Finite Group Databasebeta

← #28599857

Subgroups — #28599857

Order 15104.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837553C211yesmaximal, characteristic
2C4×#14273706C211yesmaximal, characteristic, complemented
3#21424949C212yesmaximal, complemented
4#21424782C212yesmaximal, complemented
5#21424720C212yesmaximal, complemented
6#21424454C212yesmaximal, complemented
7#21424403C211yesmaximal, characteristic, complemented
8#21424289C211yesmaximal, characteristic, complemented
9#21424027C211yesmaximal, characteristic
10C4×C22⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
11C2×C59×#390C211yesmaximal, characteristic
12C22×#11745670C2212yes—
13C2×#14273706C2211yescharacteristic
14C2×C4×Dic118C2211yescharacteristic, complemented
15#16837553C418yescomplemented
16C2×#14273658C2212yescomplemented
17C2×#14273650C2212yescomplemented
18C2×#14273648C2212yes—
19C2×#14273648C2212yes—
20C2×#14273648C2212yes—
21C2×C22⋊(C59⋊C8)C2211yescharacteristic
22C2×C22⋊(C59⋊C8)C2211yescharacteristic
23C2×C4×C59⋊C8C2212yescomplemented
24#16837308C2211yescharacteristic, complemented
25#16837300C2211yescharacteristic, complemented
26#16837286—21no—
27#16837275—21no—
28#16837262C2211yescharacteristic
29C4×Dic59⋊C4—21no—
30C4×C4⋊Dic59C2211yescharacteristic, complemented
31C4×C22⋊Dic59—21no—
32#16837164C2212yescomplemented
33#16837164C2212yescomplemented
34#16837157C2212yescomplemented
35C23×D4×C59C2211yescharacteristic
36C22×C4⋊C4×C59C2211yescharacteristic
37C22×C22⋊C4×C59—21no—
38C22×C22⋊C4×C59—21no—
39C22×C42×C59C2211yescharacteristic
40C59×#390—21no—
41C59×#390—21no—
42C59×#390C2212yes—
43C59×#390C2212yes—
44C22×Dic118C2311yescharacteristic
45C2×#11745670—42no—
46C2×#11745670—42no—
47C2×#11745670C2×C418yescomplemented
48C22×C59⋊C8C2312yes—
49C22×C59⋊C8C2312yes—
50#14273706—22no—
51#14273706C2×C414yes—
52C4×Dic118—42no—
53C2×C4⋊Dic59C2311yescharacteristic
54C2×C4⋊Dic59C2311yescharacteristic
55C2×C4⋊Dic59C2311yescharacteristic
56C2×Dic59⋊C4—21no—
57C2×Dic59⋊C4—21no—
58C2×Dic59⋊C4—21no—
59C2×C22⋊Dic59—21no—
60C2×C22⋊Dic59—21no—
61C2×C4×Dic59—21no—
62C2×C4×Dic59—21no—
63#14273658—42no—
64#14273650—42no—
65#14273648—42no—
66#14273648—42no—
67#14273648C2×C418yescomplemented
68C22⋊(C59⋊C8)—22no—
69C22⋊(C59⋊C8)C2×C414yes—
70C4×C59⋊C8—42no—
71C59⋊#95—21no—
72C59⋊#95—21no—
73C59⋊#95—21no—
74C59⋊#95—21no—
75C59⋊#95—21no—
76C25×C59—21no—
77C22×D4×C59—21no—
78C22×D4×C59—22no—
79C22×D4×C59—42no—
80C22×D4×C59—42no—
81C22×D4×C59C2×C414yes—
82C22×D4×C59C2312yes—
83C23×C4×C59—21no—
84C23×C4×C59D411yescharacteristic
85C23×C4×C59D411yescharacteristic
86C23×C4×C59C2311yescharacteristic
87C2×C4⋊C4×C59—21no—
88C2×C4⋊C4×C59—42no—
89C2×C4⋊C4×C59C2312yes—
90C2×C22⋊C4×C59—22no—
91C2×C22⋊C4×C59—22no—
92C2×C22⋊C4×C59—22no—
93C2×C22⋊C4×C59—22no—
94C2×C22⋊C4×C59—44no—
95C2×C22⋊C4×C59—44no—
96C2×C42×C59—42no—
97C2×C42×C59D411yescharacteristic
98C2×C42×C59D411yescharacteristic
99C2×C42×C59C2311yescharacteristic
100C2×C42×C59C2311yescharacteristic
101C24×C59—24no—
