Finite Group Databasebeta

← #27387319

Subgroups — #27387319

Order 13568.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133237C211yesmaximal, characteristic
2#20283359C211yesmaximal, characteristic, complemented
3C4×#13133225C211yesmaximal, characteristic, complemented
4#20275366C212yesmaximal, complemented
5#20275317C212yesmaximal, complemented
6#20275039C212yesmaximal
7#20274991C212yesmaximal, complemented
8C4×#13133152C211yesmaximal, characteristic, complemented
9C4×#13133145C211yesmaximal, characteristic, complemented
10#20274461C211yesmaximal, characteristic, complemented
11C4×#104×C53C211yesmaximal, characteristic, complemented
12C2×#13133237C2211yescharacteristic
13C2×#13133225C2212yes—
14C4×#11719945C2211yescharacteristic
15C4×#11719925C411yescharacteristic, complemented
16C4×#11719925C411yescharacteristic, complemented
17C4×#11719924C411yescharacteristic, complemented
18C4×#11719924C411yescharacteristic, complemented
19C4×#11719923C411yescharacteristic, complemented
20C4×#11719923C411yescharacteristic, complemented
21C4×#11719922C411yescharacteristic, complemented
22C4×#11719922C411yescharacteristic, complemented
23#16810684—22no—
24#16810676—21no—
25#16810676—21no—
26C2×#13133152C2212yes—
27#16810659C2211yescharacteristic
28#16810650C2212yes—
29C2×#13133145C2212yes—
30#16810619C2211yescharacteristic
31#16810617C2211yescharacteristic
32#16810590C2212yes—
33#16810581C2212yes—
34#16810580—22no—
35C4×#11719917C2211yescharacteristic
36C4×#11719912—21no—
37C2×C4×#88322C2211yescharacteristic
38C2×C4×#88322C2211yescharacteristic
39C4×C8×D53—21no—
40#16810387C2212yes—
41#16810383C2212yes—
42#16810378C2212yes—
43#16810378C2212yes—
44C4×#41×C53C2211yescharacteristic
45C2×#104×C53C2212yes—
46C4×M16×C53C2211yescharacteristic
47C43×C53C2211yescharacteristic
48C53×#339C2212yes—
49C2×#11719945C2311yescharacteristic
50#13133237—24no—
51C2×#11719925C2×C411yescharacteristic
52C2×#11719923C2×C411yescharacteristic
53C2×#11719924C2×C411yescharacteristic
54C2×#11719922C2×C411yescharacteristic
55#13133225—24no—
56#13133225—24no—
57C2×#11719917C2312yes—
58C2×#11719912—22no—
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2312yes—
62C2×C8×D53—22no—
63C4×Dic106C2×C411yescharacteristic
64C4×Dic106C2×C411yescharacteristic
65C4×D212C2×C411yescharacteristic
66C4×D212C2×C411yescharacteristic
67C4×C22⋊D53—21no—
68C4×C22⋊D53—21no—
69#13133176C2311yescharacteristic
70#13133174—21no—
71C42×D53—21no—
72#13133152—24no—
73#13133150C2×C411yescharacteristic
74#13133150C2×C411yescharacteristic
75#13133148C2×C411yescharacteristic
76#13133148C2×C411yescharacteristic
77C4⋊#88322—22no—
78C4⋊#88322—22no—
79#13133145—24no—
80#13133141C2×C411yescharacteristic
81#13133141C2×C411yescharacteristic
82#13133140C2×C411yescharacteristic
83#13133140C2×C411yescharacteristic
84#13133138C2×C411yescharacteristic
85#13133138C2×C411yescharacteristic
86#13133137C2×C411yescharacteristic
87#13133137C2×C411yescharacteristic
88C22⋊#88322—22no—
89C22⋊#88322—22no—
90#13133128—21no—
91#13133127—22no—
92C4×#88322—21no—
93C4×#88322C2×C411yescharacteristic
94C4×#88322C2×C411yescharacteristic
95C4×#88322C2×C411yescharacteristic
96C4×#88322C2×C411yescharacteristic
