Finite Group Databasebeta

← #27291938

Subgroups — #27291938

Order 13568.

822 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283232C211yesmaximal, characteristic
2C2×#16810617C211yesmaximal, characteristic
3#20283159C211yesmaximal, characteristic
4C2×#16810393C211yesmaximal, characteristic, complemented
5C2×#16810387C211yesmaximal, characteristic, complemented
6#20274412C211yesmaximal, characteristic, complemented
7C53×#1733C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C23×#88322C2211yescharacteristic
12#16810631—21no—
13#16810627—21no—
14#16810622C414yescomplemented
15#16810617—22no—
16#16810617—22no—
17#16810617—22no—
18C2×#13133140—21no—
19C2×#13133137—21no—
20#16810591—22no—
21#16810589—22no—
22#16810586—22no—
23#16810393—22no—
24#16810387—22no—
25#16810378—22no—
26C22×M16×C53C2211yescharacteristic
27C2×#117×C53C2211yescharacteristic
28C2×#117×C53C2211yescharacteristic
29C53×#339—22no—
30C2×#11719945—21no—
31C2×#11719945—21no—
32C2×#11719945—21no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945C2×C412yes—
41C2×#11719945D412yes—
42C22×Dic106D411yescharacteristic
43C22×D212D411yescharacteristic
44C22×C22⋊D53—21no—
45C22×C4×D53—21no—
46C2×#11719917—21no—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917C2×C412yes—
58C22×#88322—21no—
59C22×#88322—21no—
60C22×#88322—21no—
61C22×#88322—21no—
62C22×#88322—21no—
63C22×#88322—21no—
64C22×#88322—21no—
65C22×#88322D412yes—
66C22×#88322D412yes—
67#13133140—24no—
68#13133140—24no—
69#13133137—24no—
70#13133137—24no—
71#13133129—28no—
72C23×C4×C53C2311yescharacteristic
73C2×M16×C53—21no—
74C2×M16×C53—21no—
75C2×M16×C53—21no—
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53D412yes—
79C22×C8×C53—21no—
80#117×C53—21no—
81#117×C53—21no—
82#117×C53—21no—
83#117×C53—21no—
84#117×C53—21no—
85#117×C53—21no—
86#117×C53—21no—
87#117×C53D412yes—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53C2×C412yes—
90C2×C22⋊C4×C53—21no—
91C2×C22⋊C4×C53—21no—
92C2×C42×C53—21no—
93C2×C42×C53—21no—
94C24×C53#4111yescharacteristic
95C23×D53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—41no—
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×C22⋊D53—22no—
113C2×C22⋊D53—22no—
114C2×C22⋊D53—41no—
115C2×C22⋊D53—41no—
116C2×C22⋊D53—41no—
117C2×C22⋊D53—41no—
118C22×Dic53—21no—
119#11719945—22no—
120#11719945—22no—
121#11719945—22no—
122#11719945—22no—
123#11719945—22no—
124#11719945—22no—
125#11719945—22no—
126#11719945—22no—
127#11719945—41no—
128#11719945—41no—
129#11719945—41no—
130#11719945—41no—
131#11719945—41no—
132#11719945—41no—
133#11719945—41no—
134#11719945—41no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212C22⋊C412yes—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—41no—
144C2×C4×D53—41no—
145C2×C4×D53—41no—
146C2×C4×D53—41no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106C22⋊C412yes—
153M16×C53—22no—
154M16×C53—22no—
155M16×C53—41no—
156M16×C53—41no—
157C2×C8×C53—22no—
158C2×C8×C53—41no—
159C2×C8×C53—41no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—41no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—41no—
166C22⋊C4×C53—41no—
167C22⋊C4×C53—41no—
168C42×C53—22no—
169C42×C53—41no—
170C42×C53—41no—
171C42×C53—41no—
172#11719917—22no—
173#11719917—22no—
