Finite Group Databasebeta

← #28511345

Subgroups — #28511345

Order 15104.

2006 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273684C211yesmaximal, characteristic
2C2×#16837299C212yesmaximal, complemented
3C2×#16837261C211yesmaximal, characteristic
4C2×#16837249C211yesmaximal, characteristic
5#21422342C212yesmaximal
6#21422340C212yesmaximal
7#21422326C211yesmaximal, characteristic, complemented
8Dic59⋊#95C212yesmaximal, complemented
9Dic59⋊#95C211yesmaximal, characteristic, complemented
10#21422322C211yesmaximal, characteristic, complemented
11C59×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D59C2211yescharacteristic
13C2×#14273684—28no—
14C2×#14273684C418yescomplemented
15C22×C4⋊Dic59C2212yes—
16C22×Dic59⋊C4C2212yes—
17C22×Dic59⋊C4C2212yes—
18C22×Dic59⋊C4C2211yescharacteristic
19C22×Dic59⋊C4C2211yescharacteristic
20C22×(C2×D59)⋊C4C2212yes—
21C22×(C2×D59)⋊C4C2211yescharacteristic
22C22×(C2×D59)⋊C4C2211yescharacteristic
23C22×(C2×D59)⋊C4C2212yes—
24C22×C4×Dic59C2212yes—
25C22×C4×Dic59C2211yescharacteristic
26#16837299—28no—
27#16837261—24no—
28#16837249—24no—
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2211yescharacteristic
32C2×C59⋊#95C2211yescharacteristic
33C2×C59⋊#95C2212yes—
34C2×C59⋊#95C2212yes—
35C22×C42×C59C2211yescharacteristic
36C2×#95×C59C2212yes—
37C2×#95×C59C2212yes—
38C2×#95×C59C2211yescharacteristic
39C2×#95×C59C2211yescharacteristic
40C24×D59C2311yescharacteristic
41C23×Dic59C2312yes—
42C23×Dic59C2311yescharacteristic
43C23×Dic59C2312yes—
44C23×Dic59C2311yescharacteristic
45C23×Dic59C2311yescharacteristic
46C22×C4×D59—24no—
47C22×C4×D59—24no—
48C22×C4×D59C2×C414yes—
49C22×C4×D59D414yes—
50C22×C4×D59Q814yes—
51#14273684—232no—
52C2×C4⋊Dic59—24no—
53C2×C4⋊Dic59—24no—
54C2×C4⋊Dic59—42no—
55C2×Dic59⋊C4—24no—
56C2×Dic59⋊C4—24no—
57C2×Dic59⋊C4—24no—
58C2×Dic59⋊C4—24no—
59C2×Dic59⋊C4—24no—
60C2×Dic59⋊C4—24no—
61C2×Dic59⋊C4—42no—
62C2×Dic59⋊C4C2×C414yes—
63C2×Dic59⋊C4C2×C414yes—
64C2×Dic59⋊C4D418yes—
65C2×(C2×D59)⋊C4—24no—
66C2×(C2×D59)⋊C4—24no—
67C2×(C2×D59)⋊C4—24no—
68C2×(C2×D59)⋊C4—24no—
69C2×(C2×D59)⋊C4—24no—
70C2×(C2×D59)⋊C4—24no—
71C2×(C2×D59)⋊C4—42no—
72C2×(C2×D59)⋊C4—42no—
73C2×(C2×D59)⋊C4C2×C414yes—
74C2×(C2×D59)⋊C4C2×C414yes—
75C2×C4×Dic59—24no—
76C2×C4×Dic59—24no—
77C2×C4×Dic59—24no—
78C2×C4×Dic59—24no—
79C2×C4×Dic59C2×C414yes—
80C59⋊#95—22no—
81C59⋊#95—22no—
82C59⋊#95—24no—
83C59⋊#95—24no—
84C59⋊#95—42no—
85C59⋊#95—42no—
86C23×C4×C59C2312yes—
87C23×C4×C59C2311yescharacteristic
88C23×C4×C59C2312yes—
89C23×C4×C59C2311yescharacteristic
90C23×C4×C59C2311yescharacteristic
91C2×C42×C59—24no—
92C2×C42×C59C2×C414yes—
93#95×C59—22no—
94#95×C59—22no—
95#95×C59—24no—
96#95×C59—24no—
97C24×C59C2411yescharacteristic
