Finite Group Databasebeta

← #28524966

Subgroups — #28524966

Order 15104.

1378 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
1C2×#16837293C211yesmaximal, characteristic
2#21422359C211yesmaximal, characteristic, complemented
3#21422355C211yesmaximal, characteristic, complemented
4#21422354C212yesmaximal, complemented
5#21422340C212yesmaximal
6#21422338C211yesmaximal, characteristic
7#21422338C211yesmaximal, characteristic
8#21422334C212yesmaximal
9#21422329C212yesmaximal, complemented
10#21422323C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C59C211yesmaximal, characteristic, complemented
12C22×Dic59⋊C4C2212yes—
13C22×Dic59⋊C4C2211yescharacteristic
14C22×(C2×D59)⋊C4C2211yescharacteristic
15C22×(C2×D59)⋊C4C2212yes—
16C22×(C2×D59)⋊C4C2211yescharacteristic
17C22×(C2×D59)⋊C4C2212yes—
18C22×(C2×D59)⋊C4C2211yescharacteristic
19#16837293—24no—
20C2×C59⋊#95C2212yes—
21C2×C59⋊#95C2211yescharacteristic
22C2×C59⋊#95C2211yescharacteristic
23C2×C59⋊#95C2212yes—
24C2×C59⋊#95C2212yes—
25C2×C59⋊#95C2211yescharacteristic
26C2×C59⋊#95C2211yescharacteristic
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2212yes—
31C22×C4⋊C4×C59C2211yescharacteristic
32C22×C4⋊C4×C59C2211yescharacteristic
33C22×C42×C59C2211yescharacteristic
34C22×C42×C59C2212yes—
35C4×C4⋊C4×C59—24no—
36C2×#95×C59C2212yes—
37C24×D59C2311yescharacteristic
38C23×Dic59C2312yes—
39C23×Dic59C2311yescharacteristic
40C23×Dic59C2311yescharacteristic
41C23×Dic59C2311yescharacteristic
42C23×Dic59C2312yes—
43C2×Dic59⋊C4—22no—
44C2×Dic59⋊C4—24no—
45C2×Dic59⋊C4—42no—
46C2×Dic59⋊C4—42no—
47C2×Dic59⋊C4—42no—
48C2×(C2×D59)⋊C4—22no—
49C2×(C2×D59)⋊C4—22no—
50C2×(C2×D59)⋊C4—22no—
51C2×(C2×D59)⋊C4—22no—
52C2×(C2×D59)⋊C4—24no—
53C2×(C2×D59)⋊C4—24no—
54C2×(C2×D59)⋊C4—22no—
55C2×(C2×D59)⋊C4—22no—
56C2×(C2×D59)⋊C4—24no—
57C2×(C2×D59)⋊C4—42no—
58C2×(C2×D59)⋊C4—42no—
59C2×(C2×D59)⋊C4—42no—
60C2×(C2×D59)⋊C4—42no—
61C2×(C2×D59)⋊C4Q814yes—
62C59⋊#95—22no—
63C59⋊#95—22no—
64C59⋊#95—22no—
65C59⋊#95—24no—
66C59⋊#95—22no—
67C59⋊#95—24no—
68C59⋊#95—24no—
69C59⋊#95—24no—
70C59⋊#95—42no—
71C59⋊#95—42no—
72C59⋊#95—42no—
73C23×C4×C59C2311yescharacteristic
74C23×C4×C59C2312yes—
75C23×C4×C59C2312yes—
76C23×C4×C59C2311yescharacteristic
77C23×C4×C59C2311yescharacteristic
78C2×C4⋊C4×C59—22no—
79C2×C4⋊C4×C59—22no—
80C2×C4⋊C4×C59—24no—
81C2×C4⋊C4×C59—24no—
82C2×C42×C59—22no—
83C2×C42×C59—22no—
84C2×C42×C59—24no—
85C2×C42×C59—42no—
86C2×C42×C59—42no—
87C2×C42×C59D414yes—
88#95×C59—24no—
89C24×C59C2411yescharacteristic
90C23×D59—21no—
91C23×D59—21no—
92C23×D59—22no—
93C23×D59—22no—
94C23×D59—21no—
95C23×D59—21no—
96C23×D59—22no—
97C23×D59—22no—
98C23×D59—21no—
99C23×D59—21no—
100C23×D59C2×Q812yes—
101C22×C4×C59—21no—
102C22×C4×C59—21no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—21no—
106C22×C4×C59—21no—
107C22×C4×C59—22no—
108C22×C4×C59—22no—
109C22×C4×C59—21no—
110C22×C4×C59—21no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59C2×D412yes—
124C22×C4×C59C2×D412yes—
125C22×C4×C59C2×D412yes—
126C22×C4×C59C2×Q812yes—
127C22×C4×C59#4114yes—
128C22×C4×C59#4114yes—
129C22×C4×C59#4112yes—
130C22×C4×C59#4112yes—
131C22×C4×C59#4112yes—
132C22×Dic59—21no—
133C22×Dic59—22no—
134C22×Dic59—22no—
135C22×Dic59—22no—
136C22×Dic59—21no—
