Finite Group Databasebeta

← #28554601

Subgroups — #28554601

Order 15104.

1210 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837881C211yesmaximal, characteristic
2#21433123C212yesmaximal, complemented
3#21422438C212yesmaximal, complemented
4#21422434C211yesmaximal, characteristic, complemented
5#21422427C212yesmaximal, complemented
6#21422425C212yesmaximal, complemented
7#21422411C212yesmaximal, complemented
8#324×D59C212yesmaximal, complemented
9C59×#1465C211yesmaximal, characteristic, complemented
10C22×#11745687C411yescharacteristic
11#16837881—24no—
12#16837881—24no—
13C2×C4×Dic118C411yescharacteristic
14C2×#14273692C2211yescharacteristic
15C2×D4×Dic59C411yescharacteristic
16C2×C4×C22⋊D59C411yescharacteristic
17C2×C4×C22⋊D59C2211yescharacteristic
18C2×(C22⋊D59)⋊C4C411yescharacteristic
19C2×(C22⋊D59)⋊C4C2211yescharacteristic
20C2×#14273684C411yescharacteristic
21C2×#14273683C411yescharacteristic
22C2×#14273683C2211yescharacteristic
23C22×Dic59⋊C4C411yescharacteristic
24C22×Dic59⋊C4C2211yescharacteristic
25C2×C4⋊C4×D59C2211yescharacteristic
26#16837476—24no—
27#16837471—24no—
28#16837456C2214yes—
29#16837366C2212yes—
30C22⋊C8×D59C2214yes—
31#16837128C2212yes—
32#16837127C2214yes—
33#16837127C2214yes—
34#16837126C2212yes—
35C2×C4×D4×C59C2211yescharacteristic
36C59×#407C2212yes—
37C59×#324C2212yes—
38C59×#323C2212yes—
39C2×#11745687—24no—
40C2×#11745687—24no—
41C2×#11745687—24no—
42C22×Dic118C2×C411yescharacteristic
43C22×C22⋊D59C2×C411yescharacteristic
44C22×C22⋊D59C2×C411yescharacteristic
45C23×Dic59C2×C411yescharacteristic
46C23×Dic59C2×C411yescharacteristic
47C23×Dic59C2×C411yescharacteristic
48C23×Dic59C2×C411yescharacteristic
49C22×C4×D59C2×C411yescharacteristic
50C22×C4×D59C2×C411yescharacteristic
51C22×C4×D59C2311yescharacteristic
52C2×#11745664—22no—
53C2×C8×D59—24no—
54C4×Dic118—22no—
55C4×Dic118—22no—
56#14273692—24no—
57D4×Dic59—24no—
58C4×C22⋊D59—22no—
59C4×C22⋊D59—22no—
60C4×C22⋊D59D414yes—
61C4×C22⋊D59D414yes—
62(C22⋊D59)⋊C4—24no—
63(C22⋊D59)⋊C4—24no—
64#14273684—22no—
65#14273684—22no—
66#14273683—24no—
67#14273683—24no—
68C2×C4⋊Dic59C2×C411yescharacteristic
69C2×Dic59⋊C4—22no—
70C2×Dic59⋊C4—22no—
71C2×Dic59⋊C4—22no—
72C2×Dic59⋊C4—22no—
73C2×Dic59⋊C4C2×C411yescharacteristic
74C2×Dic59⋊C4C2×C411yescharacteristic
75C2×Dic59⋊C4C2×C411yescharacteristic
76C2×Dic59⋊C4C2×C411yescharacteristic
77C2×Dic59⋊C4C2×C411yescharacteristic
78C2×Dic59⋊C4C2×C411yescharacteristic
79C2×Dic59⋊C4C2311yescharacteristic
80C2×Dic59⋊C4C2311yescharacteristic
81C4⋊C4×D59—24no—
82C2×C22⋊Dic59C2×C411yescharacteristic
83C2×C22⋊Dic59C2×C411yescharacteristic
84C2×(C2×D59)⋊C4C2×C411yescharacteristic
85C2×(C2×D59)⋊C4C2×C411yescharacteristic
86C2×C4×Dic59C2×C411yescharacteristic
87C2×C4×Dic59C2×C411yescharacteristic
88C2×C4×Dic59C2×C411yescharacteristic
89C2×C4×Dic59C2×C411yescharacteristic
90C2×C4×Dic59C2311yescharacteristic
91Dic59⋊C8—24no—
92C22⋊(C59⋊C8)—22no—
93C22⋊(C59⋊C8)C2314yes—
94(C2×D59)⋊C8—24no—
95#14273637—24no—
96C22×D4×C59C2×C411yescharacteristic
97C23×C4×C59C2×C411yescharacteristic
98C23×C4×C59C2311yescharacteristic
99C2×M16×C59—22no—
