Finite Group Databasebeta

← #28319268

Subgroups — #28319268

Order 13568.

1694 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291391C212yesmaximal, complemented
2#20290614C212yesmaximal, complemented
3#20290579C214yesmaximal, complemented
4#20290547C212yesmaximal
5C4×#13133277C214yesmaximal, complemented
6C22×C4×C22⋊D53C211yesmaximal, characteristic
7#16811644—22no—
8#16811643—22no—
9#16811640—22no—
10#16811639—22no—
11#16811626C2214yescomplemented
12C2×#13133277C2218yes—
13#16811593C2214yes—
14C4×#11719940C2214yescomplemented
15#16811503C2218yescomplemented
16C23×C22⋊D53C2211yescharacteristic
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×#11719919C2211yescharacteristic
27C22×C22⋊Dic53C411yescharacteristic
28C22×#11719921C411yescharacteristic
29C22×C4×Dic53C2211yescharacteristic
30C24×C4×C53C411yescharacteristic
31C2×#11719940C2318yes—
32#13133284—24no—
33C4⋊#88324—24no—
34#13133277—216no—
35C22⋊#88324—24no—
36C22⋊#88324—24no—
37#13133272—24no—
38C4×#88324—24no—
39C24×D53C2×C411yescharacteristic
40C22×C22⋊D53—28no—
41C22×C22⋊D53—22no—
42C22×C22⋊D53—22no—
43C22×C22⋊D53C2×C412yes—
44C22×C22⋊D53C2312yes—
45C23×Dic53C2×C411yescharacteristic
46C23×Dic53C2×C411yescharacteristic
47C23×Dic53C2311yescharacteristic
48C22×C4×D53—21no—
49C22×C4×D53—21no—
50C22×C4×D53—21no—
51C22×C4×D53—21no—
52C22×C4×D53—44no—
53C22×C4×D53C2×C412yes—
54C22×C4×D53C2×C412yes—
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—
62C4×C22⋊D53—24no—
63C2×#11719919—22no—
64C2×#11719919—22no—
65C2×#11719919C2×C412yes—
66C2×#11719919C2312yes—
67C2×C22⋊Dic53—22no—
68C2×C22⋊Dic53—22no—
69C2×C22⋊Dic53C2×C412yes—
70C2×C22⋊Dic53C2×C412yes—
71C2×#11719921—22no—
72C2×#11719921—22no—
73C2×#11719921C2×C412yes—
74C2×#11719921C2×C412yes—
75C2×C4×Dic53C2×C411yescharacteristic
76C2×C4×Dic53C2×C411yescharacteristic
77C2×C4×Dic53D412yes—
78C2×C4×Dic53D412yes—
79C2×C4×Dic53D412yes—
80C2×C4×Dic53D412yes—
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83C25×C53C2×C411yescharacteristic
84C23×C4×C53—21no—
85C23×C4×C53—21no—
86C23×C4×C53—21no—
87C23×C4×C53—21no—
88C23×C4×C53—44no—
89C23×C4×C53C2×C411yescharacteristic
90C23×C4×C53C2×C411yescharacteristic
91C23×C4×C53C2×C412yes—
92C23×C4×C53C2×C412yes—
93C24×C53—21no—
94C24×C53—21no—
95C24×C53—24no—
96C24×C53—42no—
97C24×C53—42no—
98C24×C53C22×C411yescharacteristic
99C24×C53C22×C412yes—
100C23×D53—21no—
101C23×D53—21no—
102C23×D53—24no—
103C23×D53—42no—
104C23×D53—42no—
105C23×D53C22×C412yes—
106C22×C4×C53—24no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—44no—
114C22×C4×C53—44no—
115C22×C4×C53—44no—
116C22×C4×C53—44no—
117C22×C4×C53—44no—
118C22×C4×C53—44no—
119C22×C4×C53—42no—
120C22×C4×C53C22⋊C411yescharacteristic
121C22×C4×C53C22⋊C411yescharacteristic
122C22×C4×C53C22⋊C411yescharacteristic
123C22×C4×C53C22⋊C411yescharacteristic
124C22×C4×C53C22×C411yescharacteristic
