Finite Group Databasebeta

← #28515910

Subgroups — #28515910

Order 15104.

1782 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×#16837261C211yesmaximal, characteristic
2C2×#16837259C211yesmaximal, characteristic
3C2×#95×D59C211yesmaximal, characteristic
4#21422355C211yesmaximal, characteristic, complemented
5#21422354C212yesmaximal, complemented
6#21422353C211yesmaximal, characteristic, complemented
7#21422352C211yesmaximal, characteristic, complemented
8#21422351C212yesmaximal, complemented
9(C2×D59)⋊#95C212yesmaximal
10(C2×D59)⋊#95C212yesmaximal
11C59×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D59C2212yes—
13C23×C4×D59C2211yescharacteristic
14C22×C4⋊Dic59C2211yescharacteristic
15C22×Dic59⋊C4C2211yescharacteristic
16C22×Dic59⋊C4C2211yescharacteristic
17C22×(C2×D59)⋊C4C2212yes—
18C22×(C2×D59)⋊C4C2211yescharacteristic
19C22×(C2×D59)⋊C4C2211yescharacteristic
20#16837261—24no—
21#16837259—24no—
22C2×C59⋊#95C2212yes—
23C2×C59⋊#95C2211yescharacteristic
24C2×C59⋊#95C2212yes—
25C2×C59⋊#95C2212yes—
26C2×C59⋊#95C2212yes—
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2211yescharacteristic
32#95×D59C418yescomplemented
33C22×C4⋊C4×C59C2211yescharacteristic
34C2×#95×C59C2211yescharacteristic
35C2×#95×C59C2212yes—
36C2×#95×C59C2212yes—
37C2×#95×C59C2211yescharacteristic
38C24×D59C2311yescharacteristic
39C23×Dic59C2311yescharacteristic
40C23×Dic59C2311yescharacteristic
41C23×Dic59C2312yes—
42C23×Dic59C2312yes—
43C23×Dic59C2311yescharacteristic
44C22×C4×D59—24no—
45C22×C4×D59—24no—
46C22×C4×D59—42no—
47C22×C4×D59—42no—
48C22×C4×D59—42no—
49C22×C4×D59—42no—
50C22×C4×D59—42no—
51C22×C4×D59—42no—
52C22×C4×D59C2×C418yes—
53C22×C4×D59C2×C414yes—
54C22×C4×D59D414yes—
55C22×C4×D59Q814yes—
56C2×C4⋊Dic59—22no—
57C2×C4⋊Dic59—42no—
58C2×Dic59⋊C4—22no—
59C2×Dic59⋊C4—22no—
60C2×Dic59⋊C4—42no—
61C2×Dic59⋊C4—42no—
62C2×(C2×D59)⋊C4—24no—
63C2×(C2×D59)⋊C4—22no—
64C2×(C2×D59)⋊C4—22no—
65C2×(C2×D59)⋊C4—24no—
66C2×(C2×D59)⋊C4—24no—
67C2×(C2×D59)⋊C4—24no—
68C2×(C2×D59)⋊C4—24no—
69C59⋊#95—22no—
70C59⋊#95—24no—
71C59⋊#95—42no—
72C59⋊#95—42no—
73C59⋊#95—42no—
74C59⋊#95—42no—
75C59⋊#95—42no—
76C59⋊#95—42no—
77C59⋊#95C2×C414yes—
78C59⋊#95C2×C418yes—
79C23×C4×C59C2311yescharacteristic
80C23×C4×C59C2311yescharacteristic
81C23×C4×C59C2312yes—
82C23×C4×C59C2312yes—
83C23×C4×C59C2311yescharacteristic
84C2×C4⋊C4×C59—22no—
85C2×C4⋊C4×C59—42no—
86#95×C59—22no—
87#95×C59—42no—
88#95×C59—42no—
89#95×C59C2×C414yes—
90C24×C59C2411yescharacteristic
91C23×D59—22no—
92C23×D59—22no—
93C23×D59—21no—
94C23×D59—21no—
95C23×D59—21no—
96C23×D59—21no—
97C23×D59C22×C412yes—
98C23×D59C2×D412yes—
99C23×D59C2×Q812yes—
100C23×D59#4114yes—
101C23×D59#4114yes—
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—22no—
109C22×C4×C59—22no—
110C22×C4×C59—21no—
111C22×C4×C59—21no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
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—21no—
123C22×C4×C59—22no—
124C22×C4×C59—21no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59C22×C414yes—
128C22×C4×C59C22×C412yes—
129C22×C4×C59C2×D412yes—
130C22×C4×C59C2×Q812yes—
131C22×Dic59—22no—
132C22×Dic59—22no—
133C22×Dic59—22no—
134C22×Dic59—22no—
135C22×Dic59—22no—
136C22×Dic59—22no—
137C22×Dic59—22no—
