Finite Group Databasebeta

← #28647483

Subgroups — #28647483

Order 15104.

826 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837805C211yesmaximal, characteristic, complemented
2C2×#16837442C211yesmaximal, characteristic
3#21427787C211yesmaximal, characteristic
4#21425577C211yesmaximal, characteristic, complemented
5#21425568C211yesmaximal, characteristic, complemented
6#21425426C211yesmaximal, characteristic, complemented
7#21425424C211yesmaximal, characteristic, complemented
8#21424831C211yesmaximal, characteristic, complemented
9#21424821C211yesmaximal, characteristic, complemented
10#21424293C211yesmaximal, characteristic
11#21424284C211yesmaximal, characteristic
12#21424262C211yesmaximal, characteristic, complemented
13#21424127C211yesmaximal, characteristic
14#21423880C211yesmaximal, characteristic
15C59×#2440C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273727C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19#16837805—24no—
20#16837442—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273660C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273658C2211yescharacteristic
31C2×C22⋊(C59⋊C8)C2211yescharacteristic
32C2×#14273638C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837325—21no—
35#16837314—21no—
36#16837306—21no—
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837289—21no—
40#16837268C2211yescharacteristic
41#16837262C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4⋊Q8×C59C2211yescharacteristic
51C2×C22⋊Q8×C59—21no—
52C22×C4⋊C4×C59C2211yescharacteristic
53C59×#398—21no—
54C59×#395—21no—
55C59×#389C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#385—21no—
58C59×#384C2211yescharacteristic
59C59×#380C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118—21no—
62C2×#11745664—22no—
63C2×#11745664—22no—
64C2×#11745664Q814yescomplemented
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273727—22no—
70#14273727—41no—
71#14273686—22no—
72#14273686—41no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C4×Dic59—21no—
83#14273660—22no—
84#14273660—22no—
85#14273660—22no—
86#14273660—22no—
87#14273660—41no—
88#14273660D414yes—
89#14273658—22no—
90#14273658—22no—
91#14273658—22no—
92#14273658—41no—
93C22⋊(C59⋊C8)Q814yescomplemented
94#14273638—22no—
95C4×C59⋊C8—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×Q8×C59—21no—
102C23×C4×C59—21no—
103C23×C4×C59C2311yescharacteristic
104C4⋊Q8×C59—22no—
105C4⋊Q8×C59—41no—
106C22⋊Q8×C59—44no—
107C2×C4⋊C4×C59—22no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—41no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59D411yescharacteristic
121C2×C22⋊C4×C59D411yescharacteristic
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59D411yescharacteristic
128#95×C59D411yescharacteristic
129C24×C59C2×D411yescharacteristic
130C2×Q8×C59—42no—
131C2×Q8×C59—42no—
132C2×Q8×C59—81no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—42no—
142C22×C4×C59—42no—
143C22×C4×C59—81no—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×Q812yes—
150C22×C4×C59C2×Q812yes—
151C22×C4×C59C2411yescharacteristic
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C2×Dic118—42no—
157C2×Dic118—42no—
158C2×Dic118—81no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—42no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59—42no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59—81no—
172C4⋊C4×C59C2×D412yes—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—42no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C22⋊Dic59—42no—
181C22⋊Dic59—81no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—42no—
