Finite Group Databasebeta

← #28554603

Subgroups — #28554603

Order 15104.

1210 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×#16837884C211yesmaximal, characteristic
2#21433124C212yesmaximal, complemented
3#21422440C211yesmaximal, characteristic, complemented
4#21422433C212yesmaximal, complemented
5#21422428C212yesmaximal, complemented
6#21422424C212yesmaximal, complemented
7#21422413C212yesmaximal, complemented
8#324×D59C212yesmaximal, complemented
9C59×#1467C211yesmaximal, characteristic, complemented
10C22×#11745689C411yescharacteristic
11#16837884—24no—
12#16837884—24no—
13C2×Q8×Dic59C411yescharacteristic
14C2×C4×D236C411yescharacteristic
15C2×C4×D236C411yescharacteristic
16C2×C4×D236C2211yescharacteristic
17C2×#14273689C411yescharacteristic
18C2×#14273689C2211yescharacteristic
19C2×#14273689C2211yescharacteristic
20C2×#14273684C411yescharacteristic
21C2×#14273684C411yescharacteristic
22C2×#14273684C2211yescharacteristic
23C2×C4⋊C4×D59C411yescharacteristic
24C2×C4⋊C4×D59C2211yescharacteristic
25C2×C4⋊C4×D59C2211yescharacteristic
26#16837475—24no—
27#16837472—24no—
28#16837459C2212yes—
29#16837364C2214yes—
30C22⋊C8×D59C2214yes—
31#16837128C2212yes—
32#16837128C2212yes—
33#16837127C2214yes—
34#16837125C2214yes—
35C2×C4×Q8×C59C2211yescharacteristic
36C59×#432C2212yes—
37C59×#324C2212yes—
38C59×#324C2212yes—
39C2×#11745689—24no—
40C2×#11745689—24no—
41C2×#11745689—24no—
42C22×D236C2×C411yescharacteristic
43C22×D236C2×C411yescharacteristic
44C22×D236C2×C411yescharacteristic
45C22×C4×D59C2×C411yescharacteristic
46C22×C4×D59C2×C411yescharacteristic
47C22×C4×D59C2×C411yescharacteristic
48C22×C4×D59C2×C411yescharacteristic
49C22×C4×D59C2×C411yescharacteristic
50C22×C4×D59C2×C411yescharacteristic
51C22×C4×D59C2×C411yescharacteristic
52C22×C4×D59C2311yescharacteristic
53C22×C4×D59C2311yescharacteristic
54C2×#11745659—24no—
55C2×C8×D59—24no—
56Q8×Dic59—24no—
57C4×D236—22no—
58C4×D236—22no—
59C4×D236—22no—
60C4×D236—22no—
61C4×D236D414yes—
62C4×D236D414yes—
63#14273689—24no—
64#14273689—24no—
65#14273689—24no—
66#14273684—22no—
67#14273684—22no—
68#14273684—22no—
69#14273684—22no—
70#14273684—22no—
71#14273684—22no—
72C2×C4⋊Dic59C2×C411yescharacteristic
73C2×C4⋊Dic59C2×C411yescharacteristic
74C2×C4⋊Dic59C2×C411yescharacteristic
75C2×Dic59⋊C4C2×C411yescharacteristic
76C2×Dic59⋊C4C2×C411yescharacteristic
77C2×Dic59⋊C4C2×C411yescharacteristic
78C2×Dic59⋊C4C2311yescharacteristic
79C4⋊C4×D59—24no—
80C4⋊C4×D59—24no—
81C4⋊C4×D59—24no—
82C2×(C2×D59)⋊C4C2×C411yescharacteristic
83C2×(C2×D59)⋊C4C2×C411yescharacteristic
84C2×(C2×D59)⋊C4C2×C411yescharacteristic
85C2×(C2×D59)⋊C4C2×C411yescharacteristic
86C2×(C2×D59)⋊C4C2×C411yescharacteristic
87C2×(C2×D59)⋊C4C2311yescharacteristic
88C2×C4×Dic59C2×C411yescharacteristic
89C2×C4×Dic59C2×C411yescharacteristic
90C2×C4×Dic59C2×C411yescharacteristic
91C4⋊(C59⋊C8)—22no—
92C22⋊(C59⋊C8)C2314yes—
93(C2×D59)⋊C8—24no—
94(C2×D59)⋊C8—24no—
95#14273638—22no—
96C22×Q8×C59C2×C411yescharacteristic
97C4×Q8×C59—22no—
98C4×Q8×C59—22no—
99C2×C4⋊C4×C59C2×C411yescharacteristic
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C42×C59C2×C411yescharacteristic
103C2×C42×C59C2×C411yescharacteristic
