C2×C59—42no—
334C2×C59—82no—
335C2×C59#288311yescharacteristic
336C2×C59#289611yescharacteristic
337C2×C59#290111yescharacteristic
338#559—22no—
339#556Dic5912yescomplemented
340#555Dic5912yescomplemented
341#553—22no—
342#550Dic5912yescomplemented
343#547Dic5912yescomplemented
344#538—21no—
345#538—21no—
346#535—21no—
347#533C2×D5911yescharacteristic
348#532C2×D5911yescharacteristic
349#525C2×D5912yes—
350#525C2×D5912yes—
351#520—21no—
352C2×C4×D4C2×D5911yescharacteristic
353#435—1182no—
354#435—1182no—
355#432—591no—
356#432—591no—
357#407—592no—
358C2×C22⋊C8—592no—
359#388—1181no—
360#387—1181no—
361#386—1181no—
362#383—591no—
363#381—591no—
364#380—591no—
365C4×C22⋊C4—591no—
366C4×C22⋊C4—1181no—
367#376—591no—
368C2×#95—591no—
369#328—592no—
370#328—592no—
371#328—592no—
372#328—592no—
373#327—592no—
374#327—592no—
375#327—592no—
376#327—592no—
377C59#1762911yescharacteristic, complemented, Sylow
378C2×#41—21no—
379C2×#41C2×Dic5912yes—
380C2×#41C2×Dic5912yescomplemented
381C22×D4C22⋊D5911yescharacteristic
382C23×C4—1181no—
383C23×C4C22×D5911yescharacteristic
384C23×C4C22×D5911yescharacteristic
385C2×M16—1182no—
386C22×C8—1182no—
387C4⋊Q8C2×Dic5911yescharacteristic, complemented
388C4⋊Q8C2×Dic5911yescharacteristic, complemented
389C4⋊Q8C2×Dic5911yescharacteristic, complemented
390#127—22no—
391#127C2×Dic5911yescharacteristic, complemented
392#126—22no—
393#126—22no—
394#125—21no—
395#124—22no—
396#124—22no—
397#124—22no—
398#124—21no—
399#123—22no—
400#123—22no—
401#123—22no—
402#123—22no—
403C22⋊Q8—21no—
404C22⋊Q8C2×Dic5912yes—
405C22⋊Q8C2×Dic5912yescomplemented
406C22⋊Q8C22×D5912yes—
407#121—22no—
408#121—22no—
409#121—21no—
410#121—22no—
411#121—22no—
412#121C2×Dic5912yes—
413#121C2×Dic5912yescomplemented
414#121C22×D5912yes—
415#120—22no—
416#120—22no—
417C4×Q8—22no—
418C4×Q8—21no—
419C4×Q8C2×Dic5912yescomplemented
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—22no—
424C4×D4—22no—
425C4×D4—21no—
426C4×D4C2×Dic5912yes—
427C4×D4C2×Dic5912yes—
428C4×D4C2×Dic5912yescomplemented
429C4×D4C22⋊D5912yes—
430C4×D4C22⋊D5912yes—
431#117—21no—
432#117—22no—
433#117C2×Dic5912yes—
434#117C2×Dic5912yescomplemented
435C2×C4⋊C4—22no—
436C2×C4⋊C4—1181no—
437C2×C4⋊C4C22⋊D5911yescharacteristic
438C2×C22⋊C4—1181no—
439C2×C22⋊C4—1181no—
440C2×C22⋊C4—1181no—
441C2×C22⋊C4C22⋊D5911yescharacteristic
442C2×C22⋊C4C22⋊D5911yescharacteristic
443C2×C42—1181no—
444C2×C42—1181no—
445C2×C42C22×D5911yescharacteristic
446C4⋊C8—1181no—
447C4⋊C8—1181no—
448C22⋊C8—592no—
449C22⋊C8—592no—
450C22⋊C8—1182no—
451C22⋊C8—1182no—
452#97—1181no—
453#97—1181no—
454#95—591no—
455#95—591no—
456#95—591no—
457#95—591no—
458#95—1181no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462#95—1181no—
463C24C2×C22⋊D5911yescharacteristic
464C24C2×C22⋊D5911yescharacteristic
465#41—42no—
466#41—42no—
467#41—82no—
468#41—82no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8C22×Dic5912yes—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4C22⋊Dic5912yes—
482C2×D4C22⋊Dic5912yes—
483C2×D4C22×Dic5912yes—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—1181no—
490C22×C4—1181no—
491C22×C4—1181no—
492C22×C4—1181no—
493C22×C4—4721no—
494C22×C4—2362no—
495C22×C4—2362no—
496C22×C4C22×Dic5912yes—
497C22×C4C22×Dic5912yes—
498C22×C4C2×C22⋊D5911yescharacteristic