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C22⋊C4×C59C2×C411yescharacteristic
104C2×C22⋊C4×C59C2311yescharacteristic
105C2×C42×C59C2×C411yescharacteristic
106C22⋊C8×C59—22no—
107C22⋊C8×C59C2314yes—
108C24×C59C22×C411yescharacteristic
109C24×C59C22×C411yescharacteristic
110C23×D59C22×C411yescharacteristic
111C2×D4×C59—22no—
112C2×D4×C59—22no—
113C2×D4×C59—22no—
114C22×C4×C59—21no—
115C22×C4×C59—21no—
116C22×C4×C59—81no—
117C22×C4×C59—81no—
118C22×C4×C59C22×C411yescharacteristic
119C22×C4×C59C22×C411yescharacteristic
120C22×C4×C59C22×C411yescharacteristic
121C22×C4×C59C22×C411yescharacteristic
122C22×C4×C59C2×D412yes—
123C22×C4×C59C2×D412yes—
124C22×C4×C59C2411yescharacteristic
125C2×C22⋊D59—22no—
126C2×C22⋊D59—22no—
127C2×C22⋊D59—22no—
128C2×C22⋊D59—22no—
129C2×C22⋊D59C22⋊C414yes—
130C2×C22⋊D59M1614yescomplemented
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—81no—
140C22×Dic59—81no—
141C22×Dic59C22×C411yescharacteristic
142C22×Dic59C22×C411yescharacteristic
143C22×Dic59C22×C411yescharacteristic
144C22×Dic59C22×C411yescharacteristic
145C22×Dic59C22×C411yescharacteristic
146C22×Dic59C22×C411yescharacteristic
147C22×Dic59C22×C411yescharacteristic
148#11745687—232no—
149C2×C4×D59—22no—
150C2×C4×D59—22no—
151C2×C4×D59—22no—
152C2×C4×D59—81no—
153C2×C4×D59—81no—
154C2×C4×D59M1614yescomplemented
155C2×C4×D59C2×D414yes—
156C2×Dic118—22no—
157C2×Dic118—22no—
158C2×Dic118—22no—
159M16×C59—82no—
160C2×C8×C59—24no—
161C4⋊C4×C59—22no—
162C22⋊C4×C59—22no—
163C22⋊C4×C59—22no—
164C42×C59—21no—
165C42×C59—21no—
166C22⋊Dic59—22no—
167C22⋊Dic59C22⋊C414yes—
168(C2×D59)⋊C4—22no—
169(C2×D59)⋊C4C22⋊C412yes—
170(C2×D59)⋊C4C22⋊C412yes—
171C4⋊Dic59—22no—
172Dic59⋊C4—21no—
173Dic59⋊C4—21no—
174Dic59⋊C4—21no—
175Dic59⋊C4—21no—
176Dic59⋊C4—22no—
177Dic59⋊C4—22no—
178Dic59⋊C4—22no—
179Dic59⋊C4—22no—
180Dic59⋊C4—22no—
181Dic59⋊C4C22⋊C412yes—
182Dic59⋊C4C22⋊C412yes—
183C4×Dic59—22no—
184C4×Dic59—22no—
185C4×Dic59—22no—
186C4×Dic59—22no—
187C4×Dic59C2×D414yes—
188#11745664—82no—
189C2×C59⋊C8—24no—
190C8×D59—84no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59C2×C22⋊C412yes—
200C23×C59C2×M1612yes—
201C23×C59C23×C411yescharacteristic
202C22×D59—41no—
203C22×D59—41no—
204C22×D59C2×C22⋊C412yes—
205C22×D59C2×M1612yes—
206C22×D59C2×M1612yes—
207D4×C59—28no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—42no—
213C2×C4×C59—42no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—81no—
219C2×C4×C59—81no—
220C2×C4×C59C2×C22⋊C411yescharacteristic
221C2×C4×C59C2×C22⋊C411yescharacteristic
222C2×C4×C59C2×M1612yes—
223C2×C4×C59C22×D412yes—
224C22⋊D59—28no—
225C22⋊D59—28no—
226C2×Dic59—28no—
227C2×Dic59—28no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—82no—
241C2×Dic59—82no—
242C2×Dic59C2×C22⋊C412yes—
243C2×Dic59C2×C22⋊C412yes—
244C2×Dic59C2×C22⋊C412yes—
245C2×Dic59C2×M1612yes—
246C2×Dic59C2×M1612yes—
247C4×D59—28no—
248C4×D59—44no—
249C4×D59—82no—
250Dic118—28no—
251C8×C59—44no—
252C59⋊C8—44no—
253#17368—591nomaximal, Sylow
254C22×C59—21no—
255C22×C59—21no—
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—22no—
259C22×C59—22no—
260C22×C59—42no—
