Finite Group Databasebeta

← #28314289

Subgroups — #28314289

Order 13568.

960 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291391C211yesmaximal, characteristic
2C2×#16811644C211yesmaximal, characteristic
3#20289994C214yesmaximal
4#20289989C212yesmaximal, complemented
5#20289988C212yesmaximal, complemented
6#20289977C214yesmaximal, complemented
7C2×#16811176C211yesmaximal, characteristic
8#16811644—21no—
9#16811644—21no—
10#16811644C2212yes—
11#16811644C2212yes—
12#16811644C2212yes—
13#16811643—21no—
14#16811640C2212yes—
15#16811639—21no—
16#16811626—21no—
17C2×C4⋊#88324—21no—
18#16811593C2212yes—
19C2×C22⋊#88324C2212yes—
20C4×#11719940—21no—
21C2×#13133272—21no—
22#16811469C2218yes—
23#16811468C2218yes—
24#16811176—21no—
25#16811176—21no—
26#16811176—21no—
27#16811176—21no—
28#16811176—21no—
29#16811176—21no—
30#16811176—21no—
31#16811176—21no—
32C2×#13133216C411yescharacteristic
33C2×#13133216C411yescharacteristic
34C2×#13133208C2211yescharacteristic
35C2×#13133208C2211yescharacteristic
36C2×#13133194C412yes—
37C2×#13133194C2212yes—
38C2×C4×Dic106C412yes—
39C2×C4×Dic106C2212yes—
40C2×#13133176C411yescharacteristic
41C22×C4×Dic53C2211yescharacteristic
42C2×#117×C53C411yescharacteristic
43C2×#11719940—22no—
44C22×#88324—22no—
45#13133284—22no—
46C4⋊#88324—22no—
47C4⋊#88324—22no—
48C4⋊#88324—22no—
49C22⋊#88324—22no—
50C22⋊#88324—22no—
51C22⋊#88324C2314yes—
52C22⋊#88324C2314yes—
53#13133272—22no—
54#13133272—22no—
55#13133272—22no—
56C4×#88324—22no—
57C22×Dic106C2×C412yes—
58C22×Dic106C2×C412yes—
59C23×Dic53C2×C411yescharacteristic
60C23×Dic53C2311yescharacteristic
61#13133216—21no—
62#13133216—21no—
63#13133216—21no—
64#13133216—21no—
65#13133216—21no—
66#13133216—21no—
67#13133216—21no—
68#13133216—21no—
69#13133208—21no—
70#13133208—21no—
71#13133208—21no—
72#13133208—21no—
73#13133208—21no—
74#13133208—21no—
75#13133208—21no—
76#13133208—21no—
77#13133194—22no—
78#13133194—22no—
79#13133194—22no—
80#13133194—22no—
81#13133194—22no—
82#13133194—22no—
83#13133194—22no—
84#13133194—22no—
85C4×Dic106—22no—
86C4×Dic106—22no—
87C4×Dic106—22no—
88C4×Dic106—22no—
89C4×Dic106—22no—
90C4×Dic106—22no—
91C4×Dic106—22no—
92C4×Dic106—22no—
93#13133176—21no—
94#13133176—21no—
95#13133176—21no—
96#13133176—21no—
97C2×C4⋊Dic53C2×C411yescharacteristic
98C2×C4⋊Dic53C2×C411yescharacteristic
99C2×C4⋊Dic53C2×C412yes—
100C2×C4⋊Dic53C2312yes—
101C2×#11719919C2×C412yes—
102C2×#11719919C2×C412yes—
103C2×#11719919C2×C412yes—
104C2×#11719919C2312yes—
105C2×C22⋊Dic53C2×C412yes—
106C2×C4×Dic53C2×C412yes—
107C2×C4×Dic53C2×C411yescharacteristic
108C2×C4×Dic53C2×C411yescharacteristic
109C2×C4×Dic53D411yescharacteristic
110C2×C4×Dic53D411yescharacteristic
111C2×C4×Dic53D411yescharacteristic
112C2×C4×Dic53D411yescharacteristic
113C2×C4×Dic53D411yescharacteristic
114C2×C4×Dic53D411yescharacteristic
115C2×C4×Dic53D411yescharacteristic
116C2×C4×Dic53D411yescharacteristic
117C2×C4×Dic53C2311yescharacteristic
