Finite Group Databasebeta

← #27279657

Subgroups — #27279657

Order 13568.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20282997C211yesmaximal, characteristic
2C2×#16810358C212yesmaximal, complemented
3#20272938C211yesmaximal, characteristic, complemented
4#20272595C212yesmaximal, complemented
5C22×C22⋊C8×C53C211yesmaximal, characteristic, complemented
6C2×#13133191C411yescharacteristic, complemented
7C2×#13133191C411yescharacteristic, complemented
8C2×#13133191C2211yescharacteristic
9C2×#13133186—21no—
10C22×C22⋊Dic53—21no—
11#16810502—21no—
12#16810502—21no—
13#16810502—21no—
14#16810483C411yescharacteristic, complemented
15#16810483C411yescharacteristic, complemented
16#16810483C2211yescharacteristic
17C2×#13133124—21no—
18C2×#13133124C2212yes—
19#16810358—24no—
20C24×C4×C53C2211yescharacteristic
21C23×C8×C53—21no—
22C2×C22⋊C8×C53—21no—
23C2×C22⋊C8×C53—24no—
24C2×C22⋊C8×C53C2212yes—
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C22×C22⋊D53—21no—
28C22×C22⋊D53—21no—
29C23×Dic53—22no—
30#13133191—22no—
31#13133191—22no—
32#13133191—42no—
33#13133191—42no—
34#13133186—42no—
35#13133186—42no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C22⋊Dic53—24no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—42no—
44C2×C22⋊Dic53—42no—
45C2×#11719921—21no—
46C2×#11719921—21no—
47C2×#11719921—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—22no—
52#13133124—22no—
53#13133124—42no—
54C25×C53D411yescharacteristic
55C23×C4×C53—22no—
56C23×C4×C53—22no—
57C23×C4×C53—42no—
58C23×C4×C53—42no—
59C23×C4×C53C2×C411yescharacteristic
60C23×C4×C53C2×C411yescharacteristic
61C23×C4×C53D411yescharacteristic
62C23×C4×C53D411yescharacteristic
63C23×C4×C53D411yescharacteristic
64C23×C4×C53C2311yescharacteristic
65C22×C8×C53—24no—
66C22×C8×C53—22no—
67C22×C8×C53—42no—
68C22×C8×C53—42no—
69C22⋊C8×C53—24no—
70C22⋊C8×C53—22no—
71C22⋊C8×C53D414yescomplemented
72C24×C53—22no—
73C24×C53—22no—
74C24×C53—22no—
75C24×C53—22no—
76C24×C53—41no—
77C24×C53—41no—
78C24×C53C22⋊C411yescharacteristic
79C24×C53C22⋊C411yescharacteristic
80C24×C53D812yes—
81C24×C53SD1612yes—
82C24×C53C2×D411yescharacteristic
83C23×D53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—48no—
104C22×C4×C53—48no—
105C22×C4×C53C22⋊C411yescharacteristic
106C22×C4×C53C22⋊C411yescharacteristic
107C22×C4×C53C22×C411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D412yes—
110C22×C4×C53#4112yes—
111C2×C22⋊D53—42no—
112C2×C22⋊D53—42no—
113C2×C22⋊D53—42no—
114C2×C22⋊D53—42no—
115C2×C22⋊D53—42no—
116C2×C22⋊D53—42no—
117C2×C22⋊D53—42no—
118C2×C22⋊D53—41no—
119C2×C22⋊D53—41no—
120C2×C22⋊D53—42no—
121C22×Dic53—24no—
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—42no—
126C22×Dic53—42no—
127C22×Dic53—42no—
128C22×Dic53—42no—
129C22×Dic53—42no—
130C22×Dic53—42no—
131C2×D212—22no—
132C2×D212—41no—
133C2×D212—41no—
134C2×C8×C53—24no—
135C2×C8×C53—48no—
136C2×C8×C53—42no—
137C2×C8×C53—42no—
138C22⋊Dic53—48no—
139C22⋊Dic53—42no—
140C22⋊Dic53—42no—
141C22⋊Dic53—41no—
142C22⋊Dic53—41no—
143#11719921—42no—
144#11719921—42no—
145#11719921—41no—
146#11719921—41no—
147C4⋊Dic53—22no—
148C4⋊Dic53—41no—
149C4⋊Dic53—41no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—24no—
159C23×C53—24no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—24no—
167C23×C53—24no—
168C23×C53—42no—
169C23×C53—42no—
170C23×C53—44no—
171C23×C53—44no—
172C23×C53—44no—
173C23×C53—44no—
174C23×C53#9912yes—
175C23×C53#10212yes—
176C23×C53#10212yes—
177C23×C53#10412yes—
178C23×C53C2×C22⋊C411yescharacteristic
179C23×C53#12112yes—
180C23×C53C2×D811yescharacteristic
