Finite Group Databasebeta

← #27469452

Subgroups — #27469452

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810965C211yesmaximal, characteristic
2C2×#16810879C211yesmaximal, characteristic
3C2×#16810663C211yesmaximal, characteristic
4#20279189C211yesmaximal, characteristic
5#20279169C211yesmaximal, characteristic
6#20276264C211yesmaximal, characteristic, complemented
7#20276256C211yesmaximal, characteristic, complemented
8#20276255C211yesmaximal, characteristic, complemented
9#20276251C211yesmaximal, characteristic, complemented
10#20275583C211yesmaximal, characteristic, complemented
11#20275573C211yesmaximal, characteristic, complemented
12#20274880C211yesmaximal, characteristic, complemented
13#20274870C211yesmaximal, characteristic, complemented
14#20274501C211yesmaximal, characteristic
15C2×C53×#483C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20#16810965—24no—
21#16810879—24no—
22#16810663—24no—
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133141C2211yescharacteristic
33C2×#13133140C2211yescharacteristic
34C2×#13133127C2211yescharacteristic
35C2×C8×Dic53C2211yescharacteristic
36#16810554—22no—
37#16810551—22no—
38#16810530C2211yescharacteristic
39#16810530C2211yescharacteristic
40#16810528C2211yescharacteristic
41#16810528C2211yescharacteristic
42#16810504—22no—
43#16810496—22no—
44#16810492C2211yescharacteristic
45#16810492C2211yescharacteristic
46#16810492C2211yescharacteristic
47#16810488C2211yescharacteristic
48C4×C22⋊Dic53C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51C2×C22⋊Q8×C53C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C53×#483—24no—
54C2×#106×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C2×#11719925—22no—
61C2×#11719925—22no—
62C2×#11719925—22no—
63C2×#11719917—22no—
64C2×#11719917—22no—
65C2×#11719917D414yes—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133196—22no—
69#13133196—22no—
70#13133176—22no—
71#13133176—22no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—22no—
77C2×#11719919—22no—
78C2×#11719919—22no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53D412yes—
82C2×C4×Dic53—21no—
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85#13133151—22no—
86#13133150—22no—
87#13133150—22no—
88#13133149—22no—
89#13133148—22no—
90#13133148—22no—
91#13133148—22no—
92#13133142—22no—
93#13133142—22no—
94#13133141—22no—
95#13133140—22no—
96#13133127—22no—
97C8×Dic53—22no—
98#13133124—22no—
99#13133124—22no—
100#13133124—22no—
101#13133124—22no—
102#13133124—22no—
103#13133124D412yes—
104C22×Q8×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22⋊Q8×C53—22no—
110C22⋊Q8×C53—22no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119#106×C53—22no—
120#106×C53—22no—
121#103×C53—22no—
122#103×C53—22no—
123C22⋊C8×C53D414yes—
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×C53—42no—
140C22×C4×C53—82no—
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×Dic53C2×D412yes—
150C22×Dic53C2×D412yes—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53C2×D412yes—
162C2×C8×C53C2×D412yes—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—42no—
171C22⋊C4×C53—42no—
172C22⋊Dic53—24no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175#11719925—44no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182#11719919—42no—
183#11719919—44no—
184#11719919—82no—
185#11719919—82no—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—42no—
191#11719917—28no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322C2×D412yes—
195C2×#88322C2×D412yes—
196C2×#88322#4112yes—
197C2×#88322#4112yes—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53#12712yes—
203C23×C53C22×D411yescharacteristic
204Q8×C53—41no—
205Q8×C53—41no—
206Q8×C53—41no—
207Q8×C53—41no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—84no—
224C2×C4×C53#12712yes—
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C532−1+411yescharacteristic
230C2×C4×C532−1+411yescharacteristic
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53#13714yes—
246Dic106—41no—
247Dic106—41no—
248Dic106—41no—
249Dic106—41no—
250C8×C53—22no—
251C8×C53—22no—
252#88322—22no—
253#88322#12314yes—
254#27940—531nomaximal, Sylow
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53C2×#12711yescharacteristic
261C22×C53#54711yescharacteristic
262C22×C53#54711yescharacteristic
263C22×C53C2×#13712yes—
264C22×C53#57712yes—
265C4×C53—22no—
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—24no—
275Dic53—42no—
276Dic53—81no—
277Dic53—81no—
278Dic53—81no—
279Dic53—81no—
280C2×#485—531no—
281C2×#483D5311yescharacteristic
282C2×#470—531no—
283C2×#428—531no—
284#2502—531no—
285#2499—531no—
286#2097—531no—
287#2096—531no—
288#2096—531no—
289#2095—531no—
290#1971—531no—
291#1969—531no—
292#1847—531no—
293#1844—531no—
294#1753—531no—
295C2×C53—42no—
296C2×C53#250111yescharacteristic
297C2×C53#315211yescharacteristic
