Finite Group Databasebeta

← #28598970

Subgroups — #28598970

Order 15104.

1234 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837364C211yesmaximal, characteristic
2C2×#16837259C211yesmaximal, characteristic
3#380×D59C211yesmaximal, characteristic
4#21424291C211yesmaximal, characteristic, complemented
5#21424290C212yesmaximal, complemented
6#21424288C211yesmaximal, characteristic, complemented
7#21424284C211yesmaximal, characteristic, complemented
8#21424282C212yesmaximal, complemented
9#21424130C212yesmaximal, complemented
10#21424128C212yesmaximal, complemented
11C59×#1852C211yesmaximal, characteristic, complemented
12C23×C4×D59C2211yescharacteristic
13C22×#11745659C2212yes—
14C22×C4⋊Dic59C2211yescharacteristic
15C2×C4⋊C4×D59C2211yescharacteristic
16C2×C4⋊C4×D59C2211yescharacteristic
17C22×(C2×D59)⋊C4—21no—
18C2×C22⋊C4×D59—21no—
19C2×#14273660C2212yescomplemented
20C2×#14273659C2212yescomplemented
21C2×#14273658C2212yescomplemented
22#16837364C418yescomplemented
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×(C2×D59)⋊C8C2212yes—
25#16837262C2211yescharacteristic
26#16837261—21no—
27#16837260—21no—
28#16837259—21no—
29#16837259—21no—
30#16837259C2211yescharacteristic
31#16837259C2212yes—
32#16837259C2211yescharacteristic
33#16837259C2212yes—
34C2×C59⋊#95—21no—
35#95×D59—21no—
36#16837165C2212yescomplemented
37#16837164C2212yescomplemented
38#16837164C2212yescomplemented
39C22×C4⋊C4×C59C2211yescharacteristic
40C2×#107×C59C2212yescomplemented
41C2×C22⋊C8×C59C2211yescharacteristic
42C59×#380C2211yescharacteristic
43C59×#340C2212yescomplemented
44C24×D59D411yescharacteristic
45C23×Dic59—21no—
46C23×Dic59D411yescharacteristic
47C22×C4×D59—21no—
48C22×C4×D59—21no—
49C22×C4×D59—42no—
50C22×C4×D59—42no—
51C22×C4×D59C2×C414yes—
52C22×C4×D59D411yescharacteristic
53C22×C4×D59D411yescharacteristic
54C22×C4×D59Q814yes—
55C22×C4×D59C2311yescharacteristic
56C22×C4×D59C2311yescharacteristic
57C2×#11745659—42no—
58C2×#11745659—42no—
59C2×#11745659C2×C418yescomplemented
60C22×C59⋊C8C2312yes—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59—42no—
63C2×C4⋊Dic59C2311yescharacteristic
64C2×C4⋊Dic59C2312yes—
65C2×C4⋊Dic59C2311yescharacteristic
66C2×Dic59⋊C4—21no—
67C2×Dic59⋊C4—21no—
68C4⋊C4×D59—42no—
69C4⋊C4×D59—42no—
70C2×C22⋊Dic59—21no—
71C2×(C2×D59)⋊C4—22no—
72C2×(C2×D59)⋊C4—22no—
73C2×(C2×D59)⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×(C2×D59)⋊C4—44no—
76C22⋊C4×D59—44no—
77#14273660—42no—
78#14273659—42no—
79#14273658—42no—
80C22⋊(C59⋊C8)C2×C414yescomplemented
81(C2×D59)⋊C8C2×C418yescomplemented
82C59⋊#95—21no—
83C59⋊#95—21no—
84C59⋊#95—21no—
85C59⋊#95—22no—
86C59⋊#95—22no—
87C23×C4×C59—21no—
88C23×C4×C59C2311yescharacteristic
89C22×C8×C59C2312yes—
90C2×C4⋊C4×C59—21no—
91C2×C4⋊C4×C59—42no—
92C2×C4⋊C4×C59C2311yescharacteristic
93C2×C4⋊C4×C59C2312yes—
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C22⋊C4×C59—21no—
96#107×C59—42no—
97C22⋊C8×C59C2×C414yescomplemented
98#95×C59—21no—
99C24×C59C2×D411yescharacteristic
100C23×D59—22no—
101C23×D59—22no—
102C23×D59—22no—
103C23×D59—22no—
104C23×D59—41no—
105C23×D59—41no—
106C23×D59C4⋊C414yes—
107C23×D59C2×D411yescharacteristic
108C23×D59C2×D411yescharacteristic
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—21no—
112C22×C4×C59—21no—
113C22×C4×C59—22no—
114C22×C4×C59—42no—
115C22×C4×C59—42no—
116C22×C4×C59—42no—
117C22×C4×C59C22×C412yes—