125C22×C4×C53C22×C411yescharacteristic
126C22×C4×C53C22×C412yes—
127C2×C22⋊D53—216no—
128C2×C22⋊D53—216no—
129C2×C22⋊D53—24no—
130C2×C22⋊D53—24no—
131C2×C22⋊D53—24no—
132C2×C22⋊D53—24no—
133C2×C22⋊D53—24no—
134C2×C22⋊D53#4118yes—
135C22×Dic53—24no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53C22⋊C412yes—
141C22×Dic53C22⋊C412yes—
142C22×Dic53C22×C411yescharacteristic
143C22×Dic53C22×C412yes—
144C22×Dic53C22×C411yescharacteristic
145C22×Dic53C22×C411yescharacteristic
146C22×Dic53C2×D412yes—
147C22×Dic53C2×D412yes—
148C22×Dic53C2411yescharacteristic
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—44no—
154C2×C4×D53—44no—
155C2×C4×D53—44no—
156C2×C4×D53—44no—
157C2×C4×D53—44no—
158C2×C4×D53—44no—
159#11719940—216no—
160C2×#88324—28no—
161C22⋊Dic53—24no—
162C22⋊Dic53—24no—
163#11719921—24no—
164#11719921—24no—
165#11719919—24no—
166#11719919—24no—
167C4×Dic53C22⋊C412yes—
168C4×Dic53C22⋊C412yes—
169C4×Dic53C22⋊C412yes—
170C4×Dic53C22⋊C412yes—
171C4×Dic53C2×D412yes—
172C4×Dic53C2×D412yes—
173C4×Dic53C2×D412yes—
174C4×Dic53C2×D412yes—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—24no—
179C23×C53—24no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—416no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53C2×C22⋊C411yescharacteristic
190C23×C53C2×C22⋊C411yescharacteristic
191C23×C53#11714yes—
192C23×C53C23×C411yescharacteristic
193C22×D53—24no—
194C22×D53—24no—
195C22×D53—22no—
196C22×D53—22no—
197C22×D53—42no—
198C22×D53—416no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53—42no—
202C22×D53—42no—
203C22×D53—42no—
204C22×D53#11714yes—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—44no—
212C2×C4×C53—44no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—44no—
216C2×C4×C53—44no—
217C2×C4×C53—42no—
218C2×C4×C53#10014yescomplemented
219C2×C4×C53C2×C22⋊C411yescharacteristic
220C2×C4×C53C2×C22⋊C411yescharacteristic
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C2×C4×C53C2×C22⋊C411yescharacteristic
223C22⋊D53—216no—
224C22⋊D53—216no—
225C2×Dic53—22no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—24no—
229C2×Dic53—22no—
230C2×Dic53—42no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53C2×C22⋊C412yes—
237C2×Dic53C2×C22⋊C412yes—
238C2×Dic53C2×C22⋊C412yes—
239C2×Dic53C22×D412yes—
240C2×Dic53C2×#4114yes—
241C4×D53—44no—
242C4×D53—44no—
243C4×D53—44no—
244C4×D53—44no—
245#88324—48no—
246#20726—531nomaximal, Sylow
247C22×C53—28no—
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—24no—
255C22×C53—44no—
256C22×C53—44no—
257C22×C53—48no—
258C22×C53—48no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53C2×#10012yes—
262C22×C53#41312yes—
263C22×C53C22×C22⋊C411yescharacteristic
264C22×C53C2×#11712yes—
265C2×D53—24no—
266C2×D53—44no—
267C2×D53—48no—
268C2×D53—48no—
269C2×D53—42no—
270C2×D53—42no—
271C2×D53—44no—
272C4×C53—21no—
273C4×C53—21no—
274C4×C53—42no—
275C4×C53C2×#10012yes—
