Finite Group Databasebeta

← #28314532

Subgroups — #28314532

Order 13568.

1646 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291391C211yesmaximal, characteristic, complemented
2#20291391C211yesmaximal, characteristic, complemented
3#20290056C214yesmaximal, complemented
4#20290042C214yesmaximal, complemented
5#20290016C214yesmaximal, complemented
6C22×#13133179C211yesmaximal, characteristic
7#16811644—21no—
8#16811644C2212yescomplemented
9#16811643—21no—
10#16811643—21no—
11#16811640—21no—
12#16811640C2212yescomplemented
13#16811639C2212yescomplemented
14#16811639C2212yescomplemented
15#16811626—21no—
16#16811626C2212yescomplemented
17#16811593—21no—
18#16811593—21no—
19C4×#11719940—21no—
20C4×#11719940C2212yescomplemented
21#16811471C22116yescomplemented
22C23×D212C411yescharacteristic
23C23×C4×D53—21no—
24C2×#13133179—21no—
25C2×#13133179—21no—
26C2×#13133179—21no—
27C2×#13133179—21no—
28C2×#13133179—21no—
29C2×#13133179—21no—
30C2×#13133179C412yes—
31C2×#13133179C412yes—
32C2×#13133179C412yes—
33C2×#13133179C2212yes—
34C2×#13133179C2212yes—
35C2×#13133179C2212yes—
36C22×#11719921—21no—
37C22×C4×Dic53C2211yescharacteristic
38C22×C4⋊C4×C53C411yescharacteristic
39C2×#11719940—22no—
40C2×#11719940—22no—
41#13133284—22no—
42#13133284—22no—
43C4⋊#88324—22no—
44C4⋊#88324C2314yescomplemented
45C22⋊#88324—22no—
46C22⋊#88324—22no—
47C22⋊#88324—22no—
48C22⋊#88324—22no—
49#13133272—22no—
50#13133272—22no—
51C4×#88324—22no—
52C4×#88324C2314yescomplemented
53C24×D53—21no—
54C22×D212—21no—
55C22×D212—21no—
56C22×D212—21no—
57C22×D212—28no—
58C22×D212C2×C412yes—
59C22×D212C2×C412yes—
60C22×D212C2×C412yes—
61C23×Dic53D411yescharacteristic
62C23×Dic53D411yescharacteristic
63C22×C4×D53—22no—
64C22×C4×D53—22no—
65C22×C4×D53—22no—
66C22×C4×D53—22no—
67C22×C4×D53—22no—
68C22×C4×D53—22no—
69C22×C4×D53—48no—
70#13133179—22no—
71#13133179—22no—
72#13133179—22no—
73#13133179—22no—
74#13133179—22no—
75#13133179—22no—
76#13133179—22no—
77#13133179—22no—
78#13133179—22no—
79#13133179—22no—
80#13133179—22no—
81#13133179—22no—
82#13133179C2×C414yes—
83#13133179C2×C414yes—
84#13133179C2×C414yes—
85#13133179C2314yes—
86C2×#11719921—22no—
87C2×#11719921—22no—
88C2×#11719921—22no—
89C2×#11719921—22no—
90C2×#11719921—22no—
91C2×#11719921—22no—
92C2×C4×Dic53C2×C411yescharacteristic
93C2×C4×Dic53C2×C411yescharacteristic
94C2×C4×Dic53C2×C411yescharacteristic
95C2×C4×Dic53D411yescharacteristic
96C2×C4×Dic53D411yescharacteristic
97C2×C4×Dic53D411yescharacteristic
98C2×C4×Dic53D411yescharacteristic
99C2×C4×Dic53D411yescharacteristic
100C2×C4×Dic53D411yescharacteristic
101C2×C4×Dic53C2311yescharacteristic
102C2×C4×Dic53C2311yescharacteristic
103C2×C4×Dic53C2311yescharacteristic
104C23×C4×C53—21no—
105C23×C4×C53C2×C411yescharacteristic
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2×C412yes—
110C2×C4⋊C4×C53C2×C412yes—
111C2×C4⋊C4×C53C2×C412yes—
112C24×C53C22⋊C411yescharacteristic
113C23×D53—28no—
114C23×D53—22no—
115C23×D53—22no—
116C23×D53—22no—
117C23×D53—44no—
118C23×D53—44no—
119C22×C4×C53—24no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—44no—
