Finite Group Databasebeta

← #28556253

Subgroups — #28556253

Order 15104.

982 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433131C211yesmaximal, characteristic, complemented
2#21426739C211yesmaximal, characteristic, complemented
3#21422668C212yesmaximal, complemented
4#21422652C212yesmaximal, complemented
5#21422639C212yesmaximal, complemented
6#21422629C212yesmaximal, complemented
7#21422621C211yesmaximal, characteristic, complemented
8#21422608C211yesmaximal, characteristic, complemented
9#21422595C211yesmaximal, characteristic, complemented
10#21422582C211yesmaximal, characteristic, complemented
11C59×#3520C211yesmaximal, characteristic
12#16837480—21no—
13#16837474—22no—
14D4×C59⋊C8—21no—
15#16837459C2211yescharacteristic
16#16837455C2211yescharacteristic
17#16837366C2212yes—
18C2×C22⋊(C59⋊C8)C2212yes—
19#16837305—21no—
20#16837301—21no—
21#16837295—22no—
22#16837265C2212yes—
23#16837262C2212yes—
24C4×C4⋊Dic59C2211yescharacteristic
25#16837248C2211yescharacteristic
26#16837139C2212yes—
27#16837139C2212yes—
28#16837139C2212yes—
29#16837139C2212yes—
30#16837139C2212yes—
31#16837139C2211yescharacteristic
32#16837139C2211yescharacteristic
33#16837136C2212yes—
34#16837136C2211yescharacteristic
35#16837136C2211yescharacteristic
36C59×#556C411yescharacteristic, complemented
37C59×#554—21no—
38C59×#552C412yescomplemented
39C59×#551—22no—
40D4×Q8×C59C411yescharacteristic, complemented
41C59×#548C412yescomplemented
42C59×#547C411yescharacteristic, complemented
43C59×#547C411yescharacteristic, complemented
44C59×#546—21no—
45C59×#542—21no—
46C59×#539—22no—
47C59×#533C2211yescharacteristic
48C59×#529C2211yescharacteristic
49C59×#525C2212yes—
50C2×C22⋊Q8×C59C2212yes—
51C59×#520—21no—
52C2×C4×D4×C59C2211yescharacteristic
53C2×#11745664—22no—
54C22×C59⋊C8—22no—
55C2×C4⋊Dic59—21no—
56C2×C4⋊Dic59—22no—
57C2×C22⋊Dic59—22no—
58C2×C22⋊Dic59—22no—
59C2×C4×Dic59—21no—
60C2×C4×Dic59—21no—
61C4⋊(C59⋊C8)—21no—
62C4⋊(C59⋊C8)—21no—
63C22⋊(C59⋊C8)—22no—
64C22⋊(C59⋊C8)—22no—
65C22⋊(C59⋊C8)C2311yescharacteristic
66C22⋊(C59⋊C8)C2311yescharacteristic
67C22⋊(C59⋊C8)C2312yes—
68#14273638—21no—
69C4×C59⋊C8—21no—
70C59⋊#95—21no—
71C59⋊#95—22no—
72C59⋊#95C2311yescharacteristic
73C59⋊#95C2311yescharacteristic
74C59⋊#95C2312yes—
75C2×#41×C59—21no—
76C2×#41×C59C2×C412yes—
77C22×Q8×C59C2×C412yes—
78C22×D4×C59D411yescharacteristic
79C23×C4×C59C2312yes—
80C4⋊Q8×C59—21no—
81C4⋊Q8×C59C2×C411yescharacteristic
82C4⋊Q8×C59C2×C411yescharacteristic
83#126×C59—22no—
84#126×C59—22no—
85#126×C59—21no—
86#125×C59—22no—
87#125×C59—21no—
88#125×C59C2×C411yescharacteristic
89#124×C59—21no—
90#124×C59—22no—
91#124×C59C2×C411yescharacteristic
92#123×C59—22no—
93#123×C59—22no—
94#123×C59—21no—
95#123×C59—22no—
96#123×C59—22no—
97C22⋊Q8×C59—22no—
98C22⋊Q8×C59—22no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59—22no—
101C22⋊Q8×C59—21no—