298C2×C53#315411yescharacteristic
299C2×C53#334412yes—
300C2×C53#335612yes—
301C22×Q16—531no—
302C22×M16—531no—
303C2×C22⋊Q8—531no—
304C2×C22⋊Q8C2×D5311yescharacteristic
305C2×#117—531no—
306C22×C4⋊C4C2×D5311yescharacteristic
307#485—1064no—
308#483—24no—
309#470—1064no—
310#428—1064no—
311C2×#107—531no—
312C2×#107—531no—
313C2×#107—531no—
314C2×#106—531no—
315C2×#106—531no—
316C2×#106—531no—
317C2×#106—531no—
318C2×#106C2×D5311yescharacteristic
319C2×#106C2×D5311yescharacteristic
320C2×#103—531no—
321C2×#103—531no—
322C2×#103—531no—
323C2×#103—531no—
324C2×#103C2×D5311yescharacteristic
325C2×#103C2×D5311yescharacteristic
326C2×C22⋊C8C2×D5311yescharacteristic
327C2×#97—531no—
328C2×C4×C8—531no—
329#398—1062no—
330#397—1062no—
331#391—531no—
332#391—531no—
333#391—531no—
334#391—531no—
335#387—1062no—
336#385—1062no—
337#384—531no—
338#384—531no—
339#384—531no—
340#382—531no—
341C4×C22⋊C4—531no—
342#336—531no—
343#336—531no—
344C53#2794011yescharacteristic, complemented, Sylow
345C22×Q8—531no—
346C22×Q8C22×D5311yescharacteristic
347C23×C4—21no—
348C23×C4C22×D5311yescharacteristic
349C2×Q16—1062no—
350C2×Q16—1062no—
351C2×Q16—1062no—
352C2×M16—534no—
353C2×M16—1062no—
354C2×M16—1062no—
355C22×C8—531no—
356C22×C8—531no—
357C22×C8C22×D5311yescharacteristic
358C22×C8C22×D5311yescharacteristic
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—1062no—
362C22⋊Q8—1062no—
363#117—1062no—
364#117—1062no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1062no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—532no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1061no—
383C2×C42—531no—
384C2×C42—531no—
385C2×C42—1061no—
386#107—1062no—
387#107—1062no—
388#107—1062no—
389#106—22no—
390#106—22no—
391#106—1062no—
392#106—1062no—
393#106—1062no—
394#106—1062no—
395#103—22no—
396#103—22no—
397#103—1062no—
398#103—1062no—
399#103—1062no—
400#103—1062no—
401C22⋊C8C22⋊D5314yes—
402#97—1062no—
403C4×C8—1062no—
404#95—532no—
405#95—1062no—
406#95—1062no—
407#95—1062no—
408#95—1062no—
409#95—1062no—
410C24D4×D5311yescharacteristic
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—82no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4D4×D5311yescharacteristic
440C22×C4D4×D5312yes—
441C22×C4C2×C22⋊D5312yes—
442C22×C4C23×D5311yescharacteristic
443Q16—2124no—
444M16—1068no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—532no—
450C2×C8—532no—
451C2×C8—532no—
452C2×C8—532no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8C2×C22⋊D5312yes—
456C2×C8C2×C22⋊D5312yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—4242no—
472C4⋊C4—4242no—
473C4⋊C4—2124no—
474C4⋊C4—2122no—
475C22⋊C4—42no—
476C22⋊C4—1064no—
477C22⋊C4—2122no—
478C22⋊C4—2122no—
479C42—1061no—
480C42—1061no—
481C42—1061no—
482C42—1061no—
483C42—2122no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23#1313321312yes—
489C23C2×D4×D5311yescharacteristic
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—2121no—
495Q8—2121no—
496Q8—2121no—
497Q8—2121no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—84no—
514C2×C4—534no—
515C2×C4—1062no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2122no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2122no—
528C2×C4—2122no—
529C2×C4#1313321312yes—
530C2×C4C2×D4×D5311yescharacteristic
531C2×C4C22×C22⋊D5311yescharacteristic
532C2×C4#41×D5311yescharacteristic
533C2×C4#41×D5311yescharacteristic
534C2×C4#1313326411yescharacteristic
535C2×C4#1313326411yescharacteristic
536C8—22no—
537C8—22no—
538C8—534no—
539C8—1062no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—42no—
544C22—42no—
545C22C2×#1313321311yescharacteristic
546C22#1681127111yescharacteristic
547C22#1681127111yescharacteristic
548C22#137×D5312yes—
549C22#1681142912yes—
550C4—22no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—1064no—
556C4—4241no—
557C4—4241no—
558C4—4241no—
559C4—4241no—
560C4—2122no—
561C4#123×D5311yescharacteristic
562C4#1681119511yescharacteristic
563C4#1681127211yescharacteristic
564C4#1681127211yescharacteristic
565C2—42no—
566C2#2027918411yescharacteristic
567C2#2028517011yescharacteristic
568C2#2028518511yescharacteristic
569C2#2028717312yes—
570C2#2028732312yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#483C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#483D53
hypocenterC53#27940
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
6. #20276264C2semidirect
7. #20276256C2semidirect
8. #20276255C2semidirect
9. #20276251C2semidirect
10. #20275583C2semidirect
11. #20275573C2semidirect
12. #20274880C2semidirect
13. #20274870C2semidirect
344. C53#27940semidirect

Maximal subgroups

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

subgroupindex
C2×#168109652
C2×#168108792
C2×#168106632
#202791892
#202791692
#202762642
#202762562
#202762552
#202762512
#202755832
#202755732
#202748802
#202748702
#202745012
C2×C53×#4832
#2794053