Finite Group Databasebeta

← #27368405

Subgroups — #27368405

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285429C211yesmaximal, characteristic
2#20284520C211yesmaximal, characteristic, complemented
3Q8×#11719917C211yesmaximal, characteristic, complemented
4#20274299C211yesmaximal, characteristic, complemented
5#20274253C211yesmaximal, characteristic, complemented
6#20274241C211yesmaximal, characteristic, complemented
7#20274185C211yesmaximal, characteristic, complemented
8#20274175C211yesmaximal, characteristic, complemented
9#20274129C211yesmaximal, characteristic, complemented
10#20274117C211yesmaximal, characteristic, complemented
11#20274061C211yesmaximal, characteristic, complemented
12#20273483C211yesmaximal, characteristic, complemented
13#20273450C211yesmaximal, characteristic
14#20273446C211yesmaximal, characteristic, complemented
15C53×#3112C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811026C2211yescharacteristic
18#16810995C2211yescharacteristic
19#16810975C2211yescharacteristic
20#16810969C2211yescharacteristic
21#16810930—21no—
22#16810930—21no—
23#16810918C2211yescharacteristic
24#16810918C2211yescharacteristic
25#16810908C2211yescharacteristic
26#16810908C2211yescharacteristic
27#16810867—21no—
28#16810867—21no—
29#16810754—21no—
30#16810754—21no—
31#16810754C2211yescharacteristic
32#16810754C2211yescharacteristic
33Q8×#88322C2211yescharacteristic
34Q8×#88322C2211yescharacteristic
35#16810723—21no—
36#16810723—21no—
37#16810663—21no—
38#16810647—21no—
39#16810647C2211yescharacteristic
40#16810647C2211yescharacteristic
41#16810632—21no—
42C4×#11719917—21no—
43C4×#11719917C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810368C2211yescharacteristic
53#16810368C2211yescharacteristic
54#16810367C2211yescharacteristic
55#16810367C2211yescharacteristic
56C53×#532C2211yescharacteristic
57C2×C4×Q8×C53C2211yescharacteristic
58C53×#480—21no—
59C53×#477C2211yescharacteristic
60C53×#477C2211yescharacteristic
61C53×#460C2211yescharacteristic
62C53×#460C2211yescharacteristic
63C53×#448—21no—
64C53×#440—21no—
65C53×#440—21no—
66C53×#432C2211yescharacteristic
67C2×#11719923—21no—
68C2×#11719923—21no—
69C2×#11719917—21no—
70C2×#11719917—21no—
71C2×#11719917D411yescharacteristic
72C2×#11719917D411yescharacteristic
73#13133217C2311yescharacteristic
74#13133217C2311yescharacteristic
75#13133209—21no—
76#13133196—21no—
77#13133196—21no—
78C4×Dic106—21no—
79C4×Dic106—21no—
80#13133176—21no—
81#13133150—21no—
82#13133150—21no—
83#13133150—21no—
84#13133150—21no—
85#13133148—21no—
86#13133148—21no—
87C4⋊#88322—21no—
88C4⋊#88322—21no—
89C4⋊#88322—21no—
90C4⋊#88322—21no—
91C4⋊#88322—21no—
92C4⋊#88322—21no—
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133141—21no—
98#13133141—21no—
99#13133141—21no—
100#13133141—21no—
101#13133140—21no—
102#13133140—21no—
103#13133138—21no—
104#13133138—21no—
105#13133128—21no—
106#13133128—21no—
107#13133128D411yescharacteristic
108#13133128D411yescharacteristic
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C2×#41×C53—21no—
114C22×Q8×C53D411yescharacteristic
115C2×SD16×C53—21no—
116C2×SD16×C53—21no—
117C4⋊Q8×C53C2311yescharacteristic
118#127×C53C2311yescharacteristic
119#124×C53—21no—
120#121×C53—21no—
121#121×C53—21no—
122C4×Q8×C53—21no—
123C4×Q8×C53—21no—
124C4×Q8×C53D411yescharacteristic
125C4×Q8×C53D411yescharacteristic
126C4×Q8×C53C2311yescharacteristic
127C4×Q8×C53C2311yescharacteristic
128C4×D4×C53—21no—
129C4×D4×C53—21no—
130C2×C4⋊C4×C53—21no—
131C2×C4⋊C4×C53D411yescharacteristic
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#107×C53—21no—
135#107×C53—21no—
136C4⋊C8×C53C2311yescharacteristic
137C4⋊C8×C53C2311yescharacteristic
138#103×C53—21no—
139#103×C53—21no—
140#102×C53—21no—
141#102×C53—21no—
142C22⋊C8×C53—21no—
143#97×C53—21no—
144#41×C53—82no—
