Finite Group Databasebeta

← #27354308

Subgroups — #27354308

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284752C211yesmaximal, characteristic
2#20283646C211yesmaximal, characteristic
3#446×D53C211yesmaximal, characteristic
4#20273613C211yesmaximal, characteristic, complemented
5#20273610C211yesmaximal, characteristic, complemented
6#20273610C211yesmaximal, characteristic, complemented
7#20273568C211yesmaximal, characteristic, complemented
8#20273566C211yesmaximal, characteristic, complemented
9#20273564C211yesmaximal, characteristic, complemented
10#20273562C211yesmaximal, characteristic, complemented
11#20273488C211yesmaximal, characteristic
12#20273142C211yesmaximal, characteristic
13#20273116C211yesmaximal, characteristic
14#20272997C211yesmaximal, characteristic
15C53×#1591C211yesmaximal, characteristic, complemented
16#16811177C2211yescharacteristic
17C4×Q8×D53C2211yescharacteristic
18#16810960C411yescharacteristic
19#16810957C411yescharacteristic
20#16810917C411yescharacteristic
21#16810916C411yescharacteristic
22#16810832C411yescharacteristic
23#16810831C411yescharacteristic
24#16810780C411yescharacteristic
25#16810777C411yescharacteristic
26#16810754C2211yescharacteristic
27#16810753C2211yescharacteristic
28#16810753C2211yescharacteristic
29#16810752—21no—
30#16810751—21no—
31#16810751—21no—
32#16810750C2211yescharacteristic
33Q8×#88322C2211yescharacteristic
34#16810747—21no—
35#16810747C2211yescharacteristic
36C4×Dic212C2211yescharacteristic
37C4×#11719925C2211yescharacteristic
38C4×#11719925C2211yescharacteristic
39#16810674C2211yescharacteristic
40#16810674C2211yescharacteristic
41#16810648—21no—
42#16810645C2211yescharacteristic
43C4⋊C8×D53—21no—
44C4⋊C8×D53C2211yescharacteristic
45#16810580—21no—
46#97×D53—21no—
47C4×C8×D53C2211yescharacteristic
48#16810376C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810375C2211yescharacteristic
51#16810355C2211yescharacteristic
52#16810355C2211yescharacteristic
53Q8⋊#88322C2211yescharacteristic
54Q8⋊#88322C2211yescharacteristic
55#16810353C2211yescharacteristic
56#16810353C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810345C2211yescharacteristic
59C8×#88322C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C53×#446C2211yescharacteristic
62C4×Q16×C53C2211yescharacteristic
63C53×#335C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C82×C53C2211yescharacteristic
67C2×Q8×D53C2×C411yescharacteristic
68C2×Dic212C2×C411yescharacteristic
69C2×#11719925C2×C411yescharacteristic
70C2×#11719925C2×C411yescharacteristic
71C2×#11719912—21no—
72C2×C8×D53—21no—
73C2×C8×D53D411yescharacteristic
74C2×C8×D53D411yescharacteristic
75#13133216C2×C411yescharacteristic
76#13133215C2×C411yescharacteristic
77#13133207—21no—
78#13133197C2×C411yescharacteristic
79#13133195—21no—
80#13133193C2×C411yescharacteristic
81Q8×Dic53C2×C411yescharacteristic
82C4×Dic106—21no—
83C4×Dic106—21no—
84C4×Dic106C2311yescharacteristic
85C4×Dic106C2311yescharacteristic
86#13133182—21no—
87#13133182C2×C411yescharacteristic
88#13133174—21no—
89C4⋊C4×D53—21no—
90C4⋊C4×D53C2×C411yescharacteristic
91C42×D53—21no—
92C42×D53C2311yescharacteristic
93#13133151C2×C411yescharacteristic
94#13133150C2×C411yescharacteristic
95#13133150C2×C411yescharacteristic
96C4⋊#88322—21no—
97C4⋊#88322—21no—
98C4⋊#88322C2311yescharacteristic
99C4⋊#88322C2311yescharacteristic
100#13133146—21no—
101#13133146—21no—
102#13133146—21no—
103#13133146—21no—
104#13133146—21no—
105#13133146—21no—
106#13133146—21no—
107#13133146C2×C411yescharacteristic
108#13133146C2×C411yescharacteristic
109#13133142C2×C411yescharacteristic
110#13133142C2×C411yescharacteristic
111#13133141C2×C411yescharacteristic
112#13133141C2×C411yescharacteristic
113#13133140C2×C411yescharacteristic
114#13133140C2×C411yescharacteristic
115#13133129—21no—
116#13133129C2×C411yescharacteristic
117#13133129C2×C411yescharacteristic
118#13133128—21no—
119#13133127—21no—
