Finite Group Databasebeta

← #28319281

Subgroups — #28319281

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#20291397C212yesmaximal, complemented
2#20290616C212yesmaximal, complemented
3#20290580C212yesmaximal, complemented
4#20290579C212yesmaximal, complemented
5#20290548C212yesmaximal
6#20290465C212yesmaximal, complemented
7C4×#13133277C212yesmaximal, complemented
8C2×#16811095C211yesmaximal, characteristic
9#16811648—28no—
10#16811626C2214yescomplemented
11C2×#13133277C2214yes—
12C2×#13133277C2214yes—
13#16811595C2214yes—
14C4×#11719940C2214yescomplemented
15#16811503C2214yescomplemented
16#16811503C2214yescomplemented
17C22×D4×D53C2211yescharacteristic
18#16811095—28no—
19C2×C4×D212C411yescharacteristic
20C2×D4×Dic53C411yescharacteristic
21C2×#13133179C411yescharacteristic
22C2×C4×C22⋊D53C2211yescharacteristic
23C2×C4×C22⋊D53C2211yescharacteristic
24C2×#13133177C2211yescharacteristic
25C2×#13133177C2211yescharacteristic
26C2×#13133175C2211yescharacteristic
27C2×#13133174C2211yescharacteristic
28C22×#11719921C411yescharacteristic
29C22×#11719921C411yescharacteristic
30C2×C22⋊C4×D53C411yescharacteristic
31C2×C22⋊C4×D53C411yescharacteristic
32C2×C4×D4×C53C411yescharacteristic
33C2×#11719940C2318yes—
34#13133284—24no—
35C4⋊#88324—24no—
36#13133277—28no—
37#13133277—28no—
38#13133274—28no—
39#13133272—24no—
40C4×#88324—24no—
41C24×D53C2×C411yescharacteristic
42C24×D53C2×C411yescharacteristic
43C22×D212C2×C411yescharacteristic
44C22×C22⋊D53C2311yescharacteristic
45C22×C22⋊D53C2311yescharacteristic
46C2×D4×D53—24no—
47C2×D4×D53—24no—
48C2×D4×D53—24no—
49C23×Dic53C2×C411yescharacteristic
50C23×Dic53C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2×C411yescharacteristic
55C22×C4×D53C2311yescharacteristic
56C4×D212—24no—
57D4×Dic53—24no—
58#13133179—24no—
59C4×C22⋊D53—24no—
60C4×C22⋊D53—24no—
61#13133177—24no—
62#13133177—24no—
63#13133175D418yes—
64#13133174—24no—
65C2×C4⋊Dic53C2×C411yescharacteristic
66C2×#11719919C2311yescharacteristic
67C2×#11719919C2311yescharacteristic
68C2×C22⋊Dic53C2×C411yescharacteristic
69C2×C22⋊Dic53C2×C411yescharacteristic
70C2×#11719921—24no—
71C2×#11719921—24no—
72C2×#11719921C2×C411yescharacteristic
73C2×#11719921C2×C411yescharacteristic
74C2×#11719921C2×C411yescharacteristic
75C2×#11719921C2×C411yescharacteristic
76C2×#11719921C2×C411yescharacteristic
77C2×#11719921C2×C411yescharacteristic
78C2×#11719921C2×C411yescharacteristic
79C2×#11719921C2×C411yescharacteristic
80C22⋊C4×D53—24no—
81C22⋊C4×D53—24no—
82C2×C4×Dic53C2×C411yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C22×D4×C53C2×C411yescharacteristic
86C23×C4×C53C2×C411yescharacteristic
87C23×C4×C53C2×C411yescharacteristic
88C4×D4×C53—24no—
89C2×C4⋊C4×C53C2×C411yescharacteristic
90C2×C22⋊C4×C53C2×C411yescharacteristic
91C2×C22⋊C4×C53C2×C411yescharacteristic
92C2×C42×C53C2×C411yescharacteristic
93C24×C53C22×C411yescharacteristic
94C24×C53C22×C411yescharacteristic
95C23×D53—22no—
96C23×D53—22no—
97C23×D53—22no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—81no—
102C23×D53—81no—
103C23×D53—81no—
104C23×D53—81no—
105C23×D53C22×C411yescharacteristic
106C23×D53C22×C411yescharacteristic
107C23×D53C22×C411yescharacteristic
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—81no—
114C22×C4×C53—81no—
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C22×C411yescharacteristic
117C22×C4×C53C22×C411yescharacteristic
118C22×C4×C53C22×C411yescharacteristic
119C22×C4×C53C22×C411yescharacteristic
