Finite Group Databasebeta

← #28515605

Subgroups — #28515605

Order 15104.

1614 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837328C211yesmaximal, characteristic
2C2×#16837301C211yesmaximal, characteristic, complemented
3C2×#16837276C212yesmaximal
4C2×#16837249C211yesmaximal, characteristic
5#21422360C211yesmaximal, characteristic, complemented
6#21422354C212yesmaximal, complemented
7#21422344C212yesmaximal
8#21422342C211yesmaximal, characteristic
9#21422330C212yesmaximal, complemented
10#21422321C211yesmaximal, characteristic, complemented
11C59×#1445C211yesmaximal, characteristic, complemented
12C23×C4×D59C2211yescharacteristic
13C22×C4⋊Dic59C2212yes—
14C22×C4⋊Dic59C2211yescharacteristic
15C22×Dic59⋊C4C2212yes—
16C22×(C2×D59)⋊C4C2211yescharacteristic
17C22×(C2×D59)⋊C4C2211yescharacteristic
18C22×(C2×D59)⋊C4C2212yes—
19C22×(C2×D59)⋊C4C2212yes—
20C22×C4×Dic59C2211yescharacteristic
21C22×C4×Dic59C2211yescharacteristic
22#16837328C418yescomplemented
23#16837301—24no—
24#16837276—28no—
25#16837249—24no—
26C2×C59⋊#95C2212yes—
27C2×C59⋊#95C2211yescharacteristic
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2212yes—
32C2×C59⋊#95C2212yes—
33C2×C59⋊#95C2211yescharacteristic
34C22×C4⋊C4×C59C2211yescharacteristic
35C2×#95×C59C2211yescharacteristic
36C2×#95×C59C2212yes—
37C2×#95×C59C2212yes—
38C2×#95×C59C2211yescharacteristic
39C24×D59C2311yescharacteristic
40C23×Dic59C2312yes—
41C23×Dic59C2311yescharacteristic
42C23×Dic59C2312yes—
43C23×Dic59C2311yescharacteristic
44C23×Dic59C2311yescharacteristic
45C22×C4×D59—22no—
46C22×C4×D59—22no—
47C22×C4×D59—42no—
48C22×C4×D59—42no—
49C22×C4×D59D414yes—
50C2×C4⋊Dic59—22no—
51C2×C4⋊Dic59—24no—
52C2×C4⋊Dic59—24no—
53C2×C4⋊Dic59—24no—
54C2×C4⋊Dic59—42no—
55C2×Dic59⋊C4—24no—
56C2×Dic59⋊C4—42no—
57C2×Dic59⋊C4—42no—
58C2×(C2×D59)⋊C4—24no—
59C2×(C2×D59)⋊C4—22no—
60C2×(C2×D59)⋊C4—24no—
61C2×(C2×D59)⋊C4—22no—
62C2×(C2×D59)⋊C4—22no—
63C2×(C2×D59)⋊C4—22no—
64C2×(C2×D59)⋊C4—22no—
65C2×(C2×D59)⋊C4—24no—
66C2×(C2×D59)⋊C4—42no—
67C2×(C2×D59)⋊C4—42no—
68C2×(C2×D59)⋊C4C2×C418yes—
69C2×(C2×D59)⋊C4C2×C414yes—
70C2×C4×Dic59—22no—
71C2×C4×Dic59—42no—
72C2×C4×Dic59—42no—
73C2×C4×Dic59D414yes—
74C59⋊#95—22no—
75C59⋊#95—24no—
76C59⋊#95—24no—
77C59⋊#95—42no—
78C59⋊#95—42no—
79C59⋊#95—42no—
80C59⋊#95C2×C418yescomplemented
81C59⋊#95C2×C414yescomplemented
82C23×C4×C59C2312yes—
83C23×C4×C59C2312yes—
84C23×C4×C59C2311yescharacteristic
85C23×C4×C59C2311yescharacteristic
86C23×C4×C59C2311yescharacteristic
87C2×C4⋊C4×C59—22no—
88C2×C4⋊C4×C59—22no—
89C2×C4⋊C4×C59C2×C414yescomplemented
90#95×C59—22no—
91#95×C59—24no—
92#95×C59—24no—
93#95×C59—22no—
94C24×C59C2411yescharacteristic
95C23×D59—21no—
