Finite Group Databasebeta

← #27356119

Subgroups — #27356119

Order 13568.

564 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285298C211yesmaximal, characteristic
2#20283815C211yesmaximal, characteristic
3#20283813C211yesmaximal, characteristic
4#20273622C211yesmaximal, characteristic, complemented
5#20273621C211yesmaximal, characteristic, complemented
6#20273567C211yesmaximal, characteristic, complemented
7#20273566C211yesmaximal, characteristic, complemented
8#20273564C211yesmaximal, characteristic, complemented
9#20273532C211yesmaximal, characteristic, complemented
10#20273531C211yesmaximal, characteristic, complemented
11#20273462C211yesmaximal, characteristic
12#20273134C211yesmaximal, characteristic
13#20273134C211yesmaximal, characteristic
14#20273003C211yesmaximal, characteristic
15C53×#1584C211yesmaximal, characteristic, complemented
16#16811177C2211yescharacteristic
17#16811177C2211yescharacteristic
18#16811027C411yescharacteristic
19#16811001C411yescharacteristic
20#16810972C411yescharacteristic
21#16810972C411yescharacteristic
22#16810865C411yescharacteristic
23#16810863C411yescharacteristic
24#16810863C411yescharacteristic
25#16810861C411yescharacteristic
26#16810754C2211yescharacteristic
27#16810753C2211yescharacteristic
28#16810752—21no—
29#16810751—21no—
30#16810750C2211yescharacteristic
31Q8×#88322C2211yescharacteristic
32C8×Dic106—21no—
33C8×Dic106C2211yescharacteristic
34#16810747—21no—
35#16810747C2211yescharacteristic
36C4×Dic212C2211yescharacteristic
37C4×#11719925C2211yescharacteristic
38C4×#11719925C2211yescharacteristic
39#16810685—21no—
40#16810683—21no—
41#16810683—21no—
42#16810682—21no—
43#16810675C2211yescharacteristic
44#16810675C2211yescharacteristic
45#16810674C2211yescharacteristic
46C4×#11719912C2211yescharacteristic
47C4×C8×D53C2211yescharacteristic
48#16810376C2211yescharacteristic
49#16810373C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810355C2211yescharacteristic
52#16810355C2211yescharacteristic
53Q8⋊#88322C2211yescharacteristic
54Q8⋊#88322C2211yescharacteristic
55#16810353C2211yescharacteristic
56#16810353C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810345C2211yescharacteristic
59#16810345C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C8×Q8×C53C2211yescharacteristic
62C4×Q16×C53C2211yescharacteristic
63C53×#335C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C53×#322C2211yescharacteristic
67C2×Dic212C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719925C2×C411yescharacteristic
70C2×#11719912—21no—
71C2×C8×D53C2×C411yescharacteristic
72C2×C8×D53C2×C411yescharacteristic
73#13133216C2×C411yescharacteristic
74#13133216C2×C411yescharacteristic
75#13133215C2×C411yescharacteristic
76#13133215C2×C411yescharacteristic
77#13133207—21no—
78#13133207—21no—
79#13133197C2×C411yescharacteristic
80#13133197C2×C411yescharacteristic
81#13133195—21no—
82#13133195—21no—
83#13133193C2×C411yescharacteristic
84#13133193C2×C411yescharacteristic
85C4×Dic106—21no—
86C4×Dic106—21no—
87C4×Dic106C2311yescharacteristic
88C4×Dic106C2311yescharacteristic
89#13133174—21no—
90#13133174—21no—
91C42×D53C2311yescharacteristic
92#13133151C2×C411yescharacteristic
93#13133150C2×C411yescharacteristic
94#13133150C2×C411yescharacteristic
