Finite Group Databasebeta

← #28598982

Subgroups — #28598982

Order 15104.

1162 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837364C211yesmaximal, characteristic
2#21426478C211yesmaximal, characteristic
3C2×#16837259C211yesmaximal, characteristic
4#21424296C212yesmaximal, complemented
5#21424292C212yesmaximal, complemented
6#21424291C211yesmaximal, characteristic, complemented
7#21424289C211yesmaximal, characteristic, complemented
8#21424283C211yesmaximal, characteristic, complemented
9#21424130C212yesmaximal, complemented
10#21424128C212yesmaximal, complemented
11C59×#1852C211yesmaximal, characteristic, complemented
12C23×C4×D59C2211yescharacteristic
13C22×#11745659C2212yes—
14C2×#14273685C2211yescharacteristic
15C2×#14273685C2211yescharacteristic
16C22×C4⋊Dic59C2211yescharacteristic
17C22×(C2×D59)⋊C4—21no—
18C2×#14273660C2212yescomplemented
19C2×#14273659C2212yescomplemented
20C2×#14273658C2212yescomplemented
21#16837364C418yescomplemented
22C2×C22⋊(C59⋊C8)C2211yescharacteristic
23C2×(C2×D59)⋊C8C2212yes—
24#16837264—21no—
25#16837262C2211yescharacteristic
26#16837259—21no—
27#16837259—21no—
28#16837259C2211yescharacteristic
29#16837259C2212yes—
30#16837259C2211yescharacteristic
31#16837259C2212yes—
32#16837257—21no—
33C22⋊C4×Dic59—21no—
34#16837249—21no—
35C2×C59⋊#95—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—42no—
48C22×C4×D59—42no—
49C22×C4×D59C2×C414yes—
50C22×C4×D59D411yescharacteristic
51C22×C4×D59D411yescharacteristic
52C22×C4×D59Q814yes—
53C22×C4×D59C2311yescharacteristic
54C22×C4×D59C2311yescharacteristic
55C2×#11745659—42no—
56C2×#11745659—42no—
57C2×#11745659C2×C418yescomplemented
58C22×C59⋊C8C2312yes—
59#14273685—42no—
60#14273685—42no—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59—42no—
63C2×C4⋊Dic59C2311yescharacteristic
64C2×C4⋊Dic59C2312yes—
65C2×C4⋊Dic59C2311yescharacteristic
66C2×C22⋊Dic59—21no—
67C2×(C2×D59)⋊C4—21no—
68C2×(C2×D59)⋊C4—21no—
69C2×(C2×D59)⋊C4—21no—
70C2×(C2×D59)⋊C4—21no—
71C2×(C2×D59)⋊C4—22no—
72C2×(C2×D59)⋊C4—22no—
73C2×(C2×D59)⋊C4—44no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76#14273660—42no—
77#14273659—42no—
78#14273658—42no—
79C22⋊(C59⋊C8)C2×C414yes—
80(C2×D59)⋊C8C2×C418yescomplemented
81C59⋊#95—21no—
82C59⋊#95—22no—
83C59⋊#95—22no—
84C59⋊#95—21no—
85C59⋊#95—21no—
86C23×C4×C59—21no—
87C23×C4×C59C2311yescharacteristic
88C22×C8×C59C2312yes—
89C2×C4⋊C4×C59—21no—
90C2×C4⋊C4×C59—42no—
91C2×C4⋊C4×C59C2311yescharacteristic
92C2×C4⋊C4×C59C2312yes—
93C2×C4⋊C4×C59C2311yescharacteristic
94C2×C22⋊C4×C59—21no—
95#107×C59—42no—
96C22⋊C8×C59C2×C414yes—
97#95×C59—21no—
98C24×C59C2×D411yescharacteristic
99C23×D59—22no—
100C23×D59—22no—
101C23×D59—41no—
102C23×D59—41no—
103C23×D59—41no—
104C23×D59—41no—
105C23×D59C4⋊C414yes—
106C23×D59C2×D411yescharacteristic
107C23×D59C2×D411yescharacteristic
108C22×C4×C59—22no—
109C22×C4×C59—22no—
110C22×C4×C59—21no—
111C22×C4×C59—21no—
112C22×C4×C59—22no—
113C22×C4×C59—42no—
114C22×C4×C59—42no—
115C22×C4×C59—42no—
116C22×C4×C59C22×C412yes—
117C22×C4×C59C2×D411yescharacteristic
118C22×C4×C59C2×Q812yes—
119C22×C4×C59C2411yescharacteristic
120C22×Dic59—21no—
121C22×Dic59—21no—
122C22×Dic59—22no—
123C22×Dic59—41no—
124C22×Dic59—41no—
