Finite Group Databasebeta

← #28647482

Subgroups — #28647482

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×#16837443C211yesmaximal, characteristic
2#21427794C211yesmaximal, characteristic
3#21427783C211yesmaximal, characteristic
4#21425580C211yesmaximal, characteristic, complemented
5#21425570C211yesmaximal, characteristic, complemented
6#21425426C211yesmaximal, characteristic, complemented
7#21425421C211yesmaximal, characteristic, complemented
8#21424835C211yesmaximal, characteristic, complemented
9#21424824C211yesmaximal, characteristic, complemented
10#21424292C211yesmaximal, characteristic
11#21424282C211yesmaximal, characteristic
12#21424263C211yesmaximal, characteristic, complemented
13#21424130C211yesmaximal, characteristic
14#21423879C211yesmaximal, characteristic
15C2×C59×#500C211yesmaximal, characteristic, complemented
16C22×#11745659C2211yescharacteristic
17C2×#14273703—21no—
18C2×#14273685C2211yescharacteristic
19C2×C4⋊C4×D59C2211yescharacteristic
20#16837443—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×(C2×D59)⋊C8C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29#16837335—21no—
30#16837324—21no—
31#16837321—21no—
32#16837313—21no—
33#16837309—21no—
34#16837301C2211yescharacteristic
35#16837301C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837283—21no—
39#16837273—21no—
40#16837271C2211yescharacteristic
41#16837267C2211yescharacteristic
42#16837259C2211yescharacteristic
43#16837259C2211yescharacteristic
44C4×(C2×D59)⋊C4C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4⋊Q8×C59C2211yescharacteristic
54C2×C4⋊Q8×C59C2211yescharacteristic
55C59×#500—24no—
56C2×#107×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C2×#107×C59C2211yescharacteristic
60C2×C4×C8×C59C2211yescharacteristic
61C22×C4×D59—21no—
62C22×C4×D59C2311yescharacteristic
63C2×#11745659—22no—
64C2×#11745659—22no—
65C2×#11745659Q814yescomplemented
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273703—44no—
69#14273685—22no—
70#14273685—41no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C4⋊C4×D59—22no—
83C4⋊C4×D59—41no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4D411yescharacteristic
88C2×(C2×D59)⋊C4D411yescharacteristic
89C2×C4×Dic59—21no—
90C2×C4×Dic59C2311yescharacteristic
91#14273660—22no—
92#14273660—41no—
93#14273659—22no—
94#14273659—22no—
95#14273658—22no—
96#14273658—41no—
97(C2×D59)⋊C8Q814yescomplemented
98#14273637—22no—
99C59⋊#95—21no—
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
109C22×Q8×C59—21no—
110C22×Q8×C59—21no—
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C4⋊Q8×C59—22no—
114C4⋊Q8×C59—22no—
115C4⋊Q8×C59—41no—
116C4⋊Q8×C59—41no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#107×C59—22no—
125#107×C59—22no—
126#107×C59—22no—
127#107×C59D414yes—
128C4×C8×C59—22no—
129C23×D59C2×D411yescharacteristic
130C2×Q8×C59—42no—
131C2×Q8×C59—42no—
132C2×Q8×C59—42no—
133C2×Q8×C59—42no—
134C2×Q8×C59—81no—
135C2×Q8×C59—81no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C2×C4×D59—21no—
151C2×C4×D59—21no—
152C2×C4×D59—21no—
153C2×C4×D59—21no—
154C2×C4×D59—42no—
155C2×C4×D59—42no—
156C2×C4×D59—81no—
157C2×C4×D59C2×Q812yes—
158C2×C4×D59C2×Q812yes—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59C2×D412yes—
162C2×C8×C59C2×Q812yes—
163C2×C8×C59C2×Q812yes—
164C2×C8×C59#4112yes—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59C2×D412yes—
174C4⋊C4×C59C2×D412yes—
175C42×C59—21no—
176C42×C59—21no—
177(C2×D59)⋊C4—22no—
178(C2×D59)⋊C4—22no—
179(C2×D59)⋊C4—42no—
180(C2×D59)⋊C4—42no—
181(C2×D59)⋊C4—81no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—42no—
191C4⋊Dic59—81no—
192Dic59⋊C4—42no—
193Dic59⋊C4—42no—
194Dic59⋊C4—42no—