102C22⋊Q8×C59—22no—
103C22⋊Q8×C59C2×C412yes—
104C22⋊Q8×C59C2×C412yes—
105C22⋊Q8×C59C2×C412yes—
106C22⋊Q8×C59C2311yescharacteristic
107C22⋊Q8×C59C2312yes—
108#121×C59—22no—
109#121×C59—21no—
110#121×C59C2×C412yes—
111#121×C59C2311yescharacteristic
112#120×C59—22no—
113C4×Q8×C59—21no—
114C4×Q8×C59—21no—
115C4×Q8×C59C2×C411yescharacteristic
116C4×Q8×C59C2×C412yes—
117C4×D4×C59—21no—
118C4×D4×C59—22no—
119C4×D4×C59—22no—
120C4×D4×C59—21no—
121C4×D4×C59—22no—
122C4×D4×C59C2×C411yescharacteristic
123C4×D4×C59C2×C411yescharacteristic
124C4×D4×C59C2×C411yescharacteristic
125C4×D4×C59C2×C412yes—
126C4×D4×C59D411yescharacteristic
127C4×D4×C59D412yes—
128C4×D4×C59D411yescharacteristic
129#117×C59—21no—
130#117×C59—22no—
131#117×C59—21no—
132#117×C59C2×C412yes—
133C2×C4⋊C4×C59—22no—
134C2×C4⋊C4×C59—21no—
135C2×C4⋊C4×C59C2×C412yes—
136C2×C4⋊C4×C59D411yescharacteristic
137C2×C22⋊C4×C59—22no—
138C2×C22⋊C4×C59D412yes—
139C2×C42×C59C2311yescharacteristic
140C24×C59C2×D412yes—
141#41×C59—44no—
142#41×C59—82no—
143C2×Q8×C59—21no—
144C2×Q8×C59—22no—
145C2×Q8×C59—82no—
146C2×Q8×C59C22×C411yescharacteristic
147C2×Q8×C59C22×C412yes—
148C2×D4×C59—22no—
149C2×D4×C59—21no—
150C2×D4×C59—22no—
151C2×D4×C59—42no—
152C2×D4×C59C22⋊C411yescharacteristic
153C2×D4×C59C22⋊C411yescharacteristic
154C2×D4×C59C22⋊C412yes—
155C2×D4×C59C22×C411yescharacteristic
156C22×C4×C59—22no—
157C22×C4×C59—22no—
158C22×C4×C59—21no—
159C22×C4×C59—22no—
160C22×C4×C59—82no—
161C22×C4×C59C22×C412yes—
162C22×C4×C59C22×C412yes—
163C22×C4×C59C2×D411yescharacteristic
164C22×C4×C59C2×D411yescharacteristic
165C22×C4×C59C2×D412yes—
166C22×C4×C59C2×D412yes—
167C22×C4×C59C2×D412yes—
168C22×C4×C59C2411yescharacteristic
169C22×Dic59—21no—
170C22×Dic59—21no—
171C22×Dic59—22no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—22no—
185C4⋊C4×C59C22⋊C411yescharacteristic
186C4⋊C4×C59C22⋊C412yes—
187C4⋊C4×C59C22⋊C411yescharacteristic
188C4⋊C4×C59C22×C411yescharacteristic
189C4⋊C4×C59C22×C412yes—
190C4⋊C4×C59C22×C411yescharacteristic
191C22⋊C4×C59—22no—
192C22⋊C4×C59—22no—
193C22⋊C4×C59—22no—
194C22⋊C4×C59—22no—
195C22⋊C4×C59—21no—
196C22⋊C4×C59—22no—
197C22⋊C4×C59—21no—
198C22⋊C4×C59—21no—
199C22⋊C4×C59—22no—
200C22⋊C4×C59—21no—
201C22⋊C4×C59—21no—
202C22⋊C4×C59C22⋊C412yes—
203C22⋊C4×C59C22⋊C412yes—
204C22⋊C4×C59C22⋊C412yes—
205C22⋊C4×C59C22⋊C412yes—
206C42×C59—22no—
207C42×C59—22no—
208C42×C59—21no—
209C42×C59C22×C411yescharacteristic
210C42×C59C22×C411yescharacteristic
211C42×C59C2×D411yescharacteristic
212C42×C59C2×D411yescharacteristic
213C22⋊Dic59—82no—
214C22⋊Dic59—82no—
215C4⋊Dic59—42no—
216C4⋊Dic59—82no—
217C4×Dic59—42no—
218C4×Dic59—81no—
219#11745664—82no—
220C2×C59⋊C8—21no—
221C2×C59⋊C8—21no—
222C2×C59⋊C8—22no—
223C2×C59⋊C8—82no—
224C23×C59—22no—
225C23×C59—22no—
226C23×C59—21no—
227C23×C59—42no—
228C23×C59—42no—
229C23×C59C2×C22⋊C412yes—
230C23×C59C2×C22⋊C412yes—
