Finite Group Databasebeta

← #27354244

Subgroups — #27354244

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284752C211yesmaximal, characteristic
2#20283811C211yesmaximal, characteristic
3#20283794C211yesmaximal, characteristic
4#20273586C211yesmaximal, characteristic, complemented
5#20273584C211yesmaximal, characteristic, complemented
6#20273567C211yesmaximal, characteristic, complemented
7#20273531C211yesmaximal, characteristic, complemented
8#20273531C211yesmaximal, characteristic, complemented
9Q16×#88322C211yesmaximal, characteristic, complemented
10#20273521C211yesmaximal, characteristic, complemented
11#20273494C211yesmaximal, characteristic
12#20273160C211yesmaximal, characteristic
13#20273134C211yesmaximal, characteristic
14#20273011C211yesmaximal, characteristic
15C53×#1584C211yesmaximal, characteristic, complemented
16#16811177C2211yescharacteristic
17C4×Q8×D53C2211yescharacteristic
18#16810960C411yescharacteristic
19#16810957C411yescharacteristic
20#16810917C411yescharacteristic
21#16810916C411yescharacteristic
22#16810832C411yescharacteristic
23#16810831C411yescharacteristic
24#16810780C411yescharacteristic
25#16810777C411yescharacteristic
26#16810753C2211yescharacteristic
27#16810751—21no—
28#16810750—21no—
29Q8×#88322C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31C8×Dic106—21no—
32C8×Dic106C2211yescharacteristic
33#16810747—21no—
34#16810747C2211yescharacteristic
35#16810747C2211yescharacteristic
36C4×Dic212C2211yescharacteristic
37C4×#11719925C2211yescharacteristic
38C4×#11719925C2211yescharacteristic
39#16810685—21no—
40#16810683—21no—
41#16810679C2211yescharacteristic
42#16810675—21no—
43#16810675C2211yescharacteristic
44#16810674—21no—
45#16810674C2211yescharacteristic
46#16810674C2211yescharacteristic
47C4×#11719912C2211yescharacteristic
48#16810376C2211yescharacteristic
49#16810373C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810355C2211yescharacteristic
52#16810355C2211yescharacteristic
53Q8⋊#88322C2211yescharacteristic
54Q8⋊#88322C2211yescharacteristic
55#16810353C2211yescharacteristic
56#16810353C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810345C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C8×Q8×C53C2211yescharacteristic
62C4×Q16×C53C2211yescharacteristic
63C53×#335C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C53×#322C2211yescharacteristic
67C2×Q8×D53C2×C411yescharacteristic
68C2×Dic212C2×C411yescharacteristic
69C2×#11719925C2×C411yescharacteristic
70C2×#11719925C2×C411yescharacteristic
71C2×#11719912—21no—
72#13133216C2×C411yescharacteristic
73#13133215C2×C411yescharacteristic
74#13133207—21no—
75#13133197C2×C411yescharacteristic
76#13133195—21no—
77#13133193C2×C411yescharacteristic
78Q8×Dic53C2×C411yescharacteristic
79C4×Dic106—21no—
80C4×Dic106—21no—
81C4×Dic106C2311yescharacteristic
82C4×Dic106C2311yescharacteristic
83#13133182—21no—
84#13133182C2×C411yescharacteristic
85#13133174—21no—
86C4⋊C4×D53—21no—
87C4⋊C4×D53C2×C411yescharacteristic
88C42×D53—21no—
89C42×D53C2311yescharacteristic
90#13133151C2×C411yescharacteristic
91#13133150C2×C411yescharacteristic
92#13133150C2×C411yescharacteristic
93C4⋊#88322—21no—
94C4⋊#88322—21no—
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133146—21no—
98#13133146—21no—
99#13133146—21no—
100#13133146—21no—
101#13133146C2×C411yescharacteristic
102#13133146C2×C411yescharacteristic
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#13133129C2×C411yescharacteristic
114#13133129C2×C411yescharacteristic
115#13133129D411yescharacteristic
116#13133129D411yescharacteristic
117#13133127—21no—
118#13133127—21no—
119#13133127—21no—
120C4×#88322—21no—
121C4×#88322—21no—
122C4×#88322C2311yescharacteristic
123C4×#88322C2311yescharacteristic
124C4×#88322C2311yescharacteristic
125C8×Dic53—21no—
126C8×Dic53—21no—
127C8×Dic53—21no—
128C8×Dic53—21no—
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
146Q8×D53—22no—
147Q8×D53—22no—
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53C22×C411yescharacteristic