125C22×Dic59—42no—
126C22×Dic59—42no—
127C22×Dic59—42no—
128C22×Dic59C4⋊C414yes—
129C22×Dic59C2×D411yescharacteristic
130C22×Dic59C2×D411yescharacteristic
131C2×C4×D59—22no—
132C2×C4×D59—22no—
133C2×C4×D59—22no—
134C2×C4×D59—22no—
135C2×C4×D59—48no—
136C2×C4×D59—48no—
137C2×C4×D59—41no—
138C2×C4×D59—41no—
139C2×C4×D59—41no—
140C2×C4×D59—41no—
141C2×C4×D59C4⋊C414yes—
142C2×C4×D59C4⋊C414yes—
143C2×C4×D59C22×C412yes—
144C2×C4×D59C22×C412yes—
145C2×C4×D59C2×Q812yes—
146C2×C4×D59C2×Q812yes—
147C2×C8×C59—22no—
148C2×C8×C59—22no—
149C2×C8×C59C22×C414yes—
150C4⋊C4×C59—22no—
151C4⋊C4×C59—22no—
152C4⋊C4×C59—22no—
153C4⋊C4×C59—22no—
154C4⋊C4×C59—42no—
155C22⋊C4×C59—42no—
156C22⋊Dic59—42no—
157(C2×D59)⋊C4—42no—
158(C2×D59)⋊C4—42no—
159(C2×D59)⋊C4—44no—
160(C2×D59)⋊C4—44no—
161(C2×D59)⋊C4—42no—
162(C2×D59)⋊C4—42no—
163C4⋊Dic59—22no—
164C4⋊Dic59—22no—
165C4⋊Dic59—22no—
166C4⋊Dic59—22no—
167C4⋊Dic59—42no—
168C4×Dic59—42no—
169C4×Dic59—42no—
170C2×C59⋊C8—22no—
171C2×C59⋊C8—22no—
172C2×C59⋊C8C22×C414yes—
173#11745659—48no—
174C23×C59—21no—
175C23×C59—21no—
176C23×C59—21no—
177C23×C59—21no—
178C23×C59C2×C4⋊C412yes—
179C23×C59AGL(1,ℤ/8)12yes—
180C23×C59#13712yes—
181C23×C59C22×D411yescharacteristic
182C22×D59—21no—
183C22×D59—21no—
184C22×D59—21no—
185C22×D59—21no—
186C22×D59—21no—
187C22×D59—21no—
188C22×D59—21no—
189C22×D59—21no—
190C22×D59—21no—
191C22×D59—21no—
192C22×D59—21no—
193C22×D59—21no—
194C22×D59—28no—
195C22×D59—44no—
196C22×D59—44no—
197C22×D59—44no—
198C22×D59—44no—
199C22×D59—44no—
200C22×D59—44no—
201C22×D59C2×C4⋊C412yes—
202C22×D59C2×C4⋊C412yes—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—24no—
206C2×C4×C59—24no—
207C2×C4×C59—41no—
208C2×C4×C59—41no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—42no—
216C2×C4×C59—84no—
217C2×C4×C59C2×C4⋊C412yes—
218C2×C4×C59C23×C411yescharacteristic
219C2×C4×C59C22×Q811yescharacteristic
220C2×C4×C59H(5,𝔽2)12yes—
221C2×C4×C59H(5,𝔽2)12yes—
222C2×Dic59—21no—
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—44no—
227C2×Dic59—41no—
228C2×Dic59—41no—
229C2×Dic59—42no—
230C2×Dic59—42no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—42no—
236C2×Dic59—84no—
237C2×Dic59C2×C4⋊C412yes—
238C2×Dic59C2×C4⋊C412yes—
239C4×D59—24no—
240C4×D59—24no—
241C4×D59—24no—
242C4×D59—24no—
243C4×D59—44no—
244C4×D59—44no—
245C8×C59—24no—
246C59⋊C8—24no—
247#21944—591nomaximal, Sylow
248C22×C59—22no—
249C22×C59—22no—
250C22×C59—22no—
251C22×C59—22no—
252C22×C59—22no—
253C22×C59—22no—
254C22×C59#42814yes—
255C22×C59C22×C4⋊C411yescharacteristic
256C22×C59#53411yescharacteristic
257C22×C59#53411yescharacteristic
258C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C59C2×#13711yescharacteristic
260C22×C59#57511yescharacteristic
261C22×C59#57511yescharacteristic
262C2×D59—22no—
263C2×D59—22no—
264C2×D59—22no—
265C2×D59—22no—
266C2×D59—22no—
267C2×D59—22no—
268C2×D59—416no—
269C2×D59—41no—
270C2×D59—41no—
271C2×D59—41no—
272C2×D59—41no—
273C2×D59—41no—
274C2×D59—41no—
275C2×D59—41no—
276C2×D59—41no—
277C4×C59—22no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59—82no—
281C4×C59#51812yes—
282C4×C59#54312yes—
283Dic59—22no—
284Dic59—22no—
285Dic59—82no—
286Dic59—81no—
287Dic59—81no—
288C2×#407—591no—