174#11719917—41no—
175#11719917—41no—
176#11719917—41no—
177#11719917—41no—
178#11719917—41no—
179#11719917—41no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—41no—
188C2×#88322—41no—
189C2×#88322—41no—
190C2×#88322—41no—
191C2×#88322—41no—
192C2×#88322—41no—
193C23×C53—42no—
194C23×C53C4×D411yescharacteristic
195C23×C53#12111yescharacteristic
196C23×C53#12311yescharacteristic
197C23×C53#12311yescharacteristic
198C23×C53#12411yescharacteristic
199C23×C53#12611yescharacteristic
200C23×C53#12611yescharacteristic
201C22×D53—22no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—82no—
205C2×C4×C53—44no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—82no—
209C2×C4×C53C2×C22⋊C411yescharacteristic
210C2×C4×C53#11711yescharacteristic
211C2×C4×C53#11711yescharacteristic
212C2×C4×C53C4×D411yescharacteristic
213C2×C4×C53C4×D411yescharacteristic
214C2×C4×C53C4×D411yescharacteristic
215C2×C4×C53#12011yescharacteristic
216C2×C4×C53#12011yescharacteristic
217C2×C4×C53#12111yescharacteristic
218C2×C4×C53#12111yescharacteristic
219C2×C4×C53#12111yescharacteristic
220C2×C4×C53#12111yescharacteristic
221C2×C4×C53#12111yescharacteristic
222C2×C4×C53#12111yescharacteristic
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12311yescharacteristic
225C2×C4×C53#12311yescharacteristic
226C2×C4×C53#12311yescharacteristic
227C2×C4×C53#12311yescharacteristic
228C2×C4×C53#12311yescharacteristic
229C2×C4×C53#12311yescharacteristic
230C2×C4×C53#12311yescharacteristic
231C2×C4×C53#12311yescharacteristic
232C2×C4×C53#12411yescharacteristic
233C2×C4×C53#12411yescharacteristic
234C2×C4×C53#12411yescharacteristic
235C2×C4×C53#12611yescharacteristic
236C2×C4×C53#12611yescharacteristic
237C22⋊D53—42no—
238C22⋊D53—42no—
239C22⋊D53—42no—
240C22⋊D53—42no—
241C2×Dic53—22no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244D212—22no—
245D212—22no—
246D212—41no—
247D212—41no—
248C4×D53—42no—
249C4×D53—42no—
250C4×D53—42no—
251C4×D53—42no—
252Dic106—22no—
253Dic106—22no—
254Dic106—41no—
255Dic106—41no—
256C8×C53—42no—
257#88322—42no—
258#88322—42no—
259#88322—42no—
260#11194—531nomaximal, Sylow
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53#38611yescharacteristic
269C22×C53#38811yescharacteristic
270C22×C53#38811yescharacteristic
271C22×C53#39411yescharacteristic
272C22×C53#39711yescharacteristic
273C22×C53#39711yescharacteristic
274C22×C53#39711yescharacteristic
275C2×D53—44no—
276C2×D53—42no—
277C2×D53—82no—
278C2×D53—82no—
279C4×C53—82no—
280C4×C53—82no—
281C4×C53#38111yescharacteristic
282C4×C53#38611yescharacteristic
283C4×C53#38811yescharacteristic
284C4×C53#38811yescharacteristic
285C4×C53#39411yescharacteristic
286C4×C53#39411yescharacteristic
287C4×C53#39411yescharacteristic
288C4×C53#39711yescharacteristic
289Dic53—42no—
290#2896—531no—
291C2×#416—531no—
292#2879—531no—
293C2×#344—531no—
294C2×#341—531no—
295#1733D5311yescharacteristic, complemented
296#1729—531no—
297C2×C53—21no—
298C2×C53—21no—
299C2×C53—21no—
300C2×C53—21no—
301C2×C53—21no—
302C2×C53—21no—
303C2×C53#144011yescharacteristic
304C2×C53#188112yes—
305D53—82no—
306C22×#41—531no—
307C22×M16—531no—
308C22×M16—531no—
309C22×M16C2×D5311yescharacteristic
310C23×C8—531no—
311C2×#117C2×D5311yescharacteristic