181C23×C53C2×SD1611yescharacteristic
182C23×C53#13512yes—
183C22×D53—41no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—41no—
190C2×C4×C53—24no—
191C2×C4×C53—24no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—44no—
204C2×C4×C53—44no—
205C2×C4×C53—44no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53#9912yes—
209C2×C4×C53C4×D412yes—
210C2×C4×C53#12011yescharacteristic
211C2×C4×C53#12311yescharacteristic
212C22⋊D53—84no—
213C22⋊D53—84no—
214C22⋊D53—82no—
215C22⋊D53—82no—
216C2×Dic53—44no—
217C2×Dic53—44no—
218C2×Dic53—44no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—41no—
222C2×Dic53—41no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—82no—
229C2×Dic53—82no—
230C2×Dic53—82no—
231C2×Dic53—82no—
232D212—82no—
233C8×C53—44no—
234#9113—531nomaximal, Sylow
235C22×C53—22no—
236C22×C53—22no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—21no—
247C22×C53—22no—
248C22×C53—21no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—48no—
253C22×C53—48no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53#32714yescomplemented
259C22×C53#38611yescharacteristic
260C22×C53C2×#9911yescharacteristic
261C22×C53C2×#9911yescharacteristic
262C22×C53C2×#10211yescharacteristic
263C22×C53#41611yescharacteristic
264C22×C53C2×#10411yescharacteristic
265C22×C53#42111yescharacteristic
266C22×C53#46512yes—
267C22×C53#46612yes—
268C2×D53—81no—
269C2×D53—81no—
270C2×D53—81no—
271C2×D53—81no—
272C2×D53—81no—
273C2×D53—81no—
274C2×D53—81no—
275C4×C53—21no—
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—21no—
279C4×C53—41no—
280Dic53—84no—
281Dic53—81no—
282Dic53—81no—
283Dic53—81no—
284C22×C22⋊C8D5311yescharacteristic, complemented
285#2847—531no—
286C2×#327—532no—
287C22⋊#95—531no—
288#1294—532no—
289C2×C53—22no—
290C2×C53—24no—
291C2×C53—22no—
292C2×C53—42no—
293C2×C53C2×#32711yescharacteristic
294C2×C53#150111yescharacteristic
295C2×C53#151611yescharacteristic
296C2×C53#151911yescharacteristic
297C2×C53#187111yescharacteristic
298C2×C53#189311yescharacteristic
299C2×C53#189811yescharacteristic
300D53—161no—
301C24×C4C2×D5311yescharacteristic
302C23×C8—21no—
303C2×#121—531no—
304C2×#121—531no—
305C2×#121—531no—
306C2×#120—1061no—
307C22×C22⋊C4—1061no—
308C2×C22⋊C8—21no—
309C2×C22⋊C8—24no—
310C2×C22⋊C8C2×D5312yes—
311#386—1061no—
312#386—1061no—
313#386—1061no—
314#380—531no—
315#380—531no—
316#380—531no—
317C2×#95—532no—
318C2×#95—1061no—
319#327—1064no—
320C53#911311yescharacteristic, complemented, Sylow
321C25D21211yescharacteristic
322C22×D4—531no—
323C22×D4—1061no—
324C22×D4—1061no—
325C22×D4—1061no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—42no—
329C23×C4—42no—
330C23×C4—1062no—
331C23×C4C4×D5311yescharacteristic
332C23×C4C4×D5311yescharacteristic
333C23×C4D21211yescharacteristic
334C23×C4C22⋊D5311yescharacteristic
335C23×C4C22⋊D5311yescharacteristic
336C23×C4C22×D5311yescharacteristic
337C22×C8—24no—
338C22×C8—22no—
339C22×C8—42no—
340C22×C8—42no—
341#121—1062no—
342#121—1062no—
343#121—2122no—
344#121—2122no—
345#120—2122no—
346#120—2122no—
347C2×C4⋊C4—531no—
348C2×C4⋊C4—531no—
349C2×C4⋊C4—531no—
350C2×C22⋊C4—1064no—
351C2×C22⋊C4—1061no—
352C2×C22⋊C4—1061no—
353C2×C22⋊C4—1061no—
354C2×C22⋊C4—1061no—
355C2×C22⋊C4—1061no—
356C2×C22⋊C4—1061no—
357C2×C22⋊C4—2122no—
358C2×C22⋊C4—2122no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8C22⋊D5314yescomplemented
362#95—1061no—
363#95—1061no—
364#95—1061no—
365#95—1062no—
366#95—1062no—
367#95—2122no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—41no—
373C24—41no—
374C24—1061no—
375C24#1171991312yes—
376C24D42412yes—