118C2×C4×Dic53C2311yescharacteristic
119C23×C4×C53C2×C411yescharacteristic
120#117×C53—21no—
121#117×C53—21no—
122#117×C53—21no—
123#117×C53—21no—
124C2×C4⋊C4×C53C2×C411yescharacteristic
125C2×C4⋊C4×C53C2×C411yescharacteristic
126C2×C22⋊C4×C53C2×C412yes—
127C2×C42×C53C2×C412yes—
128C24×C53C22×C411yescharacteristic
129C22×C4×C53—24no—
130C22×C4×C53C22⋊C411yescharacteristic
131C22×C4×C53C22⋊C411yescharacteristic
132C22×C4×C53C22⋊C411yescharacteristic
133C22×C4×C53C22⋊C411yescharacteristic
134C22×C4×C53C22×C412yes—
135C22×C4×C53C22×C411yescharacteristic
136C22×C4×C53C22×C411yescharacteristic
137C22×C4×C53C22×C412yes—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53C22⋊C411yescharacteristic
141C22×Dic53C22⋊C411yescharacteristic
142C22×Dic53C22⋊C411yescharacteristic
143C22×Dic53C22⋊C411yescharacteristic
144C22×Dic53C22×C412yes—
145C22×Dic53C22×C411yescharacteristic
146C22×Dic53C22×C411yescharacteristic
147C22×Dic53C22×C411yescharacteristic
148C22×Dic53C22×C412yes—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2411yescharacteristic
154C2×Dic106—22no—
155C2×Dic106—22no—
156C2×Dic106—22no—
157C2×Dic106—22no—
158C2×Dic106—44no—
159C2×Dic106—44no—
160#11719940—44no—
161C2×#88324—24no—
162C2×#88324—24no—
163C2×#88324—44no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C42×C53—22no—
171C42×C53—22no—
172C22⋊Dic53—22no—
173C22⋊Dic53—22no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—22no—
181C4⋊Dic53—22no—
182#11719919—22no—
183#11719919—22no—
184#11719919—22no—
185#11719919—22no—
186#11719919—22no—
187#11719919—22no—
188#11719919—22no—
189#11719919—22no—
190C4×Dic53—22no—
191C4×Dic53—22no—
192C4×Dic53C22⋊C411yescharacteristic
193C4×Dic53C22⋊C411yescharacteristic
194C4×Dic53C22⋊C411yescharacteristic
195C4×Dic53C22⋊C411yescharacteristic
196C4×Dic53C22⋊C411yescharacteristic
197C4×Dic53C22⋊C411yescharacteristic
198C4×Dic53C22⋊C411yescharacteristic
199C4×Dic53C22⋊C411yescharacteristic
200C4×Dic53C2×D411yescharacteristic
201C4×Dic53C2×D411yescharacteristic
202C4×Dic53C2×D411yescharacteristic
203C4×Dic53C2×D411yescharacteristic
204C4×Dic53C2×D411yescharacteristic
205C4×Dic53C2×D411yescharacteristic
206C4×Dic53C2×D411yescharacteristic
207C4×Dic53C2×D411yescharacteristic
208C23×C53—22no—
209C23×C53—22no—
210C23×C53—41no—
211C23×C53C2×C22⋊C411yescharacteristic
212C23×C53C2×C22⋊C411yescharacteristic
213C23×C53C23×C411yescharacteristic
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—24no—
222C2×C4×C53—42no—
223C2×C4×C53#10114yes—
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53C2×C22⋊C411yescharacteristic
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—24no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—42no—
244C2×Dic53C2×C22⋊C411yescharacteristic
245C2×Dic53C2×C22⋊C411yescharacteristic
246C2×Dic53C2×C22⋊C411yescharacteristic
247C2×Dic53C2×C22⋊C411yescharacteristic
248C2×Dic53C2×C22⋊C411yescharacteristic
249C2×Dic53C2×C22⋊C411yescharacteristic
250C2×Dic53C22×D411yescharacteristic
