Finite Group Databasebeta

← #28319280

Subgroups — #28319280

Order 13568.

1382 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291395C212yesmaximal, complemented
2#20290617C212yesmaximal, complemented
3#20290580C212yesmaximal, complemented
4#20290579C212yesmaximal, complemented
5#20290547C212yesmaximal
6#20290465C212yesmaximal, complemented
7C4×#13133277C212yesmaximal, complemented
8C2×#16811096C211yesmaximal, characteristic
9#16811647—28no—
10#16811626C2214yescomplemented
11C2×#13133277C2214yes—
12C2×#13133277C2214yes—
13#16811593C2214yes—
14C4×#11719940C2214yescomplemented
15#16811503C2214yescomplemented
16#16811503C2214yescomplemented
17C23×C22⋊D53C2211yescharacteristic
18#16811096—28no—
19C2×C4×C22⋊D53C411yescharacteristic
20C2×C4×C22⋊D53C411yescharacteristic
21C2×C4×C22⋊D53C2211yescharacteristic
22C2×C4×C22⋊D53C2211yescharacteristic
23C2×#13133177C411yescharacteristic
24C2×#13133177C411yescharacteristic
25C2×#13133177C2211yescharacteristic
26C2×#13133177C2211yescharacteristic
27C2×#13133173C2211yescharacteristic
28C2×#13133173C2211yescharacteristic
29C22×C22⋊Dic53C411yescharacteristic
30C2×C22⋊C4×D53C411yescharacteristic
31C2×C22⋊C4×D53C411yescharacteristic
32C22×C22⋊C4×C53C411yescharacteristic
33C2×#11719940C2318yes—
34#13133284—24no—
35C4⋊#88324—24no—
36#13133277—28no—
37#13133277—28no—
38C22⋊#88324—28no—
39#13133272—24no—
40C4×#88324—24no—
41C24×D53C2×C411yescharacteristic
42C22×C22⋊D53—24no—
43C22×C22⋊D53—24no—
44C22×C22⋊D53—24no—
45C22×C22⋊D53C2×C411yescharacteristic
46C22×C22⋊D53C2×C411yescharacteristic
47C22×C22⋊D53C2311yescharacteristic
48C22×C22⋊D53C2311yescharacteristic
49C23×Dic53C2×C411yescharacteristic
50C23×Dic53C2×C411yescharacteristic
51C23×Dic53C2311yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2×C411yescharacteristic
55C22×C4×D53C2×C411yescharacteristic
56C4×C22⋊D53—24no—
57C4×C22⋊D53—24no—
58C4×C22⋊D53—24no—
59C4×C22⋊D53—24no—
60#13133177—24no—
61#13133177—24no—
62#13133177—24no—
63#13133177—24no—
64#13133173—24no—
65#13133173D418yes—
66C2×#11719919C2×C411yescharacteristic
67C2×#11719919C2×C411yescharacteristic
68C2×#11719919C2311yescharacteristic
69C2×#11719919C2311yescharacteristic
70C2×C22⋊Dic53—24no—
71C2×C22⋊Dic53C2×C411yescharacteristic
72C2×C22⋊Dic53C2×C411yescharacteristic
73C2×C22⋊Dic53C2×C411yescharacteristic
74C2×C22⋊Dic53C2×C411yescharacteristic
75C2×#11719921C2×C411yescharacteristic
76C2×#11719921C2×C411yescharacteristic
77C2×#11719921C2×C411yescharacteristic
78C2×#11719921C2×C411yescharacteristic
79C22⋊C4×D53—24no—
80C22⋊C4×D53—24no—
81C2×C4×Dic53C2×C411yescharacteristic
82C2×C4×Dic53C2×C411yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C25×C53C2×C411yescharacteristic
86C23×C4×C53C2×C411yescharacteristic
87C23×C4×C53C2×C411yescharacteristic
88C2×C22⋊C4×C53—24no—
89C2×C22⋊C4×C53C2×C411yescharacteristic
90C2×C22⋊C4×C53C2×C411yescharacteristic
91C2×C22⋊C4×C53C2×C411yescharacteristic
92C2×C22⋊C4×C53C2×C411yescharacteristic
93C24×C53—22no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—81no—
97C24×C53—81no—