261C22×C59—42no—
262C22×C59—42no—
263C22×C59—42no—
264C22×C59—41no—
265C22×C59—41no—
266C22×C59C22×C22⋊C411yescharacteristic
267C22×C59C22×M1611yescharacteristic
268C22×C59C22×M1611yescharacteristic
269C2×D59—22no—
270C2×D59—22no—
271C2×D59—42no—
272C2×D59—42no—
273C2×D59#40714yes—
274C4×C59—22no—
275C4×C59—22no—
276C4×C59—21no—
277C4×C59—21no—
278C4×C59—41no—
279C4×C59—41no—
280Dic59—22no—
281Dic59—22no—
282Dic59—22no—
283Dic59—22no—
284Dic59—22no—
285Dic59—41no—
286Dic59—41no—
287C2×#520—591no—
288#2973—592no—
289#1467—592no—
290#1465—591no—
291#1465D5911yescharacteristic, complemented
292#1464—592no—
293#1463—592no—
294#1459—592no—
295C2×#324—592no—
296C2×C59—21no—
297C2×C59—21no—
298C2×C59—42no—
299C2×C59—42no—
300C2×C59C2×#40711yescharacteristic
301C2×C59#288311yescharacteristic
302C2×C59#288911yescharacteristic
303D59—24no—
304C22×#41—591no—
305#520—1184no—
306#520—1184no—
307C2×C4×Q8—591no—
308C2×C4×Q8—591no—
309C2×C4×D4—591no—
310C2×C4×D4—591no—
311C2×C4×D4—591no—
312C2×C4×D4—591no—
313C2×C4×D4—591no—
314C2×C4×D4C2×D5911yescharacteristic
315C2×#117—591no—
316C2×#117—591no—
317C2×#117—591no—
318C22×C4⋊C4—591no—
319C22×C4⋊C4—591no—
320C22×C4⋊C4—591no—
321#435—1184no—
322#435—1184no—
323#432—594no—
324#407—592no—
325#407C2×D5912yes—
326C2×C22⋊C8—594no—
327#324—592no—
328#324—594no—
329#324—594no—
330#324C2×D5912yes—
331#323—592no—
332#323C2×D5912yes—
333C59#1736811yescharacteristic, complemented, Sylow
334C2×#41—1184no—
335C2×#41—1184no—
336C2×#41—1184no—
337C22×Q8—591no—
338C22×D4—591no—
339C22×D4—591no—
340C22×D4C4×D5911yescharacteristic
341C23×C4—591no—
342C23×C4—591no—
343C23×C4—591no—
344C23×C4—591no—
345C23×C4—591no—
346C23×C4—591no—
347C23×C4—591no—
348C23×C4C4×D5911yescharacteristic
349C23×C4C22×D5911yescharacteristic
350C2×M16—22no—
351C2×M16—1182no—
352C22×C8—1184no—
353C4×Q8—1182no—
354C4×Q8—1182no—
355C4×Q8—1184no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—594no—
359C4×D4—594no—
360C4×D4—1184no—
361C4×D4—1184no—
362C4×D4—1182no—
363C4×D4—1182no—
364C4×D4—1184no—
365#117—1182no—
366#117—1182no—
367#117—1184no—
368#117—1184no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—1184no—
379C2×C4⋊C4—1182no—
380C2×C4⋊C4—1182no—
381C2×C4⋊C4—1182no—
382C2×C4⋊C4—1182no—
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C22⋊C4—591no—
385C2×C22⋊C4—591no—
386C2×C22⋊C4—591no—
387C2×C22⋊C4—591no—
388C2×C22⋊C4C4×D5911yescharacteristic
389C2×C22⋊C4C22×D5911yescharacteristic
390C2×C42—591no—
391C2×C42—591no—
392C2×C42—591no—
393C2×C42—591no—
394C2×C42—591no—
395C2×C42C4×D5911yescharacteristic
396C4⋊C8—1184no—
397C22⋊C8—22no—
398C22⋊C8—594no—
399C22⋊C8—1184no—
400C22⋊C8—1182no—
401C22⋊C8C22×D5914yes—
402#97—1184no—
403C24—591no—
404C24C2×C4×D5911yescharacteristic
405C24C2×C4×D5911yescharacteristic
406#41—11832no—
407C2×Q8—1182no—
408C2×Q8—1182no—
409C2×Q8—1182no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—594no—
414C2×D4—594no—
415C2×D4—1182no—
416C2×D4—1182no—
417C2×D4—1182no—
418C2×D4—1182no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—591no—
424C22×C4—591no—
425C22×C4—591no—
426C22×C4—591no—
427C22×C4—591no—
