Finite Group Databasebeta

← #27403545

Subgroups — #27403545

Order 13568.

740 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133237C211yesmaximal, characteristic
2#20283383C211yesmaximal, characteristic, complemented
3#20283283C211yesmaximal, characteristic, complemented
4#20275427C212yesmaximal, complemented
5#20275317C212yesmaximal, complemented
6#20275040C212yesmaximal
7#20274852C211yesmaximal, characteristic, complemented
8#20274736C211yesmaximal, characteristic, complemented
9#20274584C211yesmaximal, characteristic, complemented
10#20274560C212yesmaximal, complemented
11C53×#1806C211yesmaximal, characteristic, complemented
12C2×#13133237C2211yescharacteristic
13C2×#13133224C2212yes—
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#16810683—22no—
24#16810679—21no—
25#16810675—21no—
26#16810659C2211yescharacteristic
27#16810650—22no—
28#16810650C2212yes—
29#16810647—21no—
30#16810647C2211yescharacteristic
31C2×C4⋊#88322—21no—
32C2×C4⋊#88322C2211yescharacteristic
33C2×#13133145C2212yes—
34#16810619C2211yescharacteristic
35#16810617C2211yescharacteristic
36#16810591C2212yes—
37#16810581C2212yes—
38C2×#13133128C2212yes—
39C2×C4×#88322C2211yescharacteristic
40#16810387C2212yes—
41#16810387C2212yes—
42#16810383C2212yes—
43#16810378C2212yes—
44C4×#41×C53C2211yescharacteristic
45C53×#423C2211yescharacteristic
46C2×#104×C53C2212yes—
47C4×C4⋊C4×C53C2211yescharacteristic
48C53×#339C2212yes—
49C2×#11719945C2311yescharacteristic
50#13133237—22no—
51#13133237—22no—
52C2×#11719925C2×C411yescharacteristic
53C2×#11719923C2×C411yescharacteristic
54C2×#11719924C2×C411yescharacteristic
55C2×#11719922C2×C411yescharacteristic
56#13133224—24no—
57#13133224—42no—
58C2×#11719917—22no—
59C2×#11719917C2312yes—
60C22×#88322—22no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2312yes—
64C4×Dic106C2×C411yescharacteristic
65C4×Dic106C2×C411yescharacteristic
66C4×D212C2×C411yescharacteristic
67C4×D212C2×C411yescharacteristic
68C4×C22⋊D53—21no—
69C4×C22⋊D53—21no—
70#13133176C2311yescharacteristic
71#13133174—21no—
72C42×D53—21no—
73#13133150C2×C411yescharacteristic
74#13133150C2×C411yescharacteristic
75#13133148C2×C411yescharacteristic
76#13133148C2×C411yescharacteristic
77C4⋊#88322—21no—
78C4⋊#88322—21no—
79C4⋊#88322—22no—
80C4⋊#88322—22no—
81C4⋊#88322D411yescharacteristic
82C4⋊#88322D411yescharacteristic
83C4⋊#88322D412yes—
84#13133145—42no—
85#13133141C2×C411yescharacteristic
86#13133141C2×C411yescharacteristic
87#13133140C2×C411yescharacteristic
88#13133140C2×C411yescharacteristic
89#13133138C2×C411yescharacteristic
90#13133138C2×C411yescharacteristic
91#13133137C2×C411yescharacteristic
92#13133137C2×C411yescharacteristic
93#13133129—22no—
94#13133129—22no—
95#13133128—42no—
96#13133127—22no—
97C4×#88322C2×C411yescharacteristic
98C4×#88322C2×C411yescharacteristic
99C4×#88322C2×C411yescharacteristic
100C4×#88322C2×C411yescharacteristic
101C8×Dic53—22no—
102C2×#41×C53C2311yescharacteristic
103C2×M16×C53C2312yes—
104C4×Q8×C53C2×C411yescharacteristic
105C4×Q8×C53C2×C411yescharacteristic
106C4×D4×C53—21no—
107C4×D4×C53—21no—
108C4×D4×C53C2×C411yescharacteristic
109C4×D4×C53C2×C411yescharacteristic
110#117×C53—21no—
111#117×C53C2311yescharacteristic
112C2×C4⋊C4×C53—21no—
113C2×C42×C53—21no—