98C24×C53C22×C411yescharacteristic
99C24×C53C22×C411yescharacteristic
100C24×C53C22×C411yescharacteristic
101C23×D53—22no—
102C23×D53—22no—
103C23×D53—22no—
104C23×D53—81no—
105C23×D53—81no—
106C23×D53C22×C411yescharacteristic
107C23×D53C22×C411yescharacteristic
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—81no—
111C22×C4×C53—81no—
112C22×C4×C53C22×C411yescharacteristic
113C22×C4×C53C22×C411yescharacteristic
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C22×C411yescharacteristic
116C2×C22⋊D53—232no—
117C2×C22⋊D53—22no—
118C2×C22⋊D53—22no—
119C2×C22⋊D53—22no—
120C2×C22⋊D53—22no—
121C2×C22⋊D53—22no—
122C2×C22⋊D53—22no—
123C2×C22⋊D53—22no—
124C2×C22⋊D53—22no—
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53#4114yes—
128C2×C22⋊D53#4114yes—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—81no—
134C22×Dic53—81no—
135C22×Dic53C22×C411yescharacteristic
136C22×Dic53C22×C411yescharacteristic
137C22×Dic53C22×C411yescharacteristic
138C22×Dic53C22×C411yescharacteristic
139C22×Dic53C22×C411yescharacteristic
140C22×Dic53C2×D414yes—
141C22×Dic53C2411yescharacteristic
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—81no—
147C2×C4×D53—81no—
148C2×C4×D53—81no—
149C2×C4×D53—81no—
150#11719940—28no—
151#11719940—28no—
152C2×#88324—28no—
153C22⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C22⋊C4×C53C22⋊C414yes—
157C22⋊Dic53—22no—
158C22⋊Dic53—22no—
159C22⋊Dic53—22no—
160C22⋊Dic53C22⋊C414yes—
161#11719921—22no—
162#11719921—22no—
163#11719921—22no—
164#11719921—22no—
165#11719919—22no—
166#11719919—22no—
167#11719919C22⋊C414yes—
168#11719919C22⋊C414yes—
169C4×Dic53—22no—
170C4×Dic53—22no—
171C4×Dic53C2×D414yes—
172C4×Dic53C2×D414yes—
173C23×C53—28no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—41no—
181C23×C53—41no—
182C23×C53—41no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—82no—
187C23×C53—82no—
188C23×C53—84no—
189C23×C53—84no—
190C23×C53C2×C22⋊C412yes—
191C23×C53#11712yes—
192C23×C53#11712yes—
193C23×C53C23×C411yescharacteristic
194C22×D53—21no—
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—28no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—82no—
204C22×D53—82no—
205C22×D53—84no—
206C22×D53—84no—
207C22×D53#11712yes—
208C22×D53#11712yes—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—82no—
216C2×C4×C53—82no—
217C2×C4×C53C2×C22⋊C412yes—
218C2×C4×C53C2×C22⋊C412yes—
219C22⋊D53—28no—
220C22⋊D53—28no—
221C22⋊D53—28no—
222C22⋊D53—28no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—28no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—82no—
233C2×Dic53—82no—
234C2×Dic53C2×C22⋊C412yes—
235C2×Dic53C2×C22⋊C412yes—
236C2×Dic53C2×C22⋊C412yes—
237C2×Dic53C22×D412yes—
238C2×Dic53C2×#4112yes—
239C2×Dic53C2×#4112yes—
240C4×D53—82no—
241C4×D53—82no—
242C4×D53—82no—
243C4×D53—82no—
244#88324—48no—
245#20729—531nomaximal, Sylow
246C22×C53—21no—
247C22×C53—21no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—88no—
264C22×C53C22×C22⋊C411yescharacteristic
265C22×C53C2×#11711yescharacteristic
266C22×C53C2×#11711yescharacteristic