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—21no—
147C22×Dic59—21no—
148C22×Dic59—22no—
149C22×Dic59—22no—
150C22×Dic59—22no—
151C22×Dic59—22no—
152C22×Dic59—22no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—22no—
156C22×Dic59C22×C414yes—
157C22×Dic59C22×C412yes—
158C22×Dic59C2×D412yes—
159C22×Dic59C2×Q812yes—
160C2×C4×D59—216no—
161C2×C4×D59—48no—
162C2×C4×D59—48no—
163C2×C4×D59—48no—
164C2×C4×D59—44no—
165C2×C4×D59—44no—
166C2×C4×D59—48no—
167C2×C4×D59—48no—
168C2×C4×D59—48no—
169C2×C4×D59—48no—
170C2×C4×D59—48no—
171C2×C4×D59C4⋊C4116yes—
172C4⋊C4×C59—44no—
173(C2×D59)⋊C4—48no—
174(C2×D59)⋊C4—44no—
175(C2×D59)⋊C4—44no—
176C4⋊Dic59—44no—
177Dic59⋊C4—44no—
178Dic59⋊C4—44no—
179C23×C59C23×C411yescharacteristic
180C23×C59C22×D411yescharacteristic
181C23×C59C22×Q811yescharacteristic
182C23×C59C2×#4112yes—
183C23×C59C2×#4112yes—
184C23×C59H(5,𝔽2)12yes—
185C23×C59H(5,𝔽2)11yescharacteristic
186C23×C59H(5,𝔽2)12yes—
187C23×C59H(5,𝔽2)11yescharacteristic
188C23×C592−1+411yescharacteristic
189C23×C592−1+411yescharacteristic
190C22×D59—22no—
191C22×D59—24no—
192C22×D59—22no—
193C22×D59—22no—
194C22×D59—22no—
195C22×D59—22no—
196C22×D59—22no—
197C22×D59—24no—
198C22×D59—24no—
199C22×D59—24no—
200C22×D59—24no—
201C22×D59—24no—
202C22×D59—24no—
203C22×D59—24no—
204C22×D59—24no—
205C22×D59—24no—
206C22×D59—22no—
207C22×D59—22no—
208C22×D59C2×C4⋊C414yes—
209C22×D59#11718yes—
210C22×D59#12414yes—
211C22×D59#12414yes—
212C22×D59#12518yes—
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—42no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—42no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—42no—
230C2×C4×C59—41no—
231C2×C4×C59—42no—
232C2×C4×C59—41no—
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×C59C2×C4⋊C414yes—
242C2×Dic59—24no—
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—24no—
246C2×Dic59—24no—
247C2×Dic59—24no—
248C2×Dic59—24no—
249C2×Dic59—42no—
250C2×Dic59—42no—
251C2×Dic59—42no—
252C2×Dic59—42no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—42no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—42no—
259C2×Dic59—41no—
260C2×Dic59—41no—
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×C4⋊C414yes—
271C4×D59—416no—
272C4×D59—432no—
273#14347—591nomaximal, Sylow
274C22×C59C22×C4⋊C411yescharacteristic
275C22×C59C2×#11712yes—
276C22×C59#51811yescharacteristic
277C22×C59#51812yes—
278C22×C59#51911yescharacteristic
279C22×C59C2×#12411yescharacteristic
280C22×C59C2×#12411yescharacteristic
281C22×C59C2×#12512yes—
282C22×C59#53411yescharacteristic
283C22×C59#53411yescharacteristic
284C22×C59#53711yescharacteristic
285C22×C59#53711yescharacteristic
286C22×C59#53812yes—
287C22×C59#53812yes—
288C22×C59#53812yes—
289C22×C59#53812yes—
290C22×C59#53912yes—
291C22×C59#53912yes—
292C22×C59#53912yes—
293C22×C59#53912yes—
294C22×C59#54312yes—
295C22×C59#54411yescharacteristic
296C22×C59#54411yescharacteristic
297C2×D59—28no—
298C2×D59—28no—
299C2×D59—24no—
300C2×D59—24no—
301C2×D59—24no—
302C2×D59—24no—
303C2×D59—24no—
304C2×D59—24no—
305C2×D59—28no—
306C2×D59—28no—
307C2×D59#38218yes—
308C4×C59—24no—
309C4×C59—44no—
310C4×C59—81no—
311C4×C59—81no—
312C4×C59—82no—
313Dic59—24no—
314Dic59—44no—
315Dic59—81no—
316Dic59—81no—