251C2×Dic53C22×D411yescharacteristic
252Dic106—44no—
253Dic106—44no—
254Dic106—44no—
255Dic106—44no—
256#88324—44no—
257#88324—44no—
258#17404—531nomaximal, Sylow
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—21no—
265C22×C53—21no—
266C22×C53—44no—
267C22×C53C2×#10112yes—
268C22×C53C22×C22⋊C411yescharacteristic
269C22×C53#56812yes—
270C4×C53—21no—
271C4×C53—21no—
272C4×C53—42no—
273C4×C53—42no—
274C4×C53C2×#10112yes—
275C4×C53C2×#10112yes—
276Dic53—21no—
277Dic53—21no—
278Dic53—21no—
279Dic53—21no—
280Dic53—21no—
281Dic53—21no—
282Dic53—21no—
283Dic53—42no—
284Dic53—42no—
285Dic53#41012yes—
286C2×#533—531no—
287#2920—531no—
288C2×#432—531no—
289#1472—534no—
290#1470—532no—
291#1470—532no—
292#1467—534no—
293C2×C53—22no—
294C2×C53—22no—
295C2×C53—41no—
296C2×C53#287911yescharacteristic
297C2×C53C2×#41011yescharacteristic
298C2×C53C22×#10111yescharacteristic
299#533—1061no—
300#533—1061no—
301#533—1061no—
302#533—1061no—
303#533—1061no—
304#533—1061no—
305#533—1061no—
306#533—1061no—
307C2×C4⋊Q8—531no—
308C2×C4⋊Q8—531no—
309C2×#125—531no—
310C2×#125—531no—
311C2×C22⋊Q8—532no—
312C2×C22⋊Q8—532no—
313C2×C4×Q8—532no—
314C2×C4×Q8—532no—
315C2×#117—531no—
316C2×#117#615711yescharacteristic
317C22×C42—531no—
318#432—532no—
319#432—532no—
320#432—532no—
321#432—1061no—
322#432—1061no—
323#432—1061no—
324#431—532no—
325#431—1061no—
326#423—1061no—
327C2×C4⋊C8—1061no—
328#407—532no—
329C2×C22⋊C8—532no—
330C4×M16—1061no—
331C2×#97—1061no—
332#324—538no—
333#324—538no—
334C53#1740411yescharacteristic, complemented, Sylow
335C22×Q8—532no—
336C22×Q8—532no—
337C23×C4—531no—
338C23×C4—531no—
339C23×C4C2×#615711yescharacteristic
340C2×M16—1062no—
341C22×C8—1062no—
342C4⋊Q8—1061no—
343C4⋊Q8—1061no—
344C4⋊Q8—1061no—
345C4⋊Q8—1061no—
346C4⋊Q8—1061no—
347C4⋊Q8—1061no—
348C4⋊Q8—1061no—
349C4⋊Q8—1061no—
350#125—1061no—
351#125—1061no—
352#125—1061no—
353#125—1061no—
354#125—1061no—
355#125—1061no—
356#125—1061no—
357#125—1061no—
358C22⋊Q8—1062no—
359C22⋊Q8—1062no—
360C22⋊Q8—1062no—
361C22⋊Q8—1062no—
362C22⋊Q8—1062no—
363C22⋊Q8—1062no—
364C22⋊Q8—1062no—
365C22⋊Q8—1062no—
366C4×Q8—1062no—
367C4×Q8—1062no—
368C4×Q8—1062no—
369C4×Q8—1062no—
370C4×Q8—1062no—
371C4×Q8—1062no—
372C4×Q8—1062no—
373C4×Q8—1062no—
374#117—21no—
375#117—21no—
376#117—21no—
377#117—21no—
378#117—1061no—
379#117—1061no—
380#117—1061no—
381#117—1061no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—532no—
385C2×C4⋊C4—532no—
386C2×C4⋊C4—532no—
387C2×C4⋊C4—532no—
388C2×C4⋊C4—532no—
389C2×C4⋊C4—532no—
390C2×C4⋊C4C2×#615711yescharacteristic
391C2×C4⋊C4C2×#615711yescharacteristic
392C2×C22⋊C4—532no—
393C2×C22⋊C4C2×#615712yes—
394C2×C42—531no—
395C2×C42—531no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—532no—
407C2×C42C2×#615712yes—
408C4⋊C8—1062no—
409C4⋊C8—1062no—
410C4⋊C8—1062no—
411C4⋊C8—1062no—
412C22⋊C8—534no—
413C22⋊C8—534no—
414C22⋊C8—1062no—
415C22⋊C8—1062no—
416#97—1062no—
417#97—1062no—
418#97—1062no—
419C4×C8—1062no—