276C4×C53C2×#10012yes—
277Dic53—22no—
278Dic53—22no—
279Dic53—22no—
280Dic53—42no—
281Dic53#52514yes—
282C22×C4×D4—531no—
283#2920—532no—
284#1821—532no—
285#1800—534no—
286#1783—532no—
287C4×#100—534no—
288C2×C53—22no—
289C2×C53—22no—
290C2×C53—22no—
291C2×C53—44no—
292C2×C53#228011yescharacteristic
293C2×C53C22×#10011yescharacteristic
294C2×C53C2×#41311yescharacteristic
295D53—44no—
296C23×D4—531no—
297C24×C4—531no—
298C24×C4#615711yescharacteristic
299C2×C4×D4—532no—
300C2×C4×D4—532no—
301C2×C4×D4—532no—
302C2×C4×D4—532no—
303C2×C4×D4—1062no—
304C2×C4×D4—1062no—
305C2×C4×D4—1062no—
306C2×C4×D4—1062no—
307C22×C4⋊C4—531no—
308C22×C22⋊C4—531no—
309C22×C22⋊C4—531no—
310C22×C42—531no—
311#432—1062no—
312#432—1062no—
313#431—1062no—
314#431—1062no—
315#423—534no—
316C2×#100—538no—
317#407—534no—
318C4×M16—534no—
319#343—538no—
320C53#2072611yescharacteristic, complemented, Sylow
321C25—531no—
322C25C2×#615711yescharacteristic
323C22×D4—532no—
324C22×D4—532no—
325C22×D4—1062no—
326C22×D4—1062no—
327C22×D4—1068no—
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—21no—
331C23×C4—21no—
332C23×C4—44no—
333C23×C4—531no—
334C23×C4—531no—
335C23×C4—531no—
336C23×C4—532no—
337C23×C4—532no—
338C23×C4—1061no—
339C23×C4—1061no—
340C23×C4—1061no—
341C23×C4—1061no—
342C23×C4—2124no—
343C23×C4C2×#615711yescharacteristic
344C23×C4C2×#615712yes—
345C23×C4C2×#615712yes—
346C23×C4C2×#615711yescharacteristic
347C2×M16—538no—
348C4×D4—1064no—
349C4×D4—1064no—
350C4×D4—1064no—
351C4×D4—1064no—
352C4×D4—1064no—
353C4×D4—1064no—
354C4×D4—1064no—
355C4×D4—1064no—
356C2×C4⋊C4—532no—
357C2×C4⋊C4—532no—
358C2×C4⋊C4—1062no—
359C2×C4⋊C4—1062no—
360C2×C22⋊C4—532no—
361C2×C22⋊C4—532no—
362C2×C22⋊C4—532no—
363C2×C22⋊C4—532no—
364C2×C22⋊C4—1062no—
365C2×C22⋊C4—1062no—
366C2×C22⋊C4—1062no—
367C2×C22⋊C4—1062no—
368C2×C42—532no—
369C2×C42—532no—
370C2×C42—532no—
371C2×C42—532no—
372C2×C42—531no—
373C2×C42—531no—
374C2×C42—531no—
375C2×C42—531no—
376C4⋊C8—1064no—
377C4⋊C8—1064no—
378#100—10616no—
379C22⋊C8—1064no—
380C22⋊C8—1064no—
381#97—1064no—
382C4×C8—1064no—
383C24—21no—
384C24—21no—
385C24—24no—
386C24—42no—
387C24—42no—
388C24—532no—
389C24—1064no—
390C24—1061no—
391C24—1061no—
392C24—2122no—
393C24—2122no—
394C24C22×#615711yescharacteristic
395C24C22×#615712yes—
396C2×D4—538no—
397C2×D4—1064no—
398C2×D4—1064no—
399C2×D4—1064no—
400C2×D4—1064no—
401C2×D4—10616no—
402C2×D4—10616no—
403C2×D4—1064no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—531no—
419C22×C4—532no—
420C22×C4—532no—
421C22×C4—532no—
422C22×C4—531no—
423C22×C4—532no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—532no—
427C22×C4—1064no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—2124no—
435C22×C4—2124no—
436C22×C4—2124no—
437C22×C4—2124no—
438C22×C4—2124no—
439C22×C4—2122no—
440C22×C4—2124no—
441C22×C4—2122no—
442C22×C4C22⋊#615711yescharacteristic
