Finite Group Databasebeta

← #28314064

Subgroups — #28314064

Order 13568.

1646 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291391C212yesmaximal, complemented
2#20289974C214yesmaximal, complemented
3#20289962C212yesmaximal, complemented
4#20289960C212yesmaximal, complemented
5#20289954C214yesmaximal, complemented
6C22×C4×C22⋊D53C211yesmaximal, characteristic
7#16811644C2214yescomplemented
8#16811643—22no—
9#16811640C2214yescomplemented
10#16811639—22no—
11#16811626—22no—
12#16811593C2214yes—
13C4×#11719940—22no—
14#16811467C2218yescomplemented
15#16811466C2218yescomplemented
16C23×C22⋊D53C411yescharacteristic
17C23×C4×D53C411yescharacteristic
18C2×C4×C22⋊D53—22no—
19C2×C4×C22⋊D53—22no—
20C2×C4×C22⋊D53—22no—
21C2×C4×C22⋊D53—22no—
22C2×C4×C22⋊D53C412yes—
23C2×C4×C22⋊D53C412yes—
24C2×C4×C22⋊D53C2212yes—
25C2×C4×C22⋊D53C2212yes—
26C22×#11719919C411yescharacteristic
27C22×C22⋊Dic53C2211yescharacteristic
28C22×#11719921C2211yescharacteristic
29C22×C4×Dic53C2211yescharacteristic
30C24×C4×C53C411yescharacteristic
31C2×#11719940—24no—
32#13133284—24no—
33C4⋊#88324—24no—
34C22⋊#88324—24no—
35C22⋊#88324C2318yes—
36#13133272—24no—
37C4×#88324—24no—
38C24×D53C2×C411yescharacteristic
39C22×C22⋊D53—28no—
40C22×C22⋊D53—22no—
41C22×C22⋊D53—22no—
42C22×C22⋊D53C2×C412yes—
43C22×C22⋊D53C2×C412yes—
44C23×Dic53C2×C411yescharacteristic
45C23×Dic53C2×C411yescharacteristic
46C23×Dic53C2311yescharacteristic
47C22×C4×D53—21no—
48C22×C4×D53—21no—
49C22×C4×D53—21no—
50C22×C4×D53—21no—
51C22×C4×D53—44no—
52C22×C4×D53C2×C412yes—
53C22×C4×D53C2×C412yes—
54C4×C22⋊D53—24no—
55C4×C22⋊D53—24no—
56C4×C22⋊D53—24no—
57C4×C22⋊D53—24no—
58C4×C22⋊D53—24no—
59C4×C22⋊D53—24no—
60C4×C22⋊D53—24no—
61C4×C22⋊D53—24no—
62C2×#11719919—22no—
63C2×#11719919—22no—
64C2×#11719919C2×C412yes—
65C2×#11719919C2×C412yes—
66C2×C22⋊Dic53—22no—
67C2×C22⋊Dic53—22no—
68C2×C22⋊Dic53C2×C412yes—
69C2×C22⋊Dic53C2312yes—
70C2×#11719921—22no—
71C2×#11719921—22no—
72C2×#11719921C2×C412yes—
73C2×#11719921C2312yes—
74C2×C4×Dic53C2×C411yescharacteristic
75C2×C4×Dic53C2×C411yescharacteristic
76C2×C4×Dic53D412yes—
77C2×C4×Dic53D412yes—
78C2×C4×Dic53D412yes—
79C2×C4×Dic53D412yes—
80C2×C4×Dic53C2311yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82C25×C53C2×C411yescharacteristic
83C23×C4×C53—21no—
84C23×C4×C53—21no—
85C23×C4×C53—21no—
86C23×C4×C53—21no—
87C23×C4×C53—44no—
88C23×C4×C53C2×C411yescharacteristic
89C23×C4×C53C2×C411yescharacteristic
90C23×C4×C53C2×C412yes—
91C23×C4×C53C2×C412yes—
92C24×C53—21no—
93C24×C53—21no—
94C24×C53—24no—
95C24×C53—42no—
96C24×C53—42no—
97C24×C53C22×C411yescharacteristic
98C24×C53C22×C412yes—
99C23×D53—21no—
100C23×D53—21no—
101C23×D53—24no—
102C23×D53—42no—
103C23×D53—42no—
104C23×D53C22×C412yes—
105C22×C4×C53—24no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—44no—
113C22×C4×C53—44no—
114C22×C4×C53—44no—
115C22×C4×C53—44no—
116C22×C4×C53—44no—
117C22×C4×C53—44no—
118C22×C4×C53—42no—