151C2×C4×D53C2×D411yescharacteristic
152C2×C4×D53#4111yescharacteristic
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106C22×C411yescharacteristic
156C2×Dic106C22×C411yescharacteristic
157Q16×C53—22no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53C22×C411yescharacteristic
162C2×C8×C53C22×C411yescharacteristic
163C2×C8×C53C2×D411yescharacteristic
164C2×C8×C53C2×D411yescharacteristic
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53C22×C411yescharacteristic
168C4⋊C4×C53C22×C411yescharacteristic
169C42×C53—21no—
170C42×C53—21no—
171C42×C53C2411yescharacteristic
172#11719925—22no—
173#11719925—22no—
174#11719921—21no—
175#11719921—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53C22×C411yescharacteristic
179C4⋊Dic53C22×C411yescharacteristic
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×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53C22×C411yescharacteristic
193C4×Dic53C2×D411yescharacteristic
194C4×Dic53#4111yescharacteristic
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322C22×C411yescharacteristic
199C2×#88322C22×C411yescharacteristic
200C2×#88322C2×D411yescharacteristic
201C2×#88322C2×D411yescharacteristic
202Dic212—22no—
203#11719912—42no—
204C22×D53C4×D411yescharacteristic
205Q8×C53—21no—
206Q8×C53—21no—
207Q8×C53—21no—
208Q8×C53#13112yes—
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—24no—
224C4×D53—21no—
225C4×D53—21no—
226C4×D53—21no—
227C4×D53—21no—
228C4×D53—41no—
229C4×D53—81no—
230Dic106—24no—
231Dic106—21no—
232Dic106—21no—
233Dic106—21no—
234Dic106—81no—
235Dic106#13112yes—
236C8×C53—21no—
237C8×C53—21no—
238C8×C53—21no—
239C8×C53—21no—
240C8×C53—41no—
241C8×C53—41no—
242C8×C53—41no—
243#88322—22no—
244#88322—21no—
245#88322—21no—
246#88322—21no—
247#88322—21no—
248#88322—41no—
249#88322—41no—
250#17925—531nomaximal, Sylow
251C22×C53C2×C4×D411yescharacteristic
252C2×D53—21no—
253C2×D53—21no—
254C2×D53—21no—
255C4×C53—22no—
256C4×C53—21no—
257C4×C53—41no—
258C4×C53—41no—
259C4×C53—41no—
260C4×C53C2×#13111yescharacteristic
261C4×C53C2×#13111yescharacteristic
262C4×C53#57811yescharacteristic
263C4×C53#57811yescharacteristic
264Dic53—21no—
265Dic53—22no—
266Dic53—21no—
267Dic53—21no—
268Dic53—41no—
269Dic53—41no—
270Dic53—41no—
271#3104—531no—
272#2985—531no—
273#2982—531no—
274#1587—531no—
275#1587—531no—
276#1584—531no—
277#1584D5311yescharacteristic, complemented
278#1579—531no—
279#1579—531no—
280C8×Q16—531no—
281C8×Q16—531no—
282#1572—531no—
283#1491—531no—
284#1485—531no—
285#1453—531no—
286C2×C53#292911yescharacteristic
287C2×C53#295011yescharacteristic
288C2×C53#295611yescharacteristic
289D53—41no—
290#533—531no—
291C2×C4×Q8—531no—
292#484—531no—
293#484—531no—
294#477—531no—
295#477—531no—
296#462—531no—
297#462—531no—
298#451—531no—
299#451—531no—
300#446—531no—
301#446—1061no—
302#446—1061no—
303#446C2×D5311yescharacteristic
304C8×Q8—531no—
305C8×Q8—531no—
306C8×Q8—531no—
307C8×Q8—531no—
308C8×Q8—531no—
309C8×Q8—1061no—
310C8×Q8—1061no—
311C8×Q8C2×D5311yescharacteristic
312C4×Q16—531no—
313C4×Q16—531no—
314C4×Q16—531no—
315C4×Q16C2×D5311yescharacteristic
316#433—1061no—
317#433—1061no—
318#432—531no—
319#431—531no—
320#431—531no—
321#431—531no—
322#431—1061no—
323#431—1061no—
324C4×M16—531no—
325#335—531no—
326#335C2×D5311yescharacteristic
327#334—531no—
328#334—531no—
329Q8⋊C8—531no—
330Q8⋊C8—531no—
331Q8⋊C8—531no—
332Q8⋊C8—531no—
333Q8⋊C8—531no—
334Q8⋊C8—531no—
335Q8⋊C8C2×D5311yescharacteristic
336Q8⋊C8C2×D5311yescharacteristic
337#322—531no—
338#322—531no—
339#322C2×D5311yescharacteristic
340C82—531no—
341C53#1792511yescharacteristic, complemented, Sylow
342C22×Q8—531no—
343C2×Q16—531no—
344C2×Q16—531no—
345C2×Q16—531no—
346C2×Q16C4×D5311yescharacteristic
347C2×M16—1061no—
348C4⋊Q8—531no—
349C4⋊Q8—531no—
350#125—1061no—
351C22⋊Q8—531no—
352C22⋊Q8—531no—
353C22⋊Q8—1061no—
354C4×Q8—531no—
355C4×Q8—531no—
356C4×Q8—531no—
357C4×Q8—531no—
358C4×Q8—1061no—
359C4×Q8—1061no—
360C4×Q8—1061no—