96C23×D59—21no—
97C23×D59—21no—
98C23×D59—21no—
99C23×D59—22no—
100C23×D59—22no—
101C23×D59—22no—
102C23×D59—22no—
103C23×D59C22×C412yes—
104C23×D59C2×D412yes—
105C23×D59#4112yes—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—21no—
109C22×C4×C59—21no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—21no—
116C22×C4×C59—22no—
117C22×C4×C59—21no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59C22×C414yes—
130C22×C4×C59C22×C412yes—
131C22×C4×C59C2×D412yes—
132C22×C4×C59C2×D412yes—
133C22×C4×C59#4112yes—
134C22×C4×C59#4112yes—
135C22×Dic59—22no—
136C22×Dic59—22no—
137C22×Dic59—21no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—22no—
144C22×Dic59—22no—
145C22×Dic59—22no—
146C22×Dic59—22no—
147C22×Dic59—22no—
148C22×Dic59—22no—
149C22×Dic59—22no—
150C22×Dic59—22no—
151C22×Dic59—22no—
152C22×Dic59—22no—
153C22×Dic59—22no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—22no—
158C22×Dic59C22×C412yes—
159C22×Dic59C22×C414yes—
160C22×Dic59C2×D412yes—
161C22×Dic59C2×D412yes—
162C22×Dic59C2×D412yes—
163C22×Dic59#4112yes—
164C2×C4×D59—28no—
165C2×C4×D59—48no—
166C2×C4×D59—44no—
167C2×C4×D59—44no—
168C2×C4×D59—48no—
169C4⋊C4×C59—28no—
170(C2×D59)⋊C4—28no—
171(C2×D59)⋊C4—216no—
172(C2×D59)⋊C4—28no—
173(C2×D59)⋊C4—48no—
174C4⋊Dic59—44no—
175C4⋊Dic59—48no—
176Dic59⋊C4—48no—
177C4×Dic59—44no—
178C4×Dic59—44no—
179C23×C59C23×C411yescharacteristic
180C23×C59C22×D411yescharacteristic
181C23×C59C22×D411yescharacteristic
182C23×C59C2×#4111yescharacteristic
183C23×C59C2×#4111yescharacteristic
184C23×C59C2×#4111yescharacteristic
185C23×C59C2×#4111yescharacteristic
186C23×C59H(5,𝔽2)12yes—
187C23×C59H(5,𝔽2)12yes—
188C23×C592−1+412yes—
189C23×C592−1+412yes—
190C22×D59—22no—
191C22×D59—22no—
192C22×D59—24no—
193C22×D59—24no—
194C22×D59—22no—
195C22×D59—22no—
196C22×D59—22no—
197C22×D59—22no—
198C22×D59—24no—
199C22×D59—24no—
200C22×D59—24no—
201C22×D59—24no—
202C22×D59—41no—
203C22×D59—41no—
204C22×D59—41no—
205C22×D59—41no—
206C22×D59—42no—
207C22×D59—42no—
208C22×D59—42no—
209C22×D59—42no—
210C22×D59—42no—
211C22×D59—42no—
212C22×D59C4×D414yes—
213C2×C4×C59—24no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—24no—
217C2×C4×C59—24no—
218C2×C4×C59—24no—
219C2×C4×C59—24no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—24no—
223C2×C4×C59—24no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—42no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59C4×D414yes—
242C2×Dic59—22no—
243C2×Dic59—24no—
244C2×Dic59—22no—
245C2×Dic59—24no—
246C2×Dic59—24no—
247C2×Dic59—22no—
248C2×Dic59—24no—
249C2×Dic59—24no—
250C2×Dic59—24no—
251C2×Dic59—22no—
252C2×Dic59—22no—
253C2×Dic59—22no—
254C2×Dic59—42no—
255C2×Dic59—41no—