95C4⋊#88322—21no—
96C4⋊#88322—21no—
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99#13133146—21no—
100#13133146—21no—
101#13133146—21no—
102#13133146—21no—
103#13133146D411yescharacteristic
104#13133146D411yescharacteristic
105#13133142C2×C411yescharacteristic
106#13133142C2×C411yescharacteristic
107#13133141C2×C411yescharacteristic
108#13133141C2×C411yescharacteristic
109#13133140C2×C411yescharacteristic
110#13133140C2×C411yescharacteristic
111#13133129—21no—
112#13133129—21no—
113#13133129D411yescharacteristic
114#13133129D411yescharacteristic
115#13133128—21no—
116#13133127—21no—
117#13133127—21no—
118#13133127—21no—
119C4×#88322—21no—
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C8×Dic53—21no—
124C8×Dic53—21no—
125C8×Dic53—21no—
126C8×Dic53—21no—
127C8×Dic53C2×C411yescharacteristic
128C8×Dic53C2×C411yescharacteristic
129C2×Q16×C53C2×C411yescharacteristic
130C4×Q8×C53C2311yescharacteristic
131C4×Q8×C53C2311yescharacteristic
132#107×C53C2×C411yescharacteristic
133C4⋊C8×C53—21no—
134C4⋊C8×C53—21no—
135C4⋊C8×C53C2311yescharacteristic
136C4⋊C8×C53C2311yescharacteristic
137#103×C53C2×C411yescharacteristic
138#103×C53C2×C411yescharacteristic
139#97×C53—21no—
140C4×C8×C53—21no—
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×C4×D53C22×C411yescharacteristic
147C2×C4×D53C2×D411yescharacteristic
148C2×C4×D53#4111yescharacteristic
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106C22×C411yescharacteristic
152C2×Dic106C22×C411yescharacteristic
153Q16×C53—41no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53C22×C411yescharacteristic
158C2×C8×C53C22×C411yescharacteristic
159C2×C8×C53C2×D411yescharacteristic
160C2×C8×C53C2×D411yescharacteristic
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53C22×C411yescharacteristic
164C4⋊C4×C53C22×C411yescharacteristic
165C42×C53—21no—
166C42×C53—21no—
167C42×C53C2411yescharacteristic
168#11719925—22no—
169#11719925—22no—
170#11719921—21no—
171#11719921—21no—
172#11719921—21no—
173#11719921—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53C22×C411yescharacteristic
177C4⋊Dic53C22×C411yescharacteristic
178#11719919—21no—
179#11719919—21no—
180#11719919—21no—
181#11719919—21no—
182#11719919—21no—
183#11719919—21no—
184#11719919—21no—
185#11719919—21no—
186#11719919—21no—
187#11719919—21no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53C22×C411yescharacteristic
191C4×Dic53C2×D411yescharacteristic
192C4×Dic53#4111yescharacteristic
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322C22×C411yescharacteristic
197C2×#88322C22×C411yescharacteristic
198C2×#88322C2×D411yescharacteristic
199C2×#88322C2×D411yescharacteristic
200Dic212—41no—
201#11719912—42no—
202C8×D53—22no—
203C8×D53—41no—
204C22×D53C4×D411yescharacteristic
205Q8×C53—21no—
206Q8×C53—21no—
207Q8×C53—21no—
208Q8×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53C23×C411yescharacteristic
214C2×C4×C53C22×D411yescharacteristic
215C2×C4×C53C2×#4111yescharacteristic
216C2×Dic53—21no—
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53—21no—
220C2×Dic53C4×D411yescharacteristic
221C2×Dic53C4×D411yescharacteristic
222C2×Dic53C4×D411yescharacteristic
223C4×D53—21no—
224C4×D53—21no—
225C4×D53—21no—
226C4×D53—21no—
227C4×D53—22no—
228Dic106—21no—