120#13133127—21no—
121#13133127—21no—
122C4×#88322—21no—
123C4×#88322C2311yescharacteristic
124C4×#88322C2311yescharacteristic
125C4×#88322C2311yescharacteristic
126C8×Dic53—21no—
127C8×Dic53D411yescharacteristic
128C8×Dic53D411yescharacteristic
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—
140#97×C53—21no—
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146Q8×D53—41no—
147Q8×D53—41no—
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—41no—
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—41no—
173#11719925—41no—
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—41no—
203#11719912—42no—
204C8×D53—22no—
205C8×D53—42no—
206C8×D53Q1614yescomplemented
207C22×D53C4×D411yescharacteristic
208Q8×C53—21no—
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53C23×C411yescharacteristic
217C2×C4×C53C22×D411yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53C4×D411yescharacteristic
224C2×Dic53C4×D411yescharacteristic
225C2×Dic53C4×D411yescharacteristic
226C4×D53—21no—
227C4×D53—21no—
228C4×D53—41no—
229C4×D53—42no—
230C4×D53—42no—
231C4×D53C2×Q1612yes—
232C4×D53#13512yes—
233Dic106—21no—
234Dic106—21no—
235Dic106—21no—
236Dic106—21no—
237Dic106—42no—
238Dic106—42no—
239C8×C53—21no—
240C8×C53—21no—
241C8×C53—21no—
242C8×C53—41no—
243C8×C53—41no—
244C8×C53—41no—
245C8×C53C2×Q1612yes—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—41no—
250#88322—41no—
251#88322—41no—
252#88322C2×Q1612yes—
253#17931—531nomaximal, Sylow
254C22×C53C2×C4×D411yescharacteristic
255C2×D53—21no—
256C2×D53—21no—
257C2×D53C4×Q1612yes—
258C4×C53—21no—
259C4×C53—41no—
260C4×C53—41no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53#56811yescharacteristic
264C4×C53#56811yescharacteristic
265C4×C53C22×Q1611yescharacteristic
266C4×C53C2×#13511yescharacteristic
267Dic53—21no—
268Dic53—21no—
269Dic53—41no—
270Dic53—41no—
271Dic53—41no—
272Dic53—41no—
273Dic53C4×Q1612yes—
274#3104—531no—
275#2960—531no—
276C2×#446—531no—
277#1591—531no—
278#1591—531no—
279#1591—531no—
280#1591D5311yescharacteristic, complemented
281#1584—531no—
282#1584—531no—
283#1584—531no—
284#1584—531no—
285#1571—531no—
286#1487—531no—
287#1481—531no—
288#1450—531no—
289C2×C53#293011yescharacteristic
290C2×C53C2×C4×Q1611yescharacteristic
291C2×C53#295511yescharacteristic
292D53—22no—
293#533—531no—
294C2×C4×Q8—531no—
295#484—531no—
296#484—531no—
297#477—531no—
298#477—531no—
299#462—531no—
300#462—531no—
301#451—531no—
302#451—531no—
303#446—531no—
304#446—531no—
305#446—531no—
306#446—531no—
307#446—1061no—
308#446—1061no—
309#446—1061no—
310#446C2×D5311yescharacteristic
311#446C2×D5311yescharacteristic
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8—1061no—
315C4×Q16—531no—
316C4×Q16—531no—
317C4×Q16—531no—
318C4×Q16C2×D5311yescharacteristic
319#431—531no—
320#431—531no—
321#424—1061no—
322#423—531no—
323C2×C4⋊C8—531no—
324C2×C4⋊C8—1061no—
325#405—1061no—
326C2×#97—1061no—
327C2×C4×C8—531no—
328#335—531no—
329#335—531no—
330#335—531no—
331#335C2×D5311yescharacteristic
332Q8⋊C8—531no—
333Q8⋊C8—531no—
334Q8⋊C8—531no—
335Q8⋊C8—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8C2×D5311yescharacteristic
339Q8⋊C8C2×D5311yescharacteristic
340#322—531no—
341#322—531no—
342C82—531no—
343C82C2×D5311yescharacteristic
344C53#1793111yescharacteristic, complemented, Sylow
345C22×Q8—531no—
346C2×Q16—531no—
347C2×Q16—531no—
348C2×Q16—531no—
349C2×Q16C4×D5311yescharacteristic
350C2×M16—1061no—
351C22×C8—531no—
352C22×C8—531no—
353C22×C8—1061no—
354C4⋊Q8—531no—
355C4⋊Q8—531no—
356#125—1061no—
357C22⋊Q8—531no—
358C22⋊Q8—531no—
359C22⋊Q8—1061no—
360C4×Q8—531no—
361C4×Q8—531no—
362C4×Q8—531no—
363C4×Q8—531no—
364C4×Q8—1061no—
365C4×Q8—1061no—
366C4×Q8—1061no—
367C4×Q8C22×D5311yescharacteristic
368C4×Q8C22×D5311yescharacteristic
369#117—1061no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—1061no—
372C2×C42—531no—
373C2×C42—1061no—
374#107—531no—
375#107—531no—
