Finite Group Databasebeta

← #28611487

Subgroups — #28611487

Order 15104.

942 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837658C211yesmaximal, characteristic
2C2×#16837507C211yesmaximal, characteristic, complemented
3C2×#16837442C211yesmaximal, characteristic
4#21427121C211yesmaximal, characteristic
5#21424956C211yesmaximal, characteristic, complemented
6#21424836C211yesmaximal, characteristic, complemented
7#21424737C211yesmaximal, characteristic, complemented
8#21424727C211yesmaximal, characteristic, complemented
9#21424409C211yesmaximal, characteristic
10#21424403C211yesmaximal, characteristic
11#21424363C211yesmaximal, characteristic, complemented
12#21424350C211yesmaximal, characteristic, complemented
13C4×#14273658C211yesmaximal, characteristic, complemented
14#21423880C211yesmaximal, characteristic
15C59×#2360C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C22×#11745664C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20C2×#14273686C2211yescharacteristic
21#16837658C418yescomplemented
22#16837507—24no—
23#16837442—24no—
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273651C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×#14273650C2211yescharacteristic
34C2×#14273650C2211yescharacteristic
35C2×C22⋊(C59⋊C8)C2211yescharacteristic
36C2×#14273638C2211yescharacteristic
37C2×C4×C59⋊C8C2211yescharacteristic
38#16837308C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837296—21no—
41#16837290—21no—
42#16837284—21no—
43C4×C4⋊Dic59C2211yescharacteristic
44#16837253—21no—
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837157C2211yescharacteristic
50#16837157C2211yescharacteristic
51C2×C22⋊Q8×C59C2211yescharacteristic
52C2×C4×Q8×C59C2211yescharacteristic
53C22×C4⋊C4×C59C2211yescharacteristic
54C59×#391C2211yescharacteristic
55C59×#389—21no—
56C59×#389C2211yescharacteristic
57C59×#387—21no—
58C59×#385—21no—
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C22⋊C4×C59—21no—
61C4×C22⋊C4×C59C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672C2×C414yescomplemented
66C2×#11745664—22no—
67C2×#11745664—22no—
68C2×#11745664C2×C414yes—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273706—22no—
74#14273706C2×C414yes—
75C4×Dic118—22no—
76C4×Dic118—41no—
77#14273686—22no—
78#14273686—41no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×C22⋊Dic59—21no—
86C2×C22⋊Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90#14273660—22no—
91#14273660—22no—
92#14273658—22no—
93#14273658C2×C414yescomplemented
94#14273658D414yes—
95#14273651—22no—
96#14273651—41no—
97#14273651C2×C414yescomplemented
98#14273650—22no—
99#14273650—41no—
100#14273650C2×C414yescomplemented
101C22⋊(C59⋊C8)—22no—
102#14273638—22no—
103C4×C59⋊C8—22no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C22×Q8×C59C2311yescharacteristic
109C23×C4×C59—21no—
110C23×C4×C59C2311yescharacteristic
111C22⋊Q8×C59—22no—
112C22⋊Q8×C59C2×C414yes—
113C4×Q8×C59—22no—
114C4×Q8×C59—41no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59—22no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59—21no—
120C2×C4⋊C4×C59—41no—
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C22⋊C4×C59—21no—
125C2×C22⋊C4×C59D411yescharacteristic
126C2×C22⋊C4×C59D411yescharacteristic
127C2×C42×C59—21no—
128C2×C42×C59—21no—
129C2×C42×C59C2311yescharacteristic
130C2×C42×C59C2311yescharacteristic
131#95×C59—21no—
132#95×C59—21no—
133#95×C59—21no—
134#95×C59—21no—
135#95×C59D411yescharacteristic
136#95×C59D411yescharacteristic
137C24×C59C2×D411yescharacteristic
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59C22×C412yes—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—42no—
153C22×C4×C59—42no—
154C22×C4×C59—81no—
155C22×C4×C59C22×C412yes—
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59#4112yes—
162C22×C4×C59C2411yescharacteristic
163C22×Dic59—21no—
164C22×Dic59—21no—
165C22×Dic59—21no—
166C22×Dic59—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118C22×C412yes—
173C4⋊C4×C59—24no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59—42no—
180C4⋊C4×C59—81no—
181C4⋊C4×C59C22×C412yes—
182C4⋊C4×C59C2×D412yes—
183C22⋊C4×C59—24no—