120C2×C22⋊D53—22no—
121C2×C22⋊D53—22no—
122C2×C22⋊D53—22no—
123C2×C22⋊D53—22no—
124C2×C22⋊D53#4114yes—
125C2×C22⋊D53#4114yes—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—81no—
129C22×Dic53—81no—
130C22×Dic53C22×C411yescharacteristic
131C22×Dic53C22×C411yescharacteristic
132C22×Dic53C22×C411yescharacteristic
133C22×Dic53C22×C411yescharacteristic
134C22×Dic53C2411yescharacteristic
135D4×D53—232no—
136C2×D212—22no—
137C2×D212—22no—
138C2×D212—22no—
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—81no—
146C2×C4×D53—81no—
147C2×C4×D53—81no—
148C2×C4×D53—81no—
149C2×C4×D53C2×D414yes—
150#11719940—28no—
151#11719940—28no—
152C2×#88324—28no—
153C4⋊C4×C53C22⋊C414yes—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C42×C53—22no—
157C22⋊Dic53—22no—
158C22⋊Dic53—22no—
159#11719921—22no—
160#11719921—22no—
161#11719921—22no—
162#11719921—22no—
163#11719921—22no—
164#11719921—22no—
165#11719921C22⋊C414yes—
166#11719921C22⋊C414yes—
167C4⋊Dic53C22⋊C414yes—
168#11719919—22no—
169#11719919—22no—
170C4×Dic53—22no—
171C4×Dic53C2×D414yes—
172C4×Dic53C2×D414yes—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53#11712yes—
182C23×C53#11712yes—
183C23×C53C23×C411yescharacteristic
184C22×D53—21no—
185C22×D53—21no—
186C22×D53—21no—
187C22×D53—21no—
188C22×D53—21no—
189C22×D53—21no—
190C22×D53—28no—
191C22×D53—28no—
192C22×D53—41no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—82no—
199C22×D53—82no—
200C22×D53—82no—
201C22×D53—82no—
202C22×D53—84no—
203C22×D53—84no—
204C22×D53—84no—
205C22×D53—84no—
206C22×D53C2×C22⋊C412yes—
207C22×D53#11712yes—
208C22×D53#11712yes—
209D4×C53—28no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—82no—
219C2×C4×C53—82no—
220C2×C4×C53C2×C22⋊C412yes—
221C2×C4×C53C2×C22⋊C412yes—
222C2×C4×C53C2×C22⋊C412yes—
223C22⋊D53—28no—
224C22⋊D53—28no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—82no—
232C2×Dic53—82no—
233C2×Dic53C2×C22⋊C412yes—
234C2×Dic53C2×C22⋊C412yes—
235C2×Dic53C22×D412yes—
236C2×Dic53C2×#4112yes—
237C2×Dic53C2×#4112yes—
238D212—28no—
239C4×D53—28no—
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—42no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53C22×C22⋊C411yescharacteristic
259C22×C53C2×#11711yescharacteristic
260C22×C53C2×#11711yescharacteristic
261C2×D53—22no—
262C2×D53—22no—
263C2×D53—22no—
264C2×D53—22no—
265C2×D53—22no—
266C2×D53—42no—
267C2×D53—42no—
268C2×D53—42no—
269C2×D53—42no—
270C2×D53—41no—
271C2×D53—41no—
272C2×D53—41no—
273C2×D53—41no—
274C2×D53—42no—
275C2×D53—42no—
276C2×D53—88no—
277C2×D53—88no—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—22no—
281C4×C53—41no—
282C4×C53—41no—
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#228011yescharacteristic
301C2×C53#288811yescharacteristic
302C2×C53#288811yescharacteristic
303D53—42no—
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#615711yescharacteristic
316C2×#117—531no—
317C2×#117—531no—
318C22×C22⋊C4—531no—
319C22×C22⋊C4—531no—
320C22×C22⋊C4—531no—
321C22×C22⋊C4—531no—
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—
332C25—531no—
333C22×D4—531no—
334C22×D4—531no—
335C22×D4—531no—
336C22×D4—1064no—
337C22×D4—1064no—
338C22×D4—1064no—
339C22×D4C2×#615711yescharacteristic
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—24no—
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⋊C4C2×#615711yescharacteristic
364C2×C22⋊C4—531no—
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—531no—
374C2×C22⋊C4—1064no—