114C2×C42×C53C2312yes—
115C2×C42×C53C2311yescharacteristic
116C4⋊C8×C53—21no—
117C4⋊C8×C53—21no—
118#104×C53—24no—
119#95×C53—21no—
120#41×C53—21no—
121#41×C53—21no—
122#41×C53#4112yes—
123#41×C53#4112yes—
124C2×Q8×C53C22×C411yescharacteristic
125C2×D4×C53—21no—
126C2×D4×C53C22×C411yescharacteristic
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—22no—
131C22×C4×C53#4111yescharacteristic
132C22×C4×C53#4111yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C2×C22⋊D53—21no—
135#11719945—21no—
136#11719945—21no—
137#11719945—21no—
138#11719945—21no—
139C2×D212C22×C411yescharacteristic
140C2×C4×D53—21no—
141C2×C4×D53—21no—
142C2×C4×D53—21no—
143C2×Dic106C22×C411yescharacteristic
144M16×C53—22no—
145M16×C53—22no—
146C2×C8×C53—22no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—42no—
150C4⋊C4×C53C22×C411yescharacteristic
151C4⋊C4×C53C22×C411yescharacteristic
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C22⋊C4×C53—21no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53—21no—
158C42×C53—22no—
159C42×C53—22no—
160C42×C53C22×C411yescharacteristic
161C42×C53C22×C411yescharacteristic
162C42×C53C2×D411yescharacteristic
163C42×C53C2×D411yescharacteristic
164C22⋊Dic53—21no—
165#11719925—22no—
166#11719924—22no—
167#11719923—22no—
168#11719922—22no—
169#11719921—21no—
170#11719921—21no—
171C4⋊Dic53C22×C411yescharacteristic
172C4⋊Dic53C22×C411yescharacteristic
173#11719919—21no—
174#11719919—21no—
175C4×Dic53—21no—
176C4×Dic53—21no—
177C4×Dic53—21no—
178#11719917—22no—
179#11719917—22no—
180#11719917—44no—
181C2×#88322—22no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—22no—
186C2×#88322—21no—
187C2×#88322—22no—
188C2×#88322—44no—
189C2×#88322C22×C411yescharacteristic
190C2×#88322C22×C411yescharacteristic
191C2×#88322C22×C411yescharacteristic
192C2×#88322C22×C411yescharacteristic
193C2×#88322C2×D412yes—
194C2×#88322C2×D412yes—
195C23×C53—21no—
196C23×C53C2×#4111yescharacteristic
197C22×D53—21no—
198Q8×C53—21no—
199Q8×C53#11712yes—
200D4×C53—22no—
201D4×C53—22no—
202D4×C53—21no—
203D4×C53#11712yes—
204C2×C4×C53—21no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—41no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—81no—
216C2×C4×C53C4×D411yescharacteristic
217C2×C4×C53C4×D411yescharacteristic
218C2×C4×C53C4×D411yescharacteristic
219C2×C4×C53C4×D411yescharacteristic
220C2×C4×C53C23×C411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53H(5,𝔽2)11yescharacteristic
225C2×C4×C532−1+411yescharacteristic
226C22⋊D53—41no—
227C22⋊D53—41no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231D212—21no—
232D212—21no—
233C4×D53—41no—
234C4×D53—41no—
235C4×D53—42no—
236C4×D53—81no—
237Dic106—21no—
238Dic106—21no—
239C8×C53—42no—
240#88322—21no—
241#88322—21no—
242#88322—21no—
243#88322—21no—
244#88322—22no—
245#88322—22no—
246#88322—42no—
247#22782—531nomaximal, Sylow
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—22no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53C2×C4×D411yescharacteristic
254C22×C53#55211yescharacteristic
255C22×C53#55311yescharacteristic
256C2×D53—41no—
257C2×D53—41no—
258C2×D53—41no—
259C4×C53—22no—
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—21no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—82no—
266C4×C53#44412yes—
