Finite Group Databasebeta

← #28629355

Subgroups — #28629355

Order 15104.

960 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837607C211yesmaximal, characteristic, complemented
2C2×#16837459C211yesmaximal, characteristic, complemented
3#21432448C211yesmaximal, characteristic, complemented
4#21425299C212yesmaximal, complemented
5#21425099C212yesmaximal
6#21425019C211yesmaximal, characteristic, complemented
7#21424976C212yesmaximal, complemented
8#21424854C211yesmaximal, characteristic, complemented
9#21424785C212yesmaximal, complemented
10#21424096C211yesmaximal, characteristic, complemented
11C4×#121×C59C211yesmaximal, characteristic
12C22×#11745670C2212yes—
13C2×#14273727C2211yescharacteristic
14C2×#14273706C2212yes—
15#16837607—24no—
16#16837459—24no—
17C2×#14273660C2212yes—
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273650C2212yes—
21C2×#14273650C2212yes—
22C2×#14273650C2212yes—
23C2×#14273648C2212yes—
24C2×C22⋊(C59⋊C8)C2212yes—
25C2×#14273638C2212yes—
26#16837308C2212yes—
27#16837300—21no—
28#16837300—21no—
29#16837272C2211yescharacteristic
30#16837262—21no—
31#16837262—21no—
32#16837164C2212yes—
33#16837164C2212yes—
34C2×#121×C59C2211yescharacteristic
35C2×C4×D4×C59—21no—
36C2×C4×D4×C59C2211yescharacteristic
37C22×C42×C59C2211yescharacteristic
38C59×#390C2211yescharacteristic
39C59×#389C2211yescharacteristic
40C59×#388—21no—
41C59×#386—21no—
42C59×#380C2211yescharacteristic
43C4×C4⋊C4×C59C2211yescharacteristic
44C4×C22⋊C4×C59—21no—
45C22×Dic118C2312yes—
46C2×#11745670—24no—
47C2×#11745670—24no—
48C2×#11745670—42no—
49C22×C59⋊C8C2312yes—
50C22×C59⋊C8C2312yes—
51#14273727—22no—
52#14273727—22no—
53#14273706—42no—
54#14273706—42no—
55C2×C4⋊Dic59—21no—
56C2×C4⋊Dic59—21no—
57C2×C4⋊Dic59—21no—
58C2×C4⋊Dic59—21no—
59C2×C4⋊Dic59C2312yes—
60C2×Dic59⋊C4—22no—
61C2×C22⋊Dic59—22no—
62#14273660—42no—
63C4⋊(C59⋊C8)—22no—
64C4⋊(C59⋊C8)D414yescomplemented
65#14273650—24no—
66#14273650—24no—
67#14273650—42no—
68#14273648—42no—
69C22⋊(C59⋊C8)—24no—
70#14273638—24no—
71C59⋊#95—22no—
72C59⋊#95—22no—
73C22×D4×C59—21no—
74C22×D4×C59C2311yescharacteristic
75C23×C4×C59—21no—
76C23×C4×C59D411yescharacteristic
77C23×C4×C59D411yescharacteristic
78C23×C4×C59C2311yescharacteristic
79#121×C59—22no—
80#121×C59—22no—
81C4×D4×C59—22no—
82C4×D4×C59—22no—
83C4×D4×C59—44no—
84C2×C4⋊C4×C59—21no—
85C2×C4⋊C4×C59C2311yescharacteristic
86C2×C4⋊C4×C59C2311yescharacteristic
87C2×C4⋊C4×C59C2311yescharacteristic
88C2×C22⋊C4×C59—21no—
89C2×C22⋊C4×C59—21no—
90C2×C22⋊C4×C59—21no—
91C2×C22⋊C4×C59—21no—
92C2×C22⋊C4×C59—21no—
93C2×C42×C59—22no—
94C2×C42×C59—22no—
95C2×C42×C59—21no—
96C2×C42×C59D411yescharacteristic
97C2×C42×C59D411yescharacteristic
98C2×C42×C59C2311yescharacteristic
99C2×C42×C59C2311yescharacteristic
100#95×C59—21no—
101#95×C59—21no—
102#95×C59—21no—
103C24×C59—21no—
104C24×C59C2×D411yescharacteristic
105C2×D4×C59—24no—
106C2×D4×C59—21no—
107C2×D4×C59—21no—
108C2×D4×C59—21no—
109C2×D4×C59—21no—
110C2×D4×C59—21no—
111C2×D4×C59—42no—
112C2×D4×C59—42no—
113C2×D4×C59#4112yes—
114C22×C4×C59—22no—
115C22×C4×C59—21no—
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—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—41no—
129C22×C4×C59—42no—
130C22×C4×C59—42no—
131C22×C4×C59—42no—
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—22no—
139C22×Dic59—22no—
140C2×Dic118—22no—
141C2×Dic118—22no—
142C2×Dic118—22no—
143C2×Dic118—22no—
144C2×Dic118—22no—
145C2×Dic118—22no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—42no—
152C4⋊C4×C59#4112yes—
153C22⋊C4×C59—24no—
154C22⋊C4×C59—42no—
155C22⋊C4×C59—42no—
156C22⋊C4×C59—42no—
157C22⋊C4×C59—42no—
158C42×C59—22no—
159C42×C59—22no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—42no—
163C42×C59C2×D412yes—
164C42×C59#4112yes—
165C22⋊Dic59—44no—
166#11745670—48no—
167C4⋊Dic59—22no—
168C4⋊Dic59—22no—
