Finite Group Databasebeta

← #27353307

Subgroups — #27353307

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283832C211yesmaximal, characteristic
2#20283359C211yesmaximal, characteristic
3#20273506C211yesmaximal, characteristic, complemented
4#20273503C211yesmaximal, characteristic, complemented
5#20273501C211yesmaximal, characteristic, complemented
6#20273498C211yesmaximal, characteristic, complemented
7#20273496C211yesmaximal, characteristic, complemented
8#20273493C211yesmaximal, characteristic, complemented
9#20273491C211yesmaximal, characteristic, complemented
10#20273489C211yesmaximal, characteristic, complemented
11#20273483C211yesmaximal, characteristic, complemented
12#20273480C211yesmaximal, characteristic, complemented
13#20273476C211yesmaximal, characteristic, complemented
14#20273473C211yesmaximal, characteristic, complemented
15C53×#2989C211yesmaximal, characteristic
16#16810754—21no—
17#16810701—21no—
18#16810695—21no—
19#16810695—21no—
20#16810684C412yes—
21#16810684C412yes—
22#16810680C2211yescharacteristic
23#16810680C2211yescharacteristic
24#16810676C2211yescharacteristic
25#16810676C2211yescharacteristic
26#16810676C2211yescharacteristic
27#16810676C2211yescharacteristic
28#16810676C2211yescharacteristic
29#16810650C412yes—
30#16810647C2211yescharacteristic
31#16810647C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810590C412yes—
34C4×#11719917C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810375C2211yescharacteristic
37#16810375C2211yescharacteristic
38#16810375C2211yescharacteristic
39#16810375C2211yescharacteristic
40#16810375C2211yescharacteristic
41#16810375C2211yescharacteristic
42#16810375C2211yescharacteristic
43#16810375C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810373C2211yescharacteristic
49#16810373C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810373C2211yescharacteristic
52C4×#41×C53C2211yescharacteristic
53C53×#446—21no—
54C53×#436—21no—
55C53×#435—21no—
56C53×#435—21no—
57C53×#432C2211yescharacteristic
58C53×#432C2211yescharacteristic
59C53×#431C2211yescharacteristic
60C53×#423C2211yescharacteristic
61C53×#423C2211yescharacteristic
62C2×C4⋊C8×C53C2211yescharacteristic
63C2×#11719917—21no—
64C2×#11719917—21no—
65C2×#11719917C2×C412yes—
66C22×#88322—21no—
67C22×#88322C2×C412yes—
68C4⋊#88322—21no—
69C4⋊#88322—21no—
70C4⋊#88322—21no—
71C4⋊#88322C2×C412yes—
72C4⋊#88322C2×C412yes—
73C4⋊#88322C2×C412yes—
74C4⋊#88322C2×C412yes—
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2311yescharacteristic
77C4⋊#88322C2311yescharacteristic
78C4⋊#88322C2311yescharacteristic
79C22⋊#88322—21no—
80C22⋊#88322—21no—
81C22⋊#88322—21no—
82C22⋊#88322C2×C412yes—
83C22⋊#88322C2×C412yes—
84C22⋊#88322C2×C412yes—
85C22⋊#88322C2×C412yes—
86#13133128—21no—
87#13133128—21no—
88#13133128C2×C412yes—
89C4×#88322—21no—
90C4×#88322C2×C412yes—
91C4×#88322C2311yescharacteristic
92C4×#88322C2311yescharacteristic
93C4×#88322C2311yescharacteristic
94C4×#88322C2311yescharacteristic
95C2×#41×C53C2×C411yescharacteristic
96C2×M16×C53—21no—
97C2×M16×C53—21no—
98C22×C8×C53—21no—
99C4×Q8×C53D411yescharacteristic
100C4×Q8×C53Q811yescharacteristic
101C4×D4×C53D411yescharacteristic
102C4×D4×C53D411yescharacteristic
103C4×D4×C53D411yescharacteristic