229Dic106—21no—
230Dic106—21no—
231Dic106—21no—
232Dic106—42no—
233Dic106—42no—
234C8×C53—21no—
235C8×C53—21no—
236C8×C53—21no—
237C8×C53—21no—
238C8×C53—41no—
239C8×C53—41no—
240C8×C53—41no—
241#88322—21no—
242#88322—21no—
243#88322—21no—
244#88322—41no—
245#88322—41no—
246#88322—41no—
247#88322#13112yes—
248#18090—531nomaximal, Sylow
249C22×C53C2×C4×D411yescharacteristic
250C2×D53—21no—
251C2×D53—21no—
252C2×D53—21no—
253C4×C53—21no—
254C4×C53—41no—
255C4×C53—41no—
256C4×C53—41no—
257C4×C53—41no—
258C4×C53C2×#13111yescharacteristic
259C4×C53#56811yescharacteristic
260C4×C53#57811yescharacteristic
261C4×C53#57811yescharacteristic
262Dic53—21no—
263Dic53—21no—
264Dic53—41no—
265Dic53—41no—
266Dic53—41no—
267Dic53—41no—
268Dic53#44412yes—
269#3170—531no—
270#2986—531no—
271#2986—531no—
272#1593—531no—
273#1593—531no—
274#1584—531no—
275#1584—531no—
276#1584—531no—
277#1584D5311yescharacteristic, complemented
278#1579—531no—
279#1579—531no—
280#1564—531no—
281#1485—531no—
282#1485—531no—
283#1451—531no—
284C2×C53#293511yescharacteristic
285C2×C53#295011yescharacteristic
286C2×C53C2×#44411yescharacteristic
287D53—41no—
288#533—531no—
289#533—531no—
290#500—531no—
291#494—531no—
292#487—531no—
293#487—531no—
294#467—531no—
295#467—531no—
296#467—531no—
297#467—531no—
298#446—531no—
299#446—531no—
300#446—531no—
301#446—1061no—
302#446—1061no—
303#446C2×D5311yescharacteristic
304C8×Q8—531no—
305C8×Q8—531no—
306C8×Q8—531no—
307C8×Q8—1061no—
308C8×Q8—1061no—
309C8×Q8C2×D5311yescharacteristic
310C4×Q16—531no—
311C4×Q16—531no—
312C4×Q16—531no—
313C4×Q16C2×D5311yescharacteristic
314#433—1061no—
315#433—1061no—
316#433—1061no—
317#433—1061no—
318#431—531no—
319#431—531no—
320#431—531no—
321C4×M16—531no—
322C2×C4×C8—531no—
323#335—531no—
324#335C2×D5311yescharacteristic
325#334—531no—
326#334—531no—
327Q8⋊C8—531no—
328Q8⋊C8—531no—
329Q8⋊C8—531no—
330Q8⋊C8—531no—
331Q8⋊C8—531no—
332Q8⋊C8—531no—
333Q8⋊C8C2×D5311yescharacteristic
334Q8⋊C8C2×D5311yescharacteristic
335#322—531no—
336#322—531no—
337#322—531no—
338#322C2×D5311yescharacteristic
339C53#1809011yescharacteristic, complemented, Sylow
340C2×Q16—531no—
341C2×Q16—531no—
342C2×Q16—531no—
343C2×Q16C4×D5311yescharacteristic
344C2×M16—1061no—
345C22×C8—531no—
346C22×C8—531no—
347C4⋊Q8—531no—
348C4⋊Q8—531no—
349C4⋊Q8—531no—
350C4⋊Q8—531no—
351#125—1061no—
352#125—1061no—
353C22⋊Q8—531no—
354C22⋊Q8—531no—
355C22⋊Q8—531no—
356C22⋊Q8—531no—
357C22⋊Q8—1061no—
358C22⋊Q8—1061no—
359C4×Q8—531no—
360C4×Q8—531no—
361C4×Q8—1061no—
362C4×Q8—1061no—
363C4×Q8C22×D5311yescharacteristic
364C4×Q8C22×D5311yescharacteristic
365#117—1061no—
366#117—1061no—
367C2×C42—531no—
368#107—531no—
369#107—531no—
370#107—531no—
371#107C4×D5311yescharacteristic
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—531no—
375C4⋊C8—531no—
376C4⋊C8—531no—
377C4⋊C8—531no—
378C4⋊C8—1061no—
379C4⋊C8—1061no—
380C4⋊C8—1061no—
381C4⋊C8—1061no—
382C4⋊C8—1061no—
383C4⋊C8—1061no—
384C4⋊C8C22×D5311yescharacteristic
385C4⋊C8C22×D5311yescharacteristic
386#103—531no—
387#103—531no—
388#103—531no—
389#103—531no—
390#103—531no—
391#103—531no—
392#103C4×D5311yescharacteristic
393#103C4×D5311yescharacteristic
394C22⋊C8—531no—
395C22⋊C8—531no—
396C22⋊C8—1061no—
397C22⋊C8—1061no—
398#97—21no—
399#97—1061no—