267C4×C53C2×#11711yescharacteristic
268C4×C53#52011yescharacteristic
269C4×C53#54611yescharacteristic
270C4×C53#54811yescharacteristic
271Dic53—41no—
272Dic53—41no—
273Dic53—41no—
274C4×#135—531no—
275#2925—531no—
276#2906—531no—
277#1938—532no—
278#1922—532no—
279#1876—532no—
280#1837—531no—
281#1806—531no—
282#1806D5311yescharacteristic, complemented
283#1770—531no—
284#1765—532no—
285C2×C53—21no—
286C2×C53—21no—
287C2×C53—42no—
288C2×C53#235211yescharacteristic
289C2×C53C2×#44411yescharacteristic
290C2×C53#295711yescharacteristic
291D53—81no—
292C2×#135—531no—
293C2×#131—532no—
294C4×#41—531no—
295C4×#41C2×D5311yescharacteristic
296C4×Q16—531no—
297C4×Q16—531no—
298C4×SD16—531no—
299C4×SD16—531no—
300C4×SD16—531no—
301C4×SD16—531no—
302C4×D8—531no—
303C4×D8—531no—
304#433—1062no—
305#432—1061no—
306#431—1061no—
307#427—531no—
308#424—532no—
309#424—1062no—
310#423—531no—
311#423—1061no—
312#423C2×D5311yescharacteristic
313C2×C4⋊C8—531no—
314C2×C4⋊C8—1061no—
315C2×#104—532no—
316C2×#104C2×D5312yes—
317#416—531no—
318#416—531no—
319#408—532no—
320#405—532no—
321C2×#97—532no—
322C2×C4×C8—531no—
323C4×C4⋊C4C2×D5311yescharacteristic
324#341—532no—
325#341—532no—
326#339—532no—
327#339C2×D5312yes—
328#336—532no—
329C53#2278211yescharacteristic, complemented, Sylow
330C2×#41—531no—
331C2×#41C22×D5311yescharacteristic
332#135—1062no—
333#135—1062no—
334C2×Q16—531no—
335C2×SD16—531no—
336C2×SD16—531no—
337C2×D8—531no—
338#131—1064no—
339#131—2122no—
340C2×M16—532no—
341C2×M16—1062no—
342C2×M16C22×D5312yes—
343C22×C8—532no—
344C22×C8—531no—
345C22×C8—531no—
346C22×C8—1062no—
347C4×Q8—531no—
348C4×Q8—531no—
349C4×Q8C4×D5311yescharacteristic
350C4×Q8C4×D5311yescharacteristic
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—531no—
354C4×D4—531no—
355C4×D4—1061no—
356C4×D4—1061no—
357C4×D4C4×D5311yescharacteristic
358C4×D4C4×D5311yescharacteristic
359#117—21no—
360#117—531no—
361#117—1061no—
362#117C22×D5311yescharacteristic
363C2×C4⋊C4—21no—
364C2×C42—21no—
365C2×C42—1061no—
366C2×C42C22×D5312yes—
367C2×C42C22×D5311yescharacteristic
368#107—531no—
369#107—531no—
370#106—531no—
371#106—531no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—531no—
375C4⋊C8—531no—
376C4⋊C8—532no—
377C4⋊C8—1062no—
378C4⋊C8—1062no—
379C4⋊C8—1061no—
380C4⋊C8—1061no—
381#104—24no—
382#104—2122no—
383#103—531no—
384#103—531no—
385#103—531no—
386#103—531no—
387#102—531no—
388#102—531no—
389#102—531no—
390#102—531no—
391C22⋊C8—1062no—
392C22⋊C8—1062no—
393#97—1062no—
394#97—2122no—
395C4×C8—531no—
396C4×C8—531no—
397C4×C8—531no—
398C4×C8—531no—
399C4×C8—1062no—
400#95—21no—
401#41—21no—
402#41—21no—
403#41—1061no—
404#41—1061no—
405#41—1061no—
406#41—1061no—
407#41#1171994712yes—
408#41#1171994912yes—
409C2×Q8—531no—
410C2×Q8C2×C4×D5311yescharacteristic
411C2×D4—21no—
412C2×D4—531no—
413C2×D4—1061no—
414C2×D4C2×C4×D5311yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—1061no—
420C22×C4—1061no—
421C22×C4—1061no—
422C22×C4#1171994711yescharacteristic
423C22×C4#1171994711yescharacteristic
424C22×C4C23×D5311yescharacteristic
425Q16—1062no—
426SD16—1062no—
427SD16—1062no—
428D8—1062no—
429M16—22no—
430M16—22no—
431M16—1062no—
432M16—1062no—
433M16—2124no—
434C2×C8—22no—
435C2×C8—531no—
436C2×C8—531no—