98C23×D59—22no—
99C23×D59—22no—
100C23×D59—22no—
101C23×D59—22no—
102C23×D59—21no—
103C23×D59—21no—
104C23×D59—21no—
105C23×D59—21no—
106C23×D59C22×C412yes—
107C23×D59C2×D412yes—
108C23×D59C2×Q812yes—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—21no—
113C22×C4×C59—21no—
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—21no—
121C22×C4×C59—21no—
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×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59C22×C412yes—
132C22×C4×C59C22×C412yes—
133C22×C4×C59C22×C412yes—
134C22×C4×C59C2×D414yes—
135C22×C4×C59C2×D412yes—
136C22×C4×C59C2×Q812yes—
137C22×C4×C59#4114yes—
138C22×Dic59—22no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—22no—
142C22×Dic59—22no—
143C22×Dic59—22no—
144C22×Dic59—22no—
145C22×Dic59—22no—
146C22×Dic59—22no—
147C22×Dic59—21no—
148C22×Dic59—22no—
149C22×Dic59—22no—
150C22×Dic59—21no—
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×C412yes—
160C22×Dic59C22×C412yes—
161C22×Dic59C2×D414yes—
162C22×Dic59C2×D414yes—
163C22×Dic59C2×D412yes—
164C22×Dic59C2×Q812yes—
165C22×Dic59#4114yes—
166C22×Dic59#4114yes—
167C2×C4×D59—28no—
168C2×C4×D59—28no—
169C2×C4×D59—216no—
170C2×C4×D59—216no—
171C2×C4×D59C4⋊C4116yescomplemented
172C42×C59—28no—
173(C2×D59)⋊C4—216no—
174(C2×D59)⋊C4—28no—
175(C2×D59)⋊C4—28no—
176(C2×D59)⋊C4—48no—
177C4⋊Dic59—48no—
178Dic59⋊C4—28no—
179Dic59⋊C4—48no—
180Dic59⋊C4—48no—
181Dic59⋊C4C22⋊C4116yescomplemented
182C4×Dic59—28no—
183C4×Dic59—48no—
184C23×C59C23×C411yescharacteristic
185C23×C59C22×D412yes—
186C23×C59C22×D411yescharacteristic
187C23×C59C22×Q811yescharacteristic
188C23×C59C2×#4112yes—
189C23×C59C2×#4112yes—
190C23×C59C2×#4112yes—
191C23×C59H(5,𝔽2)11yescharacteristic
192C23×C59H(5,𝔽2)11yescharacteristic
193C23×C592−1+411yescharacteristic
194C23×C592−1+411yescharacteristic
195C22×D59—22no—
196C22×D59—22no—
197C22×D59—22no—
198C22×D59—22no—
199C22×D59—22no—
200C22×D59—22no—
201C22×D59—24no—
202C22×D59—24no—
203C22×D59—24no—
204C22×D59—24no—
205C22×D59—22no—
206C22×D59—22no—
207C22×D59—22no—
208C22×D59—22no—
209C22×D59—42no—
210C22×D59—42no—
211C22×D59—42no—
212C22×D59—42no—
213C22×D59—42no—
214C22×D59—42no—
215C22×D59—42no—
216C22×D59—42no—
217C22×D59C2×C4⋊C414yes—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—24no—
223C2×C4×C59—24no—
224C2×C4×C59—24no—
225C2×C4×C59—24no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—24no—
230C2×C4×C59—24no—
231C2×C4×C59—24no—
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×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59C2×C22⋊C414yes—
246C2×C4×C59C2×C4⋊C414yes—
247C2×Dic59—22no—
248C2×Dic59—22no—
249C2×Dic59—24no—
250C2×Dic59—24no—
251C2×Dic59—22no—
252C2×Dic59—22no—
253C2×Dic59—22no—
