Finite Group Databasebeta

← #27355539

Subgroups — #27355539

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283733C211yesmaximal, characteristic, complemented
2C4×#13133237C211yesmaximal, characteristic
3#20273605C212yesmaximal, complemented
4#20273563C212yesmaximal, complemented
5#20273544C212yesmaximal, complemented
6C8×#11719923C212yesmaximal, complemented
7#20273476C211yesmaximal, characteristic, complemented
8#20273149C211yesmaximal, characteristic, complemented
9#20273147C211yesmaximal, characteristic, complemented
10M16×#88322C211yesmaximal, characteristic, complemented
11C53×#2970C211yesmaximal, characteristic, complemented
12C2×#13133237C411yescharacteristic
13C4×#11719945C2211yescharacteristic
14C8×Dic106C2212yes—
15C4×#11719925C411yescharacteristic
16C4×#11719925C2211yescharacteristic
17C4×#11719924C411yescharacteristic
18C4×#11719924C2211yescharacteristic
19C4×#11719923C411yescharacteristic
20C4×#11719923C2211yescharacteristic
21C4×#11719922C411yescharacteristic
22C4×#11719922C2211yescharacteristic
23#16810702C2212yes—
24#16810699—22no—
25#16810680C2211yescharacteristic
26#16810678—21no—
27#16810676C2211yescharacteristic
28#16810659C411yescharacteristic
29#16810619C411yescharacteristic
30#16810617C411yescharacteristic
31#16810575—21no—
32C2×C4×#88322C2211yescharacteristic
33#16810375C2212yes—
34#16810373C2212yes—
35Q8⋊#88322C2212yes—
36#16810353C2212yes—
37D4⋊#88322C2212yes—
38#16810350C2212yes—
39#16810344C2212yes—
40C8×#88322C2212yes—
41C4×#41×C53C2211yescharacteristic
42C53×#446C2212yes—
43C53×#435—22no—
44C8×D4×C53C2212yes—
45C53×#432—21no—
46C53×#432C2211yescharacteristic
47C4×M16×C53C2211yescharacteristic
48C2×#97×C53—21no—
49C2×#11719945C2×C411yescharacteristic
50#13133237—24no—
51C2×#11719925C2×C411yescharacteristic
52C2×#11719923C2×C411yescharacteristic
53C2×#11719924C2×C411yescharacteristic
54C2×#11719922C2×C411yescharacteristic
55C2×#11719912—22no—
56C22×#88322C2×C411yescharacteristic
57C22×#88322C2×C411yescharacteristic
58C4×Dic106C2×C411yescharacteristic
59C4×Dic106C2311yescharacteristic
60C4×D212C2×C411yescharacteristic
61C4×D212C2311yescharacteristic
62C4×C22⋊D53—21no—
63C4×C22⋊D53—21no—
64#13133176C2×C411yescharacteristic
65#13133174—21no—
66C42×D53—21no—
67#13133150C2×C411yescharacteristic
68#13133150C2×C411yescharacteristic
69#13133148C2×C411yescharacteristic
70#13133148C2×C411yescharacteristic
71C4⋊#88322—21no—
72C4⋊#88322C2312yes—
73#13133146—22no—
74#13133141C2×C411yescharacteristic
75#13133141C2×C411yescharacteristic
76#13133140C2×C411yescharacteristic
77#13133140C2×C411yescharacteristic
78#13133138C2×C411yescharacteristic
79#13133138C2×C411yescharacteristic
80#13133137C2×C411yescharacteristic
81#13133137C2×C411yescharacteristic
82C22⋊#88322—21no—
83C22⋊#88322—21no—
84#13133129—22no—
85#13133128—21no—
86C4×#88322C2×C411yescharacteristic
87C4×#88322C2×C411yescharacteristic
88C4×#88322C2312yes—
89C4×#88322C2311yescharacteristic
90C4×#88322C2311yescharacteristic
91C8×Dic53—22no—