118C22×C4×C59C2×D411yescharacteristic
119C22×C4×C59C2×Q812yes—
120C22×C4×C59C2411yescharacteristic
121C22×Dic59—22no—
122C22×Dic59—21no—
123C22×Dic59—21no—
124C22×Dic59—41no—
125C22×Dic59—41no—
126C22×Dic59—42no—
127C22×Dic59—42no—
128C22×Dic59—42no—
129C22×Dic59C4⋊C414yes—
130C22×Dic59C2×D411yescharacteristic
131C22×Dic59C2×D411yescharacteristic
132C2×C4×D59—22no—
133C2×C4×D59—22no—
134C2×C4×D59—22no—
135C2×C4×D59—22no—
136C2×C4×D59—48no—
137C2×C4×D59—48no—
138C2×C4×D59—41no—
139C2×C4×D59—41no—
140C2×C4×D59—41no—
141C2×C4×D59—41no—
142C2×C4×D59—42no—
143C2×C4×D59—42no—
144C2×C4×D59—42no—
145C2×C4×D59—42no—
146C2×C4×D59—42no—
147C2×C4×D59—42no—
148C2×C4×D59C4⋊C414yes—
149C2×C4×D59C4⋊C414yes—
150C2×C4×D59C22×C412yes—
151C2×C4×D59C22×C412yes—
152C2×C4×D59C2×Q812yes—
153C2×C4×D59C2×Q812yes—
154C2×C8×C59—22no—
155C2×C8×C59—22no—
156C2×C8×C59C22×C414yes—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—22no—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—42no—
162C22⋊C4×C59—42no—
163C22⋊Dic59—42no—
164(C2×D59)⋊C4—42no—
165(C2×D59)⋊C4—42no—
166(C2×D59)⋊C4—44no—
167(C2×D59)⋊C4—44no—
168C4⋊Dic59—22no—
169C4⋊Dic59—22no—
170C4⋊Dic59—22no—
171C4⋊Dic59—22no—
172C4⋊Dic59—42no—
173Dic59⋊C4—42no—
174Dic59⋊C4—42no—
175C2×C59⋊C8—22no—
176C2×C59⋊C8—22no—
177C2×C59⋊C8C22×C414yes—
178#11745659—48no—
179C23×C59—21no—
180C23×C59—21no—
181C23×C59—21no—
182C23×C59—21no—
183C23×C59C2×C4⋊C412yes—
184C23×C59AGL(1,ℤ/8)12yes—
185C23×C59#13712yes—
186C23×C59C22×D411yescharacteristic
187C22×D59—21no—
188C22×D59—21no—
189C22×D59—21no—
190C22×D59—21no—
191C22×D59—21no—
192C22×D59—21no—
193C22×D59—21no—
194C22×D59—21no—
195C22×D59—21no—
196C22×D59—21no—
197C22×D59—28no—
198C22×D59—28no—
199C22×D59—44no—
200C22×D59—44no—
201C22×D59—44no—
202C22×D59—44no—
203C22×D59—44no—
204C22×D59C2×C4⋊C412yes—
205C22×D59C2×C4⋊C412yes—
206C22×D59AGL(1,ℤ/8)12yes—
207C22×D59#13712yes—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—24no—
211C2×C4×C59—24no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—42no—
220C2×C4×C59—41no—
221C2×C4×C59—84no—
222C2×C4×C59C2×C4⋊C412yes—
223C2×C4×C59C23×C411yescharacteristic
224C2×C4×C59C22×Q811yescharacteristic
225C2×C4×C59H(5,𝔽2)12yes—
226C2×C4×C59H(5,𝔽2)12yes—
227C2×Dic59—21no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—44no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—42no—
235C2×Dic59—42no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—42no—
241C2×Dic59—84no—
242C2×Dic59C2×C4⋊C412yes—
243C2×Dic59C2×C4⋊C412yes—
244C4×D59—24no—
245C4×D59—24no—
246C4×D59—24no—
247C4×D59—24no—
248C4×D59—44no—
249C4×D59—44no—
250C4×D59—84no—
251C4×D59—84no—
252C8×C59—24no—
253C59⋊C8—24no—
254#21942—591nomaximal, Sylow
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—22no—
259C22×C59—22no—
260C22×C59—22no—
261C22×C59#42814yes—
262C22×C59C22×C4⋊C411yescharacteristic
263C22×C59#53411yescharacteristic
264C22×C59#53411yescharacteristic
265C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C59C2×#13711yescharacteristic
268C22×C59C2×#13711yescharacteristic
269C2×D59—22no—
270C2×D59—22no—
271C2×D59—22no—
272C2×D59—22no—
273C2×D59—22no—
274C2×D59—22no—
275C2×D59—22no—
276C2×D59—22no—
277C2×D59—22no—
278C2×D59—416no—
279C2×D59—41no—
280C2×D59—41no—
281C2×D59—41no—
282C2×D59—41no—
283C2×D59#42814yes—
284C4×C59—22no—
285C4×C59—81no—
286C4×C59—82no—
287C4×C59—81no—
288C4×C59#51812yes—