231C23×C59C22×D411yescharacteristic
232Q8×C59—41no—
233Q8×C59—42no—
234Q8×C59—42no—
235Q8×C59—82no—
236D4×C59—22no—
237D4×C59—22no—
238D4×C59—42no—
239D4×C59—41no—
240D4×C59—44no—
241D4×C59—41no—
242D4×C59—41no—
243D4×C59—82no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—22no—
247C2×C4×C59—21no—
248C2×C4×C59—22no—
249C2×C4×C59—21no—
250C2×C4×C59—22no—
251C2×C4×C59—21no—
252C2×C4×C59—42no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×C4×C59—41no—
256C2×C4×C59—42no—
257C2×C4×C59—42no—
258C2×C4×C59—44no—
259C2×C4×C59—42no—
260C2×C4×C59—82no—
261C2×C4×C59—82no—
262C2×C4×C59C2×C22⋊C411yescharacteristic
263C2×C4×C59C2×C22⋊C412yes—
264C2×C4×C59C2×C22⋊C411yescharacteristic
265C2×C4×C59C2×C22⋊C411yescharacteristic
266C2×C4×C59C2×C22⋊C412yes—
267C2×C4×C59C2×C22⋊C411yescharacteristic
268C2×C4×C59C23×C411yescharacteristic
269C2×C4×C59C22×D411yescharacteristic
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—42no—
273C2×Dic59—41no—
274C2×Dic59—42no—
275C2×Dic59—41no—
276C2×Dic59—42no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C59⋊C8—41no—
280C59⋊C8—42no—
281C59⋊C8—41no—
282#17600—591nomaximal, Sylow
283C22×C59—21no—
284C22×C59—21no—
285C22×C59—22no—
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—82no—
295C22×C59—82no—
296C22×C59C22×C22⋊C411yescharacteristic
297C22×C59#57711yescharacteristic
298C22×C59#57811yescharacteristic
299C4×C59—21no—
300C4×C59—21no—
301C4×C59—22no—
302C4×C59—22no—
303C4×C59—21no—
304C4×C59—21no—
305C4×C59—21no—
306C4×C59—42no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—41no—
310C4×C59—42no—
311C4×C59#42112yes—
312Dic59—81no—
313Dic59—82no—
314Dic59—81no—
315#3520D5911yescharacteristic
316#2974—591no—
317#2319—591no—
318#1520—592no—
319#1516—592no—
320#1513—592no—
321#1511—592no—
322#1509—591no—
323#1506—591no—
324#1503—591no—
325#1500—591no—
326C2×C59—21no—
327C2×C59—21no—
328C2×C59—42no—
329C2×C59—42no—
330C2×C59—81no—
331C2×C59#288311yescharacteristic
332C2×C59#289811yescharacteristic
333C2×C59C2×#42111yescharacteristic
334#556Dic5911yescharacteristic, complemented
335#554—21no—
336#552Dic5912yescomplemented
337#551—22no—
338D4×Q8Dic5911yescharacteristic, complemented
339#548Dic5912yescomplemented
340#547Dic5911yescharacteristic, complemented
341#547Dic5911yescharacteristic, complemented
342#546—21no—
343#542—21no—
344#539—22no—
345#533C2×D5911yescharacteristic
346#529C2×D5911yescharacteristic
347#525C2×D5912yes—
348C2×C22⋊Q8C2×D5912yes—
349#520—21no—
350C2×C4×D4C2×D5911yescharacteristic
351#436—1181no—
352#435—1182no—
353C8×D4—1181no—
354#432—591no—
355#431—591no—
356#407—592no—
357C2×C22⋊C8—592no—
358#390—1181no—
359#389—1181no—
360#388—1182no—
361#381—592no—
362#380—592no—
363C4×C4⋊C4—591no—
364#376—591no—
365#328—591no—
366#328—591no—
367#328—592no—
368#328—592no—
369#328—592no—
370#328—592no—
371#328—592no—
372#327—591no—
373#327—591no—
374#327—592no—