97C4×#88322D411yescharacteristic
98C4×#88322D411yescharacteristic
99C4×#88322D412yes—
100C8×Dic53—22no—
101C2×#41×C53C2311yescharacteristic
102C2×M16×C53C2312yes—
103C4×Q8×C53C2×C411yescharacteristic
104C4×Q8×C53C2×C411yescharacteristic
105C4×D4×C53—21no—
106C4×D4×C53—21no—
107C4×D4×C53C2×C411yescharacteristic
108C4×D4×C53C2×C411yescharacteristic
109#117×C53—21no—
110#117×C53C2311yescharacteristic
111C2×C42×C53—21no—
112C2×C42×C53—21no—
113C2×C42×C53—21no—
114C2×C42×C53C2312yes—
115C2×C42×C53C2311yescharacteristic
116#104×C53—24no—
117#97×C53—21no—
118C4×C8×C53—21no—
119#41×C53—22no—
120#41×C53—22no—
121C2×Q8×C53C22×C411yescharacteristic
122C2×D4×C53—21no—
123C2×D4×C53C22×C411yescharacteristic
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53#4111yescharacteristic
129C22×C4×C53#4111yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C2×C22⋊D53—21no—
132#11719945—22no—
133#11719945#4114yes—
134C2×D212C22×C411yescharacteristic
135C2×C4×D53—21no—
136C2×C4×D53—21no—
137C2×C4×D53—21no—
138C2×Dic106C22×C411yescharacteristic
139M16×C53—22no—
140M16×C53#4114yes—
141C2×C8×C53—22no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53C22×C411yescharacteristic
145C4⋊C4×C53C22×C411yescharacteristic
146C22⋊C4×C53—21no—
147C22⋊C4×C53—21no—
148C22⋊C4×C53—21no—
149C42×C53—24no—
150C42×C53—21no—
151C42×C53—21no—
152C42×C53—21no—
153C42×C53—22no—
154C42×C53—22no—
155C42×C53—42no—
156C42×C53C22×C411yescharacteristic
157C42×C53C22×C411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160C22⋊Dic53—21no—
161#11719925—22no—
162#11719924—22no—
163#11719923—22no—
164#11719922—22no—
165#11719921—21no—
166#11719921—21no—
167C4⋊Dic53C22×C411yescharacteristic
168C4⋊Dic53C22×C411yescharacteristic
169#11719919—21no—
170#11719919—21no—
171C4×Dic53—21no—
172C4×Dic53—21no—
173C4×Dic53—21no—
174#11719917—22no—
175#11719917—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322C22×C411yescharacteristic
182C2×#88322C22×C411yescharacteristic
183C2×#88322C22×C411yescharacteristic
184C2×#88322C22×C411yescharacteristic
185C2×#88322C2×D412yes—
186C2×#88322C2×D412yes—
187#11719912—28no—
188C8×D53—28no—
189C23×C53—21no—
190C23×C53C2×#4111yescharacteristic
191C22×D53—21no—
192Q8×C53—21no—
193Q8×C53—21no—
194D4×C53—21no—
195D4×C53—21no—
196D4×C53—42no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—24no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53C4×D411yescharacteristic
209C2×C4×C53C4×D411yescharacteristic
210C2×C4×C53C4×D411yescharacteristic
211C2×C4×C53C4×D411yescharacteristic
212C2×C4×C53C23×C411yescharacteristic
213C2×C4×C53C22×D411yescharacteristic
214C2×C4×C53C2×#4112yes—
215C2×C4×C53C2×#4112yes—
216C22⋊D53—24no—
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53—21no—
220D212—21no—
221D212#11712yes—
222C4×D53—24no—
223C4×D53—42no—
224C4×D53—42no—
225Dic106—21no—
226Dic106#11712yes—
227C8×C53—24no—
228#88322—21no—
229#88322—21no—
230#88322—21no—
231#88322—21no—
232#88322—22no—
233#88322—22no—
234#88322—42no—
235#21137—531nomaximal, Sylow
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—41no—
239C22×C53—41no—
240C22×C53—41no—
241C22×C53C2×C4×D411yescharacteristic
242C22×C53#52912yes—
243C2×D53—22no—
244C2×D53—41no—
