Finite Group Databasebeta

← #27469487

Subgroups — #27469487

Order 13568.

846 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C211yesmaximal, characteristic
2C2×#16810964C211yesmaximal, characteristic
3C2×#16810879C211yesmaximal, characteristic
4C2×#16810666C211yesmaximal, characteristic
5#20279200C211yesmaximal, characteristic
6#20276266C211yesmaximal, characteristic, complemented
7#20276263C211yesmaximal, characteristic, complemented
8#20276263C211yesmaximal, characteristic, complemented
9#20276254C211yesmaximal, characteristic, complemented
10#20275585C211yesmaximal, characteristic, complemented
11#20275575C211yesmaximal, characteristic, complemented
12#20274881C211yesmaximal, characteristic, complemented
13#20274877C211yesmaximal, characteristic, complemented
14#20274499C211yesmaximal, characteristic
15C53×#2499C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810964—24no—
21#16810956—24no—
22#16810879—24no—
23#16810666—24no—
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32C2×#13133148C2211yescharacteristic
33C2×#13133141C2211yescharacteristic
34C2×#13133141C2211yescharacteristic
35C2×#13133141C2211yescharacteristic
36C2×#13133140C2211yescharacteristic
37C2×#13133140C2211yescharacteristic
38C2×#13133140C2211yescharacteristic
39C2×C22⋊#88322C2211yescharacteristic
40C2×#13133128C2211yescharacteristic
41C2×C4×#88322C2211yescharacteristic
42#16810563—22no—
43#16810529C2211yescharacteristic
44#16810529C2211yescharacteristic
45#16810489C2211yescharacteristic
46#16810489C2211yescharacteristic
47#16810484—22no—
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#397—22no—
53C53×#391C2211yescharacteristic
54C53×#391C2211yescharacteristic
55C53×#387—22no—
56C53×#384C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C23×Dic53—21no—
61C2×#11719925—22no—
62C2×#11719925—22no—
63C2×#11719925—22no—
64C2×#11719917—22no—
65C2×#11719917—22no—
66C2×#11719917D414yes—
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71#13133196—22no—
72#13133196—22no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81#13133150—22no—
82#13133150—22no—
83#13133150—22no—
84#13133150—22no—
85#13133150—22no—
86#13133148—22no—
87#13133148—22no—
88#13133148—22no—
89#13133148—22no—
90#13133141—22no—
91#13133141—22no—
92#13133141—22no—
93#13133140—22no—
94#13133140—22no—
95#13133140—22no—
96C22⋊#88322D414yes—
97#13133128—22no—
98C4×#88322—22no—
99#13133124—22no—
100#13133124—22no—
101#13133124—22no—
102#13133124—22no—
103C22×Q8×C53C2311yescharacteristic
104C23×C4×C53C2311yescharacteristic
105C22⋊Q8×C53—22no—
106C22⋊Q8×C53—22no—
107#117×C53—22no—
108#117×C53—22no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—22no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53D412yes—
117C2×C42×C53—21no—
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120#95×C53—22no—
121#95×C53—22no—
122#95×C53—22no—
123#95×C53D412yes—
124C24×C53C2×D411yescharacteristic
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—44no—
150C22×Dic53—42no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—44no—
164C4⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—42no—
167C22⋊C4×C53—42no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—42no—
173C22⋊Dic53—42no—
174#11719925—44no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—42no—
182#11719919—42no—
183#11719917—28no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322C2×D412yes—
191C2×#88322C2×D412yes—
192C2×#88322C2×D412yes—
193C2×#88322C2×D412yes—
194C2×#88322#4112yes—
195C2×#88322#4112yes—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53#12712yes—
201C23×C53C22×D411yescharacteristic
202Q8×C53—41no—
203Q8×C53—41no—
204Q8×C53—41no—
205Q8×C53—41no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53#12712yes—
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C532−1+411yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—84no—
244Dic106—41no—
245Dic106—41no—
246Dic106—41no—
247Dic106—41no—
248#88322—22no—
249#88322—22no—
250#88322—22no—
251#88322#12314yes—
252#27943—531nomaximal, Sylow
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53C2×#12711yescharacteristic
259C22×C53#54711yescharacteristic
260C22×C53#54711yescharacteristic
261C22×C53#57512yes—
262C22×C53#57812yes—
263C4×C53—22no—
264C4×C53—42no—
265C4×C53—42no—
266C4×C53—81no—
267C4×C53—81no—
268C4×C53—81no—
269C4×C53—81no—
270C4×C53C2×#12311yescharacteristic
271C4×C53#53611yescharacteristic
272C4×C53#54711yescharacteristic
273C4×C53#54711yescharacteristic
274Dic53—81no—
275Dic53—81no—
276Dic53—81no—
277Dic53—81no—
278C2×#484—531no—
279C2×#485—531no—
280C2×#470—531no—
281C2×#428—531no—
282#2503—531no—
283#2499D5311yescharacteristic
284#2097—531no—
285#2097—531no—
286#2097—531no—
287#2095—531no—
288#1971—531no—
289#1969—531no—
290#1847—531no—
291#1846—531no—
292#1753—531no—
293C2×C53—42no—
294C2×C53#250111yescharacteristic
295C2×C53#315311yescharacteristic