137C22×Dic59—22no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—22no—
141C22×Dic59—22no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—22no—
145C22×Dic59—22no—
146C22×Dic59—22no—
147C22×Dic59—22no—
148C22×Dic59—22no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—22no—
152C22×Dic59—22no—
153C22×Dic59—21no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—22no—
158C22×Dic59—21no—
159C22×Dic59C2×Q812yes—
160C22×Dic59#4114yes—
161C4⋊C4×C59—28no—
162C4⋊C4×C59—28no—
163C42×C59—28no—
164C42×C59—48no—
165(C2×D59)⋊C4—28no—
166(C2×D59)⋊C4—48no—
167(C2×D59)⋊C4—44no—
168(C2×D59)⋊C4—48no—
169(C2×D59)⋊C4—44no—
170Dic59⋊C4—44no—
171Dic59⋊C4—48no—
172C23×C59C22×D411yescharacteristic
173C23×C59C22×Q811yescharacteristic
174C23×C59C2×#4112yes—
175C23×C59C2×#4111yescharacteristic
176C23×C59C2×#4111yescharacteristic
177C23×C59C2×#4112yes—
178C23×C59C2×#4112yes—
179C23×C59C2×#4111yescharacteristic
180C23×C59H(5,𝔽2)11yescharacteristic
181C23×C59H(5,𝔽2)11yescharacteristic
182C23×C59H(5,𝔽2)12yes—
183C22×D59—22no—
184C22×D59—24no—
185C22×D59—22no—
186C22×D59—22no—
187C22×D59—24no—
188C22×D59—41no—
189C22×D59—41no—
190C22×D59—41no—
191C22×D59—42no—
192C22×D59—42no—
193C22×D59—42no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—42no—
198C22×D59—42no—
199C22×D59—42no—
200C22×D59—42no—
201C22×D59—41no—
202C22×D59—41no—
203C22×D59—42no—
204C22×D59—42no—
205C22×D59—42no—
206C2×C4×C59—22no—
207C2×C4×C59—22no—
208C2×C4×C59—22no—
209C2×C4×C59—24no—
210C2×C4×C59—24no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—24no—
218C2×C4×C59—24no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—24no—
222C2×C4×C59—24no—
223C2×C4×C59—24no—
224C2×C4×C59—24no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59C22⋊Q814yes—
236C2×C4×C59#12314yes—
237C2×Dic59—22no—
238C2×Dic59—24no—
239C2×Dic59—24no—
240C2×Dic59—24no—
241C2×Dic59—24no—
242C2×Dic59—41no—
243C2×Dic59—42no—
244C2×Dic59—41no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248C2×Dic59—42no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—42no—
252C2×Dic59—42no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—42no—
258C2×Dic59—42no—
259C2×Dic59—42no—
260C2×Dic59—42no—
261C2×Dic59—42no—
262C2×Dic59—42no—
263C2×Dic59—42no—
264C2×Dic59—42no—
265C2×Dic59—42no—
266#15100—591nomaximal, Sylow
267C22×C59C2×C22⋊Q811yescharacteristic
268C22×C59C2×#12311yescharacteristic
269C22×C59#52511yescharacteristic
270C22×C59#52511yescharacteristic
271C22×C59#52912yes—
272C22×C59#52912yes—
273C22×C59#52912yes—
274C22×C59#53211yescharacteristic
275C22×C59#53312yes—
276C22×C59#53511yescharacteristic
277C22×C59#53511yescharacteristic
278C22×C59#53812yes—
279C22×C59#53812yes—
280C22×C59#55112yes—
281C22×C59#55312yes—
282C22×C59#55311yescharacteristic
283C22×C59#55311yescharacteristic
284C22×C59#55312yes—
285C22×C59#55411yescharacteristic
286C22×C59#55411yescharacteristic
287C22×C59#55412yes—
288C22×C59#55512yes—
289C22×C59#55912yes—
290C2×D59—44no—
291C2×D59—42no—
292C2×D59—44no—
293C2×D59—42no—
294C2×D59—42no—
295C2×D59—82no—
296C2×D59—82no—
297C2×D59—81no—
298C2×D59—81no—
299C2×D59—81no—
300C2×D59—81no—
301C4×C59—24no—
302C4×C59—24no—
303C4×C59—44no—
304C4×C59—44no—
305C4×C59—42no—
306Dic59—42no—
307Dic59—44no—
308Dic59—82no—
309Dic59—81no—
310Dic59—81no—
311C2×#388—591no—