92C2×#41×C53C2×C411yescharacteristic
93C2×M16×C53D412yes—
94C22×C8×C53—22no—
95C4×Q8×C53C2×C411yescharacteristic
96C4×Q8×C53C2311yescharacteristic
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53C2×C411yescharacteristic
100C4×D4×C53C2311yescharacteristic
101#117×C53—21no—
102#117×C53C2×C411yescharacteristic
103C2×C42×C53—21no—
104C2×C42×C53C2311yescharacteristic
105C4⋊C8×C53—22no—
106C4⋊C8×C53C2312yes—
107C22⋊C8×C53—22no—
108C22⋊C8×C53—21no—
109#97×C53—22no—
110#97×C53—21no—
111#97×C53D411yescharacteristic
112#97×C53D411yescharacteristic
113C4×C8×C53C2312yes—
114#41×C53—22no—
115#41×C53—41no—
116C2×Q8×C53C22×C411yescharacteristic
117C2×D4×C53—21no—
118C2×D4×C53C22×C411yescharacteristic
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53#4111yescharacteristic
125C2×C22⋊D53—21no—
126#11719945—22no—
127#11719945—41no—
128C2×D212C22×C411yescharacteristic
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×Dic106C22×C411yescharacteristic
133M16×C53—24no—
134C2×C8×C53—22no—
135C2×C8×C53—44no—
136C2×C8×C53C2×D412yes—
137C2×C8×C53C2×D412yes—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53C22×C411yescharacteristic
141C4⋊C4×C53C22×C411yescharacteristic
142C22⋊C4×C53—21no—
143C22⋊C4×C53—21no—
144C22⋊C4×C53—21no—
145C42×C53—21no—
146C42×C53—21no—
147C42×C53—21no—
148C42×C53C22×C411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53#4111yescharacteristic
151C42×C53C2411yescharacteristic
152C22⋊Dic53—21no—
153#11719925—22no—
154#11719924—22no—
155#11719923M1614yescomplemented
156#11719922—22no—
157#11719921—21no—
158#11719921—21no—
159C4⋊Dic53C22×C411yescharacteristic
160C4⋊Dic53C22×C411yescharacteristic
161#11719919—21no—
162#11719919—21no—
163C4×Dic53—21no—
164C4×Dic53—21no—
165C4×Dic53—21no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—41no—
170C2×#88322C22×C411yescharacteristic
171C2×#88322C22×C411yescharacteristic
172C2×#88322C22×C411yescharacteristic
173C2×#88322C22×C411yescharacteristic
174#11719912—44no—
175C23×C53—21no—
176C23×C53C4×D411yescharacteristic
177C22×D53—21no—
178Q8×C53—21no—
179Q8×C53—21no—
180D4×C53—21no—
181D4×C53—42no—
182D4×C53C2×M1612yes—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—22no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—41no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—81no—
193C2×C4×C53C4×D411yescharacteristic
194C2×C4×C53C4×D411yescharacteristic
195C2×C4×C53C4×D411yescharacteristic
196C2×C4×C53C23×C411yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C2×C4×C53C2×#4111yescharacteristic
199C22⋊D53—42no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203D212—21no—
204D212—21no—
205C4×D53—42no—
206C4×D53—42no—
207C4×D53—81no—
208Dic106—21no—
209Dic106C2×M1612yes—
210C8×C53—22no—
211C8×C53—42no—
212C8×C53#13514yescomplemented
213#88322—21no—
214#88322—21no—
215#88322—21no—
216#88322—42no—
217#88322—42no—
218#88322C2×M1612yes—
219#18037—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—22no—
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—41no—