169C4⋊Dic59—42no—
170C4⋊Dic59—42no—
171C4⋊Dic59—42no—
172C4⋊Dic59—42no—
173Dic59⋊C4—44no—
174C2×C59⋊C8—22no—
175C2×C59⋊C8—22no—
176C2×C59⋊C8—22no—
177C2×C59⋊C8—22no—
178C2×C59⋊C8—22no—
179C2×C59⋊C8C2×D414yes—
180C23×C59—21no—
181C23×C59—21no—
182C23×C59—21no—
183C23×C59—21no—
184C23×C59—21no—
185C23×C59—21no—
186C23×C59—22no—
187C23×C59—41no—
188C23×C59—41no—
189C23×C59—41no—
190C23×C59—41no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59#13712yes—
194C23×C59C22×D411yescharacteristic
195D4×C59—22no—
196D4×C59—22no—
197D4×C59—48no—
198D4×C59—41no—
199D4×C59—41no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—22no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—44no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—44no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—84no—
227C2×C4×C59#12112yes—
228C2×C4×C59#12112yes—
229C2×C4×C59AGL(1,ℤ/8)12yes—
230C2×C4×C59#13712yes—
231C2×C4×C59C22×D411yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59H(5,𝔽2)11yescharacteristic
236C2×C4×C59H(5,𝔽2)11yescharacteristic
237C2×Dic59—42no—
238C2×Dic59—42no—
239C2×Dic59—42no—
240C2×Dic59—42no—
241C2×Dic59—42no—
242C2×Dic59—42no—
243Dic118—42no—
244Dic118—42no—
245Dic118—42no—
246Dic118—42no—
247C59⋊C8—24no—
248C59⋊C8—42no—
249#24659—591nomaximal, Sylow
250C22×C59—22no—
251C22×C59—22no—
252C22×C59—22no—
253C22×C59—41no—
254C22×C59—41no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—42no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59C2×#12111yescharacteristic
265C22×C59#52511yescharacteristic
266C22×C59#53511yescharacteristic
267C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C59C2×#13711yescharacteristic
269C22×C59C2×#13711yescharacteristic
270C22×C59#57511yescharacteristic
271C4×C59—22no—
272C4×C59—42no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59—41no—
276C4×C59—81no—
277C4×C59—81no—
278C4×C59#46814yes—
279C4×C59#54611yescharacteristic
280C4×C59#55011yescharacteristic
281C4×C59#55311yescharacteristic
282C4×C59#55511yescharacteristic
283Dic59—82no—
284Dic59—82no—
285C2×#461—591no—
286C2×#432—591no—
287#2857—591no—
288C4×#121D5911yescharacteristic
289#2036—592no—
290#2000—592no—
291#1981—591no—
292#1973—592no—
293#1947—591no—
294#1935—592no—
295#1802—591no—
296C2×C59—42no—
297C2×C59—82no—
298C2×C59#241411yescharacteristic
299C2×C59C2×#46811yescharacteristic
300C2×C59#305611yescharacteristic
301C2×C59#311011yescharacteristic
302C2×C59#311611yescharacteristic
303C2×C59#321411yescharacteristic
304C2×C59#321811yescharacteristic
305C22×SD16—592no—
306C2×C4⋊Q8—591no—
307C2×C22⋊Q8—592no—
308C2×#121C2×D5911yescharacteristic
309C2×C4×D4—21no—
310C2×C4×D4C2×D5911yescharacteristic
311C22×C42C2×D5911yescharacteristic
312#461—1184no—
313#432—1184no—
314C2×#107—592no—
315C2×C4⋊C8—591no—
316C2×C4⋊C8—591no—
317C2×#103—592no—
318C2×#103—592no—
319C2×#103—592no—
320C2×#102—592no—
321C2×C22⋊C8—592no—
322C2×#97—592no—
323#391—592no—
324#390C2×D5911yescharacteristic
325#389—1181no—
326#389—1181no—
327#389C2×D5911yescharacteristic
328#388—21no—
329#386—21no—
330#384—591no—
331#380—1181no—
332#380—1181no—
333#380C2×D5911yescharacteristic
334C4×C4⋊C4C2×D5911yescharacteristic
335C4×C22⋊C4—21no—
336#340—592no—
337#340—592no—
338C59#2465911yescharacteristic, complemented, Sylow
339C22×Q8—592no—
340C22×D4—21no—
341C22×D4C22×D5911yescharacteristic
342C23×C4—21no—
343C23×C4C22⋊D5911yescharacteristic
344C23×C4C22⋊D5911yescharacteristic
345C23×C4C22×D5911yescharacteristic
346C2×SD16—1184no—
347C2×SD16—1184no—
348C2×SD16—2362no—
349C22×C8—592no—
350C22×C8—592no—
351C4⋊Q8—1182no—
352C4⋊Q8—1182no—
353C22⋊Q8—2362no—
354C22⋊Q8—2362no—
355#121—22no—
356#121—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—44no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—592no—
362C2×C4⋊C4—1182no—
363C2×C4⋊C4—1181no—
364C2×C4⋊C4—1181no—