312C2×C4×C4⋊C4D5911yescharacteristic, complemented
313#1444—591no—
314#1442—592no—
315#1442—591no—
316#1439—592no—
317#1439—591no—
318#1439—591no—
319#1437—592no—
320#1436—592no—
321#1434—591no—
322C2×C59#239611yescharacteristic
323C2×C59#248711yescharacteristic
324C2×C59#250011yescharacteristic
325C2×C59#257412yes—
326C2×C59#258011yescharacteristic
327C2×C59#258712yes—
328C2×C59#259312yes—
329C2×C59#259512yes—
330C2×C59#260311yescharacteristic
331C2×C59#263111yescharacteristic
332C2×C59#285311yescharacteristic
333D59—82no—
334C22×C4⋊C4—592no—
335C22×C4⋊C4—591no—
336C22×C4⋊C4C2×D5911yescharacteristic
337C22×C4⋊C4C2×D5911yescharacteristic
338C22×C22⋊C4—591no—
339C22×C22⋊C4—591no—
340C22×C22⋊C4—592no—
341C22×C22⋊C4—592no—
342C22×C22⋊C4—591no—
343C22×C42C2×D5911yescharacteristic
344C22×C42C2×D5912yes—
345#388—1184no—
346C4×C4⋊C4—24no—
347C2×#95—592no—
348C2×#95—592no—
349C2×#95—591no—
350C2×#95—591no—
351C2×#95—591no—
352C2×#95—592no—
353C2×#95—592no—
354C2×#95—591no—
355C2×#95—592no—
356C2×#95—592no—
357C2×#95—592no—
358C2×#95C2×D5912yes—
359C59#1510011yescharacteristic, complemented, Sylow
360C25—591no—
361C23×C4—591no—
362C23×C4—592no—
363C23×C4—591no—
364C23×C4—591no—
365C23×C4—592no—
366C23×C4C22×D5912yes—
367C23×C4C22×D5912yes—
368C23×C4C22×D5911yescharacteristic
369C23×C4C22×D5911yescharacteristic
370C23×C4C22×D5911yescharacteristic
371C2×C4⋊C4—24no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—1184no—
376C2×C4⋊C4—1182no—
377C2×C4⋊C4—2362no—
378C2×C4⋊C4—2362no—
379C2×C4⋊C4—2362no—
380C2×C22⋊C4—594no—
381C2×C22⋊C4—1182no—
382C2×C22⋊C4—1182no—
383C2×C22⋊C4—1184no—
384C2×C22⋊C4—1184no—
385C2×C22⋊C4—1184no—
386C2×C22⋊C4—1182no—
387C2×C22⋊C4—1182no—
388C2×C22⋊C4—1182no—
389C2×C22⋊C4—1182no—
390C2×C22⋊C4—2362no—
391C2×C22⋊C4—2362no—
392C2×C22⋊C4—2362no—
393C2×C22⋊C4—2362no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—24no—
397C2×C42—42no—
398C2×C42—42no—
399C2×C42C22⋊D5914yes—
400#95—24no—
401#95—1184no—
402#95—1184no—
403#95—1182no—
404#95—1184no—
405#95—1182no—
406#95—1184no—
407#95—1182no—
408#95—1182no—
409#95—2362no—
410#95—2362no—
411#95—2362no—
412C24—592no—
413C24—1181no—
414C24—1182no—
415C24—1182no—
416C24—1181no—
417C24—1181no—
418C24—1182no—
419C24—1181no—
420C24—1181no—
421C24—1181no—
422C24—1182no—
423C24C23×D5911yescharacteristic
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—592no—
447C22×C4—594no—
448C22×C4—1181no—
449C22×C4—1182no—
450C22×C4—1182no—
451C22×C4—1182no—
452C22×C4—1182no—
453C22×C4—1182no—
454C22×C4—1181no—
455C22×C4—1182no—
456C22×C4—1182no—
457C22×C4—1182no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1182no—
461C22×C4—1182no—
462C22×C4—1182no—
463C22×C4—1182no—
464C22×C4—1181no—
465C22×C4—1181no—
466C22×C4—1182no—
467C22×C4—1182no—
468C22×C4—1182no—
469C22×C4—1181no—
470C22×C4—1182no—
471C22×C4—1181no—
472C22×C4—1182no—
473C22×C4—1182no—
474C22×C4—1182no—
475C22×C4#1174568514yes—
476C22×C4#1174568514yes—
477C22×C4#1174568512yes—
478C22×C4#1174568512yes—
479C22×C4Q8×D5912yes—
480C22×C4#1174568912yes—
481C22×C4C2×C22⋊D5912yes—
482C22×C4C2×C22⋊D5912yes—
483C22×C4C2×C22⋊D5912yes—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—2364no—
487C4⋊C4—2368no—
488C22⋊C4—1188no—
489C22⋊C4—2368no—
490C22⋊C4—2364no—
491C22⋊C4—2368no—