317Dic59—82no—
318C2×#380—591no—
319C2×#380—591no—
320C22×#95—591no—
321#1442—592no—
322#1442—591no—
323#1442—591no—
324#1442—592no—
325#1442—591no—
326#1442D5911yescharacteristic, complemented
327C22⋊#95—592no—
328C22⋊#95—592no—
329C2×C59C2×#38211yescharacteristic
330C2×C59#231212yes—
331C2×C59#231312yes—
332C2×C59#231511yescharacteristic
333C2×C59#231611yescharacteristic
334C2×C59#263911yescharacteristic
335C2×C59#263911yescharacteristic
336C2×C59#263911yescharacteristic
337C2×C59#263911yescharacteristic
338C2×C59#264612yes—
339C2×C59#264612yes—
340D59—28no—
341C24×C4—591no—
342C24×C4—592no—
343C22×C4⋊C4—591no—
344C22×C4⋊C4—591no—
345C22×C4⋊C4—591no—
346C22×C4⋊C4C2×D5911yescharacteristic
347C22×C22⋊C4—591no—
348C22×C22⋊C4—591no—
349C22×C22⋊C4—592no—
350#380—1184no—
351#380—1184no—
352C2×#95—592no—
353C2×#95—591no—
354C2×#95—598no—
355C2×#95—592no—
356C2×#95—591no—
357C2×#95—592no—
358C2×#95—592no—
359C2×#95—592no—
360C2×#95—592no—
361C2×#95—592no—
362C2×#95—592no—
363C2×#95C2×D5911yescharacteristic
364C2×#95C2×D5912yes—
365C2×#95C2×D5912yes—
366C2×#95C2×D5911yescharacteristic
367C59#1434711yescharacteristic, complemented, Sylow
368C25—591no—
369C23×C4—591no—
370C23×C4—594no—
371C23×C4—594no—
372C23×C4—594no—
373C23×C4—592no—
374C23×C4—598no—
375C23×C4—591no—
376C23×C4—591no—
377C23×C4—592no—
378C23×C4—1184no—
379C23×C4—1184no—
380C23×C4—2362no—
381C23×C4—2362no—
382C23×C4—2362no—
383C23×C4—2362no—
384C23×C4—2362no—
385C23×C4—2362no—
386C23×C4C22×D5911yescharacteristic
387C23×C4C22×D5911yescharacteristic
388C23×C4C22×D5912yes—
389C23×C4C22×D5912yes—
390C23×C4C22×D5911yescharacteristic
391C2×C4⋊C4—22no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4—1182no—
394C2×C4⋊C4—1182no—
395C2×C4⋊C4—1182no—
396C2×C4⋊C4—2362no—
397C2×C4⋊C4—2362no—
398C2×C4⋊C4—2362no—
399C2×C22⋊C4—1182no—
400C2×C22⋊C4—1182no—
401C2×C22⋊C4—1184no—
402C2×C22⋊C4—1184no—
403C2×C22⋊C4—1184no—
404C2×C22⋊C4—1184no—
405C2×C22⋊C4—1184no—
406#95—22no—
407#95—42no—
408#95—42no—
409#95—598no—
410#95—594no—
411#95—1184no—
412#95—1182no—
413#95—2362no—
414#95—2362no—
415#95—2362no—
416#95—2362no—
417#95—2362no—
418#95—2362no—
419#95C4×D5914yes—
420C24—592no—
421C24—592no—
422C24—592no—
423C24—594no—
424C24—594no—
425C24—1182no—
426C24—1181no—
427C24—1182no—
428C24—1181no—
429C24—1181no—
430C24—1181no—
431C24C23×D5911yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—592no—
458C22×C4—592no—
459C22×C4—592no—
460C22×C4—5916no—
461C22×C4—594no—
462C22×C4—11816no—
463C22×C4—1182no—
464C22×C4—1182no—
465C22×C4—1182no—
466C22×C4—1182no—
467C22×C4—1182no—
468C22×C4—1182no—
469C22×C4—1182no—
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—1181no—
479C22×C4—1182no—
480C22×C4—1181no—
481C22×C4—1182no—
482C22×C4—1182no—
483C22×C4—1182no—
484C22×C4—1182no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4—1182no—
488C22×C4—2364no—
489C22×C4—2364no—
490C22×C4—2368no—
491C22×C4—2368no—
492C22×C4—2368no—
493C22×C4—2368no—
494C22×C4—2368no—
495C22×C4—2368no—
496C22×C4—2368no—
497C22×C4—2368no—
498C22×C4C2×C4×D5914yes—
499C22×C4C2×C4×D5912yes—
500C22×C4D4×D5912yes—
501C22×C4Q8×D5912yes—
502C4⋊C4—44no—
503C4⋊C4—2364no—
504C4⋊C4—2364no—
505C4⋊C4—2364no—
506C22⋊C4—2364no—
507C22⋊C4—2364no—
508C22⋊C4—2368no—
