Finite Group Databasebeta

← #27354090

Subgroups — #27354090

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284782C211yesmaximal, characteristic
2C8×#11719947C211yesmaximal, characteristic
3#20283644C211yesmaximal, characteristic, complemented
4#20273589C211yesmaximal, characteristic, complemented
5#20273585C211yesmaximal, characteristic, complemented
6#20273573C211yesmaximal, characteristic, complemented
7#20273569C211yesmaximal, characteristic, complemented
8C8×Dic212C211yesmaximal, characteristic, complemented
9C8×#11719925C211yesmaximal, characteristic, complemented
10C8×#11719923C211yesmaximal, characteristic, complemented
11#20273502C211yesmaximal, characteristic, complemented
12#20273151C211yesmaximal, characteristic
13#20273134C211yesmaximal, characteristic, complemented
14C8×#11719912C211yesmaximal, characteristic, complemented
15C8×SD16×C53C211yesmaximal, characteristic, complemented
16#16811177C2211yescharacteristic
17C4×#11719947C2211yescharacteristic
18#16810967C411yescharacteristic
19#16810963C411yescharacteristic
20#16810907C411yescharacteristic
21#16810901C411yescharacteristic
22#16810832C411yescharacteristic
23#16810826C411yescharacteristic
24#16810786C411yescharacteristic
25#16810782C411yescharacteristic
26#16810754C2211yescharacteristic
27#16810752—21no—
28#16810750C2211yescharacteristic
29C8×Dic106—21no—
30C8×Dic106C2211yescharacteristic
31C8×Dic106C2211yescharacteristic
32#16810747C2211yescharacteristic
33C4×Dic212C2211yescharacteristic
34C4×#11719925C2211yescharacteristic
35C4×#11719923C2211yescharacteristic
36D4×#88322C2211yescharacteristic
37C8×C22⋊D53—21no—
38#16810686—21no—
39#16810675C2211yescharacteristic
40#16810674C2211yescharacteristic
41#16810673—21no—
42#16810648—21no—
43#16810645C2211yescharacteristic
44#16810577—21no—
45C4×#11719912C2211yescharacteristic
46C2×C8×Dic53—21no—
47C4×C8×D53C2211yescharacteristic
48#16810376C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810355C2211yescharacteristic
52#16810355C2211yescharacteristic
53Q8⋊#88322C2211yescharacteristic
54#16810353C2211yescharacteristic
55#16810353C2211yescharacteristic
56D4⋊#88322C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810344C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C8×Q8×C53C2211yescharacteristic
61C4×SD16×C53C2211yescharacteristic
62C8×D4×C53C2211yescharacteristic
63C53×#334C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C82×C53C2211yescharacteristic
67C2×#11719947C2×C411yescharacteristic
68C2×Dic212C2×C411yescharacteristic
69C2×#11719925C2×C411yescharacteristic
70C2×#11719923C2×C411yescharacteristic
71C2×#11719912—21no—
72C2×#11719912—21no—
73C22×#88322—21no—
74C2×C8×D53D411yescharacteristic
75C2×C8×D53D411yescharacteristic
76#13133216C2×C411yescharacteristic
77#13133215C2×C411yescharacteristic
78#13133207—21no—
79#13133197C2×C411yescharacteristic
80#13133195—21no—
81#13133193C2×C411yescharacteristic
82C4×Dic106—21no—
83C4×Dic106C2311yescharacteristic
84C4×Dic106C2311yescharacteristic
85#13133182C2×C411yescharacteristic
86D4×Dic53C2×C411yescharacteristic
87C4×C22⋊D53—21no—
88#13133177—21no—
89#13133175C2×C411yescharacteristic
90#13133174—21no—
91#13133173—21no—
92C2×C4×Dic53—21no—
93C42×D53C2311yescharacteristic
94#13133151C2×C411yescharacteristic
95#13133150C2×C411yescharacteristic
96#13133148C2×C411yescharacteristic
97C4⋊#88322—21no—