124C22×C4×C53C22⋊C411yescharacteristic
125C22×C4×C53C22⋊C411yescharacteristic
126C22×C4×C53C22⋊C411yescharacteristic
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C22×C411yescharacteristic
130C22×Dic53—24no—
131C22×Dic53—42no—
132C22×Dic53C22⋊C411yescharacteristic
133C22×Dic53C22⋊C411yescharacteristic
134C22×Dic53C22⋊C411yescharacteristic
135C22×Dic53C22⋊C411yescharacteristic
136C22×Dic53C22⋊C411yescharacteristic
137C22×Dic53C22⋊C411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C2×D212—28no—
145C2×D212—28no—
146C2×D212—28no—
147C2×D212—22no—
148C2×D212—22no—
149C2×D212—22no—
150C2×D212—24no—
151C2×D212—28no—
152C2×D212—28no—
153C2×D212M1618yescomplemented
154C2×D212C22×C414yes—
155C2×C4×D53—24no—
156C2×C4×D53—24no—
157C2×C4×D53—24no—
158C2×C4×D53—24no—
159C2×C4×D53—48no—
160C2×C4×D53—48no—
161C2×C4×D53—48no—
162C2×C4×D53—48no—
163C2×C4×D53—48no—
164C2×C4×D53—48no—
165#11719940—44no—
166#11719940—44no—
167C2×#88324—24no—
168C2×#88324—24no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53C22×C414yes—
173#11719921—24no—
174#11719921—24no—
175#11719921—24no—
176#11719921—24no—
177C4×Dic53C22⋊C411yescharacteristic
178C4×Dic53C22⋊C411yescharacteristic
179C4×Dic53C22⋊C411yescharacteristic
180C4×Dic53C22⋊C411yescharacteristic
181C4×Dic53C22⋊C411yescharacteristic
182C4×Dic53C22⋊C411yescharacteristic
183C4×Dic53C22×C411yescharacteristic
184C4×Dic53C22×C411yescharacteristic
185C4×Dic53C22×C411yescharacteristic
186C4×Dic53C2×D411yescharacteristic
187C4×Dic53C2×D411yescharacteristic
188C4×Dic53C2×D411yescharacteristic
189C4×Dic53C2×D411yescharacteristic
190C4×Dic53C2×D411yescharacteristic
191C4×Dic53C2×D411yescharacteristic
192C4×Dic53C2411yescharacteristic
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53C2×C22⋊C411yescharacteristic
197C23×C53C2×C22⋊C411yescharacteristic
198C23×C53C2×C22⋊C411yescharacteristic
199C22×D53—28no—
200C22×D53—28no—
201C22×D53—24no—
202C22×D53—28no—
203C22×D53—44no—
204C22×D53—44no—
205C22×D53—432no—
206C22×D53—44no—
207C22×D53—44no—
208C22×D53—44no—
209C22×D53—44no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—24no—
216C2×C4×C53—44no—
217C2×C4×C53—44no—
218C2×C4×C53—44no—
219C2×C4×C53C2×C22⋊C411yescharacteristic
220C2×C4×C53C2×C22⋊C411yescharacteristic
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C2×C4×C53C2×M1614yes—
223C2×C4×C53C23×C411yescharacteristic
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53C2×C22⋊C411yescharacteristic
236C2×Dic53C2×C22⋊C411yescharacteristic
237C2×Dic53C2×C22⋊C411yescharacteristic
238C2×Dic53C2×C22⋊C411yescharacteristic
239C2×Dic53C2×C22⋊C411yescharacteristic
240C2×Dic53C2×C22⋊C411yescharacteristic
241C2×Dic53AGL(1,ℤ/8)14yes—
242C2×Dic53C22×D411yescharacteristic
243C2×Dic53C22×D411yescharacteristic
244D212—28no—
245D212—28no—
246D212—24no—
247D212—28no—
248D212C2×M1618yes—
249C4×D53—48no—
250C4×D53—48no—
251C4×D53—48no—
252C4×D53—48no—
253#88324—44no—
254#88324—44no—
255#17448—531nomaximal, Sylow
256C22×C53—28no—
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53#40712yes—
264C22×C53#41812yes—
265C22×C53#42112yes—
266C22×C53C22×C22⋊C411yescharacteristic