104C2×C42×C59C2311yescharacteristic
105C4⋊C8×C59—22no—
106C22⋊C8×C59C2314yes—
107#97×C59—22no—
108C23×D59C22×C411yescharacteristic
109C23×D59C22×C411yescharacteristic
110C23×D59C22×C411yescharacteristic
111C2×Q8×C59—22no—
112C2×Q8×C59—22no—
113C2×Q8×C59—22no—
114C22×C4×C59C22×C411yescharacteristic
115C22×C4×C59C22×C411yescharacteristic
116C22×C4×C59C22×C411yescharacteristic
117C22×C4×C59C22×C411yescharacteristic
118C22×C4×C59C22×C411yescharacteristic
119C22×C4×C59C22×C411yescharacteristic
120C22×C4×C59C2411yescharacteristic
121C22×Dic59C22×C411yescharacteristic
122C22×Dic59C22×C411yescharacteristic
123C22×Dic59C22×C411yescharacteristic
124C22×Dic59C22×C411yescharacteristic
125C22×Dic59C22×C411yescharacteristic
126#11745689—232no—
127C2×D236—22no—
128C2×D236—22no—
129C2×D236—22no—
130C2×D236—22no—
131C2×D236—22no—
132C2×D236—22no—
133C2×D236—22no—
134C2×D236C22⋊C414yes—
135C2×D236M1614yescomplemented
136C2×C4×D59—22no—
137C2×C4×D59—22no—
138C2×C4×D59—22no—
139C2×C4×D59—22no—
140C2×C4×D59—22no—
141C2×C4×D59—22no—
142C2×C4×D59—22no—
143C2×C4×D59—22no—
144C2×C4×D59—22no—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—22no—
148C2×C4×D59—81no—
149C2×C4×D59—81no—
150C2×C4×D59—81no—
151C2×C4×D59—81no—
152C2×C4×D59—81no—
153C2×C4×D59—81no—
154C2×C4×D59M1614yescomplemented
155C2×C4×D59C2×D414yes—
156C2×C4×D59C2×D414yes—
157C2×C8×C59—24no—
158C4⋊C4×C59—22no—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—22no—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—21no—
165C42×C59C2×D412yes—
166C42×C59C2×D412yes—
167(C2×D59)⋊C4—22no—
168(C2×D59)⋊C4—22no—
169(C2×D59)⋊C4—21no—
170(C2×D59)⋊C4—21no—
171(C2×D59)⋊C4—22no—
172(C2×D59)⋊C4C22⋊C412yes—
173(C2×D59)⋊C4C22⋊C412yes—
174(C2×D59)⋊C4C22⋊C412yes—
175(C2×D59)⋊C4C22⋊C412yes—
176C4⋊Dic59—22no—
177C4⋊Dic59—22no—
178C4⋊Dic59C22⋊C414yes—
179Dic59⋊C4—21no—
180Dic59⋊C4—21no—
181Dic59⋊C4—22no—
182Dic59⋊C4—22no—
183Dic59⋊C4—22no—
184C4×Dic59—22no—
185C4×Dic59—22no—
186C4×Dic59—22no—
187C2×C59⋊C8—24no—
188#11745659—84no—
189C8×D59—84no—
190C23×C59C23×C411yescharacteristic
191C22×D59—21no—
192C22×D59—21no—
193C22×D59—21no—
194C22×D59—21no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C22×D59—41no—
200C22×D59—41no—
201C22×D59C2×C22⋊C412yes—
202C22×D59C2×C22⋊C412yes—
203C22×D59C2×M1612yes—
204C22×D59C2×M1612yes—
205C22×D59C2×M1612yes—
206Q8×C59—28no—
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—21no—
214C2×C4×C59—21no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59C2×C22⋊C412yes—
220C2×C4×C59C2×C22⋊C411yescharacteristic
221C2×C4×C59C2×C22⋊C411yescharacteristic
222C2×C4×C59C2×M1612yes—
223C2×C4×C59C2×M1612yes—
224C2×C4×C59C22×D412yes—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—21no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59C2×C22⋊C412yes—
234C2×Dic59C2×C22⋊C412yes—
235C2×Dic59C2×M1612yes—
236D236—28no—
237D236—28no—
238D236—28no—
239C4×D59—28no—
240C4×D59—28no—
241C4×D59—28no—
242C4×D59—44no—
243C4×D59—44no—
244C4×D59—82no—
245C4×D59—82no—
246C4×D59—82no—
247C4×D59—82no—
248C8×C59—44no—
249C59⋊C8—44no—
250#17368—591nomaximal, Sylow
251C22×C59—21no—
252C22×C59—21no—
253C22×C59C22×C22⋊C411yescharacteristic
254C22×C59C22×M1611yescharacteristic
255C22×C59C22×M1611yescharacteristic