245C2×D53—41no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—44no—
249C4×C53—41no—
250C4×C53—41no—
251C4×C53—41no—
252C4×C53#44312yes—
253C4×C53#44412yes—
254C4×C53C2×#11711yescharacteristic
255C4×C53C4×#4111yescharacteristic
256C4×C53#52512yes—
257Dic53—22no—
258Dic53—41no—
259Dic53—41no—
260C4×#135—531no—
261#2922—531no—
262C4×#131—531no—
263#1929—532no—
264#1922—532no—
265#1876—532no—
266#1866—532no—
267C4×#108—531no—
268C4×#104—531no—
269C4×#104D5311yescharacteristic, complemented
270#1744—531no—
271C2×C53—22no—
272C2×C53—81no—
273C2×C53#229011yescharacteristic
274C2×C53C2×#44311yescharacteristic
275C2×C53C2×#44411yescharacteristic
276D53—42no—
277C2×#135—531no—
278C2×#131—532no—
279C4×#41—531no—
280C4×#41C2×D5311yescharacteristic
281C4×Q16—531no—
282C4×Q16—531no—
283C4×SD16—531no—
284C4×SD16—531no—
285C4×SD16—531no—
286C4×SD16—531no—
287C4×D8—531no—
288C4×D8—531no—
289#433—1062no—
290#431—1061no—
291#431—1061no—
292C2×#108—532no—
293#427—531no—
294#424—532no—
295C2×#104—532no—
296C2×#104C2×D5312yes—
297#416—531no—
298#416—531no—
299#408—532no—
300#405—532no—
301#405—1062no—
302C4×M16—531no—
303C4×M16—1061no—
304C4×M16C2×D5311yescharacteristic
305C2×C4×C8—531no—
306C2×C4×C8—531no—
307C2×C4×C8—1061no—
308C43C2×D5311yescharacteristic
309#341—532no—
310#339—532no—
311#339C2×D5312yes—
312#336—532no—
313#336—532no—
314C53#2113711yescharacteristic, complemented, Sylow
315C2×#41—531no—
316C2×#41C22×D5311yescharacteristic
317#135—1064no—
318C2×Q16—531no—
319C2×SD16—531no—
320C2×SD16—531no—
321C2×D8—531no—
322#131—1064no—
323#131—1064no—
324C2×M16—532no—
325C2×M16—1062no—
326C2×M16C22×D5312yes—
327C22×C8—532no—
328C22×C8—531no—
329C22×C8—531no—
330C22×C8—1062no—
331C4×Q8—531no—
332C4×Q8—531no—
333C4×Q8C4×D5311yescharacteristic
334C4×Q8C4×D5311yescharacteristic
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—531no—
338C4×D4—531no—
339C4×D4—1061no—
340C4×D4—1061no—
341C4×D4C4×D5311yescharacteristic
342C4×D4C4×D5311yescharacteristic
343#117—21no—
344#117—531no—
345#117—1061no—
346#117C22×D5311yescharacteristic
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—1061no—
351C2×C42C22×D5312yes—
352C2×C42C22×D5311yescharacteristic
353#108—1064no—
354#107—531no—
355#107—531no—
356#106—531no—
357#106—531no—
358C4⋊C8—1062no—
359C4⋊C8—1062no—
360#104—24no—
361#104—1064no—
362#103—531no—
363#103—531no—
364#103—531no—
365#103—531no—
366#102—531no—
367#102—531no—
368#102—531no—
369#102—531no—
370C22⋊C8—1062no—
371C22⋊C8—1062no—
372#97—21no—
373#97—1062no—
374#97—1061no—
375C4×C8—21no—
376C4×C8—531no—
377C4×C8—531no—
378C4×C8—531no—
379C4×C8—531no—
380C4×C8—532no—
381C4×C8—531no—
382C4×C8—531no—
383C4×C8—1062no—
384C4×C8—1061no—
385#41—22no—
386#41—22no—
387#41—534no—
388#41—1062no—
389C2×Q8—531no—
390C2×Q8C2×C4×D5311yescharacteristic
391C2×D4—21no—
392C2×D4—531no—
393C2×D4—1061no—
394C2×D4C2×C4×D5311yescharacteristic
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—1061no—
400C22×C4—1061no—
401C22×C4—1061no—
402C22×C4#1171994711yescharacteristic
403C22×C4#1171994711yescharacteristic
404C22×C4C23×D5311yescharacteristic
405Q16—1062no—
406SD16—1062no—
407SD16—1062no—
408D8—1062no—
409M16—22no—
410M16—1068no—