375C59#1760011yescharacteristic, complemented, Sylow
376C2×#41—21no—
377C2×#41C2×Dic5912yes—
378C22×Q8C2×Dic5912yes—
379C22×D4C22⋊D5911yescharacteristic
380C23×C4C22×D5912yes—
381C2×M16—1182no—
382C22×C8—1182no—
383C4⋊Q8—21no—
384C4⋊Q8C2×Dic5911yescharacteristic
385C4⋊Q8C2×Dic5911yescharacteristic
386#126—22no—
387#126—22no—
388#126—21no—
389#125—22no—
390#125—21no—
391#125C2×Dic5911yescharacteristic
392#124—21no—
393#124—22no—
394#124C2×Dic5911yescharacteristic
395#123—22no—
396#123—22no—
397#123—21no—
398#123—22no—
399#123—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—21no—
403C22⋊Q8—22no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8C2×Dic5912yes—
407C22⋊Q8C2×Dic5912yes—
408C22⋊Q8C2×Dic5912yes—
409C22⋊Q8C22×D5911yescharacteristic
410C22⋊Q8C22×D5912yes—
411#121—22no—
412#121—21no—
413#121C2×Dic5912yes—
414#121C22×D5911yescharacteristic
415#120—22no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8C2×Dic5911yescharacteristic
419C4×Q8C2×Dic5912yes—
420C4×D4—21no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—21no—
424C4×D4—22no—
425C4×D4C2×Dic5911yescharacteristic
426C4×D4C2×Dic5911yescharacteristic
427C4×D4C2×Dic5911yescharacteristic
428C4×D4C2×Dic5912yes—
429C4×D4C22⋊D5911yescharacteristic
430C4×D4C22⋊D5912yes—
431C4×D4C22⋊D5911yescharacteristic
432#117—21no—
433#117—22no—
434#117—21no—
435#117C2×Dic5912yes—
436C2×C4⋊C4—22no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—1182no—
439C2×C4⋊C4—1181no—
440C2×C4⋊C4C2×Dic5912yes—
441C2×C4⋊C4C22⋊D5911yescharacteristic
442C2×C22⋊C4—22no—
443C2×C22⋊C4—1182no—
444C2×C22⋊C4—1182no—
445C2×C22⋊C4C22⋊D5912yes—
446C2×C42—1181no—
447C2×C42—1181no—
448C2×C42C22×D5911yescharacteristic
449C4⋊C8—1181no—
450C4⋊C8—1181no—
451C22⋊C8—591no—
452C22⋊C8—592no—
453C22⋊C8—591no—
454C22⋊C8—1182no—
455C22⋊C8—1182no—
456#97—1181no—
457C4×C8—1181no—
458#95—591no—
459#95—591no—
460#95—592no—
461#95—1181no—
462#95—1182no—
463C24C2×C22⋊D5912yes—
464#41—44no—
465#41—82no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—82no—
469C2×Q8C22×Dic5911yescharacteristic
470C2×Q8C22×Dic5912yes—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—42no—
475C2×D4C22⋊Dic5911yescharacteristic
476C2×D4C22⋊Dic5911yescharacteristic
477C2×D4C22⋊Dic5912yes—
478C2×D4C22×Dic5911yescharacteristic
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—82no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1182no—
487C22×C4C22×Dic5912yes—
488C22×C4C22×Dic5912yes—
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5912yes—
492C22×C4C2×C22⋊D5912yes—
493C22×C4C2×C22⋊D5912yes—
494C22×C4C23×D5911yescharacteristic
495M16—4722no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1182no—
499C2×C8—4722no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—4722no—
514C4⋊C4—2362no—
515C4⋊C4C22⋊Dic5911yescharacteristic
516C4⋊C4C22⋊Dic5912yes—
517C4⋊C4C22⋊Dic5911yescharacteristic
