Finite Group Databasebeta

← #27469313

Subgroups — #27469313

Order 13568.

850 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810965C211yesmaximal, characteristic
2C2×#16810862C211yesmaximal, characteristic
3C2×#16810659C211yesmaximal, characteristic
4#20279169C211yesmaximal, characteristic
5#20279164C211yesmaximal, characteristic
6#20276261C211yesmaximal, characteristic, complemented
7#20276260C211yesmaximal, characteristic, complemented
8#20276256C211yesmaximal, characteristic, complemented
9#20276255C211yesmaximal, characteristic, complemented
10#20275573C211yesmaximal, characteristic, complemented
11#20275573C211yesmaximal, characteristic, complemented
12#20274876C211yesmaximal, characteristic, complemented
13#20274873C211yesmaximal, characteristic, complemented
14#20274489C211yesmaximal, characteristic
15C2×C53×#485C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20#16810965—24no—
21#16810862—24no—
22#16810659—24no—
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133149C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133141C2211yescharacteristic
33C2×#13133140C2211yescharacteristic
34C2×C8×Dic53C2211yescharacteristic
35C2×C8×Dic53C2211yescharacteristic
36#16810551—22no—
37#16810551—22no—
38#16810530C2211yescharacteristic
39#16810530C2211yescharacteristic
40#16810528C2211yescharacteristic
41#16810528C2211yescharacteristic
42#16810507—22no—
43#16810504—22no—
44#16810492C2211yescharacteristic
45#16810492C2211yescharacteristic
46#16810488C2211yescharacteristic
47#16810488C2211yescharacteristic
48C4×C22⋊Dic53C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51C2×C22⋊Q8×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C53×#485—24no—
54C2×#107×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—
63C22×#88322—22no—
64C22×#88322—22no—
65C22×#88322D414yes—
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×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53D412yes—
81C2×C4×Dic53—21no—
82C2×C4×Dic53C2311yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84#13133151—22no—
85#13133150—22no—
86#13133150—22no—
87#13133150—22no—
88#13133149—22no—
89#13133148—22no—
90#13133148—22no—
91#13133142—22no—
92#13133142—22no—
93#13133141—22no—
94#13133140—22no—
95C8×Dic53—22no—
96C8×Dic53—22no—
97#13133124—22no—
98#13133124—22no—
99#13133124—22no—
100#13133124—22no—
101#13133124—22no—
102#13133124—22no—
103#13133124D412yes—
104C22×Q8×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22⋊Q8×C53—22no—
109C22⋊Q8×C53—22no—
110#117×C53—22no—
111#117×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C42×C53—21no—
119#107×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×C53C2×D411yescharacteristic
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53C2×D412yes—
148C22×Dic53C2×D412yes—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53C2×D412yes—
160C2×C8×C53C2×D412yes—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—42no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C42×C53—42no—
171C22⋊Dic53—24no—
172C22⋊Dic53—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—
181#11719919—42no—
182#11719919—44no—
183#11719919—44no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53—42no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—28no—
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—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53#12712yes—
222C2×C4×C53C22×D411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×C4×C532−1+411yescharacteristic
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53#13514yes—
243Dic106—41no—
244Dic106—41no—
245Dic106—41no—
246Dic106—41no—
247C8×C53—22no—
248C8×C53—22no—
249#88322—22no—
250#88322#12314yes—
251#27927—531nomaximal, Sylow
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53C2×#12711yescharacteristic
258C22×C53#54711yescharacteristic
259C22×C53#54711yescharacteristic
260C22×C53C2×#13512yes—
261C22×C53#57812yes—
262C4×C53—22no—
263C4×C53—81no—
264C4×C53—81no—
265C4×C53—81no—
266C4×C53—81no—
267C4×C53C2×#12311yescharacteristic
268C4×C53#53611yescharacteristic
269C4×C53#54711yescharacteristic
270C4×C53#54711yescharacteristic
271Dic53—24no—
272Dic53—42no—
273Dic53—81no—
274Dic53—81no—
275Dic53—81no—
276Dic53—81no—
277C2×#485—531no—
278C2×#485D5311yescharacteristic
279C2×#467—531no—
280C2×#427—531no—
281#2499—531no—
282#2499—531no—
283#2097—531no—
284#2097—531no—
285#2096—531no—
286#2096—531no—
287#1969—531no—
288#1969—531no—
289#1846—531no—
290#1845—531no—
291C8×C22⋊C4—531no—
292C2×C53—42no—
293C2×C53#250111yescharacteristic
294C2×C53#314711yescharacteristic
295C2×C53#315611yescharacteristic
296C2×C53#333612yes—
297C2×C53#334212yes—
298C22×Q16—531no—
299C23×C8—531no—
300C2×C22⋊Q8—531no—
301C2×C22⋊Q8C2×D5311yescharacteristic
302C2×#117—531no—
303C2×#117C2×D5311yescharacteristic
304#485—24no—
305#485—1064no—
306#467—1064no—
307#427—1064no—
308C2×#107—531no—