104C4×D4×C53Q811yescharacteristic
105C4×D4×C53Q811yescharacteristic
106C4×D4×C53Q811yescharacteristic
107#117×C53C2×C411yescharacteristic
108#117×C53C2×C411yescharacteristic
109#117×C53C2×C411yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113C4⋊C8×C53—21no—
114C4⋊C8×C53—21no—
115C4⋊C8×C53—21no—
116C4⋊C8×C53C2311yescharacteristic
117C4⋊C8×C53C2311yescharacteristic
118C4⋊C8×C53C2311yescharacteristic
119C4⋊C8×C53C2311yescharacteristic
120C22⋊C8×C53—21no—
121C22⋊C8×C53—21no—
122C22⋊C8×C53—21no—
123#97×C53—21no—
124#97×C53—21no—
125C4×C8×C53—21no—
126#41×C53—42no—
127C2×Q8×C53C4⋊C411yescharacteristic
128C2×D4×C53C4⋊C411yescharacteristic
129C2×D4×C53C4⋊C411yescharacteristic
130C2×D4×C53C4⋊C411yescharacteristic
131C22×C4×C53C22×C411yescharacteristic
132C22×C4×C53C22×C411yescharacteristic
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×Q811yescharacteristic
138C22×C4×C53C2×Q811yescharacteristic
139C22×C4×C53C2×Q811yescharacteristic
140M16×C53—81no—
141M16×C53—81no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—81no—
147C4⋊C4×C53C4⋊C411yescharacteristic
148C4⋊C4×C53C4⋊C411yescharacteristic
149C4⋊C4×C53C4⋊C411yescharacteristic
150C4⋊C4×C53C4⋊C411yescharacteristic
151C4⋊C4×C53C4⋊C411yescharacteristic
152C4⋊C4×C53C4⋊C411yescharacteristic
153C22⋊C4×C53C4⋊C411yescharacteristic
154C22⋊C4×C53C4⋊C411yescharacteristic
155C22⋊C4×C53C4⋊C411yescharacteristic
156C22⋊C4×C53C4⋊C411yescharacteristic
157C22⋊C4×C53C4⋊C411yescharacteristic
158C22⋊C4×C53C4⋊C411yescharacteristic
159C42×C53C22×C411yescharacteristic
160C42×C53C22×C411yescharacteristic
161C42×C53C22×C411yescharacteristic
162C42×C53C2×D411yescharacteristic
163C42×C53C2×D411yescharacteristic
164C42×C53C2×D411yescharacteristic
165C42×C53C2×Q811yescharacteristic
166C42×C53C2×Q811yescharacteristic
167C42×C53C2×Q811yescharacteristic
168C42×C53C2411yescharacteristic
169#11719917—42no—
170#11719917—81no—
171#11719917—81no—
172C2×#88322—21no—
173C2×#88322—21no—
174C2×#88322—21no—
175C2×#88322—21no—
176C2×#88322—42no—
177C2×#88322—81no—
178C2×#88322C22×C412yes—
179C2×#88322C22×C412yes—
180C2×#88322C22×C412yes—
181C2×#88322C22×C412yes—
182C23×C53C2×C4⋊C411yescharacteristic
183C23×C53C2×C4⋊C411yescharacteristic
184C23×C53C2×C4⋊C411yescharacteristic
185Q8×C53—41no—
186D4×C53—41no—
187D4×C53—41no—
188D4×C53—41no—
189C2×C4×C53—22no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53C2×C4⋊C411yescharacteristic
202C2×C4×C53C2×C4⋊C411yescharacteristic
203C2×C4×C53C2×C4⋊C411yescharacteristic
204C2×C4×C53C2×C4⋊C411yescharacteristic
205C2×C4×C53C2×C4⋊C411yescharacteristic
206C2×C4×C53C2×C4⋊C411yescharacteristic
207C2×C4×C53C2×C4⋊C411yescharacteristic
208C2×C4×C53C2×C4⋊C411yescharacteristic
209C2×C4×C53C2×C4⋊C411yescharacteristic
210C2×C4×C53C23×C411yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C22×Q811yescharacteristic
213C8×C53—41no—
214C8×C53—41no—
215C8×C53—41no—
216C8×C53—41no—
217#88322—22no—
218#88322—22no—
219#88322—22no—
220#88322—22no—
221#88322—41no—
222#88322—41no—
223#88322—41no—
224#88322—41no—
225#17825—531nomaximal, Sylow
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53C22×C4⋊C411yescharacteristic
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—21no—
242C4×C53—21no—
243C4×C53—21no—
244C4×C53—21no—