119C22×C4×C53C22⋊C411yescharacteristic
120C22×C4×C53C22⋊C411yescharacteristic
121C22×C4×C53C22⋊C411yescharacteristic
122C22×C4×C53C22⋊C411yescharacteristic
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C22×C411yescharacteristic
125C22×C4×C53C22×C412yes—
126C2×C22⋊D53—216no—
127C2×C22⋊D53—216no—
128C2×C22⋊D53—24no—
129C2×C22⋊D53—24no—
130C2×C22⋊D53—24no—
131C2×C22⋊D53—24no—
132C2×C22⋊D53—24no—
133C2×C22⋊D53M1618yescomplemented
134C22×Dic53—24no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53C22⋊C412yes—
140C22×Dic53C22⋊C412yes—
141C22×Dic53C22×C411yescharacteristic
142C22×Dic53C22×C412yes—
143C22×Dic53C22×C411yescharacteristic
144C22×Dic53C22×C411yescharacteristic
145C22×Dic53C2×D412yes—
146C22×Dic53C2×D412yes—
147C22×Dic53C2411yescharacteristic
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—44no—
153C2×C4×D53—44no—
154C2×C4×D53—44no—
155C2×C4×D53—44no—
156C2×C4×D53—44no—
157C2×C4×D53—44no—
158#11719940—48no—
159C2×#88324—28no—
160C22⋊Dic53—24no—
161C22⋊Dic53—24no—
162#11719921—24no—
163#11719921—24no—
164#11719919—24no—
165#11719919—24no—
166C4×Dic53C22⋊C412yes—
167C4×Dic53C22⋊C412yes—
168C4×Dic53C22⋊C412yes—
169C4×Dic53C22⋊C412yes—
170C4×Dic53C2×D412yes—
171C4×Dic53C2×D412yes—
172C4×Dic53C2×D412yes—
173C4×Dic53C2×D412yes—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—24no—
178C23×C53—24no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—416no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53C2×C22⋊C411yescharacteristic
189C23×C53C2×C22⋊C411yescharacteristic
190C23×C53C2×M1614yes—
191C23×C53C23×C411yescharacteristic
192C22×D53—24no—
193C22×D53—24no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—42no—
197C22×D53—416no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—42no—
202C22×D53—42no—
203C22×D53C2×M1614yes—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—44no—
211C2×C4×C53—44no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—44no—
215C2×C4×C53—44no—
216C2×C4×C53—42no—
217C2×C4×C53#10014yes—
218C2×C4×C53C2×C22⋊C411yescharacteristic
219C2×C4×C53C2×C22⋊C411yescharacteristic
220C2×C4×C53C2×C22⋊C411yescharacteristic
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C22⋊D53—216no—
223C22⋊D53—216no—
224C2×Dic53—22no—
225C2×Dic53—24no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—22no—
229C2×Dic53—41no—
230C2×Dic53—42no—
231C2×Dic53—41no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53C2×C22⋊C412yes—
236C2×Dic53C2×C22⋊C412yes—
237C2×Dic53C2×C22⋊C412yes—
238C2×Dic53C2×M1614yes—
239C2×Dic53C22×D412yes—
240C4×D53—44no—
241C4×D53—44no—
242C4×D53—44no—
243C4×D53—44no—
244#88324—48no—
245#17364—531nomaximal, Sylow
246C22×C53—28no—
247C22×C53—21no—
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—24no—
254C22×C53—44no—
255C22×C53—44no—
256C22×C53—48no—
257C22×C53—48no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53#41012yes—
261C22×C53C2×#10012yes—
262C22×C53C22×C22⋊C411yescharacteristic
263C22×C53C22×M1612yes—
264C2×D53—24no—