145C2×Q8×C53—21no—
146C2×Q8×C53—22no—
147C2×Q8×C53—21no—
148C2×Q8×C53—41no—
149C2×Q8×C53C2×D411yescharacteristic
150C2×Q8×C53C2×D411yescharacteristic
151C2×D4×C53—21no—
152C2×D4×C53—21no—
153C2×D4×C53—21no—
154C2×D4×C53—41no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—41no—
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53#4111yescharacteristic
161C2×Dic106—21no—
162C2×Dic106—21no—
163SD16×C53—42no—
164SD16×C53—42no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53C2×D411yescharacteristic
175C4⋊C4×C53C2×D411yescharacteristic
176C22⋊C4×C53—41no—
177C22⋊C4×C53—41no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53C2×D411yescharacteristic
183C42×C53C2×D411yescharacteristic
184C42×C53#4111yescharacteristic
185C42×C53C2411yescharacteristic
186C22⋊Dic53—41no—
187#11719923—42no—
188#11719923—42no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193#11719919—41no—
194#11719919—41no—
195C4×Dic53—41no—
196#11719917—22no—
197#11719917—41no—
198#11719917—42no—
199#11719917—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C23×C53—41no—
209C23×C53#12111yescharacteristic
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—42no—
214Q8×C53—81no—
215Q8×C53#13712yes—
216D4×C53—41no—
217D4×C53—41no—
218D4×C53—41no—
219D4×C53—41no—
220D4×C53—81no—
221D4×C53—82no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—41no—
233C2×C4×C53—82no—
234C2×C4×C53—81no—
235C2×C4×C53#12111yescharacteristic
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×Dic53—41no—
242C2×Dic53—41no—
243Dic106—41no—
244Dic106—41no—
245Dic106—81no—
246C8×C53—41no—
247C8×C53—41no—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#88322—41no—
255#88322#13712yes—
256#19309—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53C2×#12111yescharacteristic
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—22no—
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53—81no—
273C4×C53C2×#13711yescharacteristic
274C4×C53C2×#13711yescharacteristic
275C4×C53#57511yescharacteristic
276C4×C53#57511yescharacteristic
277Dic53—81no—
278Dic53—81no—
279#3182—531no—
280#3112D5311yescharacteristic
281#3078—531no—
282Q8×M16—531no—
283#1704—531no—
284#1698—531no—
285#1696—531no—
286#1686—531no—
287#1684—531no—
288#1678—531no—
289#1676—531no—
290#1666—531no—
291#1569—531no—
292#1560—531no—
293#1559—531no—
294C2×C53—41no—
295C2×C53—161no—
296C2×C53#303711yescharacteristic
297C2×C53#305611yescharacteristic
298C2×C53#306811yescharacteristic
299#533—531no—
300#532C2×D5311yescharacteristic
301C2×C4×Q8C2×D5311yescharacteristic
302#500—531no—
303#492—531no—
304#487—531no—
305#486—531no—
306#480—21no—
307#479—1061no—
308#479—1061no—
309#477—531no—
310#477—531no—
311#477C2×D5311yescharacteristic
312#477C2×D5311yescharacteristic
313#475—531no—
314#475—531no—
315#468—1061no—
316#468—1061no—
317#460C2×D5311yescharacteristic
318#460C2×D5311yescharacteristic
319#448—21no—
320#446—531no—
321#446—531no—
322#446—1061no—
323#446—1061no—
324C8×Q8—531no—
325C8×Q8—531no—
326#440—21no—
327#440—21no—
328#440—1061no—
329#440—1061no—
330#432C2×D5311yescharacteristic
331#428—1061no—
332#423—531no—
333#423—531no—
334#423—1061no—
335#419—1061no—
336C4×M16—531no—
337C4×M16—1061no—
338#335—531no—
339#335—531no—
340#334—531no—
341#334—531no—
342#333—531no—
343#333—531no—
344#333—531no—
345#333—531no—
346#332—531no—
347#332—531no—
348#331—531no—
349#331—531no—
350C53#1930911yescharacteristic, complemented, Sylow
351C2×#41—21no—
352C22×Q8C22⋊D5311yescharacteristic
353C2×SD16—21no—
354C2×SD16—21no—
355C2×SD16—1061no—
356C2×SD16—1061no—
357C2×M16—531no—
358C2×M16—531no—
359C2×M16—1061no—
360C2×M16—1061no—
361C4⋊Q8—531no—
362C4⋊Q8—531no—
363C4⋊Q8C22×D5311yescharacteristic
364#127C22×D5311yescharacteristic
365#125—1061no—
366#124—21no—
367C22⋊Q8—1061no—
368C22⋊Q8—1061no—
369#121—21no—