437C2×C8—531no—
438C2×C8—531no—
439C2×C8—532no—
440C2×C8—532no—
441C2×C8—1061no—
442C2×C8—1062no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1062no—
446C2×C8—1061no—
447C2×C8—1062no—
448C2×C8—2124no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—531no—
453C4⋊C4—531no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4C2×C4×D5311yescharacteristic
457C4⋊C4C2×C4×D5311yescharacteristic
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—1061no—
462C22⋊C4—1061no—
463C22⋊C4—1061no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—22no—
469C42—1061no—
470C42—1061no—
471C42—1061no—
472C42C2×C4×D5311yescharacteristic
473C42C2×C4×D5311yescharacteristic
474C42D4×D5311yescharacteristic
475C42D4×D5311yescharacteristic
476C23—21no—
477C23—1061no—
478C23C2×#1171994711yescharacteristic
479Q8—21no—
480Q8—1061no—
481Q8—1061no—
482Q8#1313317512yes—
483D4—22no—
484D4—22no—
485D4—21no—
486D4—1061no—
487D4—1061no—
488D4—2121no—
489D4—2121no—
490D4#1313317512yes—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—81no—
503C2×C4—1061no—
504C2×C4—1061no—
505C2×C4—1061no—
506C2×C4—4241no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2122no—
510C2×C4#1313317711yescharacteristic
511C2×C4#1313317711yescharacteristic
512C2×C4#1313317711yescharacteristic
513C2×C4#1313317711yescharacteristic
514C2×C4C22×C4×D5311yescharacteristic
515C2×C4C2×D4×D5311yescharacteristic
516C2×C4C2×#1171994711yescharacteristic
517C2×C4C2×#1171994911yescharacteristic
518C2×C4#1313326311yescharacteristic
519C2×C4#1313326411yescharacteristic
520C8—42no—
521C8—1062no—
522C8—1062no—
523C8—1061no—
524C8—1061no—
525C8—1061no—
526C8—1061no—
527C8—2122no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—41no—
532C22—41no—
533C22—2121no—
534C22—2121no—
535C22—2121no—
536C22C2×#1313317711yescharacteristic
537C22#1681131511yescharacteristic
538C22#1681132511yescharacteristic
539C4—22no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—82no—
546C4—2121no—
547C4—2121no—
548C4—2121no—
549C4#1681074212yes—
550C4C2×#1313317511yescharacteristic
551C4#1681110311yescharacteristic
552C4#1681126411yescharacteristic
553C4#1681127911yescharacteristic
554C2—21no—
555C2—21no—
556C2—42no—
557C2—4241no—
558C2#2027764611yescharacteristic
559C2C2×#1681074211yescharacteristic
560C2#2028362811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1806C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133237C22
cosocleC22×C4×C53C24
hypercenter#1806D53
hypocenterC53#22782
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #20283383C2semidirect
3. #20283283C2semidirect
4. #20275427C2semidirect
5. #20275317C2semidirect
7. #20274852C2semidirect
8. #20274736C2semidirect
9. #20274584C2semidirect
10. #20274560C2semidirect
11. C53×#1806C2semidirect
15. C4×#11719925C4semidirect
16. C4×#11719925C4semidirect
17. C4×#11719924C4semidirect
18. C4×#11719924C4semidirect
19. C4×#11719923C4semidirect
20. C4×#11719923C4semidirect
21. C4×#11719922C4semidirect
22. C4×#11719922C4semidirect
282. #1806D53semidirect
329. C53#22782semidirect

Maximal subgroups

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

subgroupindex
C4×#131332372
#202833832
#202832832
#202754272
#202753172
#202750402
#202748522
#202747362
#202745842
#202745602
C53×#18062
#2278253