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—42no—
270C2×D53—42no—
271C2×D53—41no—
272C2×D53—41no—
273C2×D53—41no—
274C2×D53—41no—
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—88no—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—41no—
281C4×C53—41no—
282Dic53—22no—
283Dic53—22no—
284Dic53—22no—
285Dic53—41no—
286Dic53—41no—
287Dic53#52514yes—
288C2×#518—531no—
289#2921—532no—
290#1822—532no—
291#1800—532no—
292#1800—532no—
293#1783—532no—
294C4×#100—532no—
295C4×#100—532no—
296C2×C53—21no—
297C2×C53—21no—
298C2×C53—42no—
299C2×C53—42no—
300C2×C53—42no—
301C2×C53#228011yescharacteristic
302C2×C53#288811yescharacteristic
303C2×C53#288811yescharacteristic
304D53—42no—
305D53—42no—
306C23×D4—531no—
307#518—1068no—
308C2×C4×D4—531no—
309C2×C4×D4—531no—
310C2×C4×D4—531no—
311C2×C4×D4—531no—
312C2×C4×D4—531no—
313C2×C4×D4—531no—
314C2×C4×D4—531no—
315C2×C4×D4—531no—
316C2×#117—531no—
317C2×#117—531no—
318C22×C22⋊C4—531no—
319C22×C22⋊C4—531no—
320C22×C22⋊C4—531no—
321C22×C22⋊C4#615711yescharacteristic
322#433—1068no—
323#423—534no—
324C2×#100—534no—
325C2×#100—534no—
326#407—534no—
327C4×M16—534no—
328#343—534no—
329#343—534no—
330C53#2072911yescharacteristic, complemented, Sylow
331C25—531no—
332C25C2×#615711yescharacteristic
333C22×D4—531no—
334C22×D4—531no—
335C22×D4—531no—
336C22×D4—531no—
337C22×D4—1064no—
338C22×D4—1064no—
339C22×D4—1064no—
340C23×C4—531no—
341C23×C4—531no—
342C23×C4—531no—
343C23×C4—531no—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4C2×#615711yescharacteristic
348C23×C4C2×#615711yescharacteristic
349C2×M16—538no—
350C4×D4—1064no—
351C4×D4—1064no—
352C4×D4—1064no—
353C4×D4—1064no—
354C4×D4—1064no—
355C4×D4—1064no—
356C4×D4—1064no—
357C4×D4—1064no—
358#117—538no—
359#117—1064no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—531no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—531no—
366C2×C22⋊C4—531no—
367C2×C22⋊C4—531no—
368C2×C22⋊C4—531no—
369C2×C22⋊C4—531no—
370C2×C22⋊C4—531no—
371C2×C22⋊C4—531no—
372C2×C22⋊C4—531no—
373C2×C22⋊C4—1064no—
374C2×C22⋊C4—1064no—
375C2×C22⋊C4—1064no—
376C2×C22⋊C4C2×#615711yescharacteristic
377C2×C22⋊C4C2×#615711yescharacteristic
378C2×C22⋊C4C2×#615711yescharacteristic
379C2×C22⋊C4C2×#615711yescharacteristic
380C2×C42—531no—
381C2×C42—531no—
382C2×C42—531no—
383C2×C42—531no—
384C4⋊C8—1064no—
385C4⋊C8—1064no—
386#100—1068no—
387#100—1068no—
388C22⋊C8—1068no—
389#97—1064no—
390C4×C8—1064no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—81no—
395C24—81no—
396C24—531no—
397C24—531no—
398C24—1062no—
399C24—1062no—
400C24—1062no—
401C24—4241no—
402C24—4241no—
403C24C22×#615711yescharacteristic
404C24C22×#615711yescharacteristic
405C24C22×#615711yescharacteristic
406C2×D4—534no—
407C2×D4—534no—
408C2×D4—1062no—
409C2×D4—1062no—
410C2×D4—1062no—
411C2×D4—1062no—
412C2×D4—1062no—
413C2×D4—1062no—
414C2×D4—1062no—
415C2×D4—1062no—
416C2×D4—10632no—
417C2×D4—1062no—
418C2×D4—1062no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—531no—