289C4×C59#54312yes—
290Dic59—22no—
291Dic59—22no—
292Dic59—81no—
293Dic59—81no—
294Dic59—82no—
295C2×#407—591no—
296C2×#380—591no—
297C2×#380—591no—
298#1853—591no—
299#1853—591no—
300#1853—592no—
301#1852—592no—
302#1852—591no—
303#1852D5911yescharacteristic, complemented
304#1809—592no—
305#1809—592no—
306C2×C59—24no—
307C2×C59#231511yescharacteristic
308C2×C59C2×#42811yescharacteristic
309C2×C59C2×#42811yescharacteristic
310C2×C59#319211yescharacteristic
311C2×C59#319311yescharacteristic
312C2×C59#319311yescharacteristic
313C2×C59#319511yescharacteristic
314D59—24no—
315D59—42no—
316C24×C4—591no—
317C22×M16—592no—
318C22×C4⋊C4—591no—
319C22×C4⋊C4—591no—
320C22×C4⋊C4—591no—
321C22×C4⋊C4C2×D5911yescharacteristic
322C22×C22⋊C4—1181no—
323C22×C22⋊C4—1181no—
324C2×#107—592no—
325C2×#107C2×D5912yescomplemented
326C2×#106—592no—
327C2×#106—592no—
328#407—598no—
329C2×C22⋊C8—591no—
330C2×C22⋊C8—592no—
331C2×C22⋊C8C2×D5911yescharacteristic
332#380—591no—
333#380—591no—
334#380—592no—
335#380—591no—
336#380—592no—
337#380—1181no—
338#380—1181no—
339#380—1181no—
340#380—1181no—
341#380C2×D5911yescharacteristic
342C2×#95—1181no—
343C2×#95—1181no—
344#340—592no—
345#340—592no—
346#340—592no—
347#340C2×D5912yescomplemented
348C59#2194211yescharacteristic, complemented, Sylow
349C25—591no—
350C23×C4—21no—
351C23×C4—591no—
352C23×C4—591no—
353C23×C4—591no—
354C23×C4—594no—
355C23×C4—591no—
356C23×C4—591no—
357C23×C4—594no—
358C23×C4—1181no—
359C23×C4—1181no—
360C23×C4—1181no—
361C23×C4—2362no—
362C23×C4—2362no—
363C23×C4C22×D5911yescharacteristic
364C2×M16—598no—
365C2×M16—2362no—
366C2×M16—2362no—
367C22×C8—592no—
368C22×C8C22×D5912yes—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—592no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—2362no—
378C2×C4⋊C4—2362no—
379C2×C4⋊C4—2362no—
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C4⋊C4C22×D5912yes—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1182no—
385C2×C22⋊C4—1181no—
386C2×C22⋊C4—1181no—
387C2×C22⋊C4—1182no—
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—2364no—
390C2×C22⋊C4—2364no—
391#107—42no—
392#107—2362no—
393#106—2362no—
394#106—2362no—
395C22⋊C8—594no—
396C22⋊C8—598no—
397C22⋊C8C4×D5914yescomplemented
398#95—21no—
399#95—1181no—
400#95—1181no—
401#95—1182no—
402#95—1182no—
403#95—1181no—
404C24—591no—
405C24—591no—
406C24—594no—
407C24—1182no—
408C24—1182no—
409C24—1182no—
410C24—1182no—
411C24—2361no—
412C24—2361no—
413C24D4×D5911yescharacteristic
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—591no—
423C22×C4—591no—
424C22×C4—594no—
425C22×C4—592no—
426C22×C4—592no—
427C22×C4—592no—
428C22×C4—592no—
429C22×C4—594no—
430C22×C4—594no—
431C22×C4—1182no—
432C22×C4—1182no—
433C22×C4—1182no—
434C22×C4—1182no—
435C22×C4—1182no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4—2368no—
439C22×C4—2368no—
440C22×C4—2361no—
441C22×C4—2361no—
442C22×C4—2361no—
443C22×C4—2361no—
444C22×C4—2362no—
445C22×C4—2362no—
446C22×C4—2362no—
447C22×C4—2362no—
448C22×C4—2362no—
449C22×C4—2362no—
450C22×C4—2361no—
451C22×C4—2361no—
452C22×C4—2362no—
453C22×C4—2362no—
454C22×C4—2362no—
455C22×C4C2×C4×D5912yes—
456C22×C4D4×D5911yescharacteristic
457C22×C4Q8×D5912yes—
458C22×C4C23×D5911yescharacteristic
459M16—2368no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—594no—
463C2×C8—1182no—
464C2×C8—1182no—
