Finite Group Databasebeta

← #28484255

Subgroups — #28484255

Order 15104.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
2C2×#16837136C212yesmaximal, complemented
3#21422319C211yesmaximal, characteristic, complemented
4#21421974C212yesmaximal, complemented
5C59×#2847C211yesmaximal, characteristic
6C23×C59⋊C8—21no—
7C22×C22⋊Dic59—21no—
8C2×C22⋊(C59⋊C8)—24no—
9C2×C22⋊(C59⋊C8)—21no—
10C2×C22⋊(C59⋊C8)C2212yes—
11C2×C59⋊#95—21no—
12C2×C59⋊#95C2212yes—
13#16837136—24no—
14C24×C4×C59C2211yescharacteristic
15C2×#121×C59C411yescharacteristic, complemented
16C2×#121×C59C411yescharacteristic, complemented
17C2×#121×C59C2211yescharacteristic
18C2×#120×C59—21no—
19C59×#386—21no—
20C59×#386—21no—
21C59×#386—21no—
22C59×#380C411yescharacteristic, complemented
23C59×#380C411yescharacteristic, complemented
24C59×#380C2211yescharacteristic
25C23×Dic59—22no—
26C22×C59⋊C8—24no—
27C22×C59⋊C8—22no—
28C22×C59⋊C8—42no—
29C22×C59⋊C8—42no—
30C2×C22⋊Dic59—24no—
31C2×C22⋊Dic59—42no—
32C2×C22⋊Dic59—42no—
33C22⋊(C59⋊C8)—24no—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)D414yescomplemented
36C59⋊#95—22no—
37C59⋊#95—22no—
38C59⋊#95—42no—
39C25×C59D411yescharacteristic
40C22×D4×C59—21no—
41C22×D4×C59—21no—
42C22×D4×C59—21no—
43C22×D4×C59C2×C411yescharacteristic
44C23×C4×C59—22no—
45C23×C4×C59—22no—
46C23×C4×C59—42no—
47C23×C4×C59—42no—
48C23×C4×C59C2×C411yescharacteristic
49C23×C4×C59C2×C411yescharacteristic
50C23×C4×C59D411yescharacteristic
51C23×C4×C59D411yescharacteristic
52C23×C4×C59D411yescharacteristic
53C23×C4×C59C2311yescharacteristic
54#121×C59—22no—
55#121×C59—22no—
56#121×C59—42no—
57#121×C59—42no—
58#120×C59—42no—
59#120×C59—42no—
60C2×C4⋊C4×C59C2×C411yescharacteristic
61C2×C4⋊C4×C59C2×C411yescharacteristic
62C2×C4⋊C4×C59C2×C411yescharacteristic
63C2×C22⋊C4×C59—21no—
64C2×C22⋊C4×C59—21no—
65C2×C22⋊C4×C59—21no—
66C2×C22⋊C4×C59—21no—
67C2×C22⋊C4×C59—21no—
68C2×C22⋊C4×C59—21no—
69#95×C59—21no—
70#95×C59—21no—
71#95×C59—21no—
72C24×C59—22no—
73C24×C59—22no—
74C24×C59—22no—
75C24×C59—22no—
76C24×C59—21no—
77C24×C59—41no—
78C24×C59—41no—
79C24×C59C22⋊C411yescharacteristic
80C24×C59C22⋊C411yescharacteristic
81C24×C59D812yes—
82C24×C59SD1612yes—
83C24×C59C2×D411yescharacteristic
84C2×D4×C59—22no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C2×D4×C59—42no—
88C2×D4×C59—42no—
89C2×D4×C59—42no—
90C2×D4×C59—42no—
91C2×D4×C59—42no—
92C2×D4×C59—42no—
93C2×D4×C59—42no—
94C2×D4×C59—41no—
95C2×D4×C59—41no—
96C2×D4×C59—42no—
97C22×C4×C59—22no—
98C22×C4×C59—22no—
99C22×C4×C59—21no—
100C22×C4×C59—21no—
101C22×C4×C59—22no—
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—22no—
107C22×C4×C59—21no—
108C22×C4×C59—21no—
109C22×C4×C59—21no—
110C22×C4×C59—21no—
111C22×C4×C59—21no—
112C22×C4×C59—41no—
113C22×C4×C59—41no—
114C22×C4×C59—41no—
115C22×C4×C59—41no—
116C22×C4×C59—41no—
117C22×C4×C59—41no—
118C22×C4×C59—41no—
119C22×C4×C59—48no—
120C22×C4×C59—48no—
121C22×C4×C59C22⋊C411yescharacteristic
122C22×C4×C59C22⋊C411yescharacteristic
123C22×C4×C59C22×C411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D412yes—
126C22×C4×C59#4112yes—
127C22×Dic59—24no—
128C22×Dic59—42no—
129C22×Dic59—42no—
130C22×Dic59—42no—
131C22×Dic59—42no—
132C22×Dic59—42no—
133C22×Dic59—42no—
134C4⋊C4×C59—22no—
135C4⋊C4×C59—41no—
136C4⋊C4×C59—41no—
137C22⋊C4×C59—42no—
138C22⋊C4×C59—42no—
139C22⋊C4×C59—42no—
140C22⋊C4×C59—41no—
141C22⋊C4×C59—41no—
142C22⋊C4×C59—42no—
143C22⋊C4×C59—41no—
144C22⋊C4×C59—41no—
145C22⋊Dic59—48no—
146C2×C59⋊C8—24no—
147C2×C59⋊C8—48no—
148C2×C59⋊C8—42no—
149C2×C59⋊C8—42no—
150C23×C59—21no—
151C23×C59—21no—
152C23×C59—21no—
153C23×C59—21no—
154C23×C59—21no—