309C2×#107—531no—
310C2×#107—531no—
311C2×#107—531no—
312C2×#107C2×D5311yescharacteristic
313C2×#106—531no—
314C2×#106—531no—
315C2×#106—531no—
316C2×#106C2×D5311yescharacteristic
317C2×#103—531no—
318C2×#103—531no—
319C2×#103—531no—
320C2×#103—531no—
321C2×#103C2×D5311yescharacteristic
322C2×#103C2×D5311yescharacteristic
323C2×C22⋊C8C2×D5311yescharacteristic
324C2×C4×C8—531no—
325C2×C4×C8—531no—
326#397—1062no—
327#397—1062no—
328#391—531no—
329#391—531no—
330#391—531no—
331#391—531no—
332#387—1062no—
333#387—1062no—
334#384—531no—
335#384—531no—
336#382—531no—
337#382—531no—
338C4×C22⋊C4—531no—
339#336—531no—
340#336—531no—
341C53#2792711yescharacteristic, complemented, Sylow
342C22×Q8—531no—
343C22×Q8C22×D5311yescharacteristic
344C23×C4C22×D5311yescharacteristic
345C2×Q16—1062no—
346C2×Q16—1062no—
347C2×Q16—1062no—
348C22×C8—531no—
349C22×C8—534no—
350C22×C8—531no—
351C22×C8—1062no—
352C22×C8—1062no—
353C22×C8C22×D5311yescharacteristic
354C22×C8C22×D5311yescharacteristic
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—1062no—
358C22⋊Q8—1062no—
359#117—22no—
360#117—22no—
361#117—1062no—
362#117—1062no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—531no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1062no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4C22×D5311yescharacteristic
371C2×C4⋊C4C22×D5311yescharacteristic
372C2×C4⋊C4C22×D5311yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—532no—
376C2×C22⋊C4—1061no—
377C2×C22⋊C4—1061no—
378C2×C42—21no—
379C2×C42—531no—
380C2×C42—531no—
381C2×C42—1061no—
382#107—22no—
383#107—1062no—
384#107—1062no—
385#107—1062no—
386#107—1062no—
387#106—22no—
388#106—1062no—
389#106—1062no—
390#106—1062no—
391#103—22no—
392#103—22no—
393#103—1062no—
394#103—1062no—
395#103—1062no—
396#103—1062no—
397C22⋊C8C22⋊D5314yes—
398C4×C8—1062no—
399C4×C8—1062no—
400#95—532no—
401#95—1062no—
402#95—1062no—
403#95—1062no—
404#95—1062no—
405#95—1062no—
406#95—1062no—
407C24D4×D5311yescharacteristic
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—1061no—
415C2×Q8—1061no—
416C2×Q8—1061no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4D4×D5311yescharacteristic
435C22×C4D4×D5312yes—
436C22×C4C2×C22⋊D5312yes—
437C22×C4C23×D5311yescharacteristic
438Q16—2124no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—532no—
444C2×C8—532no—
445C2×C8—532no—
446C2×C8—532no—
447C2×C8—1068no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8C2×C22⋊D5312yes—
451C2×C8C2×C22⋊D5312yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—2124no—
466C4⋊C4—2124no—
467C4⋊C4—2122no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—1064no—
471C22⋊C4—2122no—
472C22⋊C4—2122no—
473C42—42no—
474C42—1061no—
475C42—1061no—
476C42—1061no—
477C42—1061no—
478C42—2122no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—22no—
483C23#1313321312yes—
484C23C2×D4×D5311yescharacteristic
485Q8—41no—
486Q8—41no—
487Q8—41no—
488Q8—41no—
489Q8—2121no—
490Q8—2121no—
491Q8—2121no—
492Q8—2121no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—534no—
509C2×C4—1062no—
510C2×C4—1062no—
511C2×C4—1062no—
512C2×C4—1062no—
513C2×C4—1062no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2122no—
517C2×C4—2122no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2122no—
523C2×C4#1313321312yes—
524C2×C4C2×D4×D5311yescharacteristic
525C2×C4C22×C22⋊D5311yescharacteristic
526C2×C4#41×D5311yescharacteristic
527C2×C4#41×D5311yescharacteristic
528C2×C4#1313326411yescharacteristic
529C2×C4#1313326411yescharacteristic
530C8—22no—
531C8—22no—
532C8—534no—
533C8—1062no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22C2×#1313321311yescharacteristic
540C22#1681127111yescharacteristic
541C22#1681127111yescharacteristic
542C22#135×D5312yes—
543C22#1681143912yes—
544C4—22no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—1064no—
550C4—4241no—
551C4—4241no—
552C4—4241no—
553C4—4241no—
554C4—2122no—
555C4#123×D5311yescharacteristic
556C4#1681119511yescharacteristic
557C4#1681127211yescharacteristic
558C4#1681127211yescharacteristic
559C2—42no—
560C2#2027918411yescharacteristic
561C2#2028513911yescharacteristic
562C2#2028520811yescharacteristic
563C2#2028707612yes—
564C2#2028714012yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#485C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#485D53
hypocenterC53#27927
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
6. #20276261C2semidirect
7. #20276260C2semidirect
8. #20276256C2semidirect
9. #20276255C2semidirect
10. #20275573C2semidirect
11. #20275573C2semidirect
12. #20274876C2semidirect
13. #20274873C2semidirect
341. C53#27927semidirect

Maximal subgroups

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

subgroupindex
C2×#168109652
C2×#168108622
C2×#168106592
#202791692
#202791642
#202762612
#202762602
#202762562
#202762552
#202755732
#202748762
#202748732
#202744892
C2×C53×#4852
#2792753