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C2×C22⋊D5911yescharacteristic
502C22×C4C2×C22⋊D5912yes—
503C22×C4C2×C22⋊D5912yes—
504C22×C4C23×D5911yescharacteristic
505M16—4722no—
506C2×C8—1182no—
507C2×C8—1182no—
508C2×C8—4722no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—4721no—
518C4⋊C4C22⋊Dic5912yes—
519C4⋊C4C22⋊Dic5912yes—
520C4⋊C4C22×Dic5912yes—
521C4⋊C4C22×Dic5912yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—4721no—
532C22⋊C4—2362no—
533C22⋊C4—2362no—
534C22⋊C4C22⋊Dic5912yes—
535C22⋊C4C22⋊Dic5912yes—
536C22⋊C4C22⋊Dic5912yes—
537C22⋊C4C22⋊Dic5912yes—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—4721no—
542C42—2362no—
543C42C22×Dic5911yescharacteristic
544C42C22×Dic5911yescharacteristic
545C42C2×C22⋊D5911yescharacteristic
546C42C2×C22⋊D5911yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23C2×C22⋊Dic5912yes—
557C23C2×C22⋊Dic5912yes—
558C23C22×C22⋊D5911yescharacteristic
559Q8—42no—
560Q8—42no—
561Q8—81no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—81no—
569D4—82no—
570D4—82no—
571D4—82no—
572D4—82no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—1182no—
590C2×C4—4724no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4C2×C22⋊Dic5911yescharacteristic
603C2×C4C2×C22⋊Dic5912yes—
604C2×C4C2×C22⋊Dic5912yes—
605C2×C4C2×C22⋊Dic5911yescharacteristic
606C2×C4C2×C22⋊Dic5911yescharacteristic
607C2×C4C2×C22⋊Dic5911yescharacteristic
608C2×C4C23×Dic5911yescharacteristic
609C2×C4C22×C22⋊D5911yescharacteristic
610C8—2362no—
611C8—2362no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22C22×C22⋊Dic5911yescharacteristic
628C22#1683820011yescharacteristic
629C22#1683820011yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—4721no—
642C4—4721no—
643C4—4721no—
644C4—2362no—
645C2—21no—
646C2—21no—
647C2—42no—
648C2—42no—
649C2—82no—
650C2#2143255711yescharacteristic
651C2#2143261511yescharacteristic
652C2#2143264311yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
Fitting, prime radicalC59×#3506C2
socleC22×C59C22×C22⋊C4
cosocleC22×C4×C59C24
hypercenter#3506D59
hypocenterC59#17629
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21433125C2semidirect
2. #21427036C2semidirect
3. #21422652C2semidirect
4. #21422647C2semidirect
5. #21422643C2semidirect
6. #21422639C2semidirect
7. #21422608C2semidirect
8. #21422604C2semidirect
9. #21422600C2semidirect
10. #21422587C2semidirect
37. C59×#556C4semidirect
38. C59×#555C4semidirect
40. C59×#550C4semidirect
41. C59×#547C4semidirect
79. C2×#41×C59C2×C4semidirect
83. C4⋊Q8×C59C2×C4semidirect
84. C4⋊Q8×C59C2×C4semidirect
85. C4⋊Q8×C59C2×C4semidirect
87. #127×C59C2×C4semidirect
101. C22⋊Q8×C59C2×C4semidirect
109. #121×C59C2×C4semidirect
115. C4×Q8×C59C2×C4semidirect
124. C4×D4×C59C2×C4semidirect
130. #117×C59C2×C4semidirect
339. #556Dic59semidirect
340. #555Dic59semidirect
342. #550Dic59semidirect
343. #547Dic59semidirect
377. C59#17629semidirect
380. C2×#41C2×Dic59semidirect
387. C4⋊Q8C2×Dic59semidirect
388. C4⋊Q8C2×Dic59semidirect
389. C4⋊Q8C2×Dic59semidirect
391. #127C2×Dic59semidirect
405. C22⋊Q8C2×Dic59semidirect
413. #121C2×Dic59semidirect
419. C4×Q8C2×Dic59semidirect
428. C4×D4C2×Dic59semidirect
434. #117C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214270362
#214226522
#214226472
#214226432
#214226392
#214226082
#214226042
#214226002
#214225872
C59×#35062
#1762959