424C22×C4—534no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—531no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—4241no—
439C22×C4—4241no—
440C22×C4—4241no—
441C22×C4—4241no—
442C22×C4—4241no—
443C22×C4—4241no—
444C22×C4C22×#615711yescharacteristic
445C22×C4C22×#615711yescharacteristic
446C22×C4C22×#615711yescharacteristic
447C22×C4C22×#615711yescharacteristic
448M16—1068no—
449M16—1068no—
450C2×C8—1068no—
451C4⋊C4—534no—
452C4⋊C4—534no—
453C4⋊C4—1062no—
454C4⋊C4—1062no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—534no—
459C22⋊C4—1062no—
460C22⋊C4—1062no—
461C22⋊C4—1062no—
462C22⋊C4—1062no—
463C22⋊C4—1062no—
464C22⋊C4—1062no—
465C22⋊C4—1062no—
466C22⋊C4C22⋊#615714yes—
467C42—534no—
468C42—534no—
469C42—1062no—
470C42—1062no—
471C23—28no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—82no—
485C23—82no—
486C23—84no—
487C23—84no—
488C23—532no—
489C23—532no—
490C23—1068no—
491C23—1061no—
492C23—1061no—
493C23—1061no—
494C23—1061no—
495C23—4242no—
496C23—4242no—
497C23—4244no—
498C23—4244no—
499C23—2121no—
500C23—2121no—
501C23—2121no—
502C23—2121no—
503C23C2×C22⋊#615712yes—
504C23#1313329612yes—
505C23#1313329612yes—
506C23C23×#615711yescharacteristic
507D4—1068no—
508D4—1068no—
509D4—1068no—
510D4—1068no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—532no—
520C2×C4—532no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—532no—
525C2×C4—1068no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—4242no—
531C2×C4—4242no—
532C2×C4—4242no—
533C2×C4—4242no—
534C2×C4—4242no—
535C2×C4—4242no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4C2×C22⋊#615712yes—
541C2×C4C2×C22⋊#615712yes—
542C8—2128no—
543C22—21no—
544C22—21no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—88no—
561C22—1062no—
562C22—1062no—
563C22—4248no—
564C22—2122no—
565C22—2122no—
566C22—2122no—
567C22—2122no—
568C22—2121no—
569C22—2121no—
570C22—2121no—
571C22—2121no—
572C22C22×C22⋊#615711yescharacteristic
573C22C2×#1313329611yescharacteristic
574C22C2×#1313329611yescharacteristic
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—534no—
580C4—1062no—
581C4—1062no—
582C4—1062no—
583C4—2121no—
584C4—2121no—
585C2—21no—
586C2—21no—
587C2—42no—
588C2—42no—
589C2—42no—
590C2—2122no—
591C2—2122no—
592C2#2029115911yescharacteristic
593C2#2029132511yescharacteristic
594C2#2029132511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C22⋊C4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC23×C22⋊D53C22
cosocleC22×Dic53C24
hypercenterC22×C22⋊C4#6157
hypocenterC53#20729
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20291395C2semidirect
2. #20290617C2semidirect
3. #20290580C2semidirect
4. #20290579C2semidirect
6. #20290465C2semidirect
7. C4×#13133277C2semidirect
10. #16811626C22semidirect
14. C4×#11719940C22semidirect
15. #16811503C22semidirect
16. #16811503C22semidirect
330. C53#20729semidirect

Maximal subgroups

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

subgroupindex
#202913952
#202906172
#202905802
#202905792
#202905472
#202904652
C4×#131332772
C2×#168110962
#2072953