370#121—21no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—1061no—
374C4×Q8—1061no—
375C4×Q8C22⋊D5311yescharacteristic
376C4×Q8C22⋊D5311yescharacteristic
377C4×Q8C22×D5311yescharacteristic
378C4×Q8C22×D5311yescharacteristic
379C4×D4—21no—
380C4×D4—21no—
381#117—1061no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4C22⋊D5311yescharacteristic
384C2×C42—21no—
385C2×C42C22×D5311yescharacteristic
386#107—21no—
387#107—21no—
388#107—1061no—
389#107—1061no—
390#107—1061no—
391#107—1061no—
392#106—1061no—
393#106—1061no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C4⋊C8—1061no—
404C4⋊C8C22×D5311yescharacteristic
405C4⋊C8C22×D5311yescharacteristic
406#103—21no—
407#103—21no—
408#103—1061no—
409#103—1061no—
410#103—1061no—
411#103—1061no—
412#103—1061no—
413#103—1061no—
414#102—21no—
415#102—21no—
416#102—1061no—
417#102—1061no—
418C22⋊C8—21no—
419#97—21no—
420#97—531no—
421#97—531no—
422#97—1061no—
423#97—1061no—
424C4×C8—531no—
425C4×C8—531no—
426C4×C8—1061no—
427C4×C8—1061no—
428#41—82no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—41no—
433C2×Q8—1061no—
434C2×Q8—1061no—
435C2×Q8C2×C22⋊D5311yescharacteristic
436C2×Q8C2×C22⋊D5311yescharacteristic
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—41no—
444C22×C4D4×D5311yescharacteristic
445C22×C4#1171994711yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447SD16—42no—
448SD16—42no—
449SD16—2122no—
450SD16—2122no—
451M16—1062no—
452M16—2121no—
453M16—2122no—
454M16—2122no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—531no—
458C2×C8—531no—
459C2×C8—531no—
460C2×C8—531no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—2121no—
477C4⋊C4—2121no—
478C4⋊C4C2×C22⋊D5311yescharacteristic
479C4⋊C4C2×C22⋊D5311yescharacteristic
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—2121no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—2121no—
488C42D4×D5311yescharacteristic
489C42#1171994711yescharacteristic
490C42C2×C22⋊D5311yescharacteristic
491C42C23×D5311yescharacteristic
492C23—41no—
493C23#1313318911yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8—81no—
499Q8—4241no—
500Q8—2121no—
501Q8—2121no—
502Q8#1313324712yes—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—81no—
508D4—82no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4#1313318911yescharacteristic
525C2×C4#1313318911yescharacteristic
526C2×C4#1313318911yescharacteristic
527C2×C4C2×D4×D5311yescharacteristic
528C2×C4C22×C22⋊D5311yescharacteristic
529C2×C4C2×#1171994711yescharacteristic
530C8—41no—
531C8—41no—
532C8—532no—
533C8—1061no—
534C8—1061no—
535C8—1061no—
536C8—2121no—
537C8—2121no—
538C8—2121no—
539C8—2121no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22C2×#1313318911yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—81no—
556C4—4241no—
557C4—4241no—
558C4#137×D5311yescharacteristic
559C4C2×#1313324711yescharacteristic
560C4#1681142111yescharacteristic
561C4#1681142211yescharacteristic
562C2—41no—
563C2—161no—
564C2#2028418411yescharacteristic
565C2#2028434711yescharacteristic
566C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3112C2
socleC22×C53C2×#121
prime radicalC53×#532C22
hypercenter#3112D53
hypocenterC53#19309
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
2. #20284520C2semidirect
3. Q8×#11719917C2semidirect
4. #20274299C2semidirect
5. #20274253C2semidirect
6. #20274241C2semidirect
7. #20274185C2semidirect
8. #20274175C2semidirect
9. #20274129C2semidirect
10. #20274117C2semidirect
11. #20274061C2semidirect
12. #20273483C2semidirect
14. #20273446C2semidirect
350. C53#19309semidirect

Maximal subgroups

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

subgroupindex
#202854292
#202845202
Q8×#117199172
#202742992
#202742532
#202742412
#202741852
#202741752
#202741292
#202741172
#202740612
#202734832
#202734502
#202734462
C53×#31122
#1930953