375C2×C22⋊C4—1064no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—1064no—
378C2×C22⋊C4C2×#615711yescharacteristic
379C2×C22⋊C4C2×#615711yescharacteristic
380C2×C42—531no—
381C2×C42—531no—
382C2×C42—531no—
383C2×C42C2×#615711yescharacteristic
384C4⋊C8—1064no—
385C4⋊C8—1064no—
386#100—1068no—
387#100—1068no—
388C22⋊C8—1068no—
389#97—1064no—
390C4×C8—1064no—
391C24—531no—
392C24—531no—
393C24—531no—
394C24—1062no—
395C24—1062no—
396C24—1062no—
397C24—1062no—
398C24—1062no—
399C24—1062no—
400C24—4241no—
401C24—4241no—
402C24—4241no—
403C24—4241no—
404C24C22×#615711yescharacteristic
405C24C22×#615711yescharacteristic
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—534no—
410C2×D4—534no—
411C2×D4—1062no—
412C2×D4—1062no—
413C2×D4—1062no—
414C2×D4—1062no—
415C2×D4—1062no—
416C2×D4—1062no—
417C2×D4—10632no—
418C2×D4—1062no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—534no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—4241no—
438C22×C4—4241no—
439C22×C4—4241no—
440C22×C4—4241no—
441C22×C4—4241no—
442C22×C4—4241no—
443C22×C4C22×#615711yescharacteristic
444C22×C4C22×#615711yescharacteristic
445C22×C4C22×#615711yescharacteristic
446C22×C4C22×#615711yescharacteristic
447C22×C4C22×#615711yescharacteristic
448M16—1068no—
449M16—1068no—
450C2×C8—1068no—
451C4⋊C4—534no—
452C4⋊C4—1062no—
453C4⋊C4—1062no—
454C4⋊C4C22⋊#615714yes—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—534no—
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⋊C4—1062no—
467C42—22no—
468C42—534no—
469C42—534no—
470C42—1062no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—532no—
480C23—532no—
481C23—532no—
482C23—1068no—
483C23—1068no—
484C23—1061no—
485C23—1061no—
486C23—1061no—
487C23—1061no—
488C23—1061no—
489C23—1061no—
490C23—4242no—
491C23—4242no—
492C23—4242no—
493C23—4242no—
494C23—4244no—
495C23—4244no—
496C23—4244no—
497C23—4244no—
498C23—2121no—
499C23—2121no—
500C23—2121no—
501C23—2121no—
502C23—2121no—
503C23—2121no—
504C23#1313329612yes—
505C23#1313329612yes—
506C23C23×#615711yescharacteristic
507D4—28no—
508D4—1068no—
509D4—1068no—
510D4—1068no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—532no—
525C2×C4—532no—
526C2×C4—1068no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—4242no—
530C2×C4—4242no—
531C2×C4—4242no—
532C2×C4—4242no—
533C2×C4—4242no—
534C2×C4—4242no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4C2×C22⋊#615712yes—
540C2×C4C2×C22⋊#615712yes—
541C2×C4C2×C22⋊#615712yes—
542C8—2128no—
543C22—21no—
544C22—21no—
545C22—22no—
546C22—22no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—1062no—
556C22—1062no—
557C22—1062no—
558C22—1062no—
559C22—1062no—
560C22—4248no—
561C22—4248no—
562C22—2122no—
563C22—2122no—
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—22no—
578C4—41no—
579C4—41no—
580C4—534no—
581C4—1062no—
582C4—1062no—
583C4—2121no—
584C4—2121no—
585C2—21no—
586C2—21no—
587C2—42no—
588C2—42no—
589C2—2122no—
590C2—2122no—
591C2—2122no—
592C2#2029115911yescharacteristic
593C2#2029132511yescharacteristic
594C2#2029132611yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×C4×D4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×D53C22
cosocleC22×Dic53C24
hypercenterC2×C4×D4#6157
hypocenterC53#20729
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20291397C2semidirect
2. #20290616C2semidirect
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
#202913972
#202906162
#202905802
#202905792
#202905482
#202904652
C4×#131332772
C2×#168110952
#2072953