256C2×Dic59—42no—
257C2×Dic59—41no—
258C2×Dic59—42no—
259C2×Dic59—42no—
260C2×Dic59—42no—
261C2×Dic59—42no—
262C2×Dic59—42no—
263C2×Dic59—42no—
264C2×Dic59—42no—
265C2×Dic59—42no—
266C2×Dic59—42no—
267C2×Dic59—42no—
268C2×Dic59—42no—
269C2×Dic59—42no—
270C2×Dic59#12114yes—
271C4×D59—416no—
272#14313—591nomaximal, Sylow
273C22×C59C2×C4×D411yescharacteristic
274C22×C59C2×C4×D411yescharacteristic
275C22×C59C4×#4111yescharacteristic
276C22×C59#52012yes—
277C22×C59#52012yes—
278C22×C59C2×#12111yescharacteristic
279C22×C59#52511yescharacteristic
280C22×C59#52911yescharacteristic
281C22×C59#53211yescharacteristic
282C22×C59#53411yescharacteristic
283C22×C59#53412yes—
284C22×C59#53511yescharacteristic
285C22×C59#53611yescharacteristic
286C22×C59#53612yes—
287C22×C59#53611yescharacteristic
288C22×C59#53812yes—
289C22×C59#53912yes—
290C22×C59#53912yes—
291C22×C59#53912yes—
292C22×C59#53912yes—
293C22×C59#54112yes—
294C22×C59#54212yes—
295C22×C59#54212yes—
296C2×D59—28no—
297C2×D59—24no—
298C2×D59—42no—
299C2×D59—42no—
300C2×D59—44no—
301C2×D59—42no—
302C2×D59—42no—
303C2×D59—42no—
304C2×D59—42no—
305C2×D59—44no—
306C2×D59—44no—
307C4×C59—24no—
308C4×C59—42no—
309C4×C59—42no—
310C4×C59—44no—
311C4×C59—82no—
312Dic59—24no—
313Dic59—44no—
314Dic59—42no—
315Dic59—42no—
316Dic59—82no—
317C2×#397—591no—
318C2×#389—591no—
319C2×#385—592no—
320C2×C4×C22⋊C4—591no—
321#1445D5911yescharacteristic, complemented
322#1444—591no—
323#1442—592no—
324#1440—592no—
325#1439—591no—
326#1436—592no—
327#1434—591no—
328C2×C59C4×#12111yescharacteristic
329C2×C59#232211yescharacteristic
330C2×C59#232212yes—
331C2×C59#232611yescharacteristic
332C2×C59#233412yes—
333C2×C59#261711yescharacteristic
334C2×C59#262111yescharacteristic
335C2×C59#263311yescharacteristic
336C2×C59#263811yescharacteristic
337C2×C59#265212yes—
338C2×C59#266312yes—
339D59—44no—
340C24×C4—591no—
341C22×C4⋊C4—592no—
342C22×C4⋊C4—591no—
343C22×C4⋊C4—592no—
344C22×C4⋊C4C2×D5911yescharacteristic
345C22×C22⋊C4—591no—
346C22×C22⋊C4—592no—
347C22×C22⋊C4—591no—
348C22×C22⋊C4—592no—
349C22×C42—591no—
350C22×C42—591no—
351#397—598no—
352#389—1184no—
353#385—1188no—
354C4×C22⋊C4—1184no—
355C2×#95—591no—
356C2×#95—592no—
357C2×#95—592no—
358C2×#95—592no—
359C2×#95—591no—
360C2×#95—592no—
361C2×#95—592no—
362C2×#95—592no—
363C2×#95C2×D5911yescharacteristic
364C2×#95C2×D5912yes—
365C2×#95C2×D5912yes—
366C2×#95C2×D5911yescharacteristic
367C59#1431311yescharacteristic, complemented, Sylow
368C25—591no—
369C23×C4—591no—
370C23×C4—592no—
371C23×C4—591no—
372C23×C4—591no—
373C23×C4—594no—
374C23×C4—592no—
375C23×C4—1182no—
376C23×C4—1182no—
377C23×C4—2362no—
378C23×C4—2362no—