400#97—1061no—
401#97—1061no—
402#97—1061no—
403C4×C8—21no—
404C4×C8—531no—
405C4×C8—531no—
406C4×C8—531no—
407C4×C8—531no—
408C4×C8—531no—
409C4×C8—1061no—
410C4×C8—1061no—
411C4×C8—1061no—
412C4×C8—1061no—
413C4×C8—1061no—
414C4×C8C22×D5311yescharacteristic
415C4×C8C22×D5311yescharacteristic
416C4×C8C22×D5311yescharacteristic
417C2×Q8—531no—
418C2×Q8—531no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C2×Q8C2×C4×D5311yescharacteristic
422C2×Q8C2×C4×D5311yescharacteristic
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426Q16—41no—
427Q16—1062no—
428Q16—1062no—
429Q16—2121no—
430M16—2122no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—531no—
435C2×C8—531no—
436C2×C8—531no—
437C2×C8—531no—
438C2×C8—1062no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—2121no—
443C2×C8C2×C4×D5311yescharacteristic
444C2×C8C2×C4×D5311yescharacteristic
445C2×C8D4×D5311yescharacteristic
446C2×C8D4×D5311yescharacteristic
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—531no—
450C4⋊C4—531no—
451C4⋊C4—1061no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4C2×C4×D5311yescharacteristic
464C4⋊C4C2×C4×D5311yescharacteristic
465C22⋊C4—1061no—
466C22⋊C4—1061no—
467C22⋊C4—1061no—
468C22⋊C4—1061no—
469C42—21no—
470C42—21no—
471C42—531no—
472C42—531no—
473C42—531no—
474C42—1061no—
475C42—1061no—
476C42C23×D5311yescharacteristic
477C23—531no—
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—1061no—
483Q8—1061no—
484Q8—1061no—
485Q8—1061no—
486Q8—2122no—
487Q8—2122no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—531no—
493C2×C4—531no—
494C2×C4—531no—
495C2×C4—1062no—
496C2×C4—1061no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1061no—
500C2×C4—1061no—
501C2×C4—1061no—
502C2×C4—1061no—
503C2×C4—1061no—
504C2×C4C22×C4×D5311yescharacteristic
505C2×C4C2×D4×D5311yescharacteristic
506C2×C4#41×D5311yescharacteristic
507C8—21no—
508C8—21no—
509C8—21no—
510C8—21no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—532no—
515C8—1061no—
516C8—1061no—
517C8—1061no—
518C8—2121no—
519C8—2121no—
520C8—2121no—
521C22—1061no—
522C22—1061no—
523C22—1061no—
524C22C4×D4×D5311yescharacteristic
525C4—21no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—532no—
531C4—1061no—
532C4—1061no—
533C4—2121no—
534C4—2121no—
535C4—2121no—
536C4—2121no—
537C4#131×D5311yescharacteristic
538C4#1681138911yescharacteristic
539C4#1681143611yescharacteristic
540C4#1681143711yescharacteristic
541C2—2121no—
542C2#2028345611yescharacteristic
543C2#2028357811yescharacteristic
544C2#444×D5311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1584C2
socleC22×C53C2×C4×D4
prime radicalC22×D53C4×D4
cosocleC42×C53C24
hypercenter#1584D53
hypocenterC53#18090
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #20273622C2semidirect
5. #20273621C2semidirect
6. #20273567C2semidirect
7. #20273566C2semidirect
8. #20273564C2semidirect
9. #20273532C2semidirect
10. #20273531C2semidirect
15. C53×#1584C2semidirect
277. #1584D53semidirect
339. C53#18090semidirect

Maximal subgroups

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

subgroupindex
#202852982
#202838152
#202838132
#202736222
#202736212
#202735672
#202735662
#202735642
#202735322
#202735312
#202734622
#202731342
#202730032
C53×#15842
#1809053