98C4⋊#88322C2311yescharacteristic
99C4⋊#88322C2311yescharacteristic
100#13133146—21no—
101#13133146—21no—
102#13133146—21no—
103#13133146—21no—
104#13133146—21no—
105#13133146C2×C411yescharacteristic
106#13133146C2×C411yescharacteristic
107#13133142C2×C411yescharacteristic
108#13133142C2×C411yescharacteristic
109#13133141C2×C411yescharacteristic
110#13133140C2×C411yescharacteristic
111#13133140C2×C411yescharacteristic
112#13133138C2×C411yescharacteristic
113C22⋊#88322—21no—
114#13133129—21no—
115#13133129C2×C411yescharacteristic
116#13133129C2×C411yescharacteristic
117#13133128—21no—
118#13133127—21no—
119#13133127—21no—
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C8×Dic53—21no—
124C8×Dic53—21no—
125C8×Dic53—21no—
126C8×Dic53—21no—
127C8×Dic53D411yescharacteristic
128C8×Dic53D411yescharacteristic
129C2×SD16×C53C2×C411yescharacteristic
130C22×C8×C53—21no—
131C4×Q8×C53C2311yescharacteristic
132C4×D4×C53C2311yescharacteristic
133#106×C53C2×C411yescharacteristic
134C4⋊C8×C53—21no—
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137#103×C53C2×C411yescharacteristic
138#102×C53C2×C411yescharacteristic
139C22⋊C8×C53—21no—
140C4×C8×C53—21no—
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×D4×C53C22×C411yescharacteristic
146C22×C4×C53—21no—
147C2×C22⋊D53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150#11719947—22no—
151#11719947—22no—
152C2×C4×D53C22×C411yescharacteristic
153C2×C4×D53C2×D411yescharacteristic
154C2×C4×D53#4111yescharacteristic
155C2×Dic106—21no—
156C2×Dic106C22×C411yescharacteristic
157C2×Dic106C22×C411yescharacteristic
158SD16×C53—22no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—42no—
163C2×C8×C53C22×C411yescharacteristic
164C2×C8×C53C22×C411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C4⋊C4×C53—21no—
168C4⋊C4×C53C22×C411yescharacteristic
169C4⋊C4×C53C22×C411yescharacteristic
170C22⋊C4×C53—21no—
171C42×C53—21no—
172C42×C53C2411yescharacteristic
173C22⋊Dic53—21no—
174#11719925—22no—
175#11719923—22no—
176#11719921—21no—
177#11719921—21no—
178#11719921—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53C22×C411yescharacteristic
181C4⋊Dic53C22×C411yescharacteristic
182#11719919—21no—
183#11719919—21no—
184#11719919—21no—
185#11719919—21no—
186#11719919—21no—
187#11719919—21no—
188#11719919—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53C22×C411yescharacteristic
194C4×Dic53C2×D411yescharacteristic
195C4×Dic53#4111yescharacteristic
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—42no—
200C2×#88322C22×C411yescharacteristic
201C2×#88322C22×C411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204Dic212—22no—
205#11719912—24no—
206#11719912—42no—
207C8×D53—22no—
208C8×D53—41no—
209C23×C53—21no—
210C22×D53C4×D411yescharacteristic
211Q8×C53—21no—
212Q8×C53—21no—
213D4×C53—21no—
214D4×C53#13112yes—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53C23×C411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C22⋊D53—24no—
224C2×Dic53—24no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—42no—
230C2×Dic53C4×D411yescharacteristic
231C2×Dic53C4×D411yescharacteristic
232C2×Dic53C4×D411yescharacteristic
233C4×D53—21no—
234C4×D53—21no—
235C4×D53—21no—
236C4×D53—21no—
237C4×D53—22no—
238Dic106—21no—
239Dic106—21no—
240Dic106—21no—
241Dic106—42no—
242Dic106#13112yes—
243C8×C53—22no—
244C8×C53—21no—
245C8×C53—21no—