195Dic59⋊C4—42no—
196Dic59⋊C4—81no—
197C4×Dic59—21no—
198C4×Dic59—21no—
199C4×Dic59—42no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8C2×Q812yes—
205C2×C59⋊C8C2×Q812yes—
206#11745659—28no—
207C23×C59C22×D411yescharacteristic
208C22×D59—21no—
209C22×D59—21no—
210C22×D59—21no—
211C22×D59—21no—
212C22×D59—21no—
213C22×D59—21no—
214C22×D59C4⋊Q812yes—
215Q8×C59—84no—
216Q8×C59—84no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59AGL(1,ℤ/8)12yes—
234C2×C4×C59#13712yes—
235C2×C4×C59C22×D411yescharacteristic
236C2×C4×C59C22×Q811yescharacteristic
237C2×C4×C59C22×Q811yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59H(5,𝔽2)11yescharacteristic
240C2×C4×C592−1+411yescharacteristic
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59C4⋊Q812yes—
262C4×D59—22no—
263C4×D59—22no—
264C4×D59—22no—
265C4×D59—22no—
266C4×D59—84no—
267C8×C59—22no—
268C8×C59C22⋊Q814yescomplemented
269C59⋊C8—22no—
270C59⋊C8—22no—
271#26074—591nomaximal, Sylow
272C22×C59C2×C4⋊Q811yescharacteristic
273C22×C59D4211yescharacteristic
274C22×C59#54711yescharacteristic
275C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C59C2×#13711yescharacteristic
277C22×C59#57611yescharacteristic
278C22×C59#57811yescharacteristic
279C2×D59—22no—
280C2×D59—22no—
281C2×D59—22no—
282C2×D59—41no—
283C2×D59—41no—
284C2×D59—41no—
285C2×D59—41no—
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—42no—
289C4×C59—42no—
290C4×C59—81no—
291C4×C59—81no—
292C4×C59C2×C22⋊Q811yescharacteristic
293C4×C59#54411yescharacteristic
294C4×C59D4×Q811yescharacteristic
295C4×C59#55411yescharacteristic
296Dic59—22no—
297Dic59—41no—
298Dic59—41no—
299Dic59—81no—
300Dic59—81no—
301Dic59—81no—
302Dic59—81no—
303C2×#500D5911yescharacteristic, complemented
304C2×#428—591no—
305#2440—591no—
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#244011yescharacteristic
319C2×C59#316411yescharacteristic
320C2×C59#316511yescharacteristic
321C2×C59#326411yescharacteristic
322C2×C59#327011yescharacteristic
323C2×C59#335311yescharacteristic
324C2×C59#335711yescharacteristic
325D59—42no—
326C22×M16—591no—
327C2×C4⋊Q8C2×D5911yescharacteristic
328C2×C4⋊Q8C2×D5911yescharacteristic
329C2×C22⋊Q8—1181no—
330C2×#117—591no—
331C22×C4⋊C4—591no—
332#500—24no—
333#428—1184no—
334C2×#107—591no—
335C2×#107—591no—
336C2×#107C2×D5911yescharacteristic
337C2×#107C2×D5911yescharacteristic
338C2×#107C2×D5911yescharacteristic
339C2×#107C2×D5911yescharacteristic
340C2×#106—591no—
341C2×#106—591no—
342C2×#106—591no—
343C2×#106—591no—
344C2×C22⋊C8—591no—
345C2×#97—591no—
346C2×C4×C8C2×D5911yescharacteristic
347#398—1181no—
348#396—1181no—
349#395—1181no—
350#393—1181no—
351#391—1181no—
352#389—591no—
353#389—591no—
354#389—591no—
355#389—591no—
356#387—1181no—
357#385—1181no—
358#384—591no—
359#382—591no—
360#380—591no—
361#380—591no—
362C4×C22⋊C4—591no—
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—21no—
374C23×C4—591no—
375C23×C4—1181no—
376C2×M16—594no—
377C2×M16—1182no—
378C2×M16—1182no—
379C22×C8—591no—
380C22×C8—591no—
381C22×C8C22×D5911yescharacteristic
382C22×C8C22×D5911yescharacteristic
383C4⋊Q8—22no—
384C4⋊Q8—22no—
385C4⋊Q8—41no—
386C4⋊Q8—41no—
387C22⋊Q8—2364no—
388#117—1182no—
389#117—2361no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—1182no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—2361no—
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C22⋊C4—591no—
410C2×C22⋊C4—591no—
411C2×C22⋊C4—1181no—
412C2×C22⋊C4—1181no—
413C2×C22⋊C4—1181no—
414C2×C42—591no—
415C2×C42—1181no—
416C2×C42C22×D5911yescharacteristic
417#107—22no—
418#107—22no—
419#107—22no—
420#107—1182no—
421#107—2361no—
422#107C22⋊D5914yes—
423#106—1182no—
424#106—1182no—