377C24#1171992111yescharacteristic
378C24#1171992111yescharacteristic
379C24C2×D21211yescharacteristic
380C2×D4—1062no—
381C2×D4—2121no—
382C2×D4—2121no—
383C2×D4—2122no—
384C2×D4—2122no—
385C2×D4—2122no—
386C2×D4—2122no—
387C2×D4—2122no—
388C2×D4—2122no—
389C2×D4—2122no—
390C2×D4—2121no—
391C2×D4—2121no—
392C2×D4—2122no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—1064no—
415C22×C4—1061no—
416C22×C4—1061no—
417C22×C4—1061no—
418C22×C4—2122no—
419C22×C4—2122no—
420C22×C4—2122no—
421C22×C4—2122no—
422C22×C4—2122no—
423C22×C4—2122no—
424C22×C4#1171992111yescharacteristic
425C22×C4#1171992111yescharacteristic
426C22×C4C2×C4×D5311yescharacteristic
427C22×C4#1171994512yes—
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C2×C22⋊D5312yes—
430C2×C8—24no—
431C2×C8—48no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—1062no—
435C4⋊C4—2121no—
436C4⋊C4—2121no—
437C22⋊C4—2122no—
438C22⋊C4—2128no—
439C22⋊C4—2122no—
440C22⋊C4—2122no—
441C22⋊C4—2121no—
442C22⋊C4—2121no—
443C22⋊C4—2122no—
444C22⋊C4—2121no—
445C22⋊C4—2121no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—2121no—
471C23—2121no—
472C23—2121no—
473C23—2121no—
474C23—2121no—
475C23—2121no—
476C23—2121no—
477C23#1313313112yes—
478C23#1313313912yes—
479C23#1313313912yes—
480C23#1313314312yes—
481C23C2×#1171992111yescharacteristic
482C23#1313319112yes—
483C23C2×D42411yescharacteristic
484C23C2×#1171991311yescharacteristic
485C23#1313323912yes—
486D4—4242no—
487D4—4244no—
488D4—4242no—
489D4—4242no—
490D4—4244no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—4242no—
510C2×C4—4242no—
511C2×C4—4242no—
512C2×C4—4242no—
513C2×C4—2124no—
514C2×C4—2124no—
515C2×C4—2124no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4#1313313112yes—
526C2×C4C4×C22⋊D5312yes—
527C2×C4#1313318611yescharacteristic
528C2×C4#1313320011yescharacteristic
529C8—44no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—22no—
540C22—21no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—4241no—
554C22—4241no—
555C22—4241no—
556C22—4241no—
557C22—4241no—
558C22—4241no—
559C22—4241no—
560C22#1681035814yescomplemented
561C22#1681050211yescharacteristic
562C22C2×#1313313111yescharacteristic
563C22C2×#1313313111yescharacteristic
564C22C2×#1313313911yescharacteristic
565C22#1681062011yescharacteristic
566C22C2×#1313314311yescharacteristic
567C22#1681063911yescharacteristic
568C22#1681085212yes—
569C22#1681085912yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—4244no—
576C4—4241no—
577C4—4241no—
578C4—4241no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—42no—
583C2—8481no—
584C2C2×#1681035811yescharacteristic
585C2#2027320711yescharacteristic
586C2#2027327211yescharacteristic
587C2#2027328411yescharacteristic
588C2#2027501711yescharacteristic
589C2#2027514311yescharacteristic
590C2#2027516811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC22×C22⋊C8×C53C2
socleC23×C53C2×C22⋊C4
prime radical#20282997C2
cosocleC23×C4×C53C23
hypercenterC22×C22⋊C8D53
hypocenterC53#9113
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16810358C2semidirect
3. #20272938C2semidirect
4. #20272595C2semidirect
5. C22×C22⋊C8×C53C2semidirect
6. C2×#13133191C4semidirect
7. C2×#13133191C4semidirect
14. #16810483C4semidirect
15. #16810483C4semidirect
71. C22⋊C8×C53D4semidirect
258. C22×C53#327semidirect
284. C22×C22⋊C8D53semidirect
320. C53#9113semidirect
361. C22⋊C8C22⋊D53semidirect
560. C22#16810358semidirect

Maximal subgroups

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

subgroupindex
#202829972
C2×#168103582
#202729382
#202725952
C22×C22⋊C8×C532
#911353