267C2×D53—28no—
268C2×D53—416no—
269C2×D53—416no—
270C2×D53—416no—
271C2×D53—44no—
272C2×D53—44no—
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—44no—
276C4×C53#40712yes—
277C4×C53C22×M1612yes—
278Dic53—21no—
279Dic53—21no—
280Dic53—21no—
281Dic53—21no—
282Dic53—21no—
283Dic53—21no—
284Dic53#41812yes—
285Dic53C2×AGL(1,ℤ/8)12yes—
286C22×C4×D4—531no—
287#2920—531no—
288#2920—531no—
289#1490—534no—
290#1486—534no—
291#1478—534no—
292C2×C53—22no—
293C2×C53—22no—
294C2×C53—22no—
295C2×C53C2×#40711yescharacteristic
296C2×C53C2×#41811yescharacteristic
297C2×C53C2×#42111yescharacteristic
298D53—48no—
299C23×D4—531no—
300C24×C4—1061no—
301C2×C4×D4—532no—
302C2×C4×D4—532no—
303C2×C4×D4—532no—
304C2×C4×D4—532no—
305C2×C4×D4—532no—
306C2×C4×D4—532no—
307C2×C4×D4—1061no—
308C2×C4×D4—1061no—
309C2×C4×D4—1061no—
310C2×C4×D4—1061no—
311C2×C4×D4—1061no—
312C2×C4×D4—1061no—
313C22×C4⋊C4#615711yescharacteristic
314C22×C22⋊C4—1061no—
315C22×C42—531no—
316#432—532no—
317#432—1061no—
318#432—1061no—
319#432—1061no—
320#431—532no—
321#431—532no—
322#431—532no—
323#431—1061no—
324#423—532no—
325#423—1061no—
326#407—1061no—
327#407—1061no—
328C4×M16—532no—
329C4×M16—1061no—
330D4⋊C8—5316no—
331C53#1744811yescharacteristic, complemented, Sylow
332C25—1061no—
333C22×D4—532no—
334C22×D4—532no—
335C22×D4—532no—
336C22×D4—1068no—
337C22×D4—1061no—
338C22×D4—1061no—
339C22×D4—1061no—
340C23×C4—21no—
341C23×C4—531no—
342C23×C4—531no—
343C23×C4—1062no—
344C23×C4—1062no—
345C23×C4—1062no—
346C23×C4—1062no—
347C23×C4—1062no—
348C23×C4—1062no—
349C23×C4—2128no—
350C23×C4C2×#615711yescharacteristic
351C2×M16—1062no—
352C2×M16—1062no—
353C4×D4—534no—
354C4×D4—534no—
355C4×D4—534no—
356C4×D4—534no—
357C4×D4—1062no—
358C4×D4—1062no—
359C4×D4—1062no—
360C4×D4—1062no—
361C4×D4—1062no—
362C4×D4—1062no—
363C4×D4—1062no—
364C4×D4—1062no—
365C4×D4—1062no—
366C4×D4—1062no—
367C4×D4—1062no—
368C4×D4—1062no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4C2×#615712yes—
373C2×C4⋊C4C2×#615712yes—
374C2×C4⋊C4C2×#615712yes—
375C2×C22⋊C4—1062no—
376C2×C22⋊C4—1062no—
377C2×C22⋊C4—1062no—
378C2×C22⋊C4—1062no—
379C2×C22⋊C4—1062no—
380C2×C22⋊C4—1062no—
381C2×C42—531no—
382C2×C42—531no—
383C2×C42—531no—
384C2×C42—531no—
385C2×C42—531no—
386C2×C42—531no—
387C2×C42—531no—
388C2×C42—531no—
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—531no—
392C2×C42—531no—
393C4⋊C8—534no—
394C4⋊C8—1062no—
395C4⋊C8—1062no—
396C4⋊C8—1062no—
397C22⋊C8—1062no—
398C22⋊C8—1062no—
399C22⋊C8—1062no—
400C22⋊C8—1062no—
401#97—1062no—
402#97—1062no—
403C4×C8—534no—
404C4×C8—1062no—
405C24—1068no—
406C24—1062no—
407C24—1062no—
408C24—1062no—
409C24—2124no—
410C24—2124no—
411C24C22⋊#615711yescharacteristic
412C2×D4—538no—
413C2×D4—534no—
414C2×D4—1068no—
415C2×D4—1068no—
416C2×D4—1068no—
417C2×D4—1068no—
418C2×D4—1068no—
419C2×D4—1062no—
420C2×D4—1062no—
421C2×D4—1062no—
422C2×D4—1064no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—531no—
429C22×C4—531no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—531no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—1064no—
441C22×C4—1064no—
442C22×C4—1064no—