428C22×C4—594no—
429C22×C4—594no—
430C22×C4—591no—
431C22×C4—591no—
432C22×C4—1182no—
433C22×C4—1182no—
434C22×C4—1182no—
435C22×C4—1182no—
436C22×C4—1182no—
437C22×C4—1182no—
438C22×C4—1182no—
439C22×C4—1182no—
440C22×C4—1182no—
441C22×C4—1182no—
442C22×C4—1182no—
443C22×C4—4721no—
444C22×C4—4721no—
445C22×C4—4721no—
446C22×C4—4721no—
447C22×C4C2×C4×D5911yescharacteristic
448C22×C4C2×C4×D5911yescharacteristic
449C22×C4C2×C4×D5911yescharacteristic
450C22×C4C2×C4×D5911yescharacteristic
451C22×C4D4×D5912yes—
452C22×C4D4×D5912yes—
453C22×C4C23×D5911yescharacteristic
454M16—82no—
455M16—4722no—
456C2×C8—24no—
457C2×C8—1184no—
458C2×C8—4724no—
459C4⋊C4—22no—
460C4⋊C4—592no—
461C4⋊C4—592no—
462C4⋊C4—1182no—
463C4⋊C4—1182no—
464C4⋊C4—1182no—
465C4⋊C4—1182no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1182no—
471C4⋊C4—1182no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—594no—
475C22⋊C4—592no—
476C22⋊C4—592no—
477C22⋊C4—1182no—
478C22⋊C4—1182no—
479C42—21no—
480C42—21no—
481C42—594no—
482C42—1182no—
483C42—1182no—
484C42—1182no—
485C42—1182no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—592no—
495C23—592no—
496C23—592no—
497C23—2361no—
498C23—2361no—
499C23C22⋊C4×D5912yes—
500C23M16×D5912yes—
501C23C22×C4×D5911yescharacteristic
502Q8—1188no—
503D4—28no—
504D4—1188no—
505D4—1188no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—592no—
519C2×C4—592no—
520C2×C4—592no—
521C2×C4—592no—
522C2×C4—592no—
523C2×C4—1188no—
524C2×C4—1188no—
525C2×C4—1188no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—4722no—
535C2×C4—4722no—
536C2×C4—4722no—
537C2×C4—2364no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4C22⋊C4×D5911yescharacteristic
543C2×C4C22⋊C4×D5911yescharacteristic
544C2×C4M16×D5912yes—
545C2×C4C2×D4×D5912yes—
546C8—44no—
547C8—2364no—
548C22—21no—
549C22—21no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22—594no—
561C22—1182no—
562C22—1182no—
563C22—2362no—
564C22—2362no—
565C22C2×C22⋊C4×D5911yescharacteristic
566C22C2×M16×D5911yescharacteristic
567C22C2×M16×D5911yescharacteristic
568C4—22no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—41no—
574C4—1182no—
575C4—1182no—
576C4—1182no—
577C4—1182no—
578C4—1182no—
579C4—2361no—
580C4—2361no—
581C2—21no—
582C2—21no—
583C2—42no—
584C2—42no—
585C2—1184no—
586C2#407×D5911yescharacteristic
587C2#2143255411yescharacteristic
588C2#2143257711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D59
commutatorC23×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1465C2
socleC22×C59C22×C22⋊C4
prime radicalC22×#11745687C4
cosocleC22×C4×C59C24
hypercenter#1465D59
hypocenterC59#17368
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #21433123C2semidirect
3. #21422438C2semidirect
4. #21422434C2semidirect
5. #21422427C2semidirect
6. #21422425C2semidirect
7. #21422411C2semidirect
8. #324×D59C2semidirect
9. C59×#1465C2semidirect
130. C2×C22⋊D59M16semidirect
154. C2×C4×D59M16semidirect
291. #1465D59semidirect
333. C59#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#168378812
#214331232
#214224382
#214224342
#214224272
#214224252
#214224112
#324×D592
C59×#14652
#1736859