376#107—531no—
377#107C4×D5311yescharacteristic
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—531no—
381C4⋊C8—531no—
382C4⋊C8—531no—
383C4⋊C8—531no—
384C4⋊C8—1061no—
385C4⋊C8—1061no—
386C4⋊C8—1061no—
387C4⋊C8—1061no—
388C4⋊C8—1061no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8C22×D5311yescharacteristic
394C4⋊C8C22×D5311yescharacteristic
395#103—531no—
396#103—531no—
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103C4×D5311yescharacteristic
402#103C4×D5311yescharacteristic
403C22⋊C8—531no—
404C22⋊C8—531no—
405C22⋊C8—1061no—
406#97—21no—
407#97—21no—
408#97—1061no—
409#97—1061no—
410#97—1061no—
411#97—1061no—
412C4×C8—531no—
413C4×C8—531no—
414C4×C8—531no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—1061no—
418C4×C8—1061no—
419C4×C8C22×D5311yescharacteristic
420C4×C8C22×D5311yescharacteristic
421C4×C8C22×D5311yescharacteristic
422C2×Q8—531no—
423C2×Q8—531no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8—2121no—
427C2×Q8—2121no—
428C2×Q8C2×C4×D5311yescharacteristic
429C2×Q8C2×C4×D5311yescharacteristic
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—1061no—
434C22×C4—1061no—
435Q16—41no—
436Q16—2121no—
437Q16—2121no—
438Q16—2121no—
439M16—2122no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—531no—
444C2×C8—534no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—1062no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—2122no—
453C2×C8C2×C4×D5311yescharacteristic
454C2×C8C2×C4×D5311yescharacteristic
455C2×C8D4×D5311yescharacteristic
456C2×C8D4×D5311yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—531no—
460C4⋊C4—531no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4C2×C4×D5311yescharacteristic
472C4⋊C4C2×C4×D5311yescharacteristic
473C22⋊C4—1061no—
474C22⋊C4—1061no—
475C42—21no—
476C42—21no—
477C42—531no—
478C42—531no—
479C42—531no—
480C42—1061no—
481C42—1061no—
482C42—1061no—
483C42—1061no—
484C42C23×D5311yescharacteristic
485C23—531no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—21no—
490Q8—1061no—
491Q8—1061no—
492Q8—1061no—
493Q8—1061no—
494Q8—2122no—
495Q8—2122no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—531no—
501C2×C4—532no—
502C2×C4—532no—
503C2×C4—531no—
504C2×C4—531no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—2121no—
512C2×C4—2122no—
513C2×C4—2122no—
514C2×C4C22×C4×D5311yescharacteristic
515C2×C4C2×D4×D5311yescharacteristic
516C2×C4#41×D5311yescharacteristic
517C8—21no—
518C8—21no—
519C8—21no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—532no—
524C8—1061no—
525C8—1061no—
526C8—1061no—
527C8—2121no—
528C8—2121no—
529C8—2121no—
530C8Q16×D5312yes—
531C22—532no—
532C22—1061no—
533C22—1061no—
534C22C4×D4×D5311yescharacteristic
535C4—21no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—532no—
541C4—1061no—
542C4—1061no—
543C4—2121no—
544C4—2121no—
545C4—2121no—
546C4—2121no—
547C4#1681138811yescharacteristic
548C4#1681138911yescharacteristic
549C4C2×Q16×D5311yescharacteristic
550C4#135×D5311yescharacteristic
551C2—1062no—
552C2#2028341511yescharacteristic
553C2C4×Q16×D5311yescharacteristic
554C2#2028361311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1591C2
socleC22×C53C2×C4×D4
prime radicalC22×D53C4×D4
cosocleC42×C53C24
hypercenter#1591D53
hypocenterC53#17931
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1591

Complements

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

NH ≅ G/Ntype
4. #20273613C2semidirect
5. #20273610C2semidirect
6. #20273610C2semidirect
7. #20273568C2semidirect
8. #20273566C2semidirect
9. #20273564C2semidirect
10. #20273562C2semidirect
15. C53×#1591C2semidirect
206. C8×D53Q16semidirect
280. #1591D53semidirect
344. C53#17931semidirect

Maximal subgroups

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

subgroupindex
#202847522
#202836462
#446×D532
#202736132
#202736102
#202735682
#202735662
#202735642
#202735622
#202734882
#202731422
#202731162
#202729972
C53×#15912
#1793153