443C22×C4—1064no—
444C22×C4—1064no—
445C22×C4—2128no—
446C22×C4—2128no—
447C22×C4—2128no—
448C22×C4—2128no—
449C22×C4—2128no—
450C22×C4—2128no—
451C22×C4—2122no—
452C22×C4C22⋊#615711yescharacteristic
453C22×C4C22⋊#615711yescharacteristic
454C22×C4C22⋊#615711yescharacteristic
455C22×C4C22×#615711yescharacteristic
456C22×C4C22×#615711yescharacteristic
457C22×C4C22×#615711yescharacteristic
458M16—2124no—
459M16—2124no—
460C2×C8—1064no—
461C2×C8—1064no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4C22×#615714yes—
466C22⋊C4—1064no—
467C22⋊C4—1064no—
468C22⋊C4—1064no—
469C22⋊C4—1064no—
470C42—531no—
471C42—531no—
472C42—531no—
473C42—531no—
474C42—531no—
475C42—531no—
476C42—531no—
477C42—531no—
478C42—531no—
479C42—531no—
480C42—531no—
481C42—531no—
482C42—531no—
483C42—531no—
484C42—531no—
485C42—531no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—1068no—
490C23—1068no—
491C23—1068no—
492C23—1064no—
493C23—2124no—
494C23—2124no—
495C23—21232no—
496C23—2124no—
497C23—2124no—
498C23—2124no—
499C23—2124no—
500C23C2×C22⋊#615711yescharacteristic
501C23C2×C22⋊#615711yescharacteristic
502C23C2×C22⋊#615711yescharacteristic
503D4—538no—
504D4—1068no—
505D4—1068no—
506D4—1068no—
507D4—1064no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—534no—
518C2×C4—531no—
519C2×C4—531no—
520C2×C4—531no—
521C2×C4—531no—
522C2×C4—531no—
523C2×C4—531no—
524C2×C4—531no—
525C2×C4—531no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—2128no—
534C2×C4—2128no—
535C2×C4—2128no—
536C2×C4—2128no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4C2×C22⋊#615711yescharacteristic
542C2×C4C2×C22⋊#615711yescharacteristic
543C2×C4C2×C22⋊#615711yescharacteristic
544C2×C4D53⋊M1614yes—
545C2×C4C23×#615711yescharacteristic
546C8—2124no—
547C8—2124no—
548C22—28no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—1068no—
556C22—21216no—
557C22—21216no—
558C22—21216no—
559C22—2124no—
560C22—2124no—
561C22#1681159512yes—
562C22#1681161512yes—
563C22#1681162212yes—
564C22C22×C22⋊#615711yescharacteristic
565C4—21no—
566C4—21no—
567C4—44no—
568C4—532no—
569C4—532no—
570C4—1061no—
571C4—1061no—
572C4—1061no—
573C4—1061no—
574C4—1061no—
575C4—1061no—
576C4#1681159512yes—
577C4C2×D53⋊M1612yes—
578C2—22no—
579C2—22no—
580C2—22no—
581C2—2128no—
582C2C2×#1681159511yescharacteristic
583C2C2×#1681161511yescharacteristic
584C2C2×#1681162211yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC23×D212C4
cosocleC4×Dic53C24
hypercenterC22×C4⋊C4#6157
hypocenterC53#17448
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20291391C2semidirect
2. #20291391C2semidirect
3. #20290056C2semidirect
4. #20290042C2semidirect
5. #20290016C2semidirect
8. #16811644C22semidirect
12. #16811640C22semidirect
13. #16811639C22semidirect
14. #16811639C22semidirect
16. #16811626C22semidirect
20. C4×#11719940C22semidirect
21. #16811471C22semidirect
44. C4⋊#88324C23semidirect
52. C4×#88324C23semidirect
153. C2×D212M16semidirect
331. C53#17448semidirect

Maximal subgroups

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

subgroupindex
#202913912
#202900562
#202900422
#202900162
C22×#131331792
#1744853