256C2×D59—22no—
257C2×D59—22no—
258C2×D59—22no—
259C2×D59—22no—
260C2×D59—22no—
261C2×D59—22no—
262C2×D59—42no—
263C2×D59—42no—
264C2×D59—42no—
265C2×D59—42no—
266C2×D59—42no—
267C2×D59—41no—
268C2×D59—41no—
269C2×D59—42no—
270C2×D59#40714yes—
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—22no—
274C4×C59—22no—
275C4×C59—21no—
276C4×C59—21no—
277C4×C59—41no—
278C4×C59—41no—
279Dic59—22no—
280Dic59—22no—
281Dic59—22no—
282Dic59—41no—
283Dic59—41no—
284C2×#520—591no—
285#2973—592no—
286#1467—591no—
287#1467D5911yescharacteristic, complemented
288#1465—592no—
289#1464—592no—
290#1463—592no—
291#1459—592no—
292C2×#324—592no—
293C2×C59—21no—
294C2×C59—21no—
295C2×C59C2×#40711yescharacteristic
296C2×C59#288311yescharacteristic
297C2×C59#288911yescharacteristic
298D59—24no—
299D59—42no—
300D59—42no—
301C22×#41—591no—
302#520—1184no—
303#520—1184no—
304C2×C4×Q8—591no—
305C2×C4×Q8C2×D5911yescharacteristic
306C2×C4×D4—591no—
307C2×C4×D4—591no—
308C2×C4×D4—591no—
309C2×C4×D4—591no—
310C2×C4×D4—591no—
311C2×C4×D4—591no—
312C2×#117—591no—
313C2×#117—591no—
314C2×#117—591no—
315C22×C4⋊C4—591no—
316C22×C4⋊C4—591no—
317C22×C4⋊C4—591no—
318#435—1184no—
319#435—1184no—
320#432—592no—
321#432C2×D5912yes—
322#407—594no—
323C2×C22⋊C8—594no—
324#324—592no—
325#324—592no—
326#324—594no—
327#324C2×D5912yes—
328#324C2×D5912yes—
329#323—594no—
330C59#1736811yescharacteristic, complemented, Sylow
331C2×#41—1184no—
332C2×#41—1184no—
333C2×#41—1184no—
334C22×Q8C4×D5911yescharacteristic
335C22×D4—591no—
336C22×D4—591no—
337C22×D4—591no—
338C23×C4—591no—
339C23×C4—591no—
340C23×C4—591no—
341C23×C4—591no—
342C23×C4—591no—
343C23×C4—591no—
344C23×C4—591no—
345C23×C4—591no—
346C23×C4—591no—
347C2×M16—1184no—
348C22×C8—1184no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—1184no—
352C4×D4—594no—
353C4×D4—594no—
354C4×D4—1182no—
355C4×D4—1182no—
356C4×D4—1182no—
357C4×D4—1182no—
358C4×D4—1184no—
359C4×D4—1184no—
360C4×D4—1184no—
361#117—1182no—
362#117—1182no—
363#117—1182no—
364#117—1182no—
365#117—1182no—
366#117—1182no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1184no—
375C2×C4⋊C4—1184no—
376C2×C4⋊C4—1184no—
377C2×C4⋊C4C4×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C22⋊C4—591no—
381C2×C22⋊C4—591no—
382C2×C22⋊C4—591no—
383C2×C22⋊C4—591no—
384C2×C22⋊C4—591no—
385C2×C22⋊C4—591no—
386C2×C42—591no—
387C2×C42—591no—
388C2×C42—591no—
389C2×C42C4×D5911yescharacteristic
390C2×C42C4×D5911yescharacteristic
391C2×C42C22×D5911yescharacteristic
392C4⋊C8—22no—
393C4⋊C8—1182no—
394C22⋊C8—594no—
395C22⋊C8—1184no—
396C22⋊C8—1184no—
397C22⋊C8C22×D5914yes—
398#97—22no—
399#97—1182no—
400C24—591no—
401C24—591no—
402C24—591no—
403#41—11832no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×D4—594no—
408C2×D4—594no—
409C2×D4—1182no—
410C2×D4—1182no—
411C2×D4—1182no—
412C2×D4—1182no—
413C2×D4—1182no—
414C2×D4—1182no—
415C2×D4—1182no—
416C22×C4—591no—
417C22×C4—591no—
418C22×C4—591no—
419C22×C4—594no—
420C22×C4—594no—
421C22×C4—594no—
422C22×C4—591no—
423C22×C4—591no—
424C22×C4—1182no—
425C22×C4—1182no—
426C22×C4—1182no—
427C22×C4—1182no—
428C22×C4—1182no—
429C22×C4—1182no—
430C22×C4—1182no—
431C22×C4—1182no—
432C22×C4—1182no—
433C22×C4—1182no—