265C2×D53—44no—
266C2×D53—48no—
267C2×D53—48no—
268C2×D53—42no—
269C2×D53—42no—
270C2×D53—44no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—42no—
274C4×C53C2×#10012yes—
275C4×C53C2×#10012yes—
276Dic53—22no—
277Dic53—22no—
278Dic53—22no—
279Dic53—42no—
280Dic53#40714yes—
281C22×C4×D4—531no—
282#2920—532no—
283#1466—534no—
284#1462—532no—
285#1462—532no—
286#1460—534no—
287C2×C53—22no—
288C2×C53—22no—
289C2×C53—22no—
290C2×C53—44no—
291C2×C53C2×#40711yescharacteristic
292C2×C53C2×#41011yescharacteristic
293C2×C53C22×#10011yescharacteristic
294D53—44no—
295C23×D4—531no—
296C24×C4—531no—
297C24×C4#615711yescharacteristic
298C2×C4×D4—532no—
299C2×C4×D4—532no—
300C2×C4×D4—532no—
301C2×C4×D4—532no—
302C2×C4×D4—1062no—
303C2×C4×D4—1062no—
304C2×C4×D4—1062no—
305C2×C4×D4—1062no—
306C22×C4⋊C4—531no—
307C22×C22⋊C4—531no—
308C22×C22⋊C4—531no—
309C22×C42—531no—
310#432—534no—
311#432—1062no—
312#431—534no—
313#431—1062no—
314#423—1062no—
315#407—534no—
316C4×M16—1062no—
317#323—538no—
318#323—538no—
319C53#1736411yescharacteristic, complemented, Sylow
320C25—531no—
321C25C2×#615711yescharacteristic
322C22×D4—532no—
323C22×D4—532no—
324C22×D4—1062no—
325C22×D4—1062no—
326C22×D4—1068no—
327C23×C4—21no—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—21no—
331C23×C4—44no—
332C23×C4—531no—
333C23×C4—531no—
334C23×C4—531no—
335C23×C4—532no—
336C23×C4—532no—
337C23×C4—1061no—
338C23×C4—1061no—
339C23×C4—1061no—
340C23×C4—1061no—
341C23×C4—2124no—
342C23×C4C2×#615711yescharacteristic
343C23×C4C2×#615712yes—
344C23×C4C2×#615712yes—
345C23×C4C2×#615711yescharacteristic
346C2×M16—1064no—
347C4×D4—1064no—
348C4×D4—1064no—
349C4×D4—1064no—
350C4×D4—1064no—
351C4×D4—1064no—
352C4×D4—1064no—
353C4×D4—1064no—
354C4×D4—1064no—
355C2×C4⋊C4—532no—
356C2×C4⋊C4—532no—
357C2×C4⋊C4—1062no—
358C2×C4⋊C4—1062no—
359C2×C22⋊C4—532no—
360C2×C22⋊C4—532no—
361C2×C22⋊C4—532no—
362C2×C22⋊C4—532no—
363C2×C22⋊C4—1062no—
364C2×C22⋊C4—1062no—
365C2×C22⋊C4—1062no—
366C2×C22⋊C4—1062no—
367C2×C42—532no—
368C2×C42—532no—
369C2×C42—532no—
370C2×C42—532no—
371C2×C42—531no—
372C2×C42—531no—
373C2×C42—531no—
374C2×C42—531no—
375C4⋊C8—1064no—
376C4⋊C8—1064no—
377C22⋊C8—538no—
378C22⋊C8—1064no—
379#97—1064no—
380C4×C8—1064no—
381C24—21no—
382C24—21no—
383C24—24no—
384C24—42no—
385C24—42no—
386C24—532no—
387C24—1064no—
388C24—1061no—
389C24—1061no—
390C24—2122no—
391C24—2122no—
392C24C22×#615711yescharacteristic
393C24C22×#615712yes—
394C2×D4—538no—
395C2×D4—1064no—
396C2×D4—1064no—
397C2×D4—1064no—
398C2×D4—1064no—
399C2×D4—10616no—
400C2×D4—10616no—
401C2×D4—1064no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—531no—
417C22×C4—532no—
418C22×C4—532no—
419C22×C4—532no—
420C22×C4—531no—
421C22×C4—532no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—532no—
425C22×C4—1064no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—2124no—
433C22×C4—2124no—
434C22×C4—2124no—
435C22×C4—2124no—
436C22×C4—2124no—
437C22×C4—2122no—