254C2×Dic59—24no—
255C2×Dic59—24no—
256C2×Dic59—24no—
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—
266C2×Dic59—42no—
267C2×Dic59—42no—
268C2×Dic59—42no—
269C2×Dic59—42no—
270C2×Dic59C2×C22⋊C414yes—
271C2×Dic59C2×C22⋊C414yes—
272C2×Dic59C2×C4⋊C414yes—
273C2×Dic59#11714yes—
274C2×Dic59#12118yes—
275C2×Dic59C22⋊Q818yes—
276C4×D59—232no—
277#13951—591nomaximal, Sylow
278C22×C59C22×C22⋊C411yescharacteristic
279C22×C59C22×C4⋊C411yescharacteristic
280C22×C59C2×#11711yescharacteristic
281C22×C59C4×#4111yescharacteristic
282C22×C59C4×#4111yescharacteristic
283C22×C59#52011yescharacteristic
284C22×C59#52011yescharacteristic
285C22×C59C2×#12112yes—
286C22×C59C2×C22⋊Q812yes—
287C22×C59#52512yes—
288C22×C59#53312yes—
289C22×C59#53511yescharacteristic
290C22×C59#53611yescharacteristic
291C22×C59#54411yescharacteristic
292C22×C59#54411yescharacteristic
293C22×C59#54612yes—
294C22×C59#54712yes—
295C22×C59#54812yes—
296C22×C59#55012yes—
297C22×C59#55212yes—
298C22×C59#55312yes—
299C22×C59#55512yes—
300C22×C59#55612yes—
301C2×D59—24no—
302C2×D59—24no—
303C2×D59—24no—
304C2×D59—42no—
305C2×D59—42no—
306C2×D59—42no—
307C2×D59—42no—
308C2×D59—44no—
309C2×D59—44no—
310C2×D59—44no—
311C2×D59—44no—
312C4×C59—24no—
313C4×C59—24no—
314C4×C59—42no—
315C4×C59—44no—
316C4×C59—82no—
317Dic59—24no—
318Dic59—42no—
319Dic59—44no—
320Dic59—82no—
321Dic59#38018yes—
322C22×#117—591no—
323C2×#389—592no—
324C2×#380—591no—
325C2×C4×C22⋊C4—591no—
326#1439—592no—
327#1439—592no—
328C4⋊#95—591no—
329C4⋊#95—592no—
330C4⋊#95—591no—
331#1434—591no—
332#1434D5911yescharacteristic, complemented
333C2×C59C2×#38011yescharacteristic
334C2×C59#229611yescharacteristic
335C2×C59#231711yescharacteristic
336C2×C59#233911yescharacteristic
337C2×C59#234111yescharacteristic
338C2×C59#234611yescharacteristic
339C2×C59#234711yescharacteristic
340C2×C59#241412yes—
341C2×C59#241512yes—
342C2×C59#243912yes—
343C2×C59#244112yes—
344D59—44no—
345C24×C4—591no—
346C2×#117—598no—
347C2×#117—1188no—
348C22×C4⋊C4—591no—
349C22×C4⋊C4—591no—
350C22×C4⋊C4—592no—
351C22×C4⋊C4—592no—
352C22×C4⋊C4—592no—
353C22×C22⋊C4—591no—
354C22×C22⋊C4—591no—
355C22×C22⋊C4—592no—
356C22×C22⋊C4—592no—
357C22×C42—591no—
358C22×C42—592no—
359C22×C42C2×D5911yescharacteristic
360#389—1188no—
361#380—1184no—
362C4×C22⋊C4—1184no—
363C2×#95—592no—
364C2×#95—591no—
365C2×#95—592no—
366C2×#95—591no—
367C2×#95—592no—
368C2×#95—592no—
369C2×#95C2×D5911yescharacteristic
370C2×#95C2×D5912yes—
371C2×#95C2×D5911yescharacteristic
372C2×#95C2×D5912yes—
373C59#1395111yescharacteristic, complemented, Sylow
374C25—591no—
375C23×C4—591no—
376C23×C4—591no—
377C23×C4—594no—
378C23×C4—594no—
379C23×C4—594no—
380C23×C4—592no—
381C23×C4—591no—
382C23×C4—592no—
