Finite Group Databasebeta

← #27355546

Subgroups — #27355546

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#20283732C211yesmaximal, characteristic, complemented
2#20283729C211yesmaximal, characteristic, complemented
3C4×#13133237C211yesmaximal, characteristic
4#20273602C212yesmaximal, complemented
5#20273568C212yesmaximal, complemented
6#20273541C212yesmaximal, complemented
7#20273518C212yesmaximal, complemented
8#20273476C211yesmaximal, characteristic, complemented
9#20273150C211yesmaximal, characteristic, complemented
10#20272996C211yesmaximal, characteristic, complemented
11C53×#1488C211yesmaximal, characteristic, complemented
12C2×#13133237C411yescharacteristic
13C4×#11719945C2211yescharacteristic
14#16810754C2212yes—
15#16810747C2212yes—
16C4×#11719925C411yescharacteristic
17C4×#11719925C2211yescharacteristic
18C4×#11719924C411yescharacteristic
19C4×#11719924C2211yescharacteristic
20C4×#11719923C411yescharacteristic
21C4×#11719923C2211yescharacteristic
22C4×#11719922C411yescharacteristic
23C4×#11719922C2211yescharacteristic
24#16810700C2212yes—
25#16810698—22no—
26#16810695—22no—
27D4×#88322C2212yes—
28#16810680—21no—
29#16810680C2211yescharacteristic
30#16810678—21no—
31#16810659C411yescharacteristic
32#16810619C411yescharacteristic
33#16810617C411yescharacteristic
34C4×#11719917C2211yescharacteristic
35#16810575—21no—
36C2×#13133128—21no—
37C2×C4×#88322C2211yescharacteristic
38#16810375C2212yes—
39#16810373C2212yes—
40#16810355C2212yes—
41#16810352C2212yes—
42#16810345C2212yes—
43C8×#88322C2212yes—
44C4×#41×C53C2211yescharacteristic
45C53×#432C2211yescharacteristic
46C53×#431C2211yescharacteristic
47C53×Q8⋊C8C2212yes—
48C53×D4⋊C8C2212yes—
49C2×#11719945C2×C411yescharacteristic
50#13133237—24no—
51C2×#11719925C2×C411yescharacteristic
52C2×#11719923C2×C411yescharacteristic
53C2×#11719924C2×C411yescharacteristic
54C2×#11719922C2×C411yescharacteristic
55C2×#11719917D412yes—
56C2×#11719912—22no—
57C22×#88322—22no—
58C22×#88322C2×C411yescharacteristic
59C22×#88322C2×C411yescharacteristic
60C4×Dic106C2×C411yescharacteristic
61C4×Dic106C2311yescharacteristic
62C4×D212C2×C411yescharacteristic
63C4×D212C2311yescharacteristic
64C4×C22⋊D53—21no—
65C4×C22⋊D53—21no—
66#13133176C2×C411yescharacteristic
67#13133174—21no—
68C42×D53—21no—
69#13133150C2×C411yescharacteristic
70#13133150C2×C411yescharacteristic
71#13133148C2×C411yescharacteristic
72#13133148C2×C411yescharacteristic
73C4⋊#88322—22no—
74C4⋊#88322C2312yes—
75#13133146—22no—
76#13133141C2×C411yescharacteristic
77#13133141C2×C411yescharacteristic
78#13133140C2×C411yescharacteristic
79#13133140C2×C411yescharacteristic
80#13133138C2×C411yescharacteristic
81#13133138C2×C411yescharacteristic
82#13133137C2×C411yescharacteristic
83#13133137C2×C411yescharacteristic
84C22⋊#88322—22no—
85C22⋊#88322—21no—
86#13133129—22no—
87#13133128—21no—
88#13133128—22no—
89#13133128D411yescharacteristic
90#13133128D411yescharacteristic
91C4×#88322C2×C411yescharacteristic
92C4×#88322C2×C411yescharacteristic
93C4×#88322C2311yescharacteristic
94C4×#88322C2311yescharacteristic
95C4×#88322C2312yes—
96C8×Dic53—22no—
97C2×#41×C53C2×C411yescharacteristic
98C4×Q8×C53C2×C411yescharacteristic
99C4×Q8×C53C2311yescharacteristic
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53C2×C411yescharacteristic
103C4×D4×C53C2311yescharacteristic
104#117×C53—21no—
105#117×C53C2×C411yescharacteristic
106C2×C42×C53—21no—
107C2×C42×C53C2311yescharacteristic
108C4⋊C8×C53—21no—
109C4⋊C8×C53C2312yes—
110C22⋊C8×C53—21no—
111C22⋊C8×C53—21no—
112#97×C53—21no—
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
133C2×C8×C53—22no—
134C2×C8×C53—22no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53C22×C411yescharacteristic
138C4⋊C4×C53C22×C411yescharacteristic
139C22⋊C4×C53—21no—
140C22⋊C4×C53—21no—
141C22⋊C4×C53—21no—
142C42×C53—21no—
143C42×C53—21no—
144C42×C53—21no—
145C42×C53C22×C411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C42×C53#4111yescharacteristic
148C42×C53C2411yescharacteristic
149C22⋊Dic53—21no—
150#11719925—22no—
151#11719924—22no—
152#11719923M1614yescomplemented
153#11719922—22no—
154#11719921—21no—
155#11719921—21no—