492C22⋊C4—2364no—
493C42—28no—
494C42—48no—
495C23—1182no—
496C23—1184no—
497C23—1182no—
498C23—1182no—
499C23—1184no—
500C23—2361no—
501C23—2361no—
502C23—2361no—
503C23—2362no—
504C23—2362no—
505C23—2362no—
506C23—2361no—
507C23—2361no—
508C23—2361no—
509C23—2362no—
510C23—2362no—
511C23—2362no—
512C23—2362no—
513C23—2361no—
514C23—2361no—
515C23—2362no—
516C23—2362no—
517C23—2362no—
518C23C22×C22⋊D5911yescharacteristic
519C23C2×Q8×D5911yescharacteristic
520C23#41×D5912yes—
521C23C2×#1174568511yescharacteristic
522C23C2×#1174568511yescharacteristic
523C23C2×#1174568512yes—
524C23C2×#1174568512yes—
525C23C2×#1174568911yescharacteristic
526C23#1427377211yescharacteristic
527C23#1427377211yescharacteristic
528C23#1427377312yes—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—1182no—
559C2×C4—1184no—
560C2×C4—1184no—
561C2×C4—1184no—
562C2×C4—1184no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2362no—
566C2×C4—2362no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2362no—
572C2×C4—2362no—
573C2×C4—2362no—
574C2×C4—2362no—
575C2×C4—2362no—
576C2×C4—2362no—
577C2×C4—2362no—
578C2×C4—2362no—
579C2×C4—2362no—
580C2×C4—2361no—
581C2×C4—2362no—
582C2×C4—2361no—
583C2×C4—2362no—
584C2×C4—2362no—
585C2×C4—2362no—
586C2×C4—2362no—
587C2×C4#1427370314yes—
588C2×C4#1427371014yes—
589C22—4721no—
590C22—4722no—
591C22—4721no—
592C22—4722no—
593C22—4721no—
594C22—4721no—
595C22—2364no—
596C22—2364no—
597C22—2362no—
598C22—2362no—
599C22—2362no—
600C22C2×#1427370311yescharacteristic
601C22C2×#1427371011yescharacteristic
602C22#1683791311yescharacteristic
603C22#1683791311yescharacteristic
604C22#1683793612yes—
605C22#1683793612yes—
606C22#1683793912yes—
607C22#1683794811yescharacteristic
608C22#1683795612yes—
609C22#1683796611yescharacteristic
610C22#1683796911yescharacteristic
611C22#1683798512yes—
612C22#1683798512yes—
613C22#1683808112yes—
614C22#1683809612yes—
615C22#1683810011yescharacteristic
616C22#1683810011yescharacteristic
617C22#1683810112yes—
618C22#1683810711yescharacteristic
619C22#1683810711yescharacteristic
620C22#1683811212yes—
621C22#1683811512yes—
622C22#1683813712yes—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—4721no—
629C4—4722no—
630C4—4721no—
631C4—2364no—
632C4—2362no—
633C2—4722no—
634C2#2142736611yescharacteristic
635C2#2142840711yescharacteristic
636C2#2142855411yescharacteristic
637C2#2142936412yes—
638C2#2142945411yescharacteristic
639C2#2142951712yes—
640C2#2142958712yes—
641C2#2142961712yes—
642C2#2142971711yescharacteristic
643C2#2142994811yescharacteristic
644C2#2143241811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D59
commutator, socle, cosocleC24×C59C24
FittingC2×C4×C4⋊C4×C59C2
prime radicalC24×D59C23
hypercenterC2×C4×C4⋊C4D59
hypocenterC59#15100
LCSG ▷ C24×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21422359C2semidirect
3. #21422355C2semidirect
4. #21422354C2semidirect
9. #21422329C2semidirect
10. #21422323C2semidirect
11. C2×C4×C4⋊C4×C59C2semidirect
312. C2×C4×C4⋊C4D59semidirect
359. C59#15100semidirect

Maximal subgroups

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

subgroupindex
C2×#168372932
#214223592
#214223552
#214223542
#214223402
#214223382
#214223342
#214223292
#214223232
C2×C4×C4⋊C4×C592
#1510059