102C24×C59—22no—
103C24×C59—42no—
104C24×C59—42no—
105C24×C59—42no—
106C24×C59—42no—
107C24×C59—42no—
108C24×C59—42no—
109C24×C59C2×D411yescharacteristic
110C2×D4×C59—24no—
111C2×D4×C59—22no—
112C2×D4×C59—22no—
113C2×D4×C59—22no—
114C2×D4×C59—22no—
115C2×D4×C59—22no—
116C2×D4×C59—48no—
117C2×D4×C59—42no—
118C2×D4×C59—42no—
119C2×D4×C59—48no—
120C2×D4×C59C22×C414yes—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—41no—
127C22×C4×C59—41no—
128C22×C4×C59—41no—
129C22×C4×C59—42no—
130C22×C4×C59—42no—
131C22×C4×C59—42no—
132C22×C4×C59SD1614yescomplemented
133C22×C4×C59C22×C412yes—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59#4112yes—
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C2×Dic118—22no—
146C2×Dic118—22no—
147C2×Dic118—21no—
148C2×Dic118C22×C412yes—
149C4⋊C4×C59—22no—
150C4⋊C4×C59—22no—
151C4⋊C4×C59—42no—
152C22⋊C4×C59—44no—
153C22⋊C4×C59—44no—
154C22⋊C4×C59—44no—
155C22⋊C4×C59—44no—
156C42×C59—22no—
157C42×C59—22no—
158C42×C59—41no—
159C42×C59—41no—
160C22⋊Dic59—24no—
161C22⋊Dic59—42no—
162#11745670—48no—
163C4⋊Dic59—21no—
164C4⋊Dic59—22no—
165C4⋊Dic59—22no—
166C4⋊Dic59C22×C412yes—
167Dic59⋊C4—24no—
168Dic59⋊C4—42no—
169Dic59⋊C4—42no—
170C4×Dic59—42no—
171C4×Dic59—42no—
172C2×C59⋊C8—22no—
173C2×C59⋊C8—22no—
174C2×C59⋊C8—22no—
175C2×C59⋊C8—22no—
176C2×C59⋊C8—22no—
177C2×C59⋊C8C22×C414yes—
178C23×C59—21no—
179C23×C59—21no—
180C23×C59—21no—
181C23×C59—21no—
182C23×C59—24no—
183C23×C59—48no—
184C23×C59—48no—
185C23×C59—42no—
186C23×C59—42no—
187C23×C59—42no—
188C23×C59—42no—
189C23×C59—42no—
190C23×C59—42no—
191C23×C59—48no—
192C23×C59—84no—
193C23×C59—84no—
194C23×C59—84no—
195C23×C59—84no—
196C23×C59C4×D412yes—
197C23×C59C2×SD1612yes—
198C23×C59#13512yes—
199C23×C59C22×D411yescharacteristic
200D4×C59—24no—
201D4×C59—24no—
202D4×C59—48no—
203D4×C59—42no—
204D4×C59—42no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—24no—
214C2×C4×C59—24no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—44no—
219C2×C4×C59—44no—
220C2×C4×C59—42no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—84no—
224C2×C4×C59C4×D412yes—
225C2×C4×C59C2×SD1612yes—
226C2×C4×C59C2×SD1612yes—
227C2×C4×C59AGL(1,ℤ/8)12yes—
228C2×C4×C59C23×C411yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59H(5,𝔽2)12yes—
231C2×C4×C59H(5,𝔽2)12yes—
232C2×Dic59—22no—
233C2×Dic59—22no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244Dic118—24no—
245Dic118—42no—
246C59⋊C8—24no—
247C59⋊C8—42no—
248#22043—591nomaximal, Sylow
249C22×C59—22no—
250C22×C59—22no—
251C22×C59—22no—
252C22×C59—22no—
253C22×C59—22no—
254C22×C59—22no—
255C22×C59—44no—
256C22×C59—44no—
257C22×C59—44no—
258C22×C59—816no—
259C22×C59—82no—
260C22×C59—82no—
261C22×C59—82no—
262C22×C59—82no—
263C22×C59C4×SD1614yes—
264C22×C59C2×C4×D411yescharacteristic
265C22×C59#53411yescharacteristic
266C22×C59#53511yescharacteristic
267C22×C59C22×SD1611yescharacteristic
268C22×C59C2×#13511yescharacteristic
269C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C59#57511yescharacteristic