246C8×C53—21no—
247C8×C53—41no—
248C8×C53—41no—
249C8×C53#13712yes—
250#88322—22no—
251#88322—21no—
252#88322—21no—
253#88322—21no—
254#88322—21no—
255#88322—41no—
256#88322—41no—
257#17913—531nomaximal, Sylow
258C22×C53—22no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×C4×D411yescharacteristic
262C2×D53—21no—
263C2×D53—21no—
264C2×D53—21no—
265C4×C53—21no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53C2×#13111yescharacteristic
270C4×C53C2×#13111yescharacteristic
271C4×C53C2×#13711yescharacteristic
272C4×C53#57511yescharacteristic
273Dic53—22no—
274Dic53—21no—
275Dic53—21no—
276Dic53—41no—
277Dic53—41no—
278Dic53—41no—
279Dic53#44312yes—
280#3107—531no—
281C8×#41—531no—
282#2960—531no—
283#1587—531no—
284#1587—531no—
285#1585—531no—
286#1585—531no—
287C8×Q16—531no—
288C8×Q16—531no—
289C8×SD16—531no—
290C8×SD16D5311yescharacteristic, complemented
291#1574—531no—
292#1489—531no—
293#1485—531no—
294C8×M16—531no—
295C2×C53—42no—
296C2×C53#292911yescharacteristic
297C2×C53C4×#13711yescharacteristic
298C2×C53C2×#44311yescharacteristic
299D53—41no—
300#533—531no—
301C4×#41—531no—
302#485—531no—
303#485—531no—
304#475—531no—
305#474—531no—
306#462—531no—
307#461—531no—
308#452—531no—
309#452—531no—
310#446—531no—
311#446—531no—
312#446—1061no—
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8—531no—
316C8×Q8—1061no—
317C8×Q8C2×D5311yescharacteristic
318C4×Q16—531no—
319C4×Q16—531no—
320C4×SD16—531no—
321C4×SD16C2×D5311yescharacteristic
322C8×D4—531no—
323C8×D4—1061no—
324C8×D4—1061no—
325C8×D4C2×D5311yescharacteristic
326#431—531no—
327#431—531no—
328#431—1061no—
329#424—1061no—
330#423—531no—
331#405—1061no—
332C4×M16—531no—
333C2×C4×C8—531no—
334C2×C4×C8—1061no—
335#335—531no—
336#335—531no—
337#334—531no—
338#334C2×D5311yescharacteristic
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342Q8⋊C8—531no—
343Q8⋊C8—531no—
344Q8⋊C8C2×D5311yescharacteristic
345D4⋊C8—531no—
346D4⋊C8C2×D5311yescharacteristic
347#322—531no—
348#322—531no—
349C82—531no—
350C82C2×D5311yescharacteristic
351C53#1791311yescharacteristic, complemented, Sylow
352C2×#41—531no—
353C2×Q16—531no—
354C2×Q16—531no—
355C2×SD16—531no—
356C2×SD16C4×D5311yescharacteristic
357C2×M16—1061no—
358C2×M16—1061no—
359C22×C8—21no—
360C22×C8—531no—
361C22×C8—531no—
362C22×C8—1061no—
363C4⋊Q8—531no—
364C4⋊Q8—531no—
365#125—1061no—
366C22⋊Q8—531no—
367C22⋊Q8—531no—
368C22⋊Q8—1061no—
369C4×Q8—531no—
370C4×Q8—531no—
371C4×Q8—531no—
372C4×Q8—1061no—
373C4×Q8C22×D5311yescharacteristic
374C4×D4—531no—
375C4×D4—1061no—
376C4×D4—1061no—
377C4×D4C22×D5311yescharacteristic
378#117—531no—
379#117—1061no—
380#117—1061no—
381C2×C42—531no—
382C2×C42—1061no—
383#107—531no—
384#107—531no—
385#106—531no—
386#106C4×D5311yescharacteristic
387C4⋊C8—21no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—531no—
391C4⋊C8—531no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
398C4⋊C8C22×D5311yescharacteristic
399C4⋊C8C22×D5311yescharacteristic
400#103—531no—
401#103—531no—
402#103—531no—
403#103—531no—
404#103—531no—
405#103C4×D5311yescharacteristic
406#102—531no—
407#102C4×D5311yescharacteristic
408C22⋊C8—21no—
409C22⋊C8—531no—
410C22⋊C8—531no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—1061no—
414#97—1061no—
415#97—1061no—
416C4×C8—21no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8C22×D5311yescharacteristic