155C23×C59—21no—
156C23×C59—21no—
157C23×C59—21no—
158C23×C59—24no—
159C23×C59—24no—
160C23×C59—21no—
161C23×C59—21no—
162C23×C59—21no—
163C23×C59—21no—
164C23×C59—21no—
165C23×C59—21no—
166C23×C59—24no—
167C23×C59—24no—
168C23×C59—42no—
169C23×C59—42no—
170C23×C59—44no—
171C23×C59—44no—
172C23×C59—44no—
173C23×C59—44no—
174C23×C59—41no—
175C23×C59—41no—
176C23×C59—41no—
177C23×C59—41no—
178C23×C59—41no—
179C23×C59—41no—
180C23×C59—41no—
181C23×C59#9912yes—
182C23×C59#10212yes—
183C23×C59#10212yes—
184C23×C59#10412yes—
185C23×C59C2×C22⋊C411yescharacteristic
186C23×C59#12112yes—
187C23×C59C2×D811yescharacteristic
188C23×C59C2×SD1611yescharacteristic
189C23×C59#13512yes—
190D4×C59—82no—
191D4×C59—84no—
192D4×C59—84no—
193D4×C59—82no—
194D4×C59—82no—
195C2×C4×C59—24no—
196C2×C4×C59—24no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
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—42no—
207C2×C4×C59—42no—
208C2×C4×C59—44no—
209C2×C4×C59—44no—
210C2×C4×C59—44no—
211C2×C4×C59—42no—
212C2×C4×C59—42no—
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—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59#9912yes—
223C2×C4×C59C4×D412yes—
224C2×C4×C59#12011yescharacteristic
225C2×C4×C59#12311yescharacteristic
226C2×Dic59—44no—
227C2×Dic59—44no—
228C2×Dic59—44no—
229C2×Dic59—82no—
230C2×Dic59—82no—
231C2×Dic59—82no—
232C2×Dic59—82no—
233C59⋊C8—44no—
234#9113—591nomaximal, Sylow
235C22×C59—22no—
236C22×C59—22no—
237C22×C59—22no—
238C22×C59—22no—
239C22×C59—22no—
240C22×C59—21no—
241C22×C59—21no—
242C22×C59—22no—
243C22×C59—22no—
244C22×C59—22no—
245C22×C59—22no—
246C22×C59—21no—
247C22×C59—22no—
248C22×C59—21no—
249C22×C59—22no—
250C22×C59—22no—
251C22×C59—22no—
252C22×C59—48no—
253C22×C59—48no—
254C22×C59—41no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59—41no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59—81no—
265C22×C59#32714yescomplemented
266C22×C59#38611yescharacteristic
267C22×C59C2×#9911yescharacteristic
268C22×C59C2×#9911yescharacteristic
269C22×C59C2×#10211yescharacteristic
270C22×C59#41611yescharacteristic
271C22×C59C2×#10411yescharacteristic
272C22×C59#42111yescharacteristic
273C22×C59#46512yes—
274C22×C59#46612yes—
275C4×C59—21no—
276C4×C59—22no—
277C4×C59—21no—
278C4×C59—22no—
279C4×C59—41no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59—81no—
283Dic59—84no—
284C22×C22⋊C8—591no—
285#2847D5911yescharacteristic
286C2×#327—592no—
287C22⋊#95—591no—
288#1294—592no—
289C2×C59—22no—
290C2×C59—24no—
291C2×C59—22no—
292C2×C59—42no—
293C2×C59—161no—
294C2×C59C2×#32711yescharacteristic
295C2×C59#150111yescharacteristic
296C2×C59#151611yescharacteristic
297C2×C59#151911yescharacteristic
298C2×C59#187111yescharacteristic
299C2×C59#189311yescharacteristic
300C2×C59#189811yescharacteristic
301C24×C4C2×D5911yescharacteristic
302C23×C8—1181no—
303C2×#121Dic5911yescharacteristic, complemented
304C2×#121Dic5911yescharacteristic, complemented
305C2×#121C2×D5911yescharacteristic
306C2×#120—21no—
307C22×C22⋊C4—1181no—
308C2×C22⋊C8—592no—
309C2×C22⋊C8—1184no—
310C2×C22⋊C8—1181no—
311#386—21no—
312#386—21no—
313#386—21no—
314#380Dic5911yescharacteristic, complemented
315#380Dic5911yescharacteristic, complemented
316#380C2×D5911yescharacteristic
317C2×#95—592no—
318C2×#95—1181no—
319#327—1184no—
320C59#911311yescharacteristic, complemented, Sylow
321C25C22⋊D5911yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4C2×Dic5911yescharacteristic
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—42no—
329C23×C4—42no—
330C23×C4—1182no—
331C23×C4C2×Dic5911yescharacteristic
332C23×C4C2×Dic5911yescharacteristic
333C23×C4C22⋊D5911yescharacteristic
