Finite Group Databasebeta

← #28605751

Subgroups — #28605751

Order 15104.

990 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#103×D59C211yesmaximal, characteristic
2#21426964C211yesmaximal, characteristic
3Q8×(C2×D59)⋊C4C211yesmaximal, characteristic
4#21424787C211yesmaximal, characteristic, complemented
5#21424785C211yesmaximal, characteristic, complemented
6#21424738C211yesmaximal, characteristic, complemented
7#21424736C211yesmaximal, characteristic, complemented
8#21424352C211yesmaximal, characteristic, complemented
9#21424181C211yesmaximal, characteristic
10#21424122C211yesmaximal, characteristic
11#21424029C211yesmaximal, characteristic
12#21423928C211yesmaximal, characteristic, complemented
13#21423926C211yesmaximal, characteristic, complemented
14#21423871C211yesmaximal, characteristic
15C59×#1863C211yesmaximal, characteristic, complemented
16C22×Q8×D59C2211yescharacteristic
17C22×C8×D59C2211yescharacteristic
18C2×Q8×Dic59C2211yescharacteristic
19C2×#14273685—21no—
20C2×C4⋊C4×D59C2211yescharacteristic
21C2×#14273661C2211yescharacteristic
22C2×#14273652C2211yescharacteristic
23C2×#14273652C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28#103×D59C418yescomplemented
29C2×(C2×D59)⋊C8C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×#14273637C2211yescharacteristic
32#16837308—21no—
33#16837308C2211yescharacteristic
34#16837307—21no—
35#16837307C2211yescharacteristic
36#16837301C2211yescharacteristic
37#16837289—21no—
38#16837283—21no—
39#16837264—21no—
40#16837259—21no—
41#16837259C2211yescharacteristic
42#16837259C2211yescharacteristic
43C4⋊C4×Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45C4×(C2×D59)⋊C4—21no—
46C4×(C2×D59)⋊C4C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×C4×Q8×C59C2211yescharacteristic
54C2×#107×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×#103×C59C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118—21no—
61C22×Dic118C2311yescharacteristic
62C2×Q8×D59—22no—
63C2×Q8×D59—22no—
64C2×Q8×D59—22no—
65C2×Q8×D59—41no—
66C2×Q8×D59—41no—
67C2×Q8×D59C2×C414yes—
68C22×C4×D59—21no—
69C22×C4×D59—21no—
70C22×C4×D59C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73C2×C8×D59—41no—
74C2×C8×D59—41no—
75C2×C8×D59C2×C414yes—
76Q8×Dic59—22no—
77Q8×Dic59—41no—
78#14273685—44no—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59—21no—
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×Dic59⋊C4—21no—
85C4⋊C4×D59—22no—
86C4⋊C4×D59C2×C414yes—
87C2×(C2×D59)⋊C4—21no—
88C2×(C2×D59)⋊C4—21no—
89C2×(C2×D59)⋊C4D411yescharacteristic
90C2×(C2×D59)⋊C4D411yescharacteristic
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59C2311yescharacteristic
95#14273661—41no—
96#14273652—41no—
97#14273652C2×C414yes—
98#14273651—22no—
99#14273651C2×C414yes—
100#14273650—41no—
101#14273650C2×C414yes—
102(C2×D59)⋊C8—22no—
103#14273638—41no—
104#14273637—41no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C59⋊#95—21no—
114C59⋊#95D411yescharacteristic
115C59⋊#95D411yescharacteristic
116C22×Q8×C59C2311yescharacteristic
117C22×C8×C59C2311yescharacteristic
118C22×C8×C59C2311yescharacteristic
119C4×Q8×C59—22no—
120C4×Q8×C59—41no—
121C2×C4⋊C4×C59—21no—
122C2×C4⋊C4×C59—21no—
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C4⋊C4×C59C2311yescharacteristic