312C2×#117C2×D5311yescharacteristic
313#419—1061no—
314#418—1061no—
315#417—534no—
316#416—1062no—
317#416—1062no—
318#416—1062no—
319C2×#103—1061no—
320C2×#102—1061no—
321#408—1062no—
322#408—1062no—
323#407—1062no—
324#344—1062no—
325#341—1062no—
326#339—22no—
327#336—1062no—
328C53#1119411yescharacteristic, complemented, Sylow
329C2×#41—532no—
330C2×#41—532no—
331C2×#41—1061no—
332C2×#41—1061no—
333C2×#41—1061no—
334C2×#41—1061no—
335C2×#41—1061no—
336C2×#41—1061no—
337C2×#41—1061no—
338C2×#41—1061no—
339C2×#41—1061no—
340C2×#41—1061no—
341C22×Q8—531no—
342C22×D4—531no—
343C22×D4—1061no—
344C23×C4—1061no—
345C23×C4C22×D5311yescharacteristic
346C2×M16—21no—
347C2×M16—21no—
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—532no—
352C2×M16—1061no—
353C2×M16—1061no—
354C2×M16—1061no—
355C2×M16—1061no—
356C2×M16—1061no—
357C2×M16—1061no—
358C2×M16—1061no—
359C2×M16—1061no—
360C2×M16—1061no—
361C2×M16—1061no—
362C2×M16—1061no—
363C2×M16C22⋊D5312yes—
364C22×C8—21no—
365C22×C8—532no—
366C22×C8—532no—
367C22×C8—1061no—
368C22×C8—1061no—
369C22×C8—1061no—
370C22×C8—1061no—
371C22×C8—1061no—
372C22×C8—1061no—
373C22×C8—1061no—
374#117—21no—
375#117—21no—
376#117—21no—
377#117—21no—
378#117—21no—
379#117—21no—
380#117—21no—
381#117D21212yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4C4×D5312yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42—21no—
388#103—1064no—
389#103—1064no—
390#102—1064no—
391#102—1064no—
392C22⋊C8—1068no—
393C24—1061no—
394C24#1171994711yescharacteristic
395#41—1062no—
396#41—1062no—
397#41—1062no—
398#41—1062no—
399#41—1062no—
400#41—1062no—
401#41—1062no—
402#41—1062no—
403#41—2121no—
404#41—2121no—
405#41—2121no—
406#41—2121no—
407#41—2121no—
408#41—2121no—
409#41—2121no—
410#41—2121no—
411C2×Q8—532no—
412C2×Q8—1061no—
413C2×Q8—1061no—
414C2×Q8—1061no—
415C2×Q8—1061no—
416C2×Q8—1061no—
417C2×D4—532no—
418C2×D4—1062no—
419C2×D4—1062no—
420C2×D4—1061no—
421C2×D4—1061no—
422C2×D4—1061no—
423C2×D4—1061no—
424C2×D4—1061no—
425C2×D4—2121no—
426C2×D4—2121no—
427C2×D4—2121no—
428C2×D4—2121no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—1061no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—2121no—
435C22×C4—2121no—
436C22×C4—2121no—
437C22×C4—2121no—
438C22×C4C2×C4×D5311yescharacteristic
439C22×C4C2×D21211yescharacteristic
440C22×C4#1171994511yescharacteristic
441C22×C4#1171994511yescharacteristic
442C22×C4#1171994511yescharacteristic
443C22×C4#1171994511yescharacteristic
444C22×C4D4×D5311yescharacteristic
445C22×C4D4×D5311yescharacteristic
446C22×C4D4×D5311yescharacteristic
447C22×C4D4×D5311yescharacteristic
448C22×C4#1171994711yescharacteristic
449C22×C4#1171994911yescharacteristic
450C22×C4#1171994911yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452M16—22no—
453M16—22no—
454M16—41no—
455M16—41no—
456M16—1062no—
457M16—1062no—
458M16—2121no—
459M16—2121no—
460M16—2121no—
461M16—2121no—
462M16—2121no—
463M16—2121no—
464C2×C8—22no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—1062no—
468C2×C8—1062no—
469C2×C8—1062no—
470C2×C8—1062no—
471C2×C8—1062no—
472C2×C8—1062no—
473C2×C8—1062no—
474C2×C8—2121no—
475C2×C8—2121no—
476C2×C8—2121no—
477C2×C8—2121no—
478C2×C8—2121no—
479C2×C8—2121no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—41no—