434C22×C4—1182no—
435C22×C4—1182no—
436C22×C4—4721no—
437C22×C4—4721no—
438C22×C4—4721no—
439C22×C4—4721no—
440C22×C4—4721no—
441C22×C4—4721no—
442C22×C4C2×C4×D5911yescharacteristic
443C22×C4C2×C4×D5911yescharacteristic
444C22×C4C2×C4×D5911yescharacteristic
445C22×C4C2×C4×D5911yescharacteristic
446C22×C4C2×C4×D5911yescharacteristic
447C22×C4C2×C4×D5911yescharacteristic
448C22×C4C23×D5911yescharacteristic
449M16—4724no—
450C2×C8—24no—
451C2×C8—1184no—
452C2×C8—4724no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—594no—
457C4⋊C4—1182no—
458C4⋊C4—1182no—
459C4⋊C4—1182no—
460C4⋊C4—1182no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1182no—
464C22⋊C4—592no—
465C22⋊C4—592no—
466C22⋊C4—592no—
467C22⋊C4—592no—
468C22⋊C4—1182no—
469C22⋊C4—1182no—
470C22⋊C4—1181no—
471C22⋊C4—1181no—
472C22⋊C4—1182no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—1182no—
478C42—1182no—
479C42—1182no—
480C42D4×D5912yes—
481C42D4×D5912yes—
482C23—592no—
483C23—592no—
484C23—592no—
485C23—592no—
486C23—592no—
487C23—1181no—
488C23—1181no—
489C23—1181no—
490C23—1181no—
491C23—2361no—
492C23—2361no—
493C23—2361no—
494C23—2361no—
495C23—2361no—
496C23—2361no—
497C23C22×C4×D5911yescharacteristic
498Q8—28no—
499D4—1188no—
500D4—1188no—
501D4—1188no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—592no—
515C2×C4—592no—
516C2×C4—592no—
517C2×C4—1188no—
518C2×C4—1188no—
519C2×C4—1188no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—4722no—
525C2×C4—4722no—
526C2×C4—4722no—
527C2×C4—4722no—
528C2×C4—2364no—
529C2×C4—2364no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4C22⋊C4×D5911yescharacteristic
535C2×C4C22⋊C4×D5911yescharacteristic
536C2×C4C22⋊C4×D5912yes—
537C2×C4M16×D5912yes—
538C2×C4M16×D5912yes—
539C2×C4C2×D4×D5912yes—
540C8—44no—
541C8—2364no—
542C22—21no—
543C22—21no—
544C22—594no—
545C22—1182no—
546C22—1182no—
547C22—1182no—
548C22—1182no—
549C22—1182no—
550C22—1182no—
551C22—2362no—
552C22—2362no—
553C22—2362no—
554C22—2362no—
555C22—2362no—
556C22—2362no—
557C22—2361no—
558C22—2361no—
559C22C2×C22⋊C4×D5911yescharacteristic
560C22C2×M16×D5911yescharacteristic
561C22C2×M16×D5911yescharacteristic
562C4—21no—
563C4—21no—
564C4—22no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—1182no—
571C4—1182no—
572C4—1182no—
573C4—2361no—
574C4—2361no—
575C2—21no—
576C2—21no—
577C2—1184no—
578C2—2362no—
579C2—2362no—
580C2#407×D5911yescharacteristic
581C2#2143255511yescharacteristic
582C2#2143257611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D59
commutatorC23×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1467C2
socleC22×C59C22×C22⋊C4
prime radicalC22×#11745689C4
cosocleC22×C4×C59C24
hypercenter#1467D59
hypocenterC59#17368
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #21433124C2semidirect
3. #21422440C2semidirect
4. #21422433C2semidirect
5. #21422428C2semidirect
6. #21422424C2semidirect
7. #21422413C2semidirect
8. #324×D59C2semidirect
9. C59×#1467C2semidirect
135. C2×D236M16semidirect
154. C2×C4×D59M16semidirect
287. #1467D59semidirect
330. C59#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#168378842
#214331242
#214224402
#214224332
#214224282
#214224242
#214224132
#324×D592
C59×#14672
#1736859