289#2280—591no—
290C2×#380—591no—
291#1854—592no—
292#1854—592no—
293#1853—591no—
294#1853—591no—
295#1852—591no—
296#1852D5911yescharacteristic, complemented
297#1809—592no—
298#1809—592no—
299C2×C59—24no—
300C2×C59#231511yescharacteristic
301C2×C59C2×#42811yescharacteristic
302C2×C59#291111yescharacteristic
303C2×C59#319411yescharacteristic
304C2×C59#319411yescharacteristic
305C2×C59#319611yescharacteristic
306C2×C59#319611yescharacteristic
307D59—42no—
308D59—42no—
309C24×C4—591no—
310C22×M16—592no—
311C2×#117—591no—
312C2×#117—591no—
313C22×C4⋊C4—591no—
314C22×C4⋊C4C2×D5911yescharacteristic
315C22×C22⋊C4—1181no—
316C2×#107—592no—
317C2×#107C2×D5912yescomplemented
318C2×#106—592no—
319C2×#106—592no—
320#407—598no—
321C2×C22⋊C8—591no—
322C2×C22⋊C8—592no—
323C2×C22⋊C8C2×D5911yescharacteristic
324#381—1181no—
325#380—591no—
326#380—592no—
327#380—591no—
328#380—591no—
329#380—592no—
330#380—1181no—
331#380—1181no—
332#380C2×D5911yescharacteristic
333#379—1181no—
334C4×C22⋊C4—1181no—
335C4×C22⋊C4—1181no—
336C2×#95—1181no—
337#340—592no—
338#340—592no—
339#340—592no—
340#340C2×D5912yescomplemented
341C59#2194411yescharacteristic, complemented, Sylow
342C25—591no—
343C23×C4—21no—
344C23×C4—591no—
345C23×C4—591no—
346C23×C4—591no—
347C23×C4—594no—
348C23×C4—591no—
349C23×C4—591no—
350C23×C4—594no—
351C23×C4—1181no—
352C23×C4—2362no—
353C23×C4—2362no—
354C23×C4C22×D5911yescharacteristic
355C2×M16—598no—
356C2×M16—2362no—
357C2×M16—2362no—
358C22×C8—592no—
359C22×C8C22×D5912yes—
360#117—2362no—
361#117—2362no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—592no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—1181no—
368C2×C4⋊C4—2362no—
369C2×C4⋊C4C22×D5912yes—
370C2×C4⋊C4C22×D5911yescharacteristic
371C2×C4⋊C4C22×D5911yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—1182no—
374C2×C22⋊C4—1182no—
375C2×C22⋊C4—1181no—
376C2×C22⋊C4—1181no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—2364no—
381C2×C42—1181no—
382C2×C42—1181no—
383#107—42no—
384#107—2362no—
385#106—2362no—
386#106—2362no—
387C22⋊C8—594no—
388C22⋊C8—598no—
389C22⋊C8C4×D5914yes—
390#95—21no—
391#95—1182no—
392#95—1182no—
393#95—1181no—
394#95—1181no—
395#95—1181no—
396C24—591no—
397C24—591no—
398C24—594no—
399C24—1182no—
400C24—1182no—
401C24—2361no—
402C24—2361no—
403C24—2361no—
404C24—2361no—
405C24D4×D5911yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—591no—
415C22×C4—591no—
416C22×C4—594no—
417C22×C4—592no—
418C22×C4—592no—
419C22×C4—592no—
420C22×C4—592no—
421C22×C4—594no—
422C22×C4—594no—
423C22×C4—1182no—
424C22×C4—1182no—
425C22×C4—1182no—
426C22×C4—1182no—
427C22×C4—1181no—
428C22×C4—1181no—
429C22×C4—1182no—
430C22×C4—2368no—
431C22×C4—2368no—
432C22×C4—2361no—
433C22×C4—2361no—
434C22×C4—2361no—
435C22×C4—2361no—
436C22×C4—2361no—
437C22×C4—2361no—
438C22×C4—2362no—
439C22×C4—2362no—
440C22×C4—2362no—
441C22×C4C2×C4×D5912yes—
442C22×C4D4×D5911yescharacteristic
443C22×C4Q8×D5912yes—
444C22×C4C23×D5911yescharacteristic
445M16—2368no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—594no—
449C2×C8—1182no—
450C2×C8—1182no—
451C2×C8C2×C4×D5914yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—1182no—
458C4⋊C4—1182no—