190C4⋊Dic59C2×D412yes—
191Dic59⋊C4—42no—
192Dic59⋊C4—81no—
193C4×Dic59—42no—
194#11745664—28no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8C2×D412yes—
202C2×C59⋊C8C2×Q812yes—
203C2×C59⋊C8C2×Q812yes—
204C2×C59⋊C8C2×Q812yes—
205C2×C59⋊C8C2×Q812yes—
206C2×C59⋊C8#4112yes—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59C4⋊Q812yes—
214C23×C59C22×D411yescharacteristic
215Q8×C59—84no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
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—41no—
243C2×C4×C59—41no—
244C2×C4×C59—84no—
245C2×C4×C59C4⋊Q812yes—
246C2×C4×C59AGL(1,ℤ/8)12yes—
247C2×C4×C59#13712yes—
248C2×C4×C59C22×D411yescharacteristic
249C2×C4×C59C22×Q811yescharacteristic
250C2×C4×C59C22×Q811yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59H(5,𝔽2)11yescharacteristic
253C2×C4×C592−1+411yescharacteristic
254C2×Dic59—22no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266Dic118—84no—
267C59⋊C8—22no—
268C59⋊C8—22no—
269C59⋊C8—22no—
270C59⋊C8C22⋊Q814yescomplemented
271#26074—591nomaximal, Sylow
272C22×C59—22no—
273C22×C59—22no—
274C22×C59—22no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59C2×C4⋊Q811yescharacteristic
280C22×C59D4211yescharacteristic
281C22×C59#54711yescharacteristic
282C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C59C2×#13711yescharacteristic
284C22×C59#57611yescharacteristic
285C22×C59#57811yescharacteristic
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—22no—
289C4×C59—41no—
290C4×C59—41no—
291C4×C59—42no—
292C4×C59—81no—
293C4×C59—81no—
294C4×C59—81no—
295C4×C59C2×C22⋊Q811yescharacteristic
296C4×C59#54411yescharacteristic
297C4×C59D4×Q811yescharacteristic
298C4×C59#55411yescharacteristic
299Dic59—42no—
300Dic59—81no—
301Dic59—81no—
302Dic59—81no—
303C2×#500—591no—
304C2×#428—591no—
305#2440D5911yescharacteristic
306#2439—591no—
307#2085—591no—
308#2083—591no—
309#2058—591no—
310#2057—591no—
311#1943—591no—
312#1941—591no—
313#1854—591no—
314#1852—591no—
315#1847—591no—
316#1809—591no—
317#1753—591no—
318C2×C59—42no—
319C2×C59#244011yescharacteristic
320C2×C59#316411yescharacteristic
321C2×C59#316511yescharacteristic
322C2×C59#326411yescharacteristic
323C2×C59#327011yescharacteristic
324C2×C59#335311yescharacteristic
325C2×C59#335711yescharacteristic
326C22×M16—591no—
327C2×C4⋊Q8—591no—
328C2×C4⋊Q8C2×D5911yescharacteristic
329C2×C22⋊Q8—21no—
330C2×#117—591no—
331C22×C4⋊C4C2×D5911yescharacteristic
332#500—1184no—
333#428—1184no—
334C2×#107—591no—
335C2×#107—591no—
336C2×#107—591no—
337C2×#107—591no—
338C2×#107—591no—
339C2×#107—591no—
340C2×#106—591no—
341C2×#106—591no—
342C2×#106—591no—
343C2×#106—591no—
344C2×C22⋊C8—591no—
345C2×#97—591no—
346C2×C4×C8—591no—
347#398—21no—
348#396—1181no—
349#395—21no—
350#393—1181no—
351#391—1181no—
352#389—591no—
353#389—591no—
354#389C2×D5911yescharacteristic
355#389C2×D5911yescharacteristic
356#387—1181no—
357#385—21no—
358#384C2×D5911yescharacteristic
359#382—591no—
360#380—591no—
361#380C2×D5911yescharacteristic
362C4×C22⋊C4C2×D5911yescharacteristic
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#336—591no—
370#336—591no—
371C59#2607411yescharacteristic, complemented, Sylow
372C22×Q8—21no—
373C22×Q8—1181no—
374C23×C4—21no—
375C23×C4C22×D5911yescharacteristic
376C2×M16—594no—
377C2×M16—1182no—
378C2×M16—1182no—
379C22×C8—591no—
380C22×C8—591no—
381C22×C8—591no—
382C22×C8—591no—
383C4⋊Q8—22no—
384C4⋊Q8—41no—
385C4⋊Q8—1182no—
386C4⋊Q8—2361no—
387C22⋊Q8—44no—
388#117—1182no—
389#117—2361no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1181no—