484C22⋊C4—22no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—22no—
489C42—41no—
490C42—41no—
491C42—41no—
492C23—42no—
493C23—1062no—
494C23—4242no—
495C23—2121no—
496C23—2121no—
497C23#1313317711yescharacteristic
498C23#1313318811yescharacteristic
499C23#1313319911yescharacteristic
500C23#1313320111yescharacteristic
501C23#1313320311yescharacteristic
502C23#1313321211yescharacteristic
503C23#1313321211yescharacteristic
504Q8—1062no—
505Q8—1062no—
506Q8—2121no—
507Q8—2121no—
508D4—1062no—
509D4—1062no—
510D4—2122no—
511D4—2122no—
512D4—2122no—
513D4—2122no—
514D4—2121no—
515D4—2121no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—82no—
520C2×C4—1062no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4—2122no—
526C2×C4—2122no—
527C2×C4C22⋊C4×D5311yescharacteristic
528C2×C4#1313317411yescharacteristic
529C2×C4#1313317511yescharacteristic
530C2×C4C4×C22⋊D5311yescharacteristic
531C2×C4#1313317911yescharacteristic
532C2×C4C4×D21211yescharacteristic
533C2×C4#1313318511yescharacteristic
534C2×C4#1313318711yescharacteristic
535C2×C4#1313318911yescharacteristic
536C2×C4#1313319011yescharacteristic
537C2×C4#1313319011yescharacteristic
538C2×C4#1313319011yescharacteristic
539C2×C4#1313319111yescharacteristic
540C2×C4#1313319211yescharacteristic
541C2×C4#1313319211yescharacteristic
542C2×C4#1313319911yescharacteristic
543C2×C4#1313319911yescharacteristic
544C2×C4#1313320011yescharacteristic
545C2×C4#1313320211yescharacteristic
546C2×C4#1313320211yescharacteristic
547C2×C4#1313320211yescharacteristic
548C2×C4#1313320211yescharacteristic
549C2×C4#1313320211yescharacteristic
550C2×C4#1313320311yescharacteristic
551C2×C4#1313320511yescharacteristic
552C2×C4#1313320611yescharacteristic
553C2×C4#1313321111yescharacteristic
554C2×C4#1313321211yescharacteristic
555C8—42no—
556C8—2122no—
557C8—2122no—
558C8—2122no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—4242no—
567C22—4242no—
568C22—2122no—
569C22—2124no—
570C22#1681049911yescharacteristic
571C22#1681051211yescharacteristic
572C22#1681051511yescharacteristic
573C22#1681053911yescharacteristic
574C22#1681054911yescharacteristic
575C22#1681055011yescharacteristic
576C22#1681055211yescharacteristic
577C4—82no—
578C4—82no—
579C4—2122no—
580C4#1681048511yescharacteristic
581C4#1681050111yescharacteristic
582C4#1681051311yescharacteristic
583C4#1681051411yescharacteristic
584C4#1681053811yescharacteristic
585C4#1681054011yescharacteristic
586C4#1681054111yescharacteristic
587C4#1681055211yescharacteristic
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—4242no—
595C2#2027296311yescharacteristic
596C2#2027507412yes—
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1733C2
socleC22×C53#386
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenter#1733D53
hypocenterC53#11194
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
4. C2×#16810393C2semidirect
5. C2×#16810387C2semidirect
6. #20274412C2semidirect
7. C53×#1733C2semidirect
14. #16810622C4semidirect
295. #1733D53semidirect
328. C53#11194semidirect

Maximal subgroups

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

subgroupindex
#202832322
C2×#168106172
#202831592
C2×#168103932
C2×#168103872
#202744122
C53×#17332
#1119453