184C22⋊C4×C59—22no—
185C22⋊C4×C59—41no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—42no—
191C42×C59—81no—
192C22⋊Dic59—24no—
193C22⋊Dic59—42no—
194#11745672—28no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59C22×C412yes—
200C4⋊Dic59C2×D412yes—
201Dic59⋊C4—24no—
202Dic59⋊C4—42no—
203Dic59⋊C4—42no—
204C4×Dic59—42no—
205C4×Dic59—42no—
206C4×Dic59—42no—
207#11745664—28no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8C22×C412yes—
217C2×C59⋊C8C22×C412yes—
218C2×C59⋊C8C2×D412yes—
219C2×C59⋊C8#4112yes—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59C4×D412yes—
227C23×C59C22×D411yescharacteristic
228Q8×C59—22no—
229Q8×C59—22no—
230Q8×C59—41no—
231Q8×C59—41no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—22no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—84no—
262C2×C4×C59C4×D412yes—
263C2×C4×C59#13712yes—
264C2×C4×C59C23×C411yescharacteristic
265C2×C4×C59C22×D411yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59H(5,𝔽2)11yescharacteristic
269C2×C4×C592−1+411yescharacteristic
270C2×Dic59—22no—
271C2×Dic59—22no—
272C2×Dic59—22no—
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—
282Dic118—22no—
283Dic118—22no—
284Dic118—41no—
285Dic118—41no—
286C59⋊C8—22no—
287C59⋊C8—41no—
288C59⋊C8—41no—
289C59⋊C8C4×D414yes—
290#23038—591nomaximal, Sylow
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59—41no—
298C22×C59C2×C4×D411yescharacteristic
299C22×C59D4211yescharacteristic
300C22×C59#54711yescharacteristic
301C22×C59C2×#13711yescharacteristic
302C22×C59#57511yescharacteristic
303C22×C59#57711yescharacteristic
304C22×C59#57811yescharacteristic
305C4×C59—22no—
306C4×C59—22no—
307C4×C59—41no—
308C4×C59—42no—
309C4×C59—42no—
310C4×C59—41no—
311C4×C59—42no—
312C4×C59—41no—
313C4×C59—81no—
314C4×C59C2×C4×D411yescharacteristic
315C4×C59#52011yescharacteristic
316C4×C59#54811yescharacteristic
317C4×C59#55311yescharacteristic
318Dic59—42no—
319Dic59—42no—
320Dic59—42no—
321Dic59—81no—
322C2×#470—591no—
323C2×#441—591no—
324C2×#428—591no—
325#2362—591no—
326#2360D5911yescharacteristic
327#1969—591no—
328#1944—591no—
329#1928—591no—
330#1927—591no—
331#1873—591no—
332#1872—591no—
333#1865—591no—
334#1863—591no—
335C4×#106—591no—
336#1753—591no—
337C2×C59—42no—
338C2×C59#236111yescharacteristic
339C2×C59C4×#13711yescharacteristic
340C2×C59#294711yescharacteristic
341C2×C59#327511yescharacteristic
342C2×C59#327611yescharacteristic
343C2×C59#334411yescharacteristic
344C2×C59#334911yescharacteristic
345C22×Q16—591no—
346C22×M16—591no—
347C2×C22⋊Q8—591no—
348C2×C22⋊Q8C2×D5911yescharacteristic
349C2×C4×Q8—591no—
350C2×C4×Q8C2×D5911yescharacteristic
351C2×#117—591no—
352C22×C4⋊C4C2×D5911yescharacteristic
353#470—598no—
354#441—1184no—
355#428—1184no—
356C2×#107—591no—
357C2×#107—591no—
358C2×#106—591no—
359C2×#106—591no—
360C2×#106—591no—
361C2×#103—591no—
362C2×#103—591no—
363C2×#103—591no—
364C2×#103—591no—
365C2×#103—591no—
366C2×#103—591no—
367C2×C22⋊C8—591no—
368C2×#97—591no—
369C2×C4×C8—591no—
370#391—591no—
371#391C2×D5911yescharacteristic
372#389—21no—
373#389—591no—
374#389C2×D5911yescharacteristic
375#388—1181no—
376#388—1181no—
377#387—21no—
378#387—1181no—
379#385—21no—
380C4×C4⋊C4—591no—
381C4×C4⋊C4—1181no—
382C4×C4⋊C4C2×D5911yescharacteristic
383C4×C22⋊C4—21no—
384C4×C22⋊C4C2×D5911yescharacteristic
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#336—591no—
390#336—591no—
391C59#2303811yescharacteristic, complemented, Sylow
392C22×Q8—591no—
393C22×Q8C22×D5911yescharacteristic
394C23×C4—21no—
395C23×C4C22×D5911yescharacteristic
396C2×Q16—594no—
397C2×Q16—1182no—
398C2×Q16—1182no—
399C2×M16—594no—
400C2×M16—1182no—
401C2×M16—1182no—
402C22×C8—591no—
403C22×C8—591no—
404C22×C8—591no—
405C22×C8—591no—
406C22⋊Q8—22no—
407C22⋊Q8—594no—
408C22⋊Q8—1182no—
409C22⋊Q8C4×D5914yes—
410C4×Q8—22no—
411C4×Q8—41no—
412C4×Q8—1182no—
413C4×Q8—2361no—
414#117—1182no—
415#117—2361no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—41no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4C22×D5911yescharacteristic
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C4⋊C4C22×D5911yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1181no—