427C4×C8C22×D5311yescharacteristic
428C4×C8C22×D5311yescharacteristic
429#41—1062no—
430#41—1062no—
431C2×Q8—531no—
432C2×Q8—531no—
433C2×Q8—1061no—
434C2×Q8C2×C4×D5311yescharacteristic
435C2×D4—1061no—
436C2×D4C2×C4×D5311yescharacteristic
437C22×C4—21no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—1061no—
442C22×C4—1061no—
443Q16—1062no—
444Q16—1062no—
445SD16—22no—
446SD16—1062no—
447M16—1064no—
448M16—2122no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—42no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—1061no—
458C2×C8—1062no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—2121no—
462C2×C8—2122no—
463C2×C8C2×C4×D5311yescharacteristic
464C2×C8C2×C4×D5311yescharacteristic
465C2×C8D4×D5311yescharacteristic
466C2×C8D4×D5311yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—531no—
469C4⋊C4—531no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4C2×C4×D5311yescharacteristic
479C4⋊C4C2×C4×D5311yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—1061no—
482C22⋊C4—1061no—
483C22⋊C4—1061no—
484C22⋊C4—1061no—
485C42—21no—
486C42—531no—
487C42—531no—
488C42—531no—
489C42—1061no—
490C42—1061no—
491C42—1061no—
492C42—1061no—
493C42C23×D5311yescharacteristic
494C23—21no—
495C23—531no—
496Q8—21no—
497Q8—21no—
498Q8—532no—
499Q8—1061no—
500Q8—1061no—
501Q8—1061no—
502Q8—2122no—
503D4—21no—
504D4—1064no—
505D4#1313322312yes—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—531no—
512C2×C4—531no—
513C2×C4—531no—
514C2×C4—1062no—
515C2×C4—1064no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—2122no—
525C2×C4C22×C4×D5311yescharacteristic
526C2×C4C2×D4×D5311yescharacteristic
527C2×C4#41×D5311yescharacteristic
528C8—22no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—41no—
533C8—41no—
534C8—1062no—
535C8—1061no—
536C8—1061no—
537C8—1061no—
538C8—1061no—
539C8—2121no—
540C8—2121no—
541C8#1313324612yes—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—1061no—
546C22—1061no—
547C22—1061no—
548C22C4×D4×D5311yescharacteristic
549C4—21no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—532no—
554C4—1061no—
555C4—1062no—
556C4—1061no—
557C4—2121no—
558C4—2121no—
559C4—2121no—
560C4#131×D5311yescharacteristic
561C4C2×#1313322311yescharacteristic
562C4C2×#1313324611yescharacteristic
563C4#1681142011yescharacteristic
564C2—42no—
565C2—2121no—
566C2#2028340411yescharacteristic
567C2C4×#1313324611yescharacteristic
568C2#443×D5311yescharacteristic
subgroupG/H
centerC2×C8D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC8×SD16×C53C2
socleC22×C53C2×C4×D4
prime radicalC2×#11719947C2×C4
cosocleC42×C53C24
hypercenterC8×SD16D53
hypocenterC53#17913
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
3. #20283644C2semidirect
4. #20273589C2semidirect
5. #20273585C2semidirect
6. #20273573C2semidirect
7. #20273569C2semidirect
8. C8×Dic212C2semidirect
9. C8×#11719925C2semidirect
10. C8×#11719923C2semidirect
11. #20273502C2semidirect
13. #20273134C2semidirect
14. C8×#11719912C2semidirect
15. C8×SD16×C53C2semidirect
290. C8×SD16D53semidirect
351. C53#17913semidirect

Maximal subgroups

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

subgroupindex
#202847822
C8×#117199472
#202836442
#202735892
#202735852
#202735732
#202735692
C8×Dic2122
C8×#117199252
C8×#117199232
#202735022
#202731512
#202731342
C8×#117199122
C8×SD16×C532
#1791353