126C2×C42×C59—21no—
127C2×C42×C59C2311yescharacteristic
128#107×C59—41no—
129#103×C59—22no—
130#103×C59C2×C414yes—
131#95×C59—21no—
132C23×D59C2×D411yescharacteristic
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59C22×C412yes—
137C2×Q8×C59#4112yes—
138C2×Q8×C59#4112yes—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2411yescharacteristic
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153Q8×D59—28no—
154Q8×D59—44no—
155Q8×D59—44no—
156Q8×D59—44no—
157C2×C4×D59—21no—
158C2×C4×D59—21no—
159C2×C4×D59—21no—
160C2×C4×D59—21no—
161C2×C4×D59—24no—
162C2×C4×D59—24no—
163C2×C4×D59—42no—
164C2×C4×D59—42no—
165C2×C4×D59—42no—
166C2×C4×D59—42no—
167C2×C4×D59C22×C412yes—
168C2×C4×D59#4112yes—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118—21no—
174C2×Dic118—24no—
175C2×Dic118—42no—
176C2×Dic118—42no—
177C2×Dic118C22×C412yes—
178C2×C8×C59—21no—
179C2×C8×C59—21no—
180C2×C8×C59—21no—
181C2×C8×C59—21no—
182C2×C8×C59—21no—
183C2×C8×C59C22×C412yes—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—21no—
187C4⋊C4×C59—21no—
188C4⋊C4×C59—21no—
189C4⋊C4×C59—42no—
190C4⋊C4×C59—42no—
191C4⋊C4×C59C22×C412yes—
192C42×C59—21no—
193C42×C59—21no—
194C42×C59—42no—
195(C2×D59)⋊C4—22no—
196(C2×D59)⋊C4—22no—
197(C2×D59)⋊C4—42no—
198(C2×D59)⋊C4—42no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—42no—
205C4⋊Dic59—42no—
206C4⋊Dic59C22×C412yes—
207Dic59⋊C4—24no—
208C4×Dic59—21no—
209C4×Dic59—21no—
210C4×Dic59—42no—
211C4×Dic59—42no—
212C4×Dic59—42no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8C22×C412yes—
219C8×D59—44no—
220C23×C59C22×D411yescharacteristic
221C22×D59—21no—
222C22×D59—21no—
223C22×D59—21no—
224C22×D59—21no—
225C22×D59C4×D412yes—
226C22×D59AGL(1,ℤ/8)12yes—
227C22×D59#13712yes—
228Q8×C59—22no—
229Q8×C59—22no—
230Q8×C59—22no—
231Q8×C59#11714yescomplemented
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59C23×C411yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59H(5,𝔽2)11yescharacteristic
255C2×C4×C59H(5,𝔽2)11yescharacteristic
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—22no—
259C2×Dic59—22no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59C4×D412yes—
277C4×D59—22no—
278C4×D59—22no—
279C4×D59—22no—
280C4×D59—22no—
281C4×D59—48no—
282C4×D59—48no—
283Dic118—22no—
284Dic118—22no—
285Dic118—48no—
286Dic118—41no—
287Dic118—41no—
288C8×C59—22no—
289C8×C59—41no—
290C59⋊C8—22no—
291C59⋊C8—41no—
292#22563—591nomaximal, Sylow
293C22×C59C2×C4×D411yescharacteristic
294C22×C59#54611yescharacteristic
295C22×C59#54611yescharacteristic
296C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C59C2×#13711yescharacteristic
298C22×C59#57811yescharacteristic
299C22×C59#57811yescharacteristic
300C2×D59—22no—
301C2×D59—22no—
302C2×D59—22no—
303C2×D59—22no—
304C2×D59—22no—
305C2×D59—22no—
306C2×D59#44014yes—
307C4×C59—24no—
308C4×C59—42no—
309C4×C59—41no—
310C4×C59—41no—
311C4×C59—81no—
312C4×C59—81no—
313C4×C59C2×#11711yescharacteristic
314C4×C59#51811yescharacteristic
315C4×C59#55111yescharacteristic
316C4×C59#55111yescharacteristic
317Dic59—22no—
318Dic59—42no—
319Dic59—42no—
320Dic59—41no—
321Dic59—41no—
322Dic59—81no—
323Dic59—81no—
324C22×#103—591no—
325#2344—591no—
326Q8×C22⋊C4—591no—
327#1935—591no—
328#1935—591no—
329#1928—591no—
330#1928—591no—
331#1863—591no—