334C23×C4C22⋊D5911yescharacteristic
335C23×C4C22⋊D5911yescharacteristic
336C23×C4C22×D5911yescharacteristic
337C22×C8—1184no—
338C22×C8—1182no—
339C22×C8—2362no—
340C22×C8—2362no—
341#121—22no—
342#121—22no—
343#121—42no—
344#121—42no—
345#120—42no—
346#120—42no—
347C2×C4⋊C4C2×Dic5911yescharacteristic
348C2×C4⋊C4C2×Dic5911yescharacteristic
349C2×C4⋊C4C2×Dic5911yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—1184no—
357C2×C22⋊C4—2362no—
358C2×C22⋊C4—2362no—
359C22⋊C8—594no—
360C22⋊C8—1184no—
361C22⋊C8—1182no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—1182no—
366#95—1182no—
367#95—2362no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—41no—
374C24—41no—
375C24#1174566912yes—
376C24#1174567012yes—
377C24C22⋊Dic5911yescharacteristic
378C24C22⋊Dic5911yescharacteristic
379C24C2×C22⋊D5911yescharacteristic
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—1184no—
418C22×C4—2362no—
419C22×C4—2362no—
420C22×C4—2362no—
421C22×C4—2362no—
422C22×C4—2362no—
423C22×C4—2362no—
424C22×C4C22⋊Dic5911yescharacteristic
425C22×C4C22⋊Dic5911yescharacteristic
426C22×C4D4×D5912yes—
427C22×C4#1174568712yes—
428C22×C4C22×Dic5911yescharacteristic
429C22×C4C2×C22⋊D5911yescharacteristic
430C2×C8—1184no—
431C2×C8—2368no—
432C2×C8—2362no—
433C2×C8—2362no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—42no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—2368no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#1427364212yes—
478C23#1427364812yes—
479C23#1427364812yes—
480C23#1427365412yes—
481C23C2×C22⋊Dic5911yescharacteristic
482C23#1427369912yes—
483C23C2×#1174566911yescharacteristic
484C23C2×#1174567011yescharacteristic
485C23#1427374712yes—
486D4—82no—
487D4—84no—
488D4—82no—
489D4—84no—
490D4—82no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—4722no—
519C2×C4—4722no—
520C2×C4—4722no—
521C2×C4—4722no—
522C2×C4—2364no—
523C2×C4—2364no—
524C2×C4—2364no—
525C2×C4#1427364212yes—
526C2×C4D4×Dic5912yes—
527C2×C4#1427369511yescharacteristic
528C2×C4#1427370811yescharacteristic
529C8—2364no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—21no—
535C22—22no—
536C22—21no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#1683713614yescomplemented
561C22#1683728211yescharacteristic
562C22C2×#1427364211yescharacteristic
563C22C2×#1427364211yescharacteristic
564C22C2×#1427364811yescharacteristic
565C22#1683739811yescharacteristic
566C22C2×#1427365411yescharacteristic
567C22#1683741911yescharacteristic
568C22#1683762912yes—
569C22#1683763612yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—4724no—
579C2—22no—
580C2—24no—
581C2—22no—
582C2—42no—
583C2—161no—
584C2C2×#1683713611yescharacteristic
585C2#2142258711yescharacteristic
586C2#2142265211yescharacteristic
587C2#2142266411yescharacteristic
588C2#2142439911yescharacteristic
589C2#2142452511yescharacteristic
590C2#2142455011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
Fitting, prime radicalC59×#2847C2
socleC23×C59C2×C22⋊C4
cosocleC23×C4×C59C23
hypercenter#2847D59
hypocenterC59#9113
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C59⋊C8)C2semidirect
2. C2×#16837136C2semidirect
3. #21422319C2semidirect
4. #21421974C2semidirect
15. C2×#121×C59C4semidirect
16. C2×#121×C59C4semidirect
22. C59×#380C4semidirect
23. C59×#380C4semidirect
35. C22⋊(C59⋊C8)D4semidirect
265. C22×C59#327semidirect
303. C2×#121Dic59semidirect
304. C2×#121Dic59semidirect
314. #380Dic59semidirect
315. #380Dic59semidirect
320. C59#9113semidirect
560. C22#16837136semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊(C59⋊C8)2
C2×#168371362
#214223192
#214219742
C59×#28472
#911359