420C24C22×#615711yescharacteristic
421C2×Q8—1062no—
422C2×Q8—1062no—
423C2×Q8—1062no—
424C2×Q8—1062no—
425C2×Q8—2124no—
426C2×Q8—2124no—
427C22×C4—24no—
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—532no—
441C22×C4—532no—
442C22×C4—2122no—
443C22×C4—2122no—
444C22×C4C22⋊#615711yescharacteristic
445C22×C4C22⋊#615711yescharacteristic
446C22×C4C22⋊#615711yescharacteristic
447C22×C4C22⋊#615711yescharacteristic
448C22×C4C22×#615712yes—
449C22×C4C22×#615711yescharacteristic
450C22×C4C22×#615711yescharacteristic
451C22×C4C22×#615712yes—
452M16—2124no—
453C2×C8—1064no—
454C2×C8—1064no—
455C2×C8—2124no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—1062no—
461C4⋊C4—1062no—
462C4⋊C4—1062no—
463C4⋊C4—1062no—
464C4⋊C4—1062no—
465C4⋊C4—1062no—
466C4⋊C4—1062no—
467C4⋊C4—1062no—
468C4⋊C4—1062no—
469C4⋊C4—1062no—
470C4⋊C4—1062no—
471C4⋊C4—1062no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—1062no—
479C22⋊C4—1062no—
480C42—22no—
481C42—22no—
482C42—531no—
483C42—531no—
484C42—531no—
485C42—531no—
486C42—531no—
487C42—531no—
488C42—531no—
489C42—531no—
490C42—531no—
491C42—531no—
492C42—531no—
493C42—531no—
494C42—531no—
495C42—531no—
496C42—531no—
497C42—531no—
498C42—1062no—
499C42—1062no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23C2×C22⋊#615711yescharacteristic
504C23C2×C22⋊#615711yescharacteristic
505C23C23×#615711yescharacteristic
506Q8—2124no—
507Q8—2124no—
508Q8—2124no—
509Q8—2124no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—531no—
520C2×C4—531no—
521C2×C4—531no—
522C2×C4—531no—
523C2×C4—531no—
524C2×C4—531no—
525C2×C4—531no—
526C2×C4—531no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1064no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2122no—
543C2×C4#1313327814yes—
544C2×C4C2×C22⋊#615711yescharacteristic
545C2×C4C2×C22⋊#615711yescharacteristic
546C2×C4C2×C22⋊#615711yescharacteristic
547C2×C4C2×C22⋊#615711yescharacteristic
548C8—2124no—
549C8—2124no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—44no—
558C22C2×#1313327812yes—
559C22C22×C22⋊#615711yescharacteristic
560C22#1681170212yes—
561C4—21no—
562C4—21no—
563C4—42no—
564C4—42no—
565C4—532no—
566C4—1061no—
567C4—1061no—
568C4—1061no—
569C4—1061no—
570C4—1061no—
571C4—1061no—
572C4—1061no—
573C4—2122no—
574C4—2122no—
575C4C2×#1313327812yes—
576C4C2×#1313327812yes—
577C2—22no—
578C2—22no—
579C2—41no—
580C2#2029130011yescharacteristic
581C2D53⋊#41011yescharacteristic
582C2C22×#1313327811yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×#117×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC22×Dic53C24
hypercenterC2×#117#6157
hypocenterC53#17404
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
4. #20289989C2semidirect
5. #20289988C2semidirect
6. #20289977C2semidirect
334. C53#17404semidirect

Maximal subgroups

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

subgroupindex
#202913912
C2×#168116442
#202899942
#202899892
#202899882
#202899772
C2×#168111762
#1740453