425#106—1182no—
426#106—2361no—
427C22⋊C8—594no—
428#97—1182no—
429C4×C8—22no—
430#95—591no—
431#95—591no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440C24—591no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—81no—
446C2×Q8—81no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—591no—
452C22×C4—592no—
453C22×C4—592no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—4721no—
465C22×C4—2362no—
466C22×C4—2362no—
467C22×C4D4×D5911yescharacteristic
468C22×C4D4×D5911yescharacteristic
469C22×C4C23×D5911yescharacteristic
470M16—1188no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—592no—
474C2×C8—592no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8Q8×D5912yes—
480C2×C8Q8×D5912yes—
481C2×C8#1174568912yes—
482C2×C8C2×C22⋊D5912yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—4721no—
500C4⋊C4—4721no—
501C4⋊C4—2362no—
502C4⋊C4—2362no—
503C4⋊C4—2362no—
504C4⋊C4—2362no—
505C4⋊C4—2362no—
506C4⋊C4C2×C22⋊D5912yes—
507C4⋊C4C2×C22⋊D5912yes—
508C22⋊C4—1182no—
509C22⋊C4—1182no—
510C22⋊C4—4721no—
511C22⋊C4—2362no—
512C22⋊C4—2362no—
513C42—21no—
514C42—21no—
515C42—1181no—
516C42—1181no—
517C42—2362no—
518C23—592no—
519C23—1181no—
520C23—1181no—
521C23—1181no—
522C23—1181no—
523C23—1181no—
524C23—1181no—
525C23C2×D4×D5911yescharacteristic
526Q8—84no—
527Q8—84no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—592no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1182no—
548C2×C4—1182no—
549C2×C4—1182no—
550C2×C4—1182no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—4724no—
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—2361no—
570C2×C4#1427375112yes—
571C2×C4#1427375912yes—
572C2×C4C22×C22⋊D5911yescharacteristic
573C2×C4C2×Q8×D5911yescharacteristic
574C2×C4C2×Q8×D5911yescharacteristic
575C2×C4C2×#1174568911yescharacteristic
576C2×C4#1427377211yescharacteristic
577C2×C4#1427377511yescharacteristic
578C8—22no—
579C8—1182no—
580C8—1182no—
581C8#1427370314yescomplemented
582C22—1182no—
583C22—1182no—
584C22—1182no—
585C22—2361no—
586C22—2361no—
587C22—2361no—
588C22—2361no—
589C22C4⋊Q8×D5911yescharacteristic
590C22D4×C22⋊D5911yescharacteristic
591C22#1683805511yescharacteristic
592C22C2×#1427375111yescharacteristic
593C22C2×#1427375911yescharacteristic
594C22#1683820311yescharacteristic
595C22#1683821511yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—1182no—
603C4—4721no—
604C4—4721no—
605C4—4721no—
606C4—4721no—
607C4—2361no—
608C4—2361no—
609C4C2×#1427370311yescharacteristic
610C4#1683802911yescharacteristic
611C4Q8×C22⋊D5911yescharacteristic
612C4#1683811311yescharacteristic
613C2—2362no—
614C2#2142779411yescharacteristic
615C2#2143462911yescharacteristic
616C2#2143463911yescharacteristic
617C2#2143557611yescharacteristic
618C2#2143566611yescharacteristic
619C2#2143667911yescharacteristic
620C2#2143672111yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#500C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
hypercenterC2×#500D59
hypocenterC59#26074
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
4. #21425580C2semidirect
5. #21425570C2semidirect
6. #21425426C2semidirect
7. #21425421C2semidirect
8. #21424835C2semidirect
9. #21424824C2semidirect
12. #21424263C2semidirect
15. C2×C59×#500C2semidirect
65. C2×#11745659Q8semidirect
97. (C2×D59)⋊C8Q8semidirect
268. C8×C59C22⋊Q8semidirect
303. C2×#500D59semidirect
371. C59#26074semidirect
581. C8#14273703semidirect

Maximal subgroups

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

subgroupindex
C2×#168374432
#214277942
#214277832
#214255802
#214255702
#214254262
#214254212
#214248352
#214248242
#214242922
#214242822
#214242632
#214241302
#214238792
C2×C59×#5002
#2607459