459C4⋊C4—1182no—
460C4⋊C4—1182no—
461C4⋊C4—2362no—
462C22⋊C4—42no—
463C22⋊C4—2364no—
464C22⋊C4—2364no—
465C22⋊C4—2362no—
466C22⋊C4—2362no—
467C22⋊C4—2362no—
468C22⋊C4—2362no—
469C22⋊C4—2362no—
470C42—2362no—
471C42—2362no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—592no—
477C23—592no—
478C23—1181no—
479C23—1181no—
480C23—1181no—
481C23—1181no—
482C23—1181no—
483C23—1181no—
484C23—1181no—
485C23—1181no—
486C23—1188no—
487C23—1181no—
488C23—1181no—
489C23—1181no—
490C23—1181no—
491C23—2364no—
492C23—2364no—
493C23—2364no—
494C23—2364no—
495C23—2364no—
496C23—2364no—
497C23C4⋊C4×D5912yes—
498C23#1427375112yes—
499C23#1427375912yes—
500C23C2×D4×D5911yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—592no—
516C2×C4—592no—
517C2×C4—1181no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1184no—
522C2×C4—1184no—
523C2×C4—1184no—
524C2×C4—1184no—
525C2×C4—4724no—
526C2×C4—2364no—
527C2×C4—2364no—
528C2×C4—2364no—
529C2×C4—2362no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2362no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2362no—
538C2×C4C4⋊C4×D5912yes—
539C2×C4C22×C4×D5911yescharacteristic
540C2×C4C2×Q8×D5911yescharacteristic
541C2×C4#1427377212yes—
542C2×C4#1427377312yes—
543C8—24no—
544C8—1184no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—1182no—
552C22—1182no—
553C22—1182no—
554C22—1182no—
555C22—1182no—
556C22—1182no—
557C22—23616no—
558C22—2361no—
559C22—2361no—
560C22—2361no—
561C22—2361no—
562C22—2361no—
563C22—2361no—
564C22—2361no—
565C22—2361no—
566C22#1683744314yes—
567C22C2×C4⋊C4×D5911yescharacteristic
568C22#1683796211yescharacteristic
569C22#1683796311yescharacteristic
570C22C2×#1427375111yescharacteristic
571C22C2×#1427375911yescharacteristic
572C22#1683819811yescharacteristic
573C22#1683820111yescharacteristic
574C4—22no—
575C4—82no—
576C4—81no—
577C4—81no—
578C4—1182no—
579C4—1182no—
580C4—4722no—
581C4—4721no—
582C4—4721no—
583C4#1683787412yes—
584C4#1683802712yes—
585C2—24no—
586C2—2362no—
587C2—2362no—
588C2#2142670711yescharacteristic
589C2C2×#1683744311yescharacteristic
590C2#2143268311yescharacteristic
591C2#2143490711yescharacteristic
592C2#2143490911yescharacteristic
593C2#2143492911yescharacteristic
594C2#2143493111yescharacteristic
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#21944
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. #21424296C2semidirect
5. #21424292C2semidirect
6. #21424291C2semidirect
7. #21424289C2semidirect
8. #21424283C2semidirect
9. #21424130C2semidirect
10. #21424128C2semidirect
11. C59×#1852C2semidirect
18. C2×#14273660C22semidirect
19. C2×#14273659C22semidirect
20. C2×#14273658C22semidirect
21. #16837364C4semidirect
36. #16837165C22semidirect
37. #16837164C22semidirect
38. #16837164C22semidirect
40. C2×#107×C59C22semidirect
43. C59×#340C22semidirect
57. C2×#11745659C2×C4semidirect
80. (C2×D59)⋊C8C2×C4semidirect
296. #1852D59semidirect
317. C2×#107C2×D59semidirect
340. #340C2×D59semidirect
341. C59#21944semidirect

Maximal subgroups

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

subgroupindex
C2×#168373642
#214264782
C2×#168372592
#214242962
#214242922
#214242912
#214242892
#214242832
#214241302
#214241282
C59×#18522
#2194459