245C4×C53—21no—
246C4×C53#56812yes—
247C4×C53#57611yescharacteristic
248C4×C53#57811yescharacteristic
249#2989—531no—
250#2989D5311yescharacteristic
251#2922—531no—
252#1574—531no—
253#1574—531no—
254#1573—531no—
255#1573—531no—
256#1572—531no—
257#1572—531no—
258#1571—531no—
259#1571—531no—
260#1569—531no—
261#1569—531no—
262#1568—531no—
263#1567—531no—
264C2×C53—41no—
265C2×C53—41no—
266C2×C53—41no—
267C2×C53#290511yescharacteristic
268C2×C53#291311yescharacteristic
269C2×C53#291611yescharacteristic
270C4×#41C2×D5311yescharacteristic
271#446—21no—
272#446—1061no—
273#436—21no—
274#436—1061no—
275#435—21no—
276#435—21no—
277#435—1061no—
278#435—1061no—
279#433—532no—
280#433—532no—
281#432—531no—
282#432—531no—
283#432C2×D5311yescharacteristic
284#432C2×D5311yescharacteristic
285#431—531no—
286#431—531no—
287#431—531no—
288#431—531no—
289#431—531no—
290#431C2×D5311yescharacteristic
291#424—532no—
292#423—531no—
293#423—531no—
294#423C2×D5311yescharacteristic
295#423C2×D5311yescharacteristic
296C2×C4⋊C8—531no—
297C2×C4⋊C8C2×D5311yescharacteristic
298#408—532no—
299C4×M16—531no—
300C2×C4×C8—531no—
301#335—531no—
302#335—531no—
303#335—531no—
304#335—531no—
305#335—531no—
306#335—531no—
307#335—531no—
308#335—531no—
309#335—531no—
310#335—531no—
311#334—531no—
312#334—531no—
313#334—531no—
314#334—531no—
315#334—531no—
316#334—531no—
317C53#1782511yescharacteristic, complemented, Sylow
318C2×#41C4×D5311yescharacteristic
319C2×M16—21no—
320C2×M16—21no—
321C2×M16—532no—
322C2×M16—1061no—
323C2×M16—1061no—
324C22×C8—21no—
325C22×C8—532no—
326C22×C8—1061no—
327C4×Q8Dic10611yescharacteristic
328C4×Q8C22⋊D5311yescharacteristic
329C4×D4Dic10611yescharacteristic
330C4×D4Dic10611yescharacteristic
331C4×D4Dic10611yescharacteristic
332C4×D4C22⋊D5311yescharacteristic
333C4×D4C22⋊D5311yescharacteristic
334C4×D4C22⋊D5311yescharacteristic
335#117C4×D5311yescharacteristic
336#117C4×D5311yescharacteristic
337#117C4×D5311yescharacteristic
338C2×C42C22×D5311yescharacteristic
339C2×C42C22×D5311yescharacteristic
340C2×C42C22×D5311yescharacteristic
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—532no—
345C4⋊C8—532no—
346C4⋊C8—532no—
347C4⋊C8—532no—
348C4⋊C8—531no—
349C4⋊C8—531no—
350C4⋊C8—531no—
351C4⋊C8—531no—
352C4⋊C8—1061no—
353C4⋊C8—1061no—
354C4⋊C8—1061no—
355C4⋊C8C22×D5311yescharacteristic
356C4⋊C8C22×D5311yescharacteristic
357C4⋊C8C22×D5311yescharacteristic
358C4⋊C8C22×D5311yescharacteristic
359C22⋊C8—21no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362C22⋊C8—532no—
363C22⋊C8—532no—
364C22⋊C8—532no—
365C22⋊C8—532no—
366C22⋊C8—1061no—
367C22⋊C8—1061no—
368C22⋊C8—1061no—
369#97—21no—
370#97—21no—
371#97—532no—
372#97—1061no—
373#97—1061no—
374C4×C8—21no—
375C4×C8—532no—
376C4×C8—531no—
377C4×C8—531no—
378C4×C8—531no—
379C4×C8—531no—
380C4×C8—1061no—
381#41—42no—
382C2×Q8#1171991911yescharacteristic
383C2×D4#1171991911yescharacteristic
384C2×D4#1171991911yescharacteristic
385C2×D4#1171991911yescharacteristic
386C22×C4C2×Dic10611yescharacteristic
387C22×C4C2×Dic10611yescharacteristic
388C22×C4C2×Dic10611yescharacteristic
389C22×C4C2×C4×D5311yescharacteristic
390C22×C4C2×C4×D5311yescharacteristic
391C22×C4C2×C4×D5311yescharacteristic
392C22×C4C2×C22⋊D5311yescharacteristic