379C23×C4C22×D5912yes—
380C23×C4C22×D5912yes—
381C23×C4C22×D5911yescharacteristic
382C23×C4C22×D5911yescharacteristic
383C23×C4C22×D5911yescharacteristic
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—1184no—
387C2×C4⋊C4—1184no—
388C2×C4⋊C4—1184no—
389C2×C4⋊C4—1184no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—2362no—
392C2×C4⋊C4—2362no—
393C2×C4⋊C4—2362no—
394C2×C4⋊C4C4×D5914yescomplemented
395C2×C22⋊C4—598no—
396C2×C22⋊C4—594no—
397C2×C22⋊C4—1182no—
398C2×C22⋊C4—1184no—
399C2×C22⋊C4—1184no—
400C2×C22⋊C4—1182no—
401C2×C22⋊C4—1182no—
402C2×C22⋊C4—1182no—
403C2×C22⋊C4—1182no—
404C2×C22⋊C4—1184no—
405C2×C22⋊C4—2362no—
406C2×C22⋊C4—2362no—
407C2×C42—594no—
408C2×C42—1182no—
409C2×C42—2362no—
410C2×C42—2362no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—22no—
415#95—598no—
416#95—594no—
417#95—1182no—
418#95—1184no—
419#95—1184no—
420#95—2362no—
421#95—2362no—
422#95—2362no—
423C24—592no—
424C24—592no—
425C24—592no—
426C24—1181no—
427C24—1181no—
428C24—1181no—
429C24—1181no—
430C24—1182no—
431C24—1182no—
432C24—1182no—
433C24—1182no—
434C24C23×D5911yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—592no—
459C22×C4—592no—
460C22×C4—592no—
461C22×C4—594no—
462C22×C4—592no—
463C22×C4—592no—
464C22×C4—1188no—
465C22×C4—1182no—
466C22×C4—1182no—
467C22×C4—1181no—
468C22×C4—1182no—
469C22×C4—1181no—
470C22×C4—1182no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—1182no—
474C22×C4—1182no—
475C22×C4—1182no—
476C22×C4—1182no—
477C22×C4—1182no—
478C22×C4—1182no—
479C22×C4—1182no—
480C22×C4—1182no—
481C22×C4—1182no—
482C22×C4—1182no—
483C22×C4—1182no—
484C22×C4—1182no—
485C22×C4—1182no—
486C22×C4—1182no—
487C22×C4—1182no—
488C22×C4—2368no—
489C22×C4—2364no—
490C22×C4—2364no—
491C22×C4—2368no—
492C22×C4C2×C4×D5914yes—
493C22×C4C2×C4×D5912yes—
494C22×C4D4×D5912yes—
495C22×C4D4×D5912yes—
496C22×C4#1174568712yes—
497C22×C4#1174568712yes—
498C4⋊C4—28no—
499C4⋊C4—2368no—
500C4⋊C4—2364no—
501C4⋊C4—2368no—
502C22⋊C4—11816no—
503C22⋊C4—1188no—
504C22⋊C4—1188no—
505C22⋊C4—2368no—
506C42—2364no—
507C42—2364no—
508C23—594no—
509C23—1182no—
510C23—1182no—
511C23—1182no—
512C23—1182no—
513C23—1184no—
514C23—1184no—
515C23—1184no—
516C23—1184no—
517C23—1182no—
518C23—1182no—
519C23—1184no—
520C23—1184no—
521C23—2361no—
522C23—2361no—
523C23—2361no—
524C23—2361no—
525C23—2362no—
526C23—2362no—
527C23—2362no—
528C23—2362no—
529C23—2362no—
530C23—2362no—
531C23C22×C4×D5911yescharacteristic
532C23C2×D4×D5911yescharacteristic
533C23C2×D4×D5911yescharacteristic
534C23#41×D5911yescharacteristic
535C23#41×D5911yescharacteristic
536C23C2×#1174568711yescharacteristic
537C23C2×#1174568711yescharacteristic
538C23#1427377312yes—