332#1863D5911yescharacteristic, complemented
333#1823—591no—
334#1808—591no—
335#1787—591no—
336#1763—591no—
337#1763—591no—
338#1752—591no—
339C2×C59#234211yescharacteristic
340C2×C59C2×#44011yescharacteristic
341C2×C59#295011yescharacteristic
342C2×C59#330811yescharacteristic
343C2×C59#330811yescharacteristic
344C2×C59#331211yescharacteristic
345C2×C59#331211yescharacteristic
346D59—24no—
347C23×Q8—591no—
348C23×C8—591no—
349C2×C4×Q8—591no—
350C2×C4×Q8C2×D5911yescharacteristic
351C2×#117—1181no—
352C22×C4⋊C4—591no—
353C2×#107—591no—
354C2×#107C2×D5911yescharacteristic
355C2×#103—591no—
356C2×#103—591no—
357C2×#103—591no—
358C2×#103—598no—
359C2×#103—591no—
360C2×#103—591no—
361C2×#103—591no—
362C2×#103C2×D5911yescharacteristic
363C2×#103C2×D5911yescharacteristic
364C2×C22⋊C8—591no—
365C2×#97—591no—
366C2×#97—591no—
367#391—591no—
368#391—591no—
369#391—1181no—
370#391—1181no—
371#389—591no—
372#389C2×D5911yescharacteristic
373#387—1181no—
374#387—1181no—
375#381—1181no—
376#380—591no—
377#380—591no—
378#380—1181no—
379C4×C4⋊C4—591no—
380C4×C4⋊C4—591no—
381C4×C22⋊C4—591no—
382C4×C22⋊C4—1181no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340C2×D5911yescharacteristic
389#336—591no—
390#336C2×D5911yescharacteristic
391C59#2256311yescharacteristic, complemented, Sylow
392C22×Q8—594no—
393C22×Q8—591no—
394C22×Q8—1181no—
395C22×Q8—1182no—
396C22×Q8—1182no—
397C22×Q8—1182no—
398C22×Q8—2361no—
399C22×Q8—2361no—
400C22×Q8C22×D5911yescharacteristic
401C23×C4—591no—
402C23×C4—1181no—
403C23×C4—1181no—
404C22×C8—591no—
405C22×C8—594no—
406C22×C8—591no—
407C22×C8—2361no—
408C22×C8—2361no—
409C22×C8C22×D5911yescharacteristic
410C22×C8C22×D5911yescharacteristic
411C4×Q8—22no—
412C4×Q8—41no—
413C4×Q8—1182no—
414C4×Q8—2361no—
415#117—2364no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—594no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—1182no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4C22×D5911yescharacteristic
427C2×C4⋊C4C22×D5911yescharacteristic
428C2×C4⋊C4C22×D5911yescharacteristic
429C2×C22⋊C4—591no—
430C2×C22⋊C4—591no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4—1181no—
433C2×C42—21no—
434C2×C42—591no—
435C2×C42—1181no—
436C2×C42—1181no—
437C2×C42—1181no—
438C2×C42C22×D5911yescharacteristic
439#107—41no—
440#107—2361no—
441#103—22no—
442#103—594no—
443#103—594no—
444#103—594no—
445#103—1182no—
446#103—2361no—
447#103—2361no—
448#103C4×D5914yes—
449C22⋊C8—1182no—
450#97—2361no—
451#97—2361no—
452#95—21no—
453#95—591no—
454#95—591no—
455#95—1181no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464C24—591no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—592no—
469C2×Q8—1184no—
470C2×Q8—1181no—
471C2×Q8—1188no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—2362no—
477C2×Q8—2362no—
478C2×Q8—2364no—
479C2×Q8—2364no—
480C2×Q8—2364no—
481C2×Q8C2×C4×D5912yes—
482C2×Q8#1174568712yes—
483C2×Q8#1174568912yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—591no—
489C22×C4—592no—
490C22×C4—592no—
491C22×C4—591no—
492C22×C4—591no—
493C22×C4—1184no—
494C22×C4—1184no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—1181no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—2362no—
504C22×C4—2362no—
505C22×C4—2362no—
506C22×C4—2362no—