393C22×C4C2×C22⋊D5311yescharacteristic
394C22×C4C2×C22⋊D5311yescharacteristic
395M16—81no—
396M16—81no—
397M16—4241no—
398M16—4241no—
399M16—2122no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—81no—
405C2×C8—532no—
406C2×C8—532no—
407C2×C8—532no—
408C2×C8—532no—
409C2×C8—1061no—
410C2×C8—1061no—
411C2×C8—1061no—
412C2×C8—1061no—
413C2×C8—4241no—
414C2×C8—2122no—
415C4⋊C4#1171991911yescharacteristic
416C4⋊C4#1171991911yescharacteristic
417C4⋊C4#1171991911yescharacteristic
418C4⋊C4#1171991911yescharacteristic
419C4⋊C4#1171991911yescharacteristic
420C4⋊C4#1171991911yescharacteristic
421C22⋊C4#1171991911yescharacteristic
422C22⋊C4#1171991911yescharacteristic
423C22⋊C4#1171991911yescharacteristic
424C22⋊C4#1171991911yescharacteristic
425C22⋊C4#1171991911yescharacteristic
426C22⋊C4#1171991911yescharacteristic
427C42C2×Dic10611yescharacteristic
428C42C2×Dic10611yescharacteristic
429C42C2×Dic10611yescharacteristic
430C42C2×C4×D5311yescharacteristic
431C42C2×C4×D5311yescharacteristic
432C42C2×C4×D5311yescharacteristic
433C42C2×C22⋊D5311yescharacteristic
434C42C2×C22⋊D5311yescharacteristic
435C42C2×C22⋊D5311yescharacteristic
436C42C23×D5311yescharacteristic
437C23C2×#1171991911yescharacteristic
438C23C2×#1171991911yescharacteristic
439C23C2×#1171991911yescharacteristic
440Q8—41no—
441D4—41no—
442D4—41no—
443D4—41no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4C2×#1171991911yescharacteristic
457C2×C4C2×#1171991911yescharacteristic
458C2×C4C2×#1171991911yescharacteristic
459C2×C4C2×#1171991911yescharacteristic
460C2×C4C2×#1171991911yescharacteristic
461C2×C4C2×#1171991911yescharacteristic
462C2×C4C2×#1171991911yescharacteristic
463C2×C4C2×#1171991911yescharacteristic
464C2×C4C2×#1171991911yescharacteristic
465C2×C4C22×C4×D5311yescharacteristic
466C2×C4C22×C22⋊D5311yescharacteristic
467C2×C4C22×Dic10611yescharacteristic
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—1062no—
473C8—1062no—
474C8—1062no—
475C8—1062no—
476C8—2121no—
477C8—2121no—
478C8—2121no—
479C8—2121no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22C22×#1171991911yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4#1681138912yes—
501C4#1681142611yescharacteristic
502C4#1681143811yescharacteristic
503C2—41no—
504C2—41no—
505C2—41no—
506C2#2028327811yescharacteristic
507C2#2028331311yescharacteristic
508C2#2028332411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#2989C2
socleC22×C53C22×C4⋊C4
prime radicalC2×#41×C53C2×C4
cosocleC42×C53C24
hypercenter#2989D53
hypocenterC53#17825
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2989

Complements

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

NH ≅ G/Ntype
3. #20273506C2semidirect
4. #20273503C2semidirect
5. #20273501C2semidirect
6. #20273498C2semidirect
7. #20273496C2semidirect
8. #20273493C2semidirect
9. #20273491C2semidirect
10. #20273489C2semidirect
11. #20273483C2semidirect
12. #20273480C2semidirect
13. #20273476C2semidirect
14. #20273473C2semidirect
317. C53#17825semidirect

Maximal subgroups

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

subgroupindex
#202838322
#202833592
#202735062
#202735032
#202735012
#202734982
#202734962
#202734932
#202734912
#202734892
#202734832
#202734802
#202734762
#202734732
C53×#29892
#1782553