411M16—1062no—
412M16—1062no—
413M16#1171994514yes—
414C2×C8—22no—
415C2×C8—531no—
416C2×C8—531no—
417C2×C8—531no—
418C2×C8—531no—
419C2×C8—532no—
420C2×C8—532no—
421C2×C8—1068no—
422C2×C8—1062no—
423C2×C8—1062no—
424C2×C8—1062no—
425C2×C8—1062no—
426C2×C8—1062no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—531no—
430C4⋊C4—531no—
431C4⋊C4—1061no—
432C4⋊C4—1061no—
433C4⋊C4C2×C4×D5311yescharacteristic
434C4⋊C4C2×C4×D5311yescharacteristic
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—1061no—
439C22⋊C4—1061no—
440C22⋊C4—1061no—
441C42—24no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—42no—
448C42—1061no—
449C42—1061no—
450C42—1061no—
451C42C2×C4×D5311yescharacteristic
452C42C2×C4×D5311yescharacteristic
453C42D4×D5311yescharacteristic
454C42D4×D5311yescharacteristic
455C23—21no—
456C23—1061no—
457C23C2×#1171994711yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—532no—
461Q8—1061no—
462D4—21no—
463D4—21no—
464D4—42no—
465D4—532no—
466D4—1064no—
467D4—1061no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—1061no—
480C2×C4—1064no—
481C2×C4—1061no—
482C2×C4—1061no—
483C2×C4—2122no—
484C2×C4—2122no—
485C2×C4#1313317711yescharacteristic
486C2×C4#1313317711yescharacteristic
487C2×C4#1313317711yescharacteristic
488C2×C4#1313317711yescharacteristic
489C2×C4C22×C4×D5311yescharacteristic
490C2×C4C2×D4×D5311yescharacteristic
491C2×C4#41×D5312yes—
492C2×C4C2×#1171994512yes—
493C8—24no—
494C8—1061no—
495C8—1061no—
496C8—1061no—
497C8—1061no—
498C8—1062no—
499C8—1062no—
500C8—2122no—
501C22—21no—
502C22—21no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—1062no—
507C22—2121no—
508C22—2121no—
509C22C2×#1313317711yescharacteristic
510C22#1681115512yes—
511C4—21no—
512C4—21no—
513C4—44no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—1062no—
518C4—2121no—
519C4—2121no—
520C4#1681073712yes—
521C4#1681074212yes—
522C4#117×D5311yescharacteristic
523C4C4×#1171994511yescharacteristic
524C4#1681113212yes—
525C2—22no—
526C2—81no—
527C2—2122no—
528C2#2027716111yescharacteristic
529C2C2×#1681073711yescharacteristic
530C2C2×#1681074211yescharacteristic
subgroupG/H
centerC42D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#104×C53C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133237C22
cosocleC22×C4×C53C24
hypercenterC4×#104D53
hypocenterC53#21137
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C42 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
2. #20283359C2semidirect
3. C4×#13133225C2semidirect
4. #20275366C2semidirect
5. #20275317C2semidirect
7. #20274991C2semidirect
8. C4×#13133152C2semidirect
9. C4×#13133145C2semidirect
10. #20274461C2semidirect
11. C4×#104×C53C2semidirect
15. C4×#11719925C4semidirect
16. C4×#11719925C4semidirect
17. C4×#11719924C4semidirect
18. C4×#11719924C4semidirect
19. C4×#11719923C4semidirect
20. C4×#11719923C4semidirect
21. C4×#11719922C4semidirect
22. C4×#11719922C4semidirect
269. C4×#104D53semidirect
314. C53#21137semidirect

Maximal subgroups

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

subgroupindex
C4×#131332372
#202833592
C4×#131332252
#202753662
#202753172
#202750392
#202749912
C4×#131331522
C4×#131331452
#202744612
C4×#104×C532
#2113753