383C23×C4—1184no—
384C23×C4—1184no—
385C23×C4C22×D5912yes—
386C23×C4C22×D5911yescharacteristic
387C23×C4C22×D5912yes—
388C23×C4C22×D5911yescharacteristic
389C23×C4C22×D5911yescharacteristic
390#117—11832no—
391C2×C4⋊C4—594no—
392C2×C4⋊C4—598no—
393C2×C4⋊C4—594no—
394C2×C4⋊C4—1184no—
395C2×C4⋊C4—1184no—
396C2×C4⋊C4—1184no—
397C2×C4⋊C4—1184no—
398C2×C4⋊C4—1184no—
399C2×C4⋊C4—1184no—
400C2×C4⋊C4—1184no—
401C2×C4⋊C4—1184no—
402C2×C4⋊C4—2362no—
403C2×C4⋊C4—2362no—
404C2×C22⋊C4—594no—
405C2×C22⋊C4—594no—
406C2×C22⋊C4—1184no—
407C2×C22⋊C4—1184no—
408C2×C22⋊C4—1184no—
409C2×C22⋊C4—1184no—
410C2×C22⋊C4—1184no—
411C2×C22⋊C4—1184no—
412C2×C22⋊C4—2362no—
413C2×C22⋊C4—2362no—
414C2×C42—24no—
415C2×C42—594no—
416C2×C42—1184no—
417C2×C42—1184no—
418C2×C42—1184no—
419C2×C42—1184no—
420C2×C42C4×D5914yes—
421#95—24no—
422#95—22no—
423#95—22no—
424#95—24no—
425#95—1184no—
426#95—1184no—
427#95—1182no—
428#95—1182no—
429#95—2362no—
430#95—2362no—
431C24—592no—
432C24—592no—
433C24—592no—
434C24—1182no—
435C24—1182no—
436C24—1182no—
437C24—1182no—
438C24—1181no—
439C24—1181no—
440C24—1181no—
441C24—1181no—
442C24C23×D5911yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—592no—
466C22×C4—592no—
467C22×C4—594no—
468C22×C4—592no—
469C22×C4—594no—
470C22×C4—592no—
471C22×C4—594no—
472C22×C4—5916no—
473C22×C4—592no—
474C22×C4—594no—
475C22×C4—1188no—
476C22×C4—11816no—
477C22×C4—11816no—
478C22×C4—1188no—
479C22×C4—1182no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—1182no—
483C22×C4—1182no—
484C22×C4—1182no—
485C22×C4—1182no—
486C22×C4—1182no—
487C22×C4—1182no—
488C22×C4—1182no—
489C22×C4—1181no—
490C22×C4—1182no—
491C22×C4—1181no—
492C22×C4—1182no—
493C22×C4—1182no—
494C22×C4—1182no—
495C22×C4—1182no—
496C22×C4—1182no—
497C22×C4—1182no—
498C22×C4—1182no—
499C22×C4C2×C4×D5912yes—
500C22×C4C2×C4×D5912yes—
501C22×C4C2×C4×D5912yes—
502C22×C4#1174568514yes—
503C22×C4D4×D5914yes—
504C22×C4D4×D5912yes—
505C22×C4Q8×D5912yes—
506C4⋊C4—5916no—
507C4⋊C4—1188no—
508C4⋊C4—2368no—
509C4⋊C4—2368no—
510C4⋊C4—2368no—
511C22⋊C4—1188no—
512C22⋊C4—11816no—
513C22⋊C4—1188no—
514C22⋊C4—2368no—
515C42—28no—
516C42—1188no—
517C42—2368no—
518C23—594no—
519C23—1184no—
520C23—1184no—
521C23—1184no—
522C23—1184no—
523C23—1182no—
524C23—1182no—
525C23—1182no—
526C23—1182no—
527C23—1182no—
528C23—1182no—
529C23—1182no—
530C23—1182no—
531C23—1182no—
532C23—1182no—
533C23—2362no—
534C23—2362no—
535C23—2362no—
536C23—2362no—
537C23—2362no—
538C23—2362no—
539C23—2362no—
540C23—2362no—
541C23C22×C4×D5911yescharacteristic
542C23C2×D4×D5911yescharacteristic
543C23C2×D4×D5912yes—
544C23C2×Q8×D5911yescharacteristic
545C23#41×D5912yes—
546C23#41×D5912yes—
547C23C2×#1174568512yes—
548C23#1427377311yescharacteristic