509C23—594no—
510C23—594no—
511C23—598no—
512C23—594no—
513C23—598no—
514C23—1182no—
515C23—1182no—
516C23—1182no—
517C23—1182no—
518C23—1184no—
519C23—1184no—
520C23—1184no—
521C23—1184no—
522C23—1184no—
523C23—1184no—
524C23—1184no—
525C23—1184no—
526C23—1184no—
527C23—1182no—
528C23—1182no—
529C23—1182no—
530C23—1184no—
531C23—1182no—
532C23C22×C4×D5911yescharacteristic
533C23C2×D4×D5911yescharacteristic
534C23C2×Q8×D5911yescharacteristic
535C23#41×D5912yes—
536C23#41×D5912yes—
537C23#1427377212yes—
538C23#1427377211yescharacteristic
539C23#1427377312yes—
540C23#1427377311yescharacteristic
541C23#1427377411yescharacteristic
542C23#1427377511yescharacteristic
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—594no—
572C2×C4—1182no—
573C2×C4—1182no—
574C2×C4—1184no—
575C2×C4—1184no—
576C2×C4—1184no—
577C2×C4—1184no—
578C2×C4—1184no—
579C2×C4—23616no—
580C2×C4—23632no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2362no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2362no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2362no—
590C2×C4—2362no—
591C2×C4—2362no—
592C2×C4—2362no—
593C2×C4—2362no—
594C2×C4—2362no—
595C2×C4—2362no—
596C2×C4—2362no—
597C2×C4—2362no—
598C2×C4—2362no—
599C2×C4—2362no—
600C2×C4—2362no—
601C2×C4—2362no—
602C2×C4C4⋊C4×D5914yes—
603C22—598no—
604C22—1188no—
605C22—1188no—
606C22—1184no—
607C22—1184no—
608C22—1184no—
609C22—1184no—
610C22—1184no—
611C22—1184no—
612C22—1188no—
613C22—1188no—
614C22C2×C4⋊C4×D5911yescharacteristic
615C22#117×D5912yes—
616C22#1683787411yescharacteristic
617C22#1683787412yes—
618C22#1683787811yescharacteristic
619C22#124×D5911yescharacteristic
620C22#124×D5911yescharacteristic
621C22#125×D5912yes—
622C22#1683796211yescharacteristic
623C22#1683796311yescharacteristic
624C22#1683798111yescharacteristic
625C22#1683798211yescharacteristic
626C22#1683798412yes—
627C22#1683798612yes—
628C22#1683798812yes—
629C22#1683799112yes—
630C22#1683799312yes—
631C22#1683799512yes—
632C22#1683799812yes—
633C22#1683800012yes—
634C22#1683802712yes—
635C22#1683802911yescharacteristic
636C22#1683803211yescharacteristic
637C4—24no—
638C4—44no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4—1184no—
643C4—4721no—
644C4—4721no—
645C4—4722no—
646C4—2364no—
647C2—1188no—
648C2#382×D5911yescharacteristic
649C2#2142669012yes—
650C2#2142669512yes—
651C2#2142670711yescharacteristic
652C2#2142671111yescharacteristic
653C2#2143002311yescharacteristic
654C2#2143002511yescharacteristic
655C2#2143002711yescharacteristic
656C2#2143002811yescharacteristic
657C2#2143006512yes—
658C2#2143006712yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1442C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenter#1442D59
hypocenterC59#14347
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
4. #21422355C2semidirect
5. #21422354C2semidirect
6. #21422353C2semidirect
7. #21422352C2semidirect
8. #21422351C2semidirect
11. C59×#1442C2semidirect
32. #95×D59C4semidirect
326. #1442D59semidirect
367. C59#14347semidirect

Maximal subgroups

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

subgroupindex
C2×#168372612
C2×#168372592
C2×#95×D592
#214223552
#214223542
#214223532
#214223522
#214223512
(C2×D59)⋊#952
C59×#14422
#1434759