507C22×C4D4×D5911yescharacteristic
508C22×C4D4×D5911yescharacteristic
509C22×C4C23×D5911yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—592no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—2364no—
522C2×C8C2×C4×D5912yes—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—592no—
531C4⋊C4—1184no—
532C4⋊C4—1181no—
533C4⋊C4—1181no—
534C4⋊C4—1181no—
535C4⋊C4—1181no—
536C4⋊C4—1181no—
537C4⋊C4—2362no—
538C4⋊C4—2362no—
539C4⋊C4C2×C4×D5912yes—
540C22⋊C4—1182no—
541C22⋊C4—1182no—
542C22⋊C4—2362no—
543C22⋊C4—2362no—
544C42—21no—
545C42—21no—
546C42—42no—
547C42—1181no—
548C42—1181no—
549C42—2362no—
550C42—2362no—
551C42—2362no—
552C23—592no—
553C23—592no—
554C23—592no—
555C23—1181no—
556C23—1181no—
557C23—1181no—
558C23—1181no—
559C23C2×D4×D5911yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—1182no—
564Q8—1182no—
565Q8—2368no—
566Q8—2361no—
567Q8—2361no—
568Q8#1427368514yescomplemented
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—592no—
588C2×C4—1181no—
589C2×C4—1181no—
590C2×C4—1182no—
591C2×C4—1182no—
592C2×C4—1182no—
593C2×C4—1182no—
594C2×C4—1181no—
595C2×C4—1181no—
596C2×C4—1181no—
597C2×C4—1181no—
598C2×C4—1181no—
599C2×C4—1181no—
600C2×C4—1182no—
601C2×C4—1182no—
602C2×C4—2368no—
603C2×C4—2368no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4C22×C4×D5911yescharacteristic
615C2×C4#41×D5911yescharacteristic
616C2×C4C2×#1174568711yescharacteristic
617C2×C4C2×#1174568911yescharacteristic
618C2×C4#1427377211yescharacteristic
619C2×C4#1427377311yescharacteristic
620C8—22no—
621C8—41no—
622C8—1182no—
623C8—2361no—
624C22—594no—
625C22—1182no—
626C22—1182no—
627C22—1182no—
628C22—1182no—
629C22—1182no—
630C22—1182no—
631C22C4×D4×D5911yescharacteristic
632C22#1683803911yescharacteristic
633C22#1683804311yescharacteristic
634C22AGL(1,ℤ/8)×D5911yescharacteristic
635C22#137×D5911yescharacteristic
636C22#1683821511yescharacteristic
637C22#1683821611yescharacteristic
638C4—24no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—81no—
643C4—81no—
644C4—1182no—
645C4—4721no—
646C4—4721no—
647C4—2362no—
648C4—2361no—
649C4—2361no—
650C4—2362no—
651C4C2×#1427368511yescharacteristic
652C4#1683787411yescharacteristic
653C4#1683807511yescharacteristic
654C4#1683807911yescharacteristic
655C2—1184no—
656C2#2142694311yescharacteristic
657C2#440×D5911yescharacteristic
658C2#2143295911yescharacteristic
659C2#2143608011yescharacteristic
660C2#2143608411yescharacteristic
661C2#2143614011yescharacteristic
662C2#2143614411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1863C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1863D59
hypocenterC59#22563
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
4. #21424787C2semidirect
5. #21424785C2semidirect
6. #21424738C2semidirect
7. #21424736C2semidirect
8. #21424352C2semidirect
12. #21423928C2semidirect
13. #21423926C2semidirect
15. C59×#1863C2semidirect
28. #103×D59C4semidirect
231. Q8×C59#117semidirect
332. #1863D59semidirect
391. C59#22563semidirect
568. Q8#14273685semidirect

Maximal subgroups

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

subgroupindex
C2×#103×D592
#214269642
Q8×(C2×D59)⋊C42
#214247872
#214247852
#214247382
#214247362
#214243522
#214241812
#214241222
#214240292
#214239282
#214239262
#214238712
C59×#18632
#2256359