225C22×C53—41no—
226C22×C53C2×C4×D411yescharacteristic
227C2×D53—41no—
228C2×D53—41no—
229C2×D53—41no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—41no—
235C4×C53—41no—
236C4×C53—41no—
237C4×C53C22×M1611yescharacteristic
238C4×C53#56811yescharacteristic
239C4×C53C2×#13512yes—
240Dic53—22no—
241Dic53—41no—
242Dic53—41no—
243#2971—531no—
244#2970D5311yescharacteristic, complemented
245C4×#135—531no—
246#1590—532no—
247#1584—532no—
248#1581—532no—
249C8×SD16—532no—
250#1568—531no—
251#1488—531no—
252#1488—531no—
253C8×M16—531no—
254C2×C53—41no—
255C2×C53—42no—
256C2×C53D4×M1611yescharacteristic
257C2×C53C4×#13511yescharacteristic
258C2×C53#295711yescharacteristic
259D53—81no—
260C2×#135—531no—
261C4×#41—531no—
262C4×#41C2×D5311yescharacteristic
263#446C2×D5312yes—
264C8×Q8—532no—
265C4×Q16—531no—
266C4×Q16—531no—
267C4×SD16—531no—
268C4×SD16—531no—
269C4×SD16—531no—
270C4×SD16—531no—
271C4×D8—531no—
272C4×D8—531no—
273#436—532no—
274#436—1062no—
275#435—22no—
276C8×D4C2×D5312yes—
277#432—21no—
278#432—531no—
279#432—1061no—
280#432C2×D5311yescharacteristic
281#431—531no—
282#427—531no—
283#416—531no—
284#416—531no—
285C4×M16—1061no—
286C4×M16C2×D5311yescharacteristic
287C2×#97—21no—
288C2×C4×C8—531no—
289#335—532no—
290#334—532no—
291Q8⋊C8—532no—
292Q8⋊C8—532no—
293D4⋊C8—532no—
294D4⋊C8—532no—
295#322—532no—
296C82—532no—
297C53#1803711yescharacteristic, complemented, Sylow
298C2×#41—531no—
299C2×#41C4×D5311yescharacteristic
300#135—1064no—
301C2×Q16—531no—
302C2×SD16—531no—
303C2×SD16—531no—
304C2×D8—531no—
305C2×M16—1062no—
306C2×M16C22⋊D5312yes—
307C22×C8—22no—
308C22×C8—531no—
309C22×C8—531no—
310C4×Q8—531no—
311C4×Q8—531no—
312C4×Q8C4×D5311yescharacteristic
313C4×Q8C22×D5311yescharacteristic
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—531no—
317C4×D4—531no—
318C4×D4—1061no—
319C4×D4—1061no—
320C4×D4C4×D5311yescharacteristic
321C4×D4C22×D5311yescharacteristic
322#117—21no—
323#117—531no—
324#117—1061no—
325#117C4×D5311yescharacteristic
326C2×C42—21no—
327C2×C42—1061no—
328C2×C42C22×D5311yescharacteristic
329#107—531no—
330#107—531no—
331#106—531no—
332#106—531no—
333C4⋊C8—22no—
334C4⋊C8—532no—
335C4⋊C8—1062no—
336C4⋊C8—1061no—
337C4⋊C8C22×D5312yes—
338#103—531no—
339#103—531no—
340#103—531no—
341#103—531no—
342#102—531no—
343#102—531no—
344#102—531no—
345#102—531no—
346C22⋊C8—21no—
347C22⋊C8—22no—
348C22⋊C8—1061no—
349C22⋊C8—1061no—
350C22⋊C8—1062no—
351#97—22no—
352#97—21no—
353#97—1061no—
354#97C22⋊D5311yescharacteristic
355#97C22⋊D5311yescharacteristic
356C4×C8—531no—
357C4×C8—531no—
358C4×C8—531no—
359C4×C8—531no—
360C4×C8—532no—
361C4×C8—1062no—
362C4×C8C22×D5312yes—
363#41—22no—
364#41—41no—
365#41—1062no—
366#41—2121no—
367C2×Q8—531no—
368C2×Q8C2×C4×D5311yescharacteristic
369C2×D4—21no—
370C2×D4—531no—
371C2×D4—1061no—
372C2×D4C2×C4×D5311yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—1061no—
377C22×C4—1061no—
378C22×C4—1061no—
379C22×C4C2×C4×D5311yescharacteristic