443C22×C4C22⋊#615711yescharacteristic
444C22×C4C22⋊#615711yescharacteristic
445C22×C4C22⋊#615711yescharacteristic
446C22×C4C22×#615711yescharacteristic
447C22×C4C22×#615711yescharacteristic
448C22×C4C22×#615712yes—
449M16—10616no—
450C2×C8—1068no—
451C4⋊C4—1064no—
452C4⋊C4—1064no—
453C22⋊C4—1064no—
454C22⋊C4—1064no—
455C22⋊C4—1064no—
456C22⋊C4—1064no—
457C42—532no—
458C42—532no—
459C42—532no—
460C42—532no—
461C42—532no—
462C42—532no—
463C42—532no—
464C42—532no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—416no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—534no—
480C23—1064no—
481C23—1064no—
482C23—1062no—
483C23—1062no—
484C23—2122no—
485C23—2122no—
486C23—2122no—
487C23—21216no—
488C23—2122no—
489C23—2122no—
490C23—2122no—
491C23C2×C22⋊#615711yescharacteristic
492C23C2×C22⋊#615711yescharacteristic
493C23#1313329614yes—
494C23C23×#615711yescharacteristic
495D4—10616no—
496D4—10616no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—532no—
511C2×C4—534no—
512C2×C4—532no—
513C2×C4—532no—
514C2×C4—532no—
515C2×C4—1064no—
516C2×C4—1064no—
517C2×C4—1064no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—2124no—
521C2×C4—2124no—
522C2×C4—2122no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2122no—
526C2×C4—2122no—
527C2×C4—2124no—
528C2×C4—2124no—
529C2×C4—2122no—
530C2×C4#1313327714yescomplemented
531C2×C4C2×C22⋊#615711yescharacteristic
532C2×C4C2×C22⋊#615711yescharacteristic
533C2×C4C2×C22⋊#615711yescharacteristic
534C2×C4C2×C22⋊#615711yescharacteristic
535C8—2128no—
536C22—28no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—24no—
544C22—44no—
545C22—48no—
546C22—48no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—1064no—
551C22—2124no—
552C22—2128no—
553C22—2128no—
554C22—2124no—
555C22—2122no—
556C22—2122no—
557C22C2×#1313327712yes—
558C22#1681160912yes—
559C22C22×C22⋊#615711yescharacteristic
560C22C2×#1313329612yes—
561C4—21no—
562C4—21no—
563C4—42no—
564C4—534no—
565C4—1062no—
566C4—1062no—
567C4—1062no—
568C4—2122no—
569C4C2×#1313327712yes—
570C4C2×#1313327712yes—
571C2—22no—
572C2—22no—
573C2—22no—
574C2—44no—
575C2—2124no—
576C2#2029115911yescharacteristic
577C2C22×#1313327711yescharacteristic
578C2C2×#1681160911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC24×C4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC23×C22⋊D53C22
cosocleC22×Dic53C24
hypercenterC24×C4#6157
hypocenterC53#20726
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. #20290614C2semidirect
3. #20290579C2semidirect
5. C4×#13133277C2semidirect
11. #16811626C22semidirect
14. C4×#11719940C22semidirect
15. #16811503C22semidirect
218. C2×C4×C53#100semidirect
320. C53#20726semidirect
530. C2×C4#13133277semidirect

Maximal subgroups

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

subgroupindex
#202913912
#202906142
#202905792
#202905472
C4×#131332772
C22×C4×C22⋊D532
#2072653