549C23#1427377311yescharacteristic
550C23#1427377411yescharacteristic
551C23#1427377411yescharacteristic
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—598no—
580C2×C4—598no—
581C2×C4—594no—
582C2×C4—594no—
583C2×C4—594no—
584C2×C4—594no—
585C2×C4—11832no—
586C2×C4—1184no—
587C2×C4—1184no—
588C2×C4—1182no—
589C2×C4—1182no—
590C2×C4—1184no—
591C2×C4—1184no—
592C2×C4—1184no—
593C2×C4—1182no—
594C2×C4—1182no—
595C2×C4—1182no—
596C2×C4—2362no—
597C2×C4—2362no—
598C2×C4—2362no—
599C2×C4—2362no—
600C2×C4—2362no—
601C2×C4—2362no—
602C2×C4—2362no—
603C2×C4—2362no—
604C2×C4—2362no—
605C2×C4—2362no—
606C2×C4—2362no—
607C2×C4—2362no—
608C2×C4—2362no—
609C2×C4C22⋊C4×D5914yes—
610C2×C4C4⋊C4×D5914yes—
611C22—1184no—
612C22—1184no—
613C22—1184no—
614C22—2362no—
615C22—2362no—
616C22—2364no—
617C22—2364no—
618C22—2364no—
619C22—2364no—
620C22—2362no—
621C22—2362no—
622C22C2×C22⋊C4×D5911yescharacteristic
623C22C2×C4⋊C4×D5911yescharacteristic
624C22#117×D5911yescharacteristic
625C22#1683787011yescharacteristic
626C22C4×#1174568511yescharacteristic
627C22#1683788211yescharacteristic
628C22#1683788211yescharacteristic
629C22#121×D5912yes—
630C22C22⋊Q8×D5912yes—
631C22#1683791112yes—
632C22#1683795612yes—
633C22#1683797011yescharacteristic
634C22#1683797211yescharacteristic
635C22#1683803211yescharacteristic
636C22#1683803211yescharacteristic
637C22#1683804412yes—
638C22#1683805012yes—
639C22#1683806112yes—
640C22#1683807312yes—
641C22#1683808512yes—
642C22#1683810312yes—
643C22#1683811512yes—
644C22#1683811912yes—
645C4—24no—
646C4—24no—
647C4—44no—
648C4—42no—
649C4—82no—
650C4—598no—
651C4—1184no—
652C4—4722no—
653C4—2364no—
654C4—2362no—
655C2—2364no—
656C2#380×D5911yescharacteristic
657C2#2142657611yescharacteristic
658C2#2142672111yescharacteristic
659C2#2142692011yescharacteristic
660C2#2142693611yescharacteristic
661C2#2142698211yescharacteristic
662C2#2142698411yescharacteristic
663C2#2142751312yes—
664C2#2142752212yes—
665C2#2142778612yes—
666C2#2142780812yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1434C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenter#1434D59
hypocenterC59#13951
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#16837299C2semidirect
7. #21422326C2semidirect
8. Dic59⋊#95C2semidirect
9. Dic59⋊#95C2semidirect
10. #21422322C2semidirect
11. C59×#1434C2semidirect
14. C2×#14273684C4semidirect
171. C2×C4×D59C4⋊C4semidirect
181. Dic59⋊C4C22⋊C4semidirect
332. #1434D59semidirect
373. C59#13951semidirect

Maximal subgroups

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

subgroupindex
C22×#142736842
C2×#168372992
C2×#168372612
C2×#168372492
#214223422
#214223402
#214223262
Dic59⋊#952
#214223222
C59×#14342
#1395159