380C22×C4#1171994511yescharacteristic
381C22×C4C2×C22⋊D5311yescharacteristic
382Q16—1062no—
383SD16—534no—
384SD16—1062no—
385D8—1062no—
386M16—24no—
387M16—2124no—
388C2×C8—22no—
389C2×C8—44no—
390C2×C8—531no—
391C2×C8—531no—
392C2×C8—531no—
393C2×C8—531no—
394C2×C8—1062no—
395C2×C8—1062no—
396C2×C8—1062no—
397C2×C8—2121no—
398C2×C8C2×C22⋊D5312yes—
399C2×C8C2×C22⋊D5312yes—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—531no—
403C4⋊C4—531no—
404C4⋊C4—1061no—
405C4⋊C4—1061no—
406C4⋊C4C2×C4×D5311yescharacteristic
407C4⋊C4C2×C4×D5311yescharacteristic
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—1061no—
412C22⋊C4—1061no—
413C22⋊C4—1061no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—1061no—
418C42—1061no—
419C42—1061no—
420C42C2×C4×D5311yescharacteristic
421C42#1171994511yescharacteristic
422C42C2×C22⋊D5311yescharacteristic
423C42C23×D5311yescharacteristic
424C23—21no—
425C23—1061no—
426C23C4×C22⋊D5311yescharacteristic
427Q8—21no—
428Q8—21no—
429Q8—532no—
430Q8—1061no—
431D4—21no—
432D4—42no—
433D4—1061no—
434D4—1061no—
435D4—2122no—
436D4M16×D5312yes—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—1061no—
448C2×C4—1061no—
449C2×C4—1061no—
450C2×C4—4241no—
451C2×C4—2122no—
452C2×C4—2122no—
453C2×C4C4×C22⋊D5311yescharacteristic
454C2×C4C4×C22⋊D5311yescharacteristic
455C2×C4C4×C22⋊D5311yescharacteristic
456C2×C4C22×C4×D5311yescharacteristic
457C2×C4C22×C22⋊D5311yescharacteristic
458C2×C4C2×#1171994511yescharacteristic
459C8—22no—
460C8—42no—
461C8—532no—
462C8—1061no—
463C8—1061no—
464C8—1061no—
465C8—2122no—
466C8—2122no—
467C8#1313323714yescomplemented
468C22—21no—
469C22—22no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—2121no—
475C22—2121no—
476C22—2121no—
477C22C2×C4×C22⋊D5311yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—1062no—
486C4—2121no—
487C4—2121no—
488C4C2×M16×D5311yescharacteristic
489C4#1681138911yescharacteristic
490C4C2×#1313323712yes—
491C2—41no—
492C2—42no—
493C2—4241no—
494C2M16×C22⋊D5311yescharacteristic
495C2C4×#1313323711yescharacteristic
496C2#2028363011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#2970C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133237C4
cosocleC42×C53C24
hypercenter#2970D53
hypocenterC53#18037
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2970

Complements

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

NH ≅ G/Ntype
1. #20283733C2semidirect
3. #20273605C2semidirect
4. #20273563C2semidirect
5. #20273544C2semidirect
6. C8×#11719923C2semidirect
7. #20273476C2semidirect
8. #20273149C2semidirect
9. #20273147C2semidirect
10. M16×#88322C2semidirect
11. C53×#2970C2semidirect
155. #11719923M16semidirect
212. C8×C53#135semidirect
244. #2970D53semidirect
297. C53#18037semidirect
467. C8#13133237semidirect

Maximal subgroups

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

subgroupindex
#202837332
C4×#131332372
#202736052
#202735632
#202735442
C8×#117199232
#202734762
#202731492
#202731472
M16×#883222
C53×#29702
#1803753