433C2×C22⋊C4—1181no—
434C2×C22⋊C4C22⋊D5911yescharacteristic
435C2×C22⋊C4C22⋊D5911yescharacteristic
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—1181no—
439C2×C42—1181no—
440C2×C42—1181no—
441C2×C42C22×D5911yescharacteristic
442C2×C42C22×D5911yescharacteristic
443#107—1182no—
444#107—1182no—
445#106—594no—
446#106—594no—
447#106—1182no—
448#103—594no—
449#103—594no—
450#103—1182no—
451#103—1182no—
452#103—2361no—
453#103—2361no—
454C22⋊C8—1182no—
455#97—1182no—
456C4×C8—1182no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464#95—1181no—
465#95C22⋊D5911yescharacteristic
466#95C22⋊D5911yescharacteristic
467C24C2×C22⋊D5911yescharacteristic
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—592no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8C2×C4×D5912yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—1181no—
492C22×C4—1181no—
493C22×C4—1181no—
494C22×C4—1181no—
495C22×C4C2×C4×D5912yes—
496C22×C4#1174568512yes—
497C22×C4C2×C22⋊D5911yescharacteristic
498C22×C4C2×C22⋊D5911yescharacteristic
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C2×C22⋊D5911yescharacteristic
502C22×C4C23×D5911yescharacteristic
503Q16—1188no—
504M16—1188no—
505C2×C8—592no—
506C2×C8—592no—
507C2×C8—592no—
508C2×C8—592no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C4⋊C4—24no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—81no—
525C4⋊C4—592no—
526C4⋊C4—592no—
527C4⋊C4—1184no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—2362no—
533C4⋊C4—2362no—
534C4⋊C4C2×C4×D5912yes—
535C4⋊C4D4×D5912yes—
536C22⋊C4—24no—
537C22⋊C4—22no—
538C22⋊C4—41no—
539C22⋊C4—1184no—
540C22⋊C4—2362no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—42no—
546C42—81no—
547C42—2362no—
548C42—2362no—
549C42—2362no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23C4×C22⋊D5912yes—
557C23C22×C22⋊D5911yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562Q8—1182no—
563Q8—1182no—
564Q8—2361no—
565Q8—2361no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—1182no—
597C2×C4—1182no—
598C2×C4—1182no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4C4×C22⋊D5912yes—
609C2×C4#1427375812yes—
610C2×C4C22×C4×D5911yescharacteristic
611C2×C4C2×D4×D5911yescharacteristic
612C2×C4#41×D5911yescharacteristic
613C2×C4C2×#1174568511yescharacteristic
614C2×C4#1427377211yescharacteristic
615C2×C4#1427377411yescharacteristic
616C8—594no—
617C8—1182no—
618C8—2361no—
619C8—2361no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22C2×C4×C22⋊D5911yescharacteristic
628C22D4×C22⋊D5911yescharacteristic
629C22#1683805111yescharacteristic
630C22C2×#1427375811yescharacteristic
631C22#1683820011yescharacteristic
632C22#1683821311yescharacteristic
633C22#1683821711yescharacteristic
634C4—22no—
635C4—22no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—81no—
643C4—4721no—
644C4—2362no—
645C4—2362no—
646C4—2362no—
647C4C4×D4×D5911yescharacteristic
648C4#1683788111yescharacteristic
649C4#1683806111yescharacteristic
650C4#1683810111yescharacteristic
651C2—42no—
652C2#2142711311yescharacteristic
653C2C4×#1427375811yescharacteristic
654C2#2143293511yescharacteristic
655C2#2143572311yescharacteristic
656C2#2143574211yescharacteristic
657C2#2143654511yescharacteristic
658C2#2143662611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2360C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#2360D59
hypocenterC59#23038
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. C2×#16837507C2semidirect
5. #21424956C2semidirect
6. #21424836C2semidirect
7. #21424737C2semidirect
8. #21424727C2semidirect
11. #21424363C2semidirect
12. #21424350C2semidirect
13. C4×#14273658C2semidirect
21. #16837658C4semidirect
65. C2×#11745672C2×C4semidirect
93. #14273658C2×C4semidirect
97. #14273651C2×C4semidirect
100. #14273650C2×C4semidirect
391. C59#23038semidirect

Maximal subgroups

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

subgroupindex
C2×#168376582
C2×#168375072
C2×#168374422
#214271212
#214249562
#214248362
#214247372
#214247272
#214244092
#214244032
#214243632
#214243502
C4×#142736582
#214238802
C59×#23602
#2303859