296C2×C53#315611yescharacteristic
297C2×C53#334912yes—
298C2×C53#335212yes—
299C22×Q16—531no—
300C22×M16—531no—
301C2×C22⋊Q8—531no—
302C2×C22⋊Q8C2×D5311yescharacteristic
303C2×#117C2×D5311yescharacteristic
304C22×C4⋊C4—531no—
305#485—1064no—
306#484—1064no—
307#470—1064no—
308#428—1064no—
309C2×#107—531no—
310C2×#107—531no—
311C2×#107—531no—
312C2×#107—531no—
313C2×#107—531no—
314C2×#106—531no—
315C2×#106—531no—
316C2×#106—531no—
317C2×#106—531no—
318C2×#103—531no—
319C2×#103—531no—
320C2×#103—531no—
321C2×#103—531no—
322C2×#103—531no—
323C2×#103—531no—
324C2×C22⋊C8—531no—
325C2×#97—531no—
326C2×C4×C8—531no—
327#400—1062no—
328#397—22no—
329#391—531no—
330#391—531no—
331#391C2×D5311yescharacteristic
332#391C2×D5311yescharacteristic
333#387—22no—
334#384C2×D5311yescharacteristic
335#382—531no—
336#382—531no—
337#382C2×D5311yescharacteristic
338#381—1062no—
339C4×C22⋊C4C2×D5311yescharacteristic
340#336—531no—
341#336—531no—
342C53#2794311yescharacteristic, complemented, Sylow
343C22×Q8—531no—
344C22×Q8C22×D5311yescharacteristic
345C23×C4—1061no—
346C23×C4C22×D5311yescharacteristic
347C2×Q16—1062no—
348C2×Q16—1062no—
349C2×Q16—1062no—
350C2×M16—534no—
351C2×M16—1062no—
352C2×M16—1062no—
353C22×C8—531no—
354C22×C8—531no—
355C22×C8—531no—
356C22×C8—531no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—1062no—
360C22⋊Q8—1062no—
361#117—22no—
362#117—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—1062no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C4⋊C4C22×D5311yescharacteristic
374C2×C4⋊C4C22×D5311yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—1061no—
378C2×C22⋊C4C22⋊D5312yes—
379C2×C42—21no—
380C2×C42C22×D5311yescharacteristic
381C2×C42C22×D5311yescharacteristic
382#107—1062no—
383#107—1062no—
384#107—1062no—
385#107—1062no—
386#107—1062no—
387#106—1062no—
388#106—1062no—
389#106—1062no—
390#106—1062no—
391#103—1062no—
392#103—1062no—
393#103—1062no—
394#103—1062no—
395#103—1062no—
396#103—1062no—
397C22⋊C8—534no—
398#97—1062no—
399C4×C8—1062no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—1062no—
404#95—1062no—
405#95—1062no—
406#95—1062no—
407#95C22⋊D5312yes—
408C24C2×C22⋊D5311yescharacteristic
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—1061no—
416C2×Q8—1061no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—2124no—
434C22×C4—2122no—
435C22×C4D4×D5312yes—
436C22×C4D4×D5312yes—
437C22×C4C2×C22⋊D5312yes—
438C22×C4C2×C22⋊D5312yes—
439C22×C4C2×C22⋊D5311yescharacteristic
440C22×C4C23×D5311yescharacteristic
441Q16—2124no—
442M16—1068no—
443C2×C8—532no—
444C2×C8—532no—
445C2×C8—532no—
446C2×C8—532no—
447C2×C8—532no—
448C2×C8—532no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—2122no—
470C4⋊C4—2122no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—2122no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—42no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23#1313321312yes—
485C23C22×C22⋊D5311yescharacteristic
486Q8—41no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—2121no—
491Q8—2121no—
492Q8—2121no—
493Q8—2121no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—4244no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2122no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4#1313321312yes—
526C2×C4C2×D4×D5311yescharacteristic
527C2×C4C2×D4×D5311yescharacteristic
528C2×C4#41×D5311yescharacteristic
529C2×C4#41×D5311yescharacteristic
530C2×C4#1313326411yescharacteristic
531C2×C4#1313326411yescharacteristic
532C8—534no—
533C8—1062no—
534C8—1062no—
535C8—1062no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—42no—
540C22—42no—
541C22C2×#1313321311yescharacteristic
542C22#1681127211yescharacteristic
543C22#1681127211yescharacteristic
544C22#1681142112yes—
545C22#1681143812yes—
546C4—22no—
547C4—42no—
548C4—42no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4—81no—
553C4—4241no—
554C4—4241no—
555C4—4241no—
556C4—4241no—
557C4#123×D5311yescharacteristic
558C4#1681119511yescharacteristic
559C4#1681127111yescharacteristic
560C4#1681127111yescharacteristic
561C2—42no—
562C2#2027918411yescharacteristic
563C2#2028518311yescharacteristic
564C2#2028520611yescharacteristic
565C2#2028723912yes—
566C2#2028728412yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2499C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2499D53
hypocenterC53#27943
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2499

Complements

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

NH ≅ G/Ntype
6. #20276266C2semidirect
7. #20276263C2semidirect
8. #20276263C2semidirect
9. #20276254C2semidirect
10. #20275585C2semidirect
11. #20275575C2semidirect
12. #20274881C2semidirect
13. #20274877C2semidirect
342. C53#27943semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168109642
C2×#168108792
C2×#168106662
#202792002
#202762662
#202762632
#202762542
#202755852
#202755752
#202748812
#202748772
#202744992
C53×#24992
#2794353