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—22no—
274C4×C59—22no—
275C4×C59—41no—
276C4×C59—82no—
277C4×C59#51812yes—
278C4×C59#53812yes—
279Dic59—42no—
280Dic59—42no—
281Dic59—81no—
282Dic59—81no—
283C2×#450—591no—
284C4×C22⋊Q8—591no—
285C2×#390D5911yescharacteristic
286#1968—592no—
287#1934—592no—
288#1926—592no—
289#1880—592no—
290#1872—591no—
291#1853—591no—
292#1786—591no—
293C4×C22⋊C8—591no—
294C2×C59—24no—
295C2×C59—84no—
296C2×C59#232111yescharacteristic
297C2×C59C2×C4×SD1611yescharacteristic
298C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
299C2×C59#318711yescharacteristic
300C2×C59#318811yescharacteristic
301C2×C59#320811yescharacteristic
302C2×C59#321111yescharacteristic
303C23×D4C2×D5911yescharacteristic
304C22×SD16—592no—
305C2×C22⋊Q8—591no—
306C2×C4×Q8—591no—
307C22×C4⋊C4C2×D5911yescharacteristic
308C22×C22⋊C4—21no—
309C22×C22⋊C4—21no—
310C22×C42C2×D5911yescharacteristic
311#450—598no—
312C2×#106—592no—
313C2×#103—592no—
314C2×#102—592no—
315C2×#102—592no—
316C2×#102—592no—
317C2×C22⋊C8—591no—
318C2×C22⋊C8—591no—
319C2×C4×C8—592no—
320#391—591no—
321#390—21no—
322#390—21no—
323#390C2×D5912yes—
324#390C2×D5912yes—
325#389—591no—
326#387—1181no—
327#385—1181no—
328#380—591no—
329C4×C4⋊C4—591no—
330C4×C4⋊C4—1181no—
331C4×C22⋊C4—1181no—
332#340—592no—
333#340—592no—
334#336—592no—
335C59#2204311yescharacteristic, complemented, Sylow
336C25—21no—
337C22×Q8—591no—
338C22×D4—22no—
339C22×D4—21no—
340C22×D4—42no—
341C22×D4—42no—
342C22×D4C4×D5914yes—
343C22×D4C22×D5912yes—
344C23×C4—21no—
345C23×C4C22⋊D5911yescharacteristic
346C23×C4C22⋊D5911yescharacteristic
347C23×C4C22×D5911yescharacteristic
348C2×SD16—598no—
349C2×SD16—2362no—
350C2×SD16—2362no—
351C22×C8—592no—
352C22×C8—592no—
353C22⋊Q8—594no—
354C22⋊Q8—1182no—
355C4×Q8—2362no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4—591no—
359C2×C4⋊C4—591no—
360C2×C4⋊C4—591no—
361C2×C4⋊C4—1181no—
362C2×C4⋊C4—1181no—
363C2×C4⋊C4—1181no—
364C2×C4⋊C4C22×D5912yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—1181no—
372C2×C22⋊C4—1181no—
373C2×C42—42no—
374C2×C42—1181no—
375C2×C42—1181no—
376C2×C42C22⋊D5911yescharacteristic
377C2×C42C22⋊D5911yescharacteristic
378C2×C42C22×D5911yescharacteristic
379C2×C42C22×D5911yescharacteristic
380#106—2362no—
381#103—2362no—
382#102—598no—
383#102—2362no—
384#102—2362no—
385C22⋊C8—594no—
386C22⋊C8—1182no—
387C4×C8—2362no—
388#95—1181no—
389#95—1181no—
390#95—1181no—
391#95—1181no—
392#95—1181no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24C2×C22⋊D5911yescharacteristic
402C2×Q8—592no—
403C2×Q8—1181no—
404C2×Q8—1182no—
405C2×Q8—1182no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C2×C4×D5914yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4#1174567014yescomplemented
433C22×C4C2×C4×D5912yes—
434C22×C4#1174568512yes—
435C22×C4C2×C22⋊D5911yescharacteristic
436C22×C4C2×C22⋊D5911yescharacteristic
437C22×C4C2×C22⋊D5911yescharacteristic
438C22×C4C2×C22⋊D5911yescharacteristic