438C22×C4—2124no—
439C22×C4—2122no—
440C22×C4C22⋊#615711yescharacteristic
441C22×C4C22⋊#615711yescharacteristic
442C22×C4C22⋊#615711yescharacteristic
443C22×C4C22⋊#615711yescharacteristic
444C22×C4C22×#615711yescharacteristic
445C22×C4C22×#615711yescharacteristic
446C22×C4C22×#615712yes—
447M16—2128no—
448C2×C8—1068no—
449C4⋊C4—1064no—
450C4⋊C4—1064no—
451C22⋊C4—1064no—
452C22⋊C4—1064no—
453C22⋊C4—1064no—
454C22⋊C4—1064no—
455C42—532no—
456C42—532no—
457C42—532no—
458C42—532no—
459C42—532no—
460C42—532no—
461C42—532no—
462C42—532no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—416no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—534no—
478C23—1064no—
479C23—1064no—
480C23—1062no—
481C23—1062no—
482C23—2122no—
483C23—2122no—
484C23—2122no—
485C23—21216no—
486C23—2122no—
487C23—2122no—
488C23—2122no—
489C23C2×C22⋊#615711yescharacteristic
490C23C2×C22⋊#615711yescharacteristic
491C23D53⋊M1614yes—
492C23C23×#615711yescharacteristic
493D4—10616no—
494D4—10616no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—532no—
509C2×C4—534no—
510C2×C4—532no—
511C2×C4—532no—
512C2×C4—532no—
513C2×C4—1064no—
514C2×C4—1064no—
515C2×C4—1064no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—2124no—
519C2×C4—2124no—
520C2×C4—2121no—
521C2×C4—2122no—
522C2×C4—2121no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4—2124no—
526C2×C4—2124no—
527C2×C4—2122no—
528C2×C4#1313327714yes—
529C2×C4C2×C22⋊#615711yescharacteristic
530C2×C4C2×C22⋊#615711yescharacteristic
531C2×C4C2×C22⋊#615711yescharacteristic
532C2×C4C2×C22⋊#615711yescharacteristic
533C8—2128no—
534C22—28no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—24no—
542C22—44no—
543C22—48no—
544C22—48no—
545C22—44no—
546C22—42no—
547C22—42no—
548C22—1064no—
549C22—2124no—
550C22—2128no—
551C22—2128no—
552C22—2124no—
553C22—2122no—
554C22—2122no—
555C22#1681160212yes—
556C22C2×#1313327712yes—
557C22C22×C22⋊#615711yescharacteristic
558C22C2×D53⋊M1612yes—
559C4—21no—
560C4—21no—
561C4—42no—
562C4—534no—
563C4—1062no—
564C4—1062no—
565C4—1062no—
566C4—2122no—
567C4C2×#1313327712yes—
568C4C2×#1313327712yes—
569C2—22no—
570C2—22no—
571C2—22no—
572C2—44no—
573C2—2124no—
574C2D53⋊#40711yescharacteristic
575C2C2×#1681160211yescharacteristic
576C2C22×#1313327711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC24×C4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC23×C22⋊D53C4
cosocleC22×Dic53C24
hypercenterC24×C4#6157
hypocenterC53#17364
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #20291391C2semidirect
2. #20289974C2semidirect
3. #20289962C2semidirect
4. #20289960C2semidirect
5. #20289954C2semidirect
7. #16811644C22semidirect
9. #16811640C22semidirect
14. #16811467C22semidirect
15. #16811466C22semidirect
133. C2×C22⋊D53M16semidirect
319. C53#17364semidirect

Maximal subgroups

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

subgroupindex
#202913912
#202899742
#202899622
#202899602
#202899542
C22×C4×C22⋊D532
#1736453