361C4×Q8C22×D5311yescharacteristic
362C4×Q8C22×D5311yescharacteristic
363#117—1061no—
364C2×C4⋊C4—531no—
365C2×C4⋊C4—1061no—
366C2×C42—531no—
367C2×C42—1061no—
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—531no—
379C4⋊C8—531no—
380C4⋊C8—1061no—
381C4⋊C8—1061no—
382C4⋊C8—1061no—
383C4⋊C8—1061no—
384C4⋊C8—1061no—
385C4⋊C8—1061no—
386C4⋊C8C22×D5311yescharacteristic
387C4⋊C8C22×D5311yescharacteristic
388#103—531no—
389#103—531no—
390#103—531no—
391#103—531no—
392#103—531no—
393#103—531no—
394#103C4×D5311yescharacteristic
395#103C4×D5311yescharacteristic
396C22⋊C8—531no—
397C22⋊C8—531no—
398C22⋊C8—531no—
399C22⋊C8—531no—
400C22⋊C8—1061no—
401C22⋊C8—1061no—
402#97—21no—
403#97—1061no—
404#97—1061no—
405#97—1061no—
406C4×C8—21no—
407C4×C8—531no—
408C4×C8—531no—
409C4×C8—531no—
410C4×C8—1061no—
411C4×C8—1061no—
412C4×C8—1061no—
413C4×C8—1061no—
414C4×C8—1061no—
415C4×C8—1061no—
416C4×C8C22×D5311yescharacteristic
417C4×C8C22×D5311yescharacteristic
418C4×C8C22×D5311yescharacteristic
419C2×Q8—531no—
420C2×Q8—531no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1062no—
424C2×Q8—1062no—
425C2×Q8C2×C4×D5311yescharacteristic
426C2×Q8C2×C4×D5311yescharacteristic
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—531no—
430C22×C4—1061no—
431C22×C4—1061no—
432Q16—22no—
433Q16—1062no—
434Q16—1062no—
435Q16—1062no—
436M16—2122no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—531no—
441C2×C8—531no—
442C2×C8—531no—
443C2×C8—531no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8C2×C4×D5311yescharacteristic
448C2×C8C2×C4×D5311yescharacteristic
449C2×C8D4×D5311yescharacteristic
450C2×C8D4×D5311yescharacteristic
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—531no—
454C4⋊C4—531no—
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⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4C2×C4×D5311yescharacteristic
466C4⋊C4C2×C4×D5311yescharacteristic
467C22⋊C4—1061no—
468C22⋊C4—1061no—
469C42—21no—
470C42—21no—
471C42—531no—
472C42—531no—
473C42—531no—
474C42—1061no—
475C42—1061no—
476C42—1061no—
477C42—1061no—
478C42C23×D5311yescharacteristic
479C23—531no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—532no—
484Q8—1064no—
485Q8—1061no—
486Q8—1061no—
487Q8—1061no—
488Q8—4241no—
489Q8#1313322512yes—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—531no—
495C2×C4—531no—
496C2×C4—531no—
497C2×C4—1064no—
498C2×C4—1061no—
499C2×C4—1061no—
500C2×C4—1061no—
501C2×C4—1061no—
502C2×C4—1061no—
503C2×C4—1061no—
504C2×C4—1061no—
505C2×C4—1061no—
506C2×C4—4241no—
507C2×C4—2121no—
508C2×C4C22×C4×D5311yescharacteristic
509C2×C4C2×D4×D5311yescharacteristic
510C2×C4#41×D5311yescharacteristic
511C8—21no—
512C8—21no—
513C8—21no—
514C8—21no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—1062no—
519C8—1061no—
520C8—1061no—
521C8—1061no—
522C8—1061no—
523C8—2121no—
524C8—2121no—
525C22—1061no—
526C22—1061no—
527C22—1061no—
528C22C4×D4×D5311yescharacteristic
529C4—22no—
530C4—21no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—1061no—
535C4—1062no—
536C4—1061no—
537C4—1061no—
538C4—2121no—
539C4—2121no—
540C4—2121no—
541C4#131×D5311yescharacteristic
542C4C2×#1313322511yescharacteristic
543C4#1681143611yescharacteristic
544C4#1681143711yescharacteristic
545C2—2121no—
546C2#2028340611yescharacteristic
547C2#2028357811yescharacteristic
548C2#2028362311yescharacteristic
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#17925
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. #20273586C2semidirect
5. #20273584C2semidirect
6. #20273567C2semidirect
7. #20273531C2semidirect
8. #20273531C2semidirect
9. Q16×#88322C2semidirect
10. #20273521C2semidirect
15. C53×#1584C2semidirect
277. #1584D53semidirect
341. C53#17925semidirect

Maximal subgroups

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

subgroupindex
#202847522
#202838112
#202837942
#202735862
#202735842
#202735672
#202735312
Q16×#883222
#202735212
#202734942
#202731602
#202731342
#202730112
C53×#15842
#1792553