518C4⋊C4C22×Dic5911yescharacteristic
519C4⋊C4C22×Dic5912yes—
520C4⋊C4C22×Dic5911yescharacteristic
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—21no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—4722no—
533C22⋊C4—4722no—
534C22⋊C4C22⋊Dic5912yes—
535C22⋊C4C22⋊Dic5912yes—
536C22⋊C4C22⋊Dic5912yes—
537C22⋊C4C22⋊Dic5912yes—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—4721no—
542C42—2362no—
543C42C22×Dic5911yescharacteristic
544C42C22×Dic5911yescharacteristic
545C42C2×C22⋊D5911yescharacteristic
546C42C2×C22⋊D5911yescharacteristic
547C23—22no—
548C23—22no—
549C23—21no—
550C23—42no—
551C23—42no—
552C23C2×C22⋊Dic5912yes—
553C23C2×C22⋊Dic5912yes—
554C23C22×C22⋊D5911yescharacteristic
555Q8—41no—
556Q8—42no—
557Q8—42no—
558Q8—82no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—41no—
563D4—44no—
564D4—41no—
565D4—41no—
566D4—82no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2362no—
589C2×C4—2361no—
590C2×C4—2362no—
591C2×C4—2362no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4C2×C22⋊Dic5911yescharacteristic
595C2×C4C2×C22⋊Dic5912yes—
596C2×C4C2×C22⋊Dic5911yescharacteristic
597C2×C4C2×C22⋊Dic5911yescharacteristic
598C2×C4C2×C22⋊Dic5912yes—
599C2×C4C2×C22⋊Dic5911yescharacteristic
600C2×C4C23×Dic5911yescharacteristic
601C2×C4C22×C22⋊D5911yescharacteristic
602C8—2361no—
603C8—2362no—
604C8—2361no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—82no—
617C22—82no—
618C22C22×C22⋊Dic5911yescharacteristic
619C22#1683821011yescharacteristic
620C22#1683821711yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—4721no—
634C4—4722no—
635C4—4721no—
636C4#1683741912yes—
637C2—21no—
638C2—21no—
639C2—42no—
640C2—42no—
641C2—81no—
642C2#2143255711yescharacteristic
643C2#2143262911yescharacteristic
644C2C2×#1683741911yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
Fitting, prime radicalC59×#3520C2
socleC22×C59C22×C22⋊C4
cosocleC22×C4×C59C24
hypercenter#3520D59
hypocenterC59#17600
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3520

Complements

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

NH ≅ G/Ntype
1. #21433131C2semidirect
2. #21426739C2semidirect
3. #21422668C2semidirect
4. #21422652C2semidirect
5. #21422639C2semidirect
6. #21422629C2semidirect
7. #21422621C2semidirect
8. #21422608C2semidirect
9. #21422595C2semidirect
10. #21422582C2semidirect
36. C59×#556C4semidirect
38. C59×#552C4semidirect
40. D4×Q8×C59C4semidirect
41. C59×#548C4semidirect
42. C59×#547C4semidirect
43. C59×#547C4semidirect
334. #556Dic59semidirect
336. #552Dic59semidirect
338. D4×Q8Dic59semidirect
339. #548Dic59semidirect
340. #547Dic59semidirect
341. #547Dic59semidirect
375. C59#17600semidirect

Maximal subgroups

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

subgroupindex
#214331312
#214267392
#214226682
#214226522
#214226392
#214226292
#214226212
#214226082
#214225952
#214225822
C59×#35202
#1760059