365C2×C4⋊C4—1181no—
366C2×C4⋊C4—1181no—
367C2×C4⋊C4C22×D5911yescharacteristic
368C2×C4⋊C4C22×D5911yescharacteristic
369C2×C4⋊C4C22×D5911yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1182no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42C22⋊D5911yescharacteristic
380C2×C42C22⋊D5911yescharacteristic
381C2×C42C22×D5911yescharacteristic
382C2×C42C22×D5911yescharacteristic
383#107—2362no—
384C4⋊C8—594no—
385C4⋊C8—1182no—
386#103—1184no—
387#103—1184no—
388#103—2362no—
389#102—2362no—
390C22⋊C8—1184no—
391#97—1184no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—1182no—
396#95—1182no—
397C24—21no—
398C24C2×C22⋊D5911yescharacteristic
399C2×Q8—1182no—
400C2×Q8—1182no—
401C2×Q8—1182no—
402C2×Q8—1182no—
403C2×Q8—1182no—
404C2×Q8—1182no—
405C2×D4—24no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4#1174568512yes—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—1182no—
433C22×C4—1182no—
434C22×C4C2×C22⋊D5911yescharacteristic
435C22×C4C2×C22⋊D5911yescharacteristic
436C22×C4C2×C22⋊D5911yescharacteristic
437C22×C4C2×C22⋊D5911yescharacteristic
438C22×C4C2×C22⋊D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440SD16—2368no—
441C2×C8—594no—
442C2×C8—1182no—
443C2×C8—1182no—
444C2×C8—1182no—
445C2×C8—1182no—
446C2×C8—1182no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—1182no—
454C4⋊C4—1182no—
455C4⋊C4—2364no—
456C4⋊C4—2362no—
457C4⋊C4—2362no—
458C4⋊C4—2362no—
459C4⋊C4—2362no—
460C4⋊C4#1174568512yes—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—2364no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42D4×D5912yes—
473C42#1174568712yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23#1427375712yes—
488C23C22×C22⋊D5911yescharacteristic
489Q8—2362no—
490Q8—2362no—
491Q8—2362no—
492Q8—2362no—
493D4—22no—
494D4—22no—
495D4—48no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—84no—
525C2×C4—2362no—
526C2×C4—2362no—
527C2×C4—2362no—
528C2×C4—2362no—
529C2×C4—2362no—
530C2×C4—2362no—
531C2×C4#1427369912yes—
532C2×C4#1427369912yes—
533C2×C4#1427375312yes—
534C2×C4#1427375712yes—
535C2×C4C2×D4×D5911yescharacteristic
536C2×C4C2×#1174568711yescharacteristic
537C2×C4C2×#1174568511yescharacteristic
538C2×C4C2×#1174568511yescharacteristic
539C2×C4#1427377311yescharacteristic
540C2×C4#1427377311yescharacteristic
541C8—1184no—
542C8—2362no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1427369911yescharacteristic
558C22#1683791311yescharacteristic
559C22#1683796611yescharacteristic
560C22C2×#1427375311yescharacteristic
561C22C2×#1427375711yescharacteristic
562C22C2×#1427375711yescharacteristic
563C22#1683820011yescharacteristic
564C4—22no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—4722no—
572C4—4722no—
573C4#1683764614yes—
574C4#1683804811yescharacteristic
575C4#1683807311yescharacteristic
576C4#1683809811yescharacteristic
577C4#1683811711yescharacteristic
578C2—42no—
579C2—82no—
580C2#2142751411yescharacteristic
581C2C2×#1683764611yescharacteristic
582C2#2143372811yescharacteristic
583C2#2143419511yescharacteristic
584C2#2143425811yescharacteristic
585C2#2143508711yescharacteristic
586C2#2143512011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC4×#121×C59C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenterC4×#121D59
hypocenterC59#24659
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16837607C2semidirect
2. C2×#16837459C2semidirect
3. #21432448C2semidirect
4. #21425299C2semidirect
6. #21425019C2semidirect
7. #21424976C2semidirect
8. #21424854C2semidirect
9. #21424785C2semidirect
10. #21424096C2semidirect
64. C4⋊(C59⋊C8)D4semidirect
338. C59#24659semidirect

Maximal subgroups

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

subgroupindex
C2×#168376072
C2×#168374592
#214324482
#214252992
#214250992
#214250192
#214249762
#214248542
#214247852
#214240962
C4×#121×C592
#2465959