539C23#1427377312yes—
540C23#1427377412yes—
541C23#1427377512yes—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—594no—
571C2×C4—1182no—
572C2×C4—1184no—
573C2×C4—1182no—
574C2×C4—1184no—
575C2×C4—1182no—
576C2×C4—1182no—
577C2×C4—1182no—
578C2×C4—1184no—
579C2×C4—1182no—
580C2×C4—1184no—
581C2×C4—1184no—
582C2×C4—1184no—
583C2×C4—23616no—
584C2×C4—2362no—
585C2×C4—2362no—
586C2×C4—2362no—
587C2×C4—2362no—
588C2×C4—2362no—
589C2×C4—2362no—
590C2×C4—2362no—
591C2×C4—2362no—
592C2×C4—2361no—
593C2×C4—2362no—
594C2×C4—2361no—
595C2×C4—2362no—
596C2×C4—2362no—
597C2×C4—2362no—
598C2×C4—2362no—
599C2×C4—2362no—
600C2×C4(C22⋊D59)⋊C414yes—
601C22—1188no—
602C22—1184no—
603C22—2362no—
604C22—2362no—
605C22—2362no—
606C22—2362no—
607C22—2362no—
608C22—2364no—
609C22—2364no—
610C22—2362no—
611C22—2364no—
612C22C4×D4×D5911yescharacteristic
613C22C2×(C22⋊D59)⋊C411yescharacteristic
614C22C4×#1174568711yescharacteristic
615C22#1683788212yes—
616C22#1683788412yes—
617C22#121×D5911yescharacteristic
618C22#1683791211yescharacteristic
619C22#1683793111yescharacteristic
620C22#1683795011yescharacteristic
621C22#1683796311yescharacteristic
622C22#1683796312yes—
623C22#1683797011yescharacteristic
624C22#1683797411yescharacteristic
625C22#1683797512yes—
626C22#1683797711yescharacteristic
627C22#1683799112yes—
628C22#1683799312yes—
629C22#1683799612yes—
630C22#1683799612yes—
631C22#1683800012yes—
632C22#1683801112yes—
633C22#1683802512yes—
634C22#1683802512yes—
635C4—24no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—82no—
640C4—1184no—
641C4—4722no—
642C4—2362no—
643C4—2364no—
644C4—2362no—
645C2—2364no—
646C2#2142661211yescharacteristic
647C2#2142676211yescharacteristic
648C2#2142676912yes—
649C2#2142681011yescharacteristic
650C2#2142687612yes—
651C2#2142983111yescharacteristic
652C2#2142986411yescharacteristic
653C2#2142996911yescharacteristic
654C2#2143002111yescharacteristic
655C2#2143014112yes—
656C2#2143024912yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1445C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenter#1445D59
hypocenterC59#14313
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
2. C2×#16837301C2semidirect
5. #21422360C2semidirect
6. #21422354C2semidirect
9. #21422330C2semidirect
10. #21422321C2semidirect
11. C59×#1445C2semidirect
22. #16837328C4semidirect
80. C59⋊#95C2×C4semidirect
81. C59⋊#95C2×C4semidirect
89. C2×C4⋊C4×C59C2×C4semidirect
321. #1445D59semidirect
367. C59#14313semidirect
394. C2×C4⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168373282
C2×#168373012
C2×#168372762
C2×#168372492
#214223602
#214223542
#214223442
#214223422
#214223302
#214223212
C59×#14452
#1431359