465C2×C8C2×C4×D5914yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—1182no—
472C4⋊C4—1182no—
473C4⋊C4—1182no—
474C4⋊C4—1182no—
475C4⋊C4—2362no—
476C4⋊C4—2362no—
477C4⋊C4—2362no—
478C22⋊C4—42no—
479C22⋊C4—2364no—
480C22⋊C4—2364no—
481C22⋊C4—2362no—
482C22⋊C4—2362no—
483C22⋊C4—2362no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—592no—
489C23—592no—
490C23—592no—
491C23—592no—
492C23—1188no—
493C23—1181no—
494C23—1181no—
495C23—1181no—
496C23—1181no—
497C23—1181no—
498C23—1181no—
499C23—1181no—
500C23—1181no—
501C23—1188no—
502C23—1181no—
503C23—1181no—
504C23—2364no—
505C23—2364no—
506C23—2364no—
507C23—2364no—
508C23—2364no—
509C23C4⋊C4×D5912yes—
510C23#1427375112yes—
511C23#1427375912yes—
512C23C2×D4×D5911yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—592no—
528C2×C4—592no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1184no—
534C2×C4—1184no—
535C2×C4—1184no—
536C2×C4—1184no—
537C2×C4—4724no—
538C2×C4—4724no—
539C2×C4—4724no—
540C2×C4—2364no—
541C2×C4—2364no—
542C2×C4—2364no—
543C2×C4—2362no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2362no—
549C2×C4—2361no—
550C2×C4—2362no—
551C2×C4—2361no—
552C2×C4C4⋊C4×D5912yes—
553C2×C4C22×C4×D5911yescharacteristic
554C2×C4C2×Q8×D5911yescharacteristic
555C2×C4#1427377212yes—
556C2×C4#1427377312yes—
557C8—24no—
558C8—1184no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—594no—
566C22—1182no—
567C22—1182no—
568C22—1182no—
569C22—1182no—
570C22—1182no—
571C22—1182no—
572C22—1182no—
573C22—1182no—
574C22—1182no—
575C22—23616no—
576C22—2361no—
577C22—2361no—
578C22—2361no—
579C22—2361no—
580C22#1683744314yes—
581C22C2×C4⋊C4×D5911yescharacteristic
582C22#1683796211yescharacteristic
583C22#1683796311yescharacteristic
584C22AGL(1,ℤ/8)×D5911yescharacteristic
585C22C2×#1427375111yescharacteristic
586C22#137×D5911yescharacteristic
587C22C2×#1427375911yescharacteristic
588C4—22no—
589C4—82no—
590C4—81no—
591C4—81no—
592C4—1182no—
593C4—1182no—
594C4—4721no—
595C4—4722no—
596C4—4721no—
597C4#1683787412yes—
598C4#1683802712yes—
599C2—24no—
600C2—1184no—
601C2—2362no—
602C2#2142670711yescharacteristic
603C2#428×D5911yescharacteristic
604C2C2×#1683744311yescharacteristic
605C2#2143489411yescharacteristic
606C2#2143490111yescharacteristic
607C2#2143490411yescharacteristic
608C2#2143491811yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1852C2
socleC23×C59C22×D4
prime radicalC24×D59D4
cosocleC22×C4×C59C24
hypercenter#1852D59
hypocenterC59#21942
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
4. #21424291C2semidirect
5. #21424290C2semidirect
6. #21424288C2semidirect
7. #21424284C2semidirect
8. #21424282C2semidirect
9. #21424130C2semidirect
10. #21424128C2semidirect
11. C59×#1852C2semidirect
19. C2×#14273660C22semidirect
20. C2×#14273659C22semidirect
21. C2×#14273658C22semidirect
22. #16837364C4semidirect
36. #16837165C22semidirect
37. #16837164C22semidirect
38. #16837164C22semidirect
40. C2×#107×C59C22semidirect
43. C59×#340C22semidirect
59. C2×#11745659C2×C4semidirect
80. C22⋊(C59⋊C8)C2×C4semidirect
81. (C2×D59)⋊C8C2×C4semidirect
97. C22⋊C8×C59C2×C4semidirect
303. #1852D59semidirect
325. C2×#107C2×D59semidirect
347. #340C2×D59semidirect
348. C59#21942semidirect
397. C22⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168373642
C2×#168372592
#380×D592
#214242912
#214242902
#214242882
#214242842
#214242822
#214241302
#214241282
C59×#18522
#2194259