156C4⋊Dic53C22×C411yescharacteristic
157C4⋊Dic53C22×C411yescharacteristic
158#11719919—21no—
159#11719919—21no—
160C4×Dic53—21no—
161C4×Dic53—21no—
162C4×Dic53—21no—
163#11719917—24no—
164C2×#88322—22no—
165C2×#88322—22no—
166C2×#88322—41no—
167C2×#88322—44no—
168C2×#88322C22×C411yescharacteristic
169C2×#88322C22×C411yescharacteristic
170C2×#88322C22×C411yescharacteristic
171C2×#88322C22×C411yescharacteristic
172C2×#88322C2×D412yes—
173C2×#88322C2×D412yes—
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—42no—
211C8×C53—42no—
212#88322—21no—
213#88322—21no—
214#88322—21no—
215#88322—22no—
216#88322—42no—
217#88322C2×M1612yes—
218#88322#13514yescomplemented
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#2970—531no—
245C4×#135—531no—
246#1590—532no—
247#1584—532no—
248#1581—532no—
249C8×SD16—532no—
250#1568—531no—
251#1488—531no—
252#1488D5311yescharacteristic, complemented
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#446—532no—
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—1062no—
276C8×D4—532no—
277#432—531no—
278#432—1061no—
279#432—1061no—
280#432C2×D5311yescharacteristic
281#431C2×D5311yescharacteristic
282#427—531no—
283#416—531no—
284#416—531no—
285C4×M16—531no—
286C4×M16—1061no—
287C2×#97—1061no—
288C2×C4×C8—531no—
289#335—532no—
290#334—532no—
291Q8⋊C8—532no—
292Q8⋊C8C2×D5312yes—
293D4⋊C8—532no—
294D4⋊C8C2×D5312yes—
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—532no—
306C2×M16—1062no—
307C22×C8—531no—
308C22×C8—531no—
309C22×C8—1062no—
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—21no—
334C4⋊C8—532no—
335C4⋊C8—1062no—
336C4⋊C8—1062no—
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—21no—
348C22⋊C8—1062no—
349C22⋊C8—1062no—
350C22⋊C8—1061no—
351#97—21no—
352#97—531no—
353#97—531no—
354#97—1062no—
355#97—1061no—
356C4×C8—531no—
357C4×C8—531no—
358C4×C8—531no—
359C4×C8—532no—
360C4×C8—531no—
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×C4D4×D5311yescharacteristic
381C22×C4#1171994711yescharacteristic
382Q16—1062no—
383SD16—534no—
384SD16—1062no—
385D8—1062no—
386M16—1064no—
387M16—2124no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—531no—
391C2×C8—532no—
392C2×C8—531no—
393C2×C8—531no—
394C2×C8—531no—
395C2×C8—532no—
396C2×C8—1062no—
397C2×C8—1062no—
398C2×C8—2124no—
399C2×C8—2121no—
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
421C42D4×D5311yescharacteristic
422C42#1171994711yescharacteristic
423C42C23×D5311yescharacteristic
424C23—21no—
425C23—1061no—
426C23#1313317711yescharacteristic
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×C4#1313317711yescharacteristic
454C2×C4#1313317711yescharacteristic
455C2×C4#1313317711yescharacteristic
456C2×C4C22×C4×D5311yescharacteristic
457C2×C4C2×D4×D5311yescharacteristic
458C2×C4C2×#1171994711yescharacteristic
459C8—42no—
460C8—42no—
461C8—532no—
462C8—534no—
463C8—1061no—
464C8—1061no—
465C8—1061no—
466C8—1062no—
467C8—2122no—
468C22—21no—
469C22—22no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—2121no—
475C22—2121no—
476C22—2121no—
477C22C2×#1313317711yescharacteristic
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
490C4#135×D5312yes—
491C2—41no—
492C2—42no—
493C2—4241no—
494C2#2028344711yescharacteristic
495C2#2028348311yescharacteristic
496C2#2028362811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1488C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133237C4
cosocleC42×C53C24
hypercenter#1488D53
hypocenterC53#18037
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1488

Complements

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

NH ≅ G/Ntype
1. #20283732C2semidirect
2. #20283729C2semidirect
4. #20273602C2semidirect
5. #20273568C2semidirect
6. #20273541C2semidirect
7. #20273518C2semidirect
8. #20273476C2semidirect
9. #20273150C2semidirect
10. #20272996C2semidirect
11. C53×#1488C2semidirect
152. #11719923M16semidirect
218. #88322#135semidirect
252. #1488D53semidirect
297. C53#18037semidirect

Maximal subgroups

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

subgroupindex
#202837322
#202837292
C4×#131332372
#202736022
#202735682
#202735412
#202735182
#202734762
#202731502
#202729962
C53×#14882
#1803753