411C2×C22⋊C4—1181no—
412C2×C22⋊C4C22⋊D5911yescharacteristic
413C2×C22⋊C4C22⋊D5911yescharacteristic
414C2×C42—1181no—
415C2×C42C22×D5911yescharacteristic
416C2×C42C22×D5911yescharacteristic
417#107—594no—
418#107—1182no—
419#107—1182no—
420#107—1182no—
421#107—1182no—
422#107—2361no—
423#106—1182no—
424#106—1182no—
425#106—1182no—
426#106—2361no—
427C22⋊C8—594no—
428#97—1182no—
429C4×C8—1182no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95C22⋊D5911yescharacteristic
439#95C22⋊D5911yescharacteristic
440C24C2×C22⋊D5911yescharacteristic
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—81no—
444C2×Q8—4721no—
445C2×Q8—2362no—
446C2×Q8—2362no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4Q8×D5912yes—
463C22×C4Q8×D5912yes—
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C2×C22⋊D5911yescharacteristic
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C2×C22⋊D5911yescharacteristic
468C22×C4C2×C22⋊D5911yescharacteristic
469C22×C4C23×D5911yescharacteristic
470M16—1188no—
471C2×C8—592no—
472C2×C8—592no—
473C2×C8—592no—
474C2×C8—592no—
475C2×C8—592no—
476C2×C8—592no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—592no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—4721no—
505C4⋊C4—2362no—
506C4⋊C4—2362no—
507C4⋊C4D4×D5912yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—42no—
511C22⋊C4—4721no—
512C22⋊C4—2362no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—2362no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23#1427372512yes—
525C23C22×C22⋊D5911yescharacteristic
526Q8—84no—
527Q8—4724no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—84no—
557C2×C4—1182no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4#1427372512yes—
570C2×C4#1427375312yes—
571C2×C4#1427375812yes—
572C2×C4C2×D4×D5911yescharacteristic
573C2×C4C2×Q8×D5911yescharacteristic
574C2×C4C2×Q8×D5911yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C2×C4#1427377211yescharacteristic
577C2×C4#1427377411yescharacteristic
578C8—594no—
579C8—1182no—
580C8—1182no—
581C8—1182no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1427372511yescharacteristic
590C22D4×C22⋊D5911yescharacteristic
591C22#1683805111yescharacteristic
592C22C2×#1427375311yescharacteristic
593C22C2×#1427375811yescharacteristic
594C22#1683820511yescharacteristic
595C22#1683821711yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—4721no—
606C4—4721no—
607C4—4721no—
608C4—2362no—
609C4C22⋊Q8×D5911yescharacteristic
610C4#1683803111yescharacteristic
611C4#1683806511yescharacteristic
612C4#1683810611yescharacteristic
613C2—42no—
614C2#2142779511yescharacteristic
615C2#2143463011yescharacteristic
616C2#2143463811yescharacteristic
617C2#2143557211yescharacteristic
618C2#2143566711yescharacteristic
619C2#2143668011yescharacteristic
620C2#2143672211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2440C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2440D59
hypocenterC59#26074
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
1. C2×#16837805C2semidirect
4. #21425577C2semidirect
5. #21425568C2semidirect
6. #21425426C2semidirect
7. #21425424C2semidirect
8. #21424831C2semidirect
9. #21424821C2semidirect
12. #21424262C2semidirect
64. C2×#11745664Q8semidirect
93. C22⋊(C59⋊C8)Q8semidirect
270. C59⋊C8C22⋊Q8semidirect
371. C59#26074semidirect

Maximal subgroups

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

subgroupindex
C2×#168378052
C2×#168374422
#214277872
#214255772
#214255682
#214254262
#214254242
#214248312
#214248212
#214242932
#214242842
#214242622
#214241272
#214238802
C59×#24402
#2607459