439C22×C4C2×C22⋊D5911yescharacteristic
440C22×C4C23×D5911yescharacteristic
441SD16—2368no—
442C2×C8—594no—
443C2×C8—1182no—
444C2×C8—1182no—
445C2×C8—1182no—
446C2×C8—1182no—
447C2×C8—1182no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—592no—
452C4⋊C4—1184no—
453C4⋊C4—1181no—
454C4⋊C4—1182no—
455C4⋊C4—1182no—
456C4⋊C4—2362no—
457C4⋊C4—2362no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—1184no—
463C22⋊C4—2362no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—2362no—
469C42—2362no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—48no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23C4×C22⋊D5912yes—
489C23C2×#1174567012yes—
490C23#1427374712yes—
491C23C22×C22⋊D5911yescharacteristic
492Q8—1184no—
493Q8—2362no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—84no—
518C2×C4—1182no—
519C2×C4—1182no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4C4×C22⋊D5912yes—
531C2×C4C2×#1174567012yes—
532C2×C4C2×#1174567012yes—
533C2×C4#1427375312yes—
534C2×C4C22×C4×D5911yescharacteristic
535C2×C4C2×#1174568511yescharacteristic
536C2×C4#1427377212yes—
537C2×C4#1427377212yes—
538C8—1184no—
539C8—2362no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22C4×#1174567014yes—
555C22C2×C4×C22⋊D5911yescharacteristic
556C22#1683796111yescharacteristic
557C22#1683796911yescharacteristic
558C22C22×#1174567011yescharacteristic
559C22C2×#1427374711yescharacteristic
560C22C2×#1427375311yescharacteristic
561C22#1683820011yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—82no—
568C4—4721no—
569C4—4721no—
570C4—2362no—
571C4—2362no—
572C4#1683787512yes—
573C4#1683799012yes—
574C2—24no—
575C2—84no—
576C2#2142675411yescharacteristic
577C2C2×C4×#1174567011yescharacteristic
578C2C4×#1427375311yescharacteristic
579C2#2143485311yescharacteristic
580C2#2143486111yescharacteristic
581C2#2143503711yescharacteristic
582C2#2143506111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#390C2
socleC23×C59C22×D4
prime radicalC23×D4×C59C22
cosocleC22×C4×C59C24
hypercenterC2×#390D59
hypocenterC59#22043
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C4×#14273706C2semidirect
3. #21424949C2semidirect
4. #21424782C2semidirect
5. #21424720C2semidirect
6. #21424454C2semidirect
7. #21424403C2semidirect
8. #21424289C2semidirect
10. C4×C22⋊(C59⋊C8)C2semidirect
14. C2×C4×Dic118C22semidirect
15. #16837553C4semidirect
16. C2×#14273658C22semidirect
17. C2×#14273650C22semidirect
23. C2×C4×C59⋊C8C22semidirect
24. #16837308C22semidirect
25. #16837300C22semidirect
30. C4×C4⋊Dic59C22semidirect
32. #16837164C22semidirect
33. #16837164C22semidirect
34. #16837157C22semidirect
47. C2×#11745670C2×C4semidirect
67. #14273648C2×C4semidirect
132. C22×C4×C59SD16semidirect
335. C59#22043semidirect
432. C22×C4#11745670semidirect

Maximal subgroups

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

subgroupindex
C2×#168375532
C4×#142737062
#214249492
#214247822
#214247202
#214244542
#214244032
#214242892
#214